TSTP Solution File: SEU950^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU950^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n114.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:25 EDT 2014
% Result : Timeout 300.09s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU950^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n114.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 11:45:06 CDT 2014
% % CPUTime : 300.09
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x27b5dd0>, <kernel.DependentProduct object at 0x27b54d0>) of role type named h
% Using role type
% Declaring h:((fofType->Prop)->fofType)
% FOF formula ((forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)), ((((eq fofType) (h Xx)) (h Xy))->(((eq (fofType->Prop)) Xx) Xy)))->((ex (fofType->(fofType->Prop))) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))) of role conjecture named cTHM573_pme
% Conjecture to prove = ((forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)), ((((eq fofType) (h Xx)) (h Xy))->(((eq (fofType->Prop)) Xx) Xy)))->((ex (fofType->(fofType->Prop))) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['((forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)), ((((eq fofType) (h Xx)) (h Xy))->(((eq (fofType->Prop)) Xx) Xy)))->((ex (fofType->(fofType->Prop))) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))))']
% Parameter fofType:Type.
% Parameter h:((fofType->Prop)->fofType).
% Trying to prove ((forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)), ((((eq fofType) (h Xx)) (h Xy))->(((eq (fofType->Prop)) Xx) Xy)))->((ex (fofType->(fofType->Prop))) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) (fun (x:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x X)) Y))))))
% Found (eta_expansion_dep00 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (fofType->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x1))) as proof of ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% 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)) (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x X)) Y))))))
% 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)) (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x X)) Y))))))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))
% Found eq_ref00:=(eq_ref0 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))):(((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found (eq_ref0 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((eq (fofType->Prop)) (x0 x)) Y)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((eq (fofType->Prop)) (x0 x)) Y)))
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) (fun (x:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x X)) Y))))))
% Found (eta_expansion_dep00 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (x0 x2)):(((eq (fofType->Prop)) (x0 x2)) (x0 x2))
% Found (eq_ref0 (x0 x2)) as proof of (Xx x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (Xx x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (Xx x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (Xx x2)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x2))) as proof of ((ex fofType) Xx)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (fofType->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (b x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (b x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (b x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (b x1)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x1))) as proof of ((ex fofType) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (fofType->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (f x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f x1)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x1))) as proof of ((ex fofType) f)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (fofType->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (f x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f x1)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x1))) as proof of ((ex fofType) f)
% Found eq_ref00:=(eq_ref0 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))
% Found (eq_ref0 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b x)))
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (h Xx0)):(((eq fofType) (h Xx0)) (h Xx0))
% Found (eq_ref0 (h Xx0)) as proof of (((eq fofType) (h Xx0)) (h Xx))
% Found ((eq_ref fofType) (h Xx0)) as proof of (((eq fofType) (h Xx0)) (h Xx))
% Found ((eq_ref fofType) (h Xx0)) as proof of (((eq fofType) (h Xx0)) (h Xx))
% Found ((eq_ref fofType) (h Xx0)) as proof of (((eq fofType) (h Xx0)) (h Xx))
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 (h Xx0)):(((eq fofType) (h Xx0)) (h Xx0))
% Found (eq_ref0 (h Xx0)) as proof of (((eq fofType) (h Xx0)) (h Xx))
% Found ((eq_ref fofType) (h Xx0)) as proof of (((eq fofType) (h Xx0)) (h Xx))
% Found ((eq_ref fofType) (h Xx0)) as proof of (((eq fofType) (h Xx0)) (h Xx))
% Found ((eq_ref fofType) (h Xx0)) as proof of (((eq fofType) (h Xx0)) (h Xx))
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% 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)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a 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)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% 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)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a 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)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a 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)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a 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)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (fofType->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (fofType->Prop)) Xx) b)
% Found ((eq_ref (fofType->Prop)) Xx) as proof of (((eq (fofType->Prop)) Xx) b)
% Found ((eq_ref (fofType->Prop)) Xx) as proof of (((eq (fofType->Prop)) Xx) b)
% Found ((eq_ref (fofType->Prop)) Xx) as proof of (((eq (fofType->Prop)) Xx) b)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq (fofType->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (fofType->Prop)) Xx) b)
% Found ((eq_ref (fofType->Prop)) Xx) as proof of (((eq (fofType->Prop)) Xx) b)
% Found ((eq_ref (fofType->Prop)) Xx) as proof of (((eq (fofType->Prop)) Xx) b)
% Found ((eq_ref (fofType->Prop)) Xx) as proof of (((eq (fofType->Prop)) Xx) b)
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h 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)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (Xx 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)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (Xx 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)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (Xx 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)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (Xx x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (fofType->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x1))) as proof of ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (fofType->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x1))) as proof of ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (fofType->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x1))) as proof of ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b x)))
% Found x1:(P2 (f x0))
% Instantiate: b:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (b x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (b x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b x)))
% Found x1:(P2 (f x0))
% Instantiate: b:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (b x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (b x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (h Xx0)):(((eq fofType) (h Xx0)) (h Xx0))
% Found (eq_ref0 (h Xx0)) as proof of (((eq fofType) (h Xx0)) (h Xx))
% Found ((eq_ref fofType) (h Xx0)) as proof of (((eq fofType) (h Xx0)) (h Xx))
% Found ((eq_ref fofType) (h Xx0)) as proof of (((eq fofType) (h Xx0)) (h Xx))
% Found ((eq_ref fofType) (h Xx0)) as proof of (((eq fofType) (h Xx0)) (h Xx))
% Found eq_ref00:=(eq_ref0 (h Xx0)):(((eq fofType) (h Xx0)) (h Xx0))
% Found (eq_ref0 (h Xx0)) as proof of (((eq fofType) (h Xx0)) (h Xx))
% Found ((eq_ref fofType) (h Xx0)) as proof of (((eq fofType) (h Xx0)) (h Xx))
% Found ((eq_ref fofType) (h Xx0)) as proof of (((eq fofType) (h Xx0)) (h Xx))
% Found ((eq_ref fofType) (h Xx0)) as proof of (((eq fofType) (h Xx0)) (h Xx))
% 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (fofType->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (a x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (a x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (a x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (a x1)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x1))) as proof of ((ex fofType) a)
% Found eq_ref00:=(eq_ref0 Xx):(((eq (fofType->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (fofType->Prop)) Xx) b)
% Found ((eq_ref (fofType->Prop)) Xx) as proof of (((eq (fofType->Prop)) Xx) b)
% Found ((eq_ref (fofType->Prop)) Xx) as proof of (((eq (fofType->Prop)) Xx) b)
% Found ((eq_ref (fofType->Prop)) Xx) as proof of (((eq (fofType->Prop)) Xx) b)
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (fofType->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (fofType->Prop)) Xx) b)
% Found ((eq_ref (fofType->Prop)) Xx) as proof of (((eq (fofType->Prop)) Xx) b)
% Found ((eq_ref (fofType->Prop)) Xx) as proof of (((eq (fofType->Prop)) Xx) b)
% Found ((eq_ref (fofType->Prop)) Xx) as proof of (((eq (fofType->Prop)) Xx) b)
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 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)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a 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)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% 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)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a 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)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a 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)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a 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)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% 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)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (Xx 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)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (Xx x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (Xx 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)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (Xx x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->(fofType->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b0)
% Found ((eq_ref (fofType->(fofType->Prop))) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b0)
% Found ((eq_ref (fofType->(fofType->Prop))) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b0)
% Found ((eq_ref (fofType->(fofType->Prop))) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eta_expansion0 Prop) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b)
% Found (((eta_expansion fofType) Prop) f) as proof of (((eq (fofType->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) b)
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->(fofType->Prop))) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b)
% Found ((eta_expansion0 (fofType->Prop)) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b)
% Found (((eta_expansion fofType) (fofType->Prop)) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b)
% Found (((eta_expansion fofType) (fofType->Prop)) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b)
% Found (((eta_expansion fofType) (fofType->Prop)) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->(fofType->Prop))) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> (fofType->Prop))) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> (fofType->Prop))) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> (fofType->Prop))) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> (fofType->Prop))) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))):(((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found (eq_ref0 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) (fun (x:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x X)) Y))))))
% Found (eta_expansion_dep00 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eta_expansion_dep0 (fun (x2:(fofType->(fofType->Prop)))=> Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))):(((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) (fun (x:fofType)=> (((eq (fofType->Prop)) (x0 x)) Y)))
% Found (eta_expansion_dep00 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found eq_ref00:=(eq_ref0 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))):(((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found (eq_ref0 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))):(((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) (fun (x:fofType)=> (((eq (fofType->Prop)) (x0 x)) Y)))
% Found (eta_expansion_dep00 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq (fofType->Prop)) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f x1))) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f x1))) as proof of (forall (x:fofType), (((eq (fofType->Prop)) (f x)) (x0 x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq (fofType->Prop)) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f x1))) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f x1))) as proof of (forall (x:fofType), (((eq (fofType->Prop)) (f x)) (x0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq (fofType->Prop)) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq (fofType->Prop)) (f0 x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f0 x1)) as proof of (((eq (fofType->Prop)) (f0 x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f0 x1)) as proof of (((eq (fofType->Prop)) (f0 x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f0 x1))) as proof of (((eq (fofType->Prop)) (f0 x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f0 x1))) as proof of (forall (x:fofType), (((eq (fofType->Prop)) (f0 x)) (x0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq (fofType->Prop)) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq (fofType->Prop)) (f0 x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f0 x1)) as proof of (((eq (fofType->Prop)) (f0 x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f0 x1)) as proof of (((eq (fofType->Prop)) (f0 x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f0 x1))) as proof of (((eq (fofType->Prop)) (f0 x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f0 x1))) as proof of (forall (x:fofType), (((eq (fofType->Prop)) (f0 x)) (x0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq (fofType->Prop)) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq (fofType->Prop)) (f0 x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f0 x1)) as proof of (((eq (fofType->Prop)) (f0 x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f0 x1)) as proof of (((eq (fofType->Prop)) (f0 x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f0 x1))) as proof of (((eq (fofType->Prop)) (f0 x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f0 x1))) as proof of (forall (x:fofType), (((eq (fofType->Prop)) (f0 x)) (x0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq (fofType->Prop)) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq (fofType->Prop)) (f0 x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f0 x1)) as proof of (((eq (fofType->Prop)) (f0 x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f0 x1)) as proof of (((eq (fofType->Prop)) (f0 x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f0 x1))) as proof of (((eq (fofType->Prop)) (f0 x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f0 x1))) as proof of (forall (x:fofType), (((eq (fofType->Prop)) (f0 x)) (x0 x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((eq (fofType->Prop)) (x0 x)) Y)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((eq (fofType->Prop)) (x0 x)) Y)))
% 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 (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (((eq (fofType->Prop)) (x0 x)) Y)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (((eq (fofType->Prop)) (x0 x)) Y)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (((eq (fofType->Prop)) (x0 x)) Y)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (((eq (fofType->Prop)) (x0 x)) Y)))
% Found eq_ref00:=(eq_ref0 ((f x1) y)):(((eq Prop) ((f x1) y)) ((f x1) y))
% Found (eq_ref0 ((f x1) y)) as proof of (((eq Prop) ((f x1) y)) ((x0 x1) y))
% Found ((eq_ref Prop) ((f x1) y)) as proof of (((eq Prop) ((f x1) y)) ((x0 x1) y))
% Found ((eq_ref Prop) ((f x1) y)) as proof of (((eq Prop) ((f x1) y)) ((x0 x1) y))
% Found (fun (y:fofType)=> ((eq_ref Prop) ((f x1) y))) as proof of (((eq Prop) ((f x1) y)) ((x0 x1) y))
% Found (fun (x1:fofType) (y:fofType)=> ((eq_ref Prop) ((f x1) y))) as proof of (forall (y:fofType), (((eq Prop) ((f x1) y)) ((x0 x1) y)))
% Found (fun (x1:fofType) (y:fofType)=> ((eq_ref Prop) ((f x1) y))) as proof of (forall (x:fofType) (y:fofType), (((eq Prop) ((f x) y)) ((x0 x) y)))
% Found eq_ref00:=(eq_ref0 ((f0 x1) y)):(((eq Prop) ((f0 x1) y)) ((f0 x1) y))
% Found (eq_ref0 ((f0 x1) y)) as proof of (((eq Prop) ((f0 x1) y)) ((x0 x1) y))
% Found ((eq_ref Prop) ((f0 x1) y)) as proof of (((eq Prop) ((f0 x1) y)) ((x0 x1) y))
% Found ((eq_ref Prop) ((f0 x1) y)) as proof of (((eq Prop) ((f0 x1) y)) ((x0 x1) y))
% Found (fun (y:fofType)=> ((eq_ref Prop) ((f0 x1) y))) as proof of (((eq Prop) ((f0 x1) y)) ((x0 x1) y))
% Found (fun (x1:fofType) (y:fofType)=> ((eq_ref Prop) ((f0 x1) y))) as proof of (forall (y:fofType), (((eq Prop) ((f0 x1) y)) ((x0 x1) y)))
% Found (fun (x1:fofType) (y:fofType)=> ((eq_ref Prop) ((f0 x1) y))) as proof of (forall (x:fofType) (y:fofType), (((eq Prop) ((f0 x) y)) ((x0 x) y)))
% Found eq_ref00:=(eq_ref0 ((f0 x1) y)):(((eq Prop) ((f0 x1) y)) ((f0 x1) y))
% Found (eq_ref0 ((f0 x1) y)) as proof of (((eq Prop) ((f0 x1) y)) ((x0 x1) y))
% Found ((eq_ref Prop) ((f0 x1) y)) as proof of (((eq Prop) ((f0 x1) y)) ((x0 x1) y))
% Found ((eq_ref Prop) ((f0 x1) y)) as proof of (((eq Prop) ((f0 x1) y)) ((x0 x1) y))
% Found (fun (y:fofType)=> ((eq_ref Prop) ((f0 x1) y))) as proof of (((eq Prop) ((f0 x1) y)) ((x0 x1) y))
% Found (fun (x1:fofType) (y:fofType)=> ((eq_ref Prop) ((f0 x1) y))) as proof of (forall (y:fofType), (((eq Prop) ((f0 x1) y)) ((x0 x1) y)))
% Found (fun (x1:fofType) (y:fofType)=> ((eq_ref Prop) ((f0 x1) y))) as proof of (forall (x:fofType) (y:fofType), (((eq Prop) ((f0 x) y)) ((x0 x) y)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b x)))
% Found x1:(P2 (f x0))
% Instantiate: b:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (b x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (b x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b x)))
% Found x1:(P2 (f x0))
% Instantiate: b:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (b x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (b x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b x)))
% Found x1:(P2 (f x0))
% Instantiate: b:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (b x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (b x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b x)))
% Found x1:(P2 (f x0))
% Instantiate: b:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (b x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (b x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))):(((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found (eq_ref0 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (fofType->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (((eq (fofType->Prop)) (x0 x1)) Y)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x1))) as proof of ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found eta_expansion000:=(eta_expansion00 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))):(((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) (fun (x:fofType)=> (((eq (fofType->Prop)) (x0 x)) Y)))
% Found (eta_expansion00 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found ((eta_expansion0 Prop) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found eq_ref00:=(eq_ref0 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))):(((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found (eq_ref0 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))):(((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) (fun (x:fofType)=> (((eq (fofType->Prop)) (x0 x)) Y)))
% Found (eta_expansion00 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found ((eta_expansion0 Prop) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((eq (fofType->Prop)) (x0 x)) Y)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((eq (fofType->Prop)) (x0 x)) Y)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (((eq (fofType->Prop)) (x0 x)) Y)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (((eq (fofType->Prop)) (x0 x)) Y)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (((eq (fofType->Prop)) (x0 x)) Y)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (((eq (fofType->Prop)) (x0 x)) Y)))
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((fofType->(fofType->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((fofType->(fofType->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((fofType->(fofType->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% 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) Xx)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) Xx)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) Xx)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) Xx)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) Xx)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) Xx)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) Xx)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) Xx)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) Xx)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))
% Found (eq_ref0 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b1)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b1)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b1)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b1)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% 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 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% 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) f)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) 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) f)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b 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)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (Xx 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)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (Xx x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found x1:(P2 (f x0))
% Instantiate: b:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (b x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (b x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b x)))
% Found x1:(P2 (f x0))
% Instantiate: b:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (b x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (b x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b x)))
% Found x3:(P2 (f x2))
% Instantiate: Xx:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (Xx x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (Xx x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (Xx x)))
% Found x3:(P2 (f x2))
% Instantiate: Xx:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (Xx x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (Xx x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (Xx x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found x2:(P2 (f x1))
% Instantiate: b:=f:(fofType->Prop)
% Found (fun (x2:(P2 (f x1)))=> x2) as proof of (P2 (b x1))
% Found (fun (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of ((P2 (f x1))->(P2 (b x1)))
% Found (fun (x1:fofType) (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType) (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found x2:(P2 (f x1))
% Instantiate: b:=f:(fofType->Prop)
% Found (fun (x2:(P2 (f x1)))=> x2) as proof of (P2 (b x1))
% Found (fun (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of ((P2 (f x1))->(P2 (b x1)))
% Found (fun (x1:fofType) (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType) (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))
% Found (eq_ref0 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))
% Found (eq_ref0 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found x2:(P2 (f x1))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x2:(P2 (f x1)))=> x2) as proof of (P2 (f0 x1))
% Found (fun (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of ((P2 (f x1))->(P2 (f0 x1)))
% Found (fun (x1:fofType) (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found (fun (x1:fofType) (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found x2:(P2 (f x1))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x2:(P2 (f x1)))=> x2) as proof of (P2 (f0 x1))
% Found (fun (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of ((P2 (f x1))->(P2 (f0 x1)))
% Found (fun (x1:fofType) (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found (fun (x1:fofType) (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 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)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a 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)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a 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)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a 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)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a x)))
% Found x1:(P2 (f x0))
% Instantiate: a:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (a x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (a x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a x)))
% Found x1:(P2 (f x0))
% Instantiate: a:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (a x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (a x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a x)))
% Found x1:(P2 (f x0))
% Instantiate: a:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (a x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (a x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a x)))
% Found x1:(P2 (f x0))
% Instantiate: a:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (a x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (a x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (a x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h a))
% Found eq_ref00:=(eq_ref0 (x0 x3)):(((eq (fofType->Prop)) (x0 x3)) (x0 x3))
% Found (eq_ref0 (x0 x3)) as proof of (Xx0 x3)
% Found ((eq_ref (fofType->Prop)) (x0 x3)) as proof of (Xx0 x3)
% Found ((eq_ref (fofType->Prop)) (x0 x3)) as proof of (Xx0 x3)
% Found ((eq_ref (fofType->Prop)) (x0 x3)) as proof of (Xx0 x3)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x3))) as proof of ((ex fofType) Xx0)
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))))
% 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) Xx)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) Xx)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) Xx)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) Xx)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) Xx)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) Xx)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) Xx)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) Xx)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) Xx)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->(fofType->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b0)
% Found ((eq_ref (fofType->(fofType->Prop))) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b0)
% Found ((eq_ref (fofType->(fofType->Prop))) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b0)
% Found ((eq_ref (fofType->(fofType->Prop))) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b0)
% Found eq_ref00:=(eq_ref0 (x0 x2)):(((eq (fofType->Prop)) (x0 x2)) (x0 x2))
% Found (eq_ref0 (x0 x2)) as proof of (Xx x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (Xx x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (Xx x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (Xx x2)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x2))) as proof of ((ex fofType) Xx)
% Found eq_ref00:=(eq_ref0 (x0 x2)):(((eq (fofType->Prop)) (x0 x2)) (x0 x2))
% Found (eq_ref0 (x0 x2)) as proof of (b x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (b x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (b x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (b x2)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x2))) as proof of ((ex fofType) b)
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->(fofType->Prop))) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b0)
% Found ((eq_ref (fofType->(fofType->Prop))) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b0)
% Found ((eq_ref (fofType->(fofType->Prop))) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b0)
% Found ((eq_ref (fofType->(fofType->Prop))) x0) as proof of (((eq (fofType->(fofType->Prop))) x0) b0)
% Found eq_ref00:=(eq_ref0 (x0 x2)):(((eq (fofType->Prop)) (x0 x2)) (x0 x2))
% Found (eq_ref0 (x0 x2)) as proof of (Xx x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (Xx x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (Xx x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (Xx x2)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x2))) as proof of ((ex fofType) Xx)
% Found eq_ref00:=(eq_ref0 (x0 x2)):(((eq (fofType->Prop)) (x0 x2)) (x0 x2))
% Found (eq_ref0 (x0 x2)) as proof of (Xx x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (Xx x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (Xx x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (Xx x2)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x2))) as proof of ((ex fofType) Xx)
% Found eq_ref00:=(eq_ref0 (x0 x2)):(((eq (fofType->Prop)) (x0 x2)) (x0 x2))
% Found (eq_ref0 (x0 x2)) as proof of (f x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (f x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (f x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (f x2)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x2))) as proof of ((ex fofType) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 (x0 x2)):(((eq (fofType->Prop)) (x0 x2)) (x0 x2))
% Found (eq_ref0 (x0 x2)) as proof of (f x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (f x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (f x2)
% Found ((eq_ref (fofType->Prop)) (x0 x2)) as proof of (f x2)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x2))) as proof of ((ex fofType) f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found eta_expansion000:=(eta_expansion00 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))):(((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) (fun (x:fofType)=> (((eq (fofType->Prop)) (x0 x)) Y)))
% Found (eta_expansion00 (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found ((eta_expansion0 Prop) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (a x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (a x)))
% 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 eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 x)))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) 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) f)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq (fofType->Prop)) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f x1))) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f x1))) as proof of (forall (x:fofType), (((eq (fofType->Prop)) (f x)) (x0 x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq (fofType->Prop)) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f x1))) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f x1))) as proof of (forall (x:fofType), (((eq (fofType->Prop)) (f x)) (x0 x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->(fofType->Prop))) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->(fofType->Prop))) a) b0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> (fofType->Prop))) a) as proof of (((eq (fofType->(fofType->Prop))) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> (fofType->Prop))) a) as proof of (((eq (fofType->(fofType->Prop))) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> (fofType->Prop))) a) as proof of (((eq (fofType->(fofType->Prop))) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> (fofType->Prop))) a) as proof of (((eq (fofType->(fofType->Prop))) a) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->(fofType->Prop))) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->(fofType->Prop))) b0) x0)
% Found ((eta_expansion0 (fofType->Prop)) b0) as proof of (((eq (fofType->(fofType->Prop))) b0) x0)
% Found (((eta_expansion fofType) (fofType->Prop)) b0) as proof of (((eq (fofType->(fofType->Prop))) b0) x0)
% Found (((eta_expansion fofType) (fofType->Prop)) b0) as proof of (((eq (fofType->(fofType->Prop))) b0) x0)
% Found (((eta_expansion fofType) (fofType->Prop)) b0) as proof of (((eq (fofType->(fofType->Prop))) b0) x0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) f)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h Xx0))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h Xx0))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h Xx0))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h Xx0))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq (fofType->Prop)) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f x1))) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f x1))) as proof of (forall (x:fofType), (((eq (fofType->Prop)) (f x)) (x0 x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq (fofType->Prop)) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found ((eq_ref (fofType->Prop)) (f x1)) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f x1))) as proof of (((eq (fofType->Prop)) (f x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref (fofType->Prop)) (f x1))) as proof of (forall (x:fofType), (((eq (fofType->Prop)) (f x)) (x0 x)))
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((fofType->(fofType->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h b))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h Xx0))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h Xx0))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h Xx0))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h Xx0))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) (fun (x:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (x X)) Y))))))
% Found (eta_expansion_dep00 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b1)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b1)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b1)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b1)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b1)
% Found eq_ref00:=(eq_ref0 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))
% Found (eq_ref0 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b1)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b1)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b1)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b1)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->(fofType->Prop))) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->(fofType->Prop))) a) b)
% Found ((eta_expansion0 (fofType->Prop)) a) as proof of (((eq (fofType->(fofType->Prop))) a) b)
% Found (((eta_expansion fofType) (fofType->Prop)) a) as proof of (((eq (fofType->(fofType->Prop))) a) b)
% Found (((eta_expansion fofType) (fofType->Prop)) a) as proof of (((eq (fofType->(fofType->Prop))) a) b)
% Found (((eta_expansion fofType) (fofType->Prop)) a) as proof of (((eq (fofType->(fofType->Prop))) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->(fofType->Prop))) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->(fofType->Prop))) b) x0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> (fofType->Prop))) b) as proof of (((eq (fofType->(fofType->Prop))) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> (fofType->Prop))) b) as proof of (((eq (fofType->(fofType->Prop))) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> (fofType->Prop))) b) as proof of (((eq (fofType->(fofType->Prop))) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> (fofType->Prop))) b) as proof of (((eq (fofType->(fofType->Prop))) b) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->(fofType->Prop))) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->(fofType->Prop))) a) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> (fofType->Prop))) a) as proof of (((eq (fofType->(fofType->Prop))) a) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> (fofType->Prop))) a) as proof of (((eq (fofType->(fofType->Prop))) a) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> (fofType->Prop))) a) as proof of (((eq (fofType->(fofType->Prop))) a) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> (fofType->Prop))) a) as proof of (((eq (fofType->(fofType->Prop))) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->(fofType->Prop))) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->(fofType->Prop))) b) x0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> (fofType->Prop))) b) as proof of (((eq (fofType->(fofType->Prop))) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> (fofType->Prop))) b) as proof of (((eq (fofType->(fofType->Prop))) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> (fofType->Prop))) b) as proof of (((eq (fofType->(fofType->Prop))) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> (fofType->Prop))) b) as proof of (((eq (fofType->(fofType->Prop))) b) x0)
% Found eq_ref00:=(eq_ref0 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))
% Found (eq_ref0 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y))))))
% Found (eq_ref0 (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xg:(fofType->(fofType->Prop)))=> (forall (Y:(fofType->Prop)), ((ex fofType) (fun (X:fofType)=> (((eq (fofType->Prop)) (Xg X)) Y)))))) b0)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((eq (fofType->Prop)) (x0 x)) Y)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (((eq (fofType->Prop)) (x0 x1)) Y))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((eq (fofType->Prop)) (x0 x)) Y)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((fofType->(fofType->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((fofType->(fofType->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (fofType->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (b x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (b x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (b x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (b x1)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x1))) as proof of ((ex fofType) b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (fofType->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (b x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (b x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (b x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (b x1)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x1))) as proof of ((ex fofType) b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (fofType->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (f x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f x1)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x1))) as proof of ((ex fofType) f)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (fofType->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (f x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f x1)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x1))) as proof of ((ex fofType) f)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) b0)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b0)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b0)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b0)
% 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq fofType) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found ((eq_ref fofType) (h Xx)) as proof of (((eq fofType) (h Xx)) (h f))
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> (((eq (fofType->Prop)) (x0 X)) Y)))
% 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 x)))
% Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found eq_ref00:=(eq_ref0 ((f x1) y)):(((eq Prop) ((f x1) y)) ((f x1) y))
% Found (eq_ref0 ((f x1) y)) as proof of (((eq Prop) ((f x1) y)) ((x0 x1) y))
% Found ((eq_ref Prop) ((f x1) y)) as proof of (((eq Prop) ((f x1) y)) ((x0 x1) y))
% Found ((eq_ref Prop) ((f x1) y)) as proof of (((eq Prop) ((f x1) y)) ((x0 x1) y))
% Found (fun (y:fofType)=> ((eq_ref Prop) ((f x1) y))) as proof of (((eq Prop) ((f x1) y)) ((x0 x1) y))
% Found (fun (x1:fofType) (y:fofType)=> ((eq_ref Prop) ((f x1) y))) as proof of (forall (y:fofType), (((eq Prop) ((f x1) y)) ((x0 x1) y)))
% Found (fun (x1:fofType) (y:fofType)=> ((eq_ref Prop) ((f x1) y))) as proof of (forall (x:fofType) (y:fofType), (((eq Prop) ((f x) y)) ((x0 x) y)))
% Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) b)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 ((f x1) y)):(((eq Prop) ((f x1) y)) ((f x1) y))
% Found (eq_ref0 ((f x1) y)) as proof of (((eq Prop) ((f x1) y)) ((x0 x1) y))
% Found ((eq_ref Prop) ((f x1) y)) as proof of (((eq Prop) ((f x1) y)) ((x0 x1) y))
% Found ((eq_ref Prop) ((f x1) y)) as proof of (((eq Prop) ((f x1) y)) ((x0 x1) y))
% Found (fun (y:fofType)=> ((eq_ref Prop) ((f x1) y))) as proof of (((eq Prop) ((f x1) y)) ((x0 x1) y))
% Found (fun (x1:fofType) (y:fofType)=> ((eq_ref Prop) ((f x1) y))) as proof of (forall (y:fofType), (((eq Prop) ((f x1) y)) ((x0 x1) y)))
% Found (fun (x1:fofType) (y:fofType)=> ((eq_ref Prop) ((f x1) y))) as proof of (forall (x:fofType) (y:fofType), (((eq Prop) ((f x) y)) ((x0 x) y)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) f)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% 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)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (Xx 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)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (Xx x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eq_ref00:=(eq_ref0 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 f):(((eq ((fofType->(fofType->Prop))->Prop)) f) (fun (x:(fofType->(fofType->Prop)))=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found ((eta_expansion0 Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f) b0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) (fun (x:(fofType->(fofType->Prop)))=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 f0):(((eq ((fofType->(fofType->Prop))->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) f0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) f0) b)
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 x)))
% Found x3:(P2 (f x2))
% Instantiate: Xx:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (Xx x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (Xx x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (Xx x)))
% Found x3:(P2 (f x2))
% Instantiate: Xx:=f:(fofType->Prop)
% Found (fun (x3:(P2 (f x2)))=> x3) as proof of (P2 (Xx x2))
% Found (fun (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of ((P2 (f x2))->(P2 (Xx x2)))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (((eq Prop) (f x2)) (Xx x2))
% Found (fun (x2:fofType) (P2:(Prop->Prop)) (x3:(P2 (f x2)))=> x3) as proof of (forall (x:fofType), (((eq Prop) (f x)) (Xx x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 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)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (fofType->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (f0 x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f0 x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f0 x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f0 x1)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x1))) as proof of ((ex fofType) f0)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq (fofType->Prop)) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (f0 x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f0 x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f0 x1)
% Found ((eq_ref (fofType->Prop)) (x0 x1)) as proof of (f0 x1)
% Found (ex_intro100 ((eq_ref (fofType->Prop)) (x0 x1))) as proof of ((ex fofType) f0)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found x2:(P2 (f x1))
% Instantiate: b:=f:(fofType->Prop)
% Found (fun (x2:(P2 (f x1)))=> x2) as proof of (P2 (b x1))
% Found (fun (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of ((P2 (f x1))->(P2 (b x1)))
% Found (fun (x1:fofType) (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType) (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found x2:(P2 (f x1))
% Instantiate: b:=f:(fofType->Prop)
% Found (fun (x2:(P2 (f x1)))=> x2) as proof of (P2 (b x1))
% Found (fun (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of ((P2 (f x1))->(P2 (b x1)))
% Found (fun (x1:fofType) (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:fofType) (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of (forall (x:fofType), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (f0 x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found x2:(P2 (f x1))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x2:(P2 (f x1)))=> x2) as proof of (P2 (f0 x1))
% Found (fun (P2:(Prop->Prop)) (x2:(P2 (f x1)))=> x2) as proof of ((P2 (f x1))->(P2 (f0 x1)))
% Found (fun (x1:fofType) (P2:(P
% EOF
%------------------------------------------------------------------------------