%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU797^2 : TPTP v6.1.0. Released v3.7.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n091.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 13:33:09 EDT 2014

% Result   : Timeout 300.11s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU797^2 : TPTP v6.1.0. Released v3.7.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n091.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 11:30:41 CDT 2014
% % CPUTime  : 300.11 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x13b1d88>, <kernel.DependentProduct object at 0x17e9128>) of role type named in_type
% Using role type
% Declaring in:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0x156a908>, <kernel.DependentProduct object at 0x17e9b90>) of role type named image1_type
% Using role type
% Declaring image1:(fofType->((fofType->fofType)->fofType))
% FOF formula (<kernel.Constant object at 0x13b14d0>, <kernel.Sort object at 0x1644dd0>) of role type named image1Equiv_type
% Using role type
% Declaring image1Equiv:Prop
% FOF formula (((eq Prop) image1Equiv) (forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), ((iff ((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))))) of role definition named image1Equiv
% A new definition: (((eq Prop) image1Equiv) (forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), ((iff ((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy))))))))
% Defined: image1Equiv:=(forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), ((iff ((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))))
% FOF formula (image1Equiv->(forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))->((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))))) of role conjecture named image1I
% Conjecture to prove = (image1Equiv->(forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))->((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(image1Equiv->(forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))->((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))))']
% Parameter fofType:Type.
% Parameter in:(fofType->(fofType->Prop)).
% Parameter image1:(fofType->((fofType->fofType)->fofType)).
% Definition image1Equiv:=(forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), ((iff ((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy))))))):Prop.
% Trying to prove (image1Equiv->(forall (A:fofType) (Xf:(fofType->fofType)) (Xx:fofType), (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))->((in Xx) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))))
% Found eq_ref00:=(eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->fofType)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->fofType)) a) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a) as proof of (((eq (fofType->fofType)) a) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a) as proof of (((eq (fofType->fofType)) a) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a) as proof of (((eq (fofType->fofType)) a) (fun (Xy:fofType)=> (Xf Xy)))
% Found eq_ref00:=(eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->fofType)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->fofType)) a) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eta_expansion0 fofType) a) as proof of (((eq (fofType->fofType)) a) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion fofType) fofType) a) as proof of (((eq (fofType->fofType)) a) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion fofType) fofType) a) as proof of (((eq (fofType->fofType)) a) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion fofType) fofType) a) as proof of (((eq (fofType->fofType)) a) (fun (Xy:fofType)=> (Xf Xy)))
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->fofType)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->fofType)) a) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a) as proof of (((eq (fofType->fofType)) a) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a) as proof of (((eq (fofType->fofType)) a) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a) as proof of (((eq (fofType->fofType)) a) (fun (Xy:fofType)=> (Xf Xy)))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) (fun (Xy:fofType)=> (Xf 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 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) (fun (Xy:fofType)=> (Xf 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 x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a:=Xx:fofType;b:=(Xf x1):fofType
% Found x4 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) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found x4:(((eq fofType) Xx) (Xf x1))
% Found x4 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) 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 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) 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 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) b) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) b) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) 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 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found eq_ref00:=(eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eta_expansion0 fofType) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf 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) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->fofType)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eta_expansion0 fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Found x4 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) 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) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eta_expansion_dep0 (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found x40:=(x4 (fun (x5:fofType)=> (P0 b))):((P0 b)->(P0 b))
% Found (x4 (fun (x5:fofType)=> (P0 b))) as proof of (P1 b)
% Found (x4 (fun (x5:fofType)=> (P0 b))) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->fofType)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->fofType)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eta_expansion_dep0 (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->fofType)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->fofType)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eta_expansion_dep0 (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf 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 x4:(((eq fofType) Xx) (Xf x1))
% Found x4 as proof of (((eq fofType) Xx) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xx)->(P0 Xx))
% Found (eq_ref00 P0) as proof of (P1 Xx)
% Found ((eq_ref0 Xx) P0) as proof of (P1 Xx)
% Found (((eq_ref fofType) Xx) P0) as proof of (P1 Xx)
% Found (((eq_ref fofType) Xx) P0) as proof of (P1 Xx)
% Found eq_ref00:=(eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) 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 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) b) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) b) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) b) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) b) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) b) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) 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_ref000:=(eq_ref00 P0):((P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))->(P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))
% Found (eq_ref00 P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf 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 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf 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 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found eq_ref00:=(eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found eq_ref00:=(eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found eq_ref00:=(eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found eq_ref00:=(eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) 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 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Found x4 as proof of (((eq fofType) Xx) b)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Found x4 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) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->fofType)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))->(P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))
% Found (eq_ref00 P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->fofType)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Found x4 as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eta_expansion0 fofType) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->fofType)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eta_expansion0 fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf 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 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eq_ref00:=(eq_ref0 ((image1 A) a)):(((eq fofType) ((image1 A) a)) ((image1 A) a))
% Found (eq_ref0 ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found ((eq_ref fofType) ((image1 A) a)) as proof of (((eq fofType) ((image1 A) a)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eta_expansion_dep0 (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Found x4 as proof of (((eq fofType) Xx) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% 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 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Found x4 as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->fofType)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found x40:=(x4 (fun (x5:fofType)=> (P0 b))):((P0 b)->(P0 b))
% Found (x4 (fun (x5:fofType)=> (P0 b))) as proof of (P1 b)
% Found (x4 (fun (x5:fofType)=> (P0 b))) as proof of (P1 b)
% Found x40:=(x4 (fun (x5:fofType)=> (P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))):((P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))->(P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))
% Found (x4 (fun (x5:fofType)=> (P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (x4 (fun (x5:fofType)=> (P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eta_expansion0 fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eta_expansion0 fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eta_expansion0 fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->fofType)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->fofType)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% 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 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eta_expansion0 fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found (((eta_expansion fofType) fofType) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->fofType)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->fofType)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) a)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found x40:=(x4 (fun (x5:fofType)=> (P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))):((P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))->(P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))
% Found (x4 (fun (x5:fofType)=> (P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (x4 (fun (x5:fofType)=> (P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->fofType)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eq_ref (fofType->fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found eq_ref00:=(eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Found x4 as proof of (((eq fofType) Xx) b)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) 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) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found eq_ref000:=(eq_ref00 P0):((P0 Xx)->(P0 Xx))
% Found (eq_ref00 P0) as proof of (P1 Xx)
% Found ((eq_ref0 Xx) P0) as proof of (P1 Xx)
% Found (((eq_ref fofType) Xx) P0) as proof of (P1 Xx)
% Found (((eq_ref fofType) Xx) P0) as proof of (P1 Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->fofType)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found ((eta_expansion_dep0 (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> fofType)) a0) as proof of (((eq (fofType->fofType)) a0) (fun (Xy:fofType)=> (Xf Xy)))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) 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 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) 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 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) 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 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) 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 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref000:=(eq_ref00 P0):((P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))->(P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))
% Found (eq_ref00 P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf 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 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) 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 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf 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 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) 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 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b:=(Xf x1):fofType;b0:=Xx:fofType
% Found x4 as proof of (((eq fofType) b0) b)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Found x4 as proof of (((eq fofType) Xx) b)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Found x4 as proof of (((eq fofType) Xx) b)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) 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 x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b:=(Xf x1):fofType;b0:=Xx:fofType
% Found x4 as proof of (((eq fofType) b0) b)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b:=(Xf x1):fofType;b0:=Xx:fofType
% Found x4 as proof of (((eq fofType) b0) b)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) 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 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 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))->(P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))
% Found (eq_ref00 P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found eq_ref000:=(eq_ref00 P00):((P00 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))->(P00 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))
% Found (eq_ref00 P00) as proof of (P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P00) as proof of (P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P00) as proof of (P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P00) as proof of (P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))->(P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))
% Found (eq_ref00 P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))->(P0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))))
% Found (eq_ref00 P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) P0) as proof of (P1 ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found x4:(((eq fofType) Xx) (Xf x1))
% Found x4 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 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf 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 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) 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 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a:=Xx:fofType;b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Found x4 as proof of (((eq fofType) Xx) b)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Found x4 as proof of (((eq fofType) Xx) b)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Found x4 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) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found eq_ref00:=(eq_ref0 a0):(((eq fofType) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% Found ((eq_ref fofType) a0) as proof of (((eq fofType) a0) b)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a0:=Xx:fofType;b:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a0:=Xx:fofType;b:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a0) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a0) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: b0:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found x40:=(x4 (fun (x5:fofType)=> (P0 b))):((P0 b)->(P0 b))
% Found (x4 (fun (x5:fofType)=> (P0 b))) as proof of (P1 b)
% Found (x4 (fun (x5:fofType)=> (P0 b))) as proof of (P1 b)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Found x4 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) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf 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 (((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) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a0:=Xx:fofType;b:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a0) b)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a0:=Xx:fofType;b:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a0) 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 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a0:=Xx:fofType;b:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found x4:(((eq fofType) Xx) (Xf x1))
% Instantiate: a0:=Xx:fofType;b:=(Xf x1):fofType
% Found x4 as proof of (((eq fofType) a0) b)
% Found x4:(((eq fofType) Xx) (Xf x1))
% Found x4 as proof of (((eq fofType) Xx) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found x5:(P0 b)
% Instantiate: b0:=b:fofType
% Found x5 as proof of (P1 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((image1 A) a))
% Found eq_ref00:=(eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))):(((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) ((image1 A) (fun (Xy:fofType)=> (Xf Xy))))
% Found (eq_ref0 ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) b0)
% Found ((eq_ref fofType) ((image1 A) (fun (Xy:fofType)=> (Xf Xy)))) as proof of (((eq fofType) ((image1 A) 
% EOF
%------------------------------------------------------------------------------