TSTP Solution File: SEU945^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU945^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n096.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 300.10s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU945^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n096.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 11:44:36 CDT 2014
% % CPUTime : 300.10
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula ((ex (fofType->(fofType->Prop))) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) of role conjecture named cTHM3_pme
% Conjecture to prove = ((ex (fofType->(fofType->Prop))) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['((ex (fofType->(fofType->Prop))) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy)))))']
% Parameter fofType:Type.
% Trying to prove ((ex (fofType->(fofType->Prop))) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) (fun (x:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (eta_expansion00 (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) b)
% Found ((eta_expansion0 Prop) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xx))):((P Xx)->(P Xx))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xx))) as proof of (P0 Xx)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xx))):((P Xx)->(P Xx))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xx))) as proof of (P0 Xx)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xx))) as proof of (P0 Xx)
% 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 (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy)))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy)))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy)))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy)))))
% 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 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 x1:(P Xx)
% Instantiate: b:=Xx:fofType
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xx))):((P Xx)->(P Xx))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xx))) as proof of (P0 Xx)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref fofType) b) as proof of (P b)
% Found ((eq_ref fofType) b) as proof of (P b)
% Found ((eq_ref fofType) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found x1:(P Xy)
% Instantiate: b:=Xy:fofType
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b))):((P b)->(P b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (((eq_trans000 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (((((eq_trans0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (((eq_trans000 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (((((eq_trans0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found x1:(P Xx)
% Instantiate: b:=Xx:fofType
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xx))):((P Xx)->(P Xx))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xx))) as proof of (P0 Xx)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref fofType) b) as proof of (P b)
% Found ((eq_ref fofType) b) as proof of (P b)
% Found ((eq_ref fofType) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found x1:(P Xy)
% Instantiate: b:=Xy:fofType
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b))):((P b)->(P b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b))):((P b)->(P b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found (((eq_trans000 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found ((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found (((((eq_trans0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found ((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found ((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found (((eq_trans000 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found ((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found (((((eq_trans0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found ((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found ((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found x1:(P Xy)
% Instantiate: b:=Xy:fofType
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b))):((P b)->(P b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b))):((P b)->(P b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found x1:(P Xx)
% Instantiate: a:=Xx:fofType
% Found x1 as proof of (P0 a)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref fofType) b) as proof of (P1 b)
% Found ((eq_ref fofType) b) as proof of (P1 b)
% Found ((eq_ref fofType) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref fofType) b) as proof of (P1 b)
% Found ((eq_ref fofType) b) as proof of (P1 b)
% Found ((eq_ref fofType) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found x1:(P1 Xy)
% Instantiate: b:=Xy:fofType
% Found x1 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found x1:(P1 Xy)
% Instantiate: b:=Xy:fofType
% Found x1 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found x00:=(x0 (fun (x2:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x2:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x2:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found x1:(P b)
% Instantiate: b0:=b:fofType
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found x1:(P Xy)
% Instantiate: a:=Xy:fofType
% Found x1 as proof of (P0 a)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P0 Xy))):((P0 Xy)->(P0 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P0 Xy))) as proof of (P1 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P0 Xy))) as proof of (P1 Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found x1:(P b)
% Found x1 as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found x0:(P0 (f x))
% Instantiate: b:=(f x):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found x0:(P0 (f x))
% Instantiate: b:=(f x):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_trans000 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((((eq_trans0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_trans000 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((((eq_trans0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b1)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b1)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b1)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq fofType) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq fofType) b1) Xy)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xy)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xy)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xy)
% Found x1:(P Xy)
% Instantiate: b0:=Xy:fofType
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found x1:(P Xy)
% Instantiate: b0:=Xy:fofType
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b0))):((P b0)->(P b0))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (P Xx)
% Found ((eq_ref fofType) Xx) as proof of (P Xx)
% Found ((eq_ref fofType) Xx) as proof of (P Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found x02:(P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_trans000 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((((eq_trans0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found x02:(P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_trans000 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((((eq_trans0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found x1:(P Xy)
% Found x1 as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Instantiate: b:=(forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))):Prop
% Found x0 as proof of (P1 b)
% Found x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Instantiate: b:=(forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_sym0100 ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found ((eq_sym0100 ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) b))=> ((eq_sym010 x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((eq_sym01 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (fun (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of (P0 (f x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of ((P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))->(P0 (f x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found (eq_sym000 (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_sym00 (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (((eq_sym0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((((eq_sym Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((eq_sym Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_sym0100 ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found ((eq_sym0100 ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) b))=> ((eq_sym010 x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((eq_sym01 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (fun (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of (P0 (f x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of ((P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))->(P0 (f x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found (eq_sym000 (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_sym00 (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (((eq_sym0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((((eq_sym Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((eq_sym Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found x1:(P b)
% Instantiate: b0:=b:fofType
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (P b0)
% Found ((eq_ref fofType) b) as proof of (P b0)
% Found ((eq_ref fofType) b) as proof of (P b0)
% Found ((eq_ref fofType) b) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq fofType) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b0))):((P b0)->(P b0))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (P Xy)
% Found ((eq_ref fofType) Xy) as proof of (P Xy)
% Found ((eq_ref fofType) Xy) as proof of (P Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b))):((P b)->(P b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq fofType) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b00)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b00)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b00)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq fofType) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b0))):((P b0)->(P b0))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found x1:(P Xy)
% Instantiate: b:=Xy:fofType
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b))):((P b)->(P b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b1):(((eq fofType) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 b1):(((eq fofType) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b1)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b1)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b1)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq fofType) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq fofType) b00) Xy)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) Xy)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) Xy)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b00)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b00)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b00)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b00)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found ((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found (((((eq_trans0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found (eq_sym0000 ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (eq_sym0000 ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> ((eq_sym000 x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> (((eq_sym00 (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> ((((eq_sym0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> (((((eq_sym Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) ((((eq_sym Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> (((((eq_sym Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) ((((eq_sym Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found x1:(P Xx)
% Instantiate: a:=Xx:fofType
% Found x1 as proof of (P0 a)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b))):((P b)->(P b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b))):((P b)->(P b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b1):(((eq fofType) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq fofType) b1) Xy)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xy)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xy)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b1)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b1)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b1)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b1)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref fofType) b) as proof of (P1 b)
% Found ((eq_ref fofType) b) as proof of (P1 b)
% Found ((eq_ref fofType) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref fofType) b) as proof of (P1 b)
% Found ((eq_ref fofType) b) as proof of (P1 b)
% Found ((eq_ref fofType) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found x1:(P1 Xy)
% Instantiate: b:=Xy:fofType
% Found x1 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found x1:(P1 Xy)
% Instantiate: b:=Xy:fofType
% Found x1 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found x00:=(x0 (fun (x2:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x2:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x2:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) (fun (x:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (eta_expansion_dep00 (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xf:(fofType->(fofType->Prop)))=> (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (Xf Xx)) (Xf Xy))->(((eq fofType) Xx) Xy))))) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) 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) 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 x1:(P b)
% Instantiate: b0:=b:fofType
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found x1:(P b)
% Instantiate: b0:=b:fofType
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found x1:(P Xy)
% Instantiate: a:=Xy:fofType
% Found x1 as proof of (P0 a)
% Found x1:(P Xy)
% Instantiate: a:=Xy:fofType
% Found x1 as proof of (P0 a)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P0 Xy))):((P0 Xy)->(P0 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P0 Xy))) as proof of (P1 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P0 Xy))) as proof of (P1 Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found x1:(P b)
% Found x1 as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found x0:(P0 (f x))
% Instantiate: b:=(f x):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found x0:(P0 (f x))
% Instantiate: b:=(f x):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))->(P (f x))))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))->(P (f x))))
% Found ((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))->(P (f x))))
% Found (((((eq_trans0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))->(P (f x))))
% Found ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))->(P (f x))))
% Found ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))->(P (f x))))
% Found ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found (eq_sym000 ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found ((eq_sym00 (f x)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found (((eq_sym0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found ((((eq_sym Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) ((((eq_sym Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found ((((eq_sym Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) ((((eq_sym Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found x1:(P1 Xy)
% Instantiate: b:=Xy:fofType
% Found x1 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found x1:(P1 Xy)
% Instantiate: b:=Xy:fofType
% Found x1 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b1)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b1)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b1)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq fofType) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq fofType) b1) Xy)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xy)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xy)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xy)
% Found x00:=(x0 (fun (x2:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x2:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x2:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_trans000 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((((eq_trans0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_trans000 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((((eq_trans0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found x1:(P Xy)
% Instantiate: b0:=Xy:fofType
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found x1:(P b)
% Instantiate: b0:=b:fofType
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found x1:(P b)
% Instantiate: b0:=b:fofType
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found x1:(P Xy)
% Instantiate: b0:=Xy:fofType
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b0))):((P b0)->(P b0))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b0))):((P b0)->(P b0))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (P Xx)
% Found ((eq_ref fofType) Xx) as proof of (P Xx)
% Found ((eq_ref fofType) Xx) as proof of (P Xx)
% Found x1:(P b)
% Found x1 as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (P Xx)
% Found ((eq_ref fofType) Xx) as proof of (P Xx)
% Found ((eq_ref fofType) Xx) as proof of (P Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found x02:(P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_trans000 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((((eq_trans0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found x02:(P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_trans000 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((((eq_trans0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (x00:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((((eq_trans Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (fun (x02:(P0 (f x)))=> x02))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P0 Xy))):((P0 Xy)->(P0 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P0 Xy))) as proof of (P1 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P0 Xy))) as proof of (P1 Xy)
% Found x1:(P b)
% Found x1 as proof of (P0 Xy)
% Found x1:(P b)
% Found x1 as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Instantiate: b:=(forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_sym0100 ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found ((eq_sym0100 ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) b))=> ((eq_sym010 x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((eq_sym01 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (fun (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of (P0 (f x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of ((P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))->(P0 (f x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found (eq_sym000 (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found ((eq_sym00 (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found (((eq_sym0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> ((((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found ((((eq_sym Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((eq_sym Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found ((((eq_sym Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((fun (x00:(((eq Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))=> (((((eq_sym Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq fofType) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq fofType) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b0))):((P b0)->(P b0))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (P Xy)
% Found ((eq_ref fofType) Xy) as proof of (P Xy)
% Found ((eq_ref fofType) Xy) as proof of (P Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (P Xy)
% Found ((eq_ref fofType) Xy) as proof of (P Xy)
% Found ((eq_ref fofType) Xy) as proof of (P Xy)
% Found x1:(P Xy)
% Instantiate: b0:=Xy:fofType
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found x1:(P Xy)
% Instantiate: b0:=Xy:fofType
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found x1:(P Xy)
% Found x1 as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found x1:(P Xx)
% Instantiate: b0:=Xx:fofType
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found x1:(P Xx)
% Instantiate: b0:=Xx:fofType
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found x1:(P b)
% Instantiate: b0:=b:fofType
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (P b0)
% Found ((eq_ref fofType) Xx) as proof of (P b0)
% Found ((eq_ref fofType) Xx) as proof of (P b0)
% Found ((eq_ref fofType) Xx) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (P b0)
% Found ((eq_ref fofType) Xx) as proof of (P b0)
% Found ((eq_ref fofType) Xx) as proof of (P b0)
% Found ((eq_ref fofType) Xx) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq fofType) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq fofType) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:(fofType->Prop))=> (P1 b))) as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b00)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b00)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b00)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq fofType) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b00)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b00)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b00)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq fofType) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b0))):((P b0)->(P b0))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b0))):((P b0)->(P b0))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))
% Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xx))):((P Xx)->(P Xx))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xx))) as proof of (P0 Xx)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref000:=(eq_ref00 P):((P Xx)->(P Xx))
% Found (eq_ref00 P) as proof of (P0 Xx)
% Found ((eq_ref0 Xx) P) as proof of (P0 Xx)
% Found (((eq_ref fofType) Xx) P) as proof of (P0 Xx)
% Found (((eq_ref fofType) Xx) P) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:(fofType->Prop))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found x1:(P b)
% Instantiate: b0:=b:fofType
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found ((eq_ref fofType) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (P b0)
% Found ((eq_ref fofType) b) as proof of (P b0)
% Found ((eq_ref fofType) b) as proof of (P b0)
% Found ((eq_ref fofType) b) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b0))):((P b0)->(P b0))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (P Xy)
% Found ((eq_ref fofType) Xy) as proof of (P Xy)
% Found ((eq_ref fofType) Xy) as proof of (P Xy)
% Found eq_ref00:=(eq_ref0 b1):(((eq fofType) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq fofType) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq fofType) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq fofType) b1) Xy)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xy)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xy)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b1)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b1)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b1)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq fofType) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b1)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b1)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b1)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq fofType) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b00)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b00)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b00)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq fofType) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b00)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b00)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b00)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b00)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq fofType) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found ((eq_ref fofType) b1) as proof of (((eq fofType) b1) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) b00)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b00)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b00)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq fofType) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found ((eq_ref fofType) b00) as proof of (((eq fofType) b00) b)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b0))):((P b0)->(P b0))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found x00:=(x0 (fun (x1:(fofType->Prop))=> (P b0))):((P b0)->(P b0))
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found (x0 (fun (x1:(fofType->Prop))=> (P b0))) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found ((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found (((((eq_trans0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found (eq_sym0000 ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (eq_sym0000 ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> ((eq_sym000 x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> (((eq_sym00 (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> ((((eq_sym0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> (((((eq_sym Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) ((((eq_sym Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> (((((eq_sym Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) ((((eq_sym Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> (((((eq_sym Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) ((((eq_sym Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found ((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found (((((eq_trans0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found (eq_sym0000 ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (eq_sym0000 ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> ((eq_sym000 x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> (((eq_sym00 (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> ((((eq_sym0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> (((((eq_sym Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) ((((eq_sym Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> (((((eq_sym Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) ((((eq_sym Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> (((((eq_sym Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) ((((eq_sym Prop) (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))->(P (f x))))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))->(P (f x))))
% Found ((((eq_trans00 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))->(P (f x))))
% Found (((((eq_trans0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))->(P (f x))))
% Found ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))->(P (f x))))
% Found ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))->(P (f x))))
% Found ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x))
% Found (eq_sym0000 ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found (eq_sym0000 ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> ((eq_sym000 x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> (((eq_sym00 (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> ((((eq_sym0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy))))) (((eq_sym0 (f x)) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))))
% Found ((fun (x0:(((eq Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)))=> (((((eq_sym Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) x0) P0)) ((((((eq_trans Prop) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (forall (Xx:fofType) (Xy:fofType), ((((eq (fofType->Prop)) (x Xx)) (x Xy))->(((eq fofType) Xx) Xy)))) (f x)) ((eq_ref Prop) (forall (Xx:fofTy
% EOF
%------------------------------------------------------------------------------