%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU794^2 : TPTP v6.1.0. Released v3.7.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n093.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:08 EDT 2014

% Result   : Timeout 300.10s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU794^2 : TPTP v6.1.0. Released v3.7.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n093.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:30:16 CDT 2014
% % CPUTime  : 300.10 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0xd6c560>, <kernel.DependentProduct object at 0xd8c878>) of role type named in_type
% Using role type
% Declaring in:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0xfe3950>, <kernel.DependentProduct object at 0xd8c518>) of role type named exu_type
% Using role type
% Declaring exu:((fofType->Prop)->Prop)
% FOF formula (((eq ((fofType->Prop)->Prop)) exu) (fun (Xphi:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))))) of role definition named exu
% A new definition: (((eq ((fofType->Prop)->Prop)) exu) (fun (Xphi:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))))
% Defined: exu:=(fun (Xphi:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy)))))))
% FOF formula (<kernel.Constant object at 0xd6c290>, <kernel.DependentProduct object at 0xd8c3b0>) of role type named subset_type
% Using role type
% Declaring subset:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0xd6c560>, <kernel.Sort object at 0xe3cea8>) of role type named subsetI1_type
% Using role type
% Declaring subsetI1:Prop
% FOF formula (((eq Prop) subsetI1) (forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((subset A) B)))) of role definition named subsetI1
% A new definition: (((eq Prop) subsetI1) (forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((subset A) B))))
% Defined: subsetI1:=(forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((subset A) B)))
% FOF formula (<kernel.Constant object at 0xd6c560>, <kernel.Sort object at 0xe3cea8>) of role type named setextsub_type
% Using role type
% Declaring setextsub:Prop
% FOF formula (((eq Prop) setextsub) (forall (A:fofType) (B:fofType), (((subset A) B)->(((subset B) A)->(((eq fofType) A) B))))) of role definition named setextsub
% A new definition: (((eq Prop) setextsub) (forall (A:fofType) (B:fofType), (((subset A) B)->(((subset B) A)->(((eq fofType) A) B)))))
% Defined: setextsub:=(forall (A:fofType) (B:fofType), (((subset A) B)->(((subset B) A)->(((eq fofType) A) B))))
% FOF formula (<kernel.Constant object at 0xd8c518>, <kernel.Sort object at 0xe3cea8>) of role type named image1Ex_type
% Using role type
% Declaring image1Ex:Prop
% FOF formula (((eq Prop) image1Ex) (forall (A:fofType) (Xf:(fofType->fofType)), ((ex fofType) (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy))))))))))) of role definition named image1Ex
% A new definition: (((eq Prop) image1Ex) (forall (A:fofType) (Xf:(fofType->fofType)), ((ex fofType) (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))))))))
% Defined: image1Ex:=(forall (A:fofType) (Xf:(fofType->fofType)), ((ex fofType) (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy))))))))))
% FOF formula (subsetI1->(setextsub->(image1Ex->(forall (A:fofType) (Xf:(fofType->fofType)), (exu (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy))))))))))))) of role conjecture named image1Ex1
% Conjecture to prove = (subsetI1->(setextsub->(image1Ex->(forall (A:fofType) (Xf:(fofType->fofType)), (exu (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy))))))))))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(subsetI1->(setextsub->(image1Ex->(forall (A:fofType) (Xf:(fofType->fofType)), (exu (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))))))))))']
% Parameter fofType:Type.
% Parameter in:(fofType->(fofType->Prop)).
% Definition exu:=(fun (Xphi:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xphi Xx)) (forall (Xy:fofType), ((Xphi Xy)->(((eq fofType) Xx) Xy))))))):((fofType->Prop)->Prop).
% Parameter subset:(fofType->(fofType->Prop)).
% Definition subsetI1:=(forall (A:fofType) (B:fofType), ((forall (Xx:fofType), (((in Xx) A)->((in Xx) B)))->((subset A) B))):Prop.
% Definition setextsub:=(forall (A:fofType) (B:fofType), (((subset A) B)->(((subset B) A)->(((eq fofType) A) B)))):Prop.
% Definition image1Ex:=(forall (A:fofType) (Xf:(fofType->fofType)), ((ex fofType) (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))))))):Prop.
% Trying to prove (subsetI1->(setextsub->(image1Ex->(forall (A:fofType) (Xf:(fofType->fofType)), (exu (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))))))))))
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) Xy)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) Xy)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) Xy)
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((eq_ref fofType) x2)) as proof of (((eq fofType) x2) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of ((P x2)->(P Xy))
% Found ((eq_ref0 x2) P) as proof of ((P x2)->(P Xy))
% Found (((eq_ref fofType) x2) P) as proof of ((P x2)->(P Xy))
% Found (((eq_ref fofType) x2) P) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> (((eq_ref fofType) x2) P)) as proof of (((eq fofType) x2) Xy)
% Found x3:(P x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x3:(P x2))=> x3) as proof of (P Xy)
% Found (fun (P:(fofType->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop)) (x3:(P x2))=> x3) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found (eq_sym000 ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found ((eq_sym00 x2) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found (((eq_sym0 Xy) x2) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x2) Xy)
% Found x4:((in Xx) x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x4:((in Xx) x2))=> x4) as proof of ((in Xx) Xy)
% Found (fun (Xx:fofType) (x4:((in Xx) x2))=> x4) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType) (x4:((in Xx) x2))=> x4) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) Xy)))
% Found (x30 (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found ((x3 Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) Xy)->((in Xx) Xy))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found ((eq_ref0 Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (((eq_ref fofType) Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) Xy)->((in Xx) x2)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x3 x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x010 (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (((eq fofType) x2) Xy)
% Found (((x01 Xy) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (((eq fofType) x2) Xy)
% Found ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (((eq fofType) x2) Xy)
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_ref fofType) x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_ref fofType) x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_ref fofType) x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((eq_ref fofType) x2)) as proof of (((eq fofType) x2) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of ((P x2)->(P Xy))
% Found ((eq_ref0 x2) P) as proof of ((P x2)->(P Xy))
% Found (((eq_ref fofType) x2) P) as proof of ((P x2)->(P Xy))
% Found (((eq_ref fofType) x2) P) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x2) P)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> (((eq_ref fofType) x2) P)) as proof of (((eq fofType) x2) Xy)
% Found x3:(P x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x3:(P x2))=> x3) as proof of (P Xy)
% Found (fun (P:(fofType->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop)) (x3:(P x2))=> x3) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop)) (x3:(P x2))=> x3) as proof of (((eq fofType) x2) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref fofType) x2) P) as proof of (P0 x2)
% Found (((eq_ref fofType) x2) P) as proof of (P0 x2)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref fofType) x2) P) as proof of (P0 x2)
% Found (((eq_ref fofType) x2) P) as proof of (P0 x2)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found ((fun (x3:(((eq fofType) Xy) x2))=> ((eq_sym000 x3) P)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found ((fun (x3:(((eq fofType) Xy) x2))=> (((eq_sym00 x2) x3) P)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found ((fun (x3:(((eq fofType) Xy) x2))=> ((((eq_sym0 Xy) x2) x3) P)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found ((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) P)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) P)) ((eq_ref fofType) Xy))) as proof of ((P x2)->(P Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> ((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) P)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) Xy) x2))=> ((eq_sym000 x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) Xy) x2))=> (((eq_sym00 x2) x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) Xy) x2))=> ((((eq_sym0 Xy) x2) x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P))) as proof of ((P x2)->(P Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> (((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) Xy)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) Xy)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) Xy)
% Found (fun (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2)) as proof of (((eq fofType) x2) Xy)
% Found (fun (x3:(((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2)) as proof of ((((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy))->(((eq fofType) x2) Xy))
% Found (fun (x3:(((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2)) as proof of ((((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))->((((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy))->(((eq fofType) x2) Xy)))
% Found (and_rect00 (fun (x3:(((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2))) as proof of (((eq fofType) x2) Xy)
% Found ((and_rect0 (((eq fofType) x2) Xy)) (fun (x3:(((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2))) as proof of (((eq fofType) x2) Xy)
% Found (((fun (P:Type) (x3:((((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))->((((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy))->P)))=> (((((and_rect (((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy))) P) x3) x000)) (((eq fofType) x2) Xy)) (fun (x3:(((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_trans0000 ((eq_ref fofType) x2)) ((eq_ref fofType) b)) as proof of (((eq fofType) x2) Xy)
% Found (((eq_trans000 Xy) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found ((((fun (b:fofType)=> ((eq_trans00 b) Xy)) Xy) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found ((((fun (b:fofType)=> (((eq_trans0 x2) b) Xy)) Xy) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found ((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found (eq_sym000 ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_sym00 x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (((eq_sym0 Xy) x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) Xy)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found ((x3 Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) Xy)->((in Xx) Xy))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found ((eq_ref0 Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (((eq_ref fofType) Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) Xy)->((in Xx) x2)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x3 x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x0100 (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P x2)->(P Xy))
% Found ((x0100 (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((x010 x3) x4) P)) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> ((((x01 Xy) x3) x4) P)) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) P)) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) P)) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of ((P x2)->(P Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) P)) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) Xy)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found ((x3 Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) Xy)->((in Xx) Xy))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found ((eq_ref0 Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (((eq_ref fofType) Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) Xy)->((in Xx) x2)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x3 x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x0100 (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((x0100 (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((x010 x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> ((((x01 Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P))) as proof of ((P x2)->(P Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))):(((eq Prop) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy))))
% Found (eq_ref0 (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) as proof of (((eq Prop) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) as proof of (((eq Prop) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) as proof of (((eq Prop) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) as proof of (((eq Prop) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) b)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) Xy)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found ((x3 Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) Xy)->((in Xx) Xy))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found ((eq_ref0 Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (((eq_ref fofType) Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) Xy)->((in Xx) x2)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x3 x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x010 (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (((x01 Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of (((eq fofType) x2) Xy)
% Found x3:((in Xx0) x2)
% Instantiate: x2:=A:fofType;x4:=Xx0:fofType
% Found x3 as proof of ((in x4) A)
% Found x3:((in Xx0) x2)
% Instantiate: x2:=A:fofType;x4:=Xx0:fofType
% Found x3 as proof of ((in x4) A)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref fofType) x2) P) as proof of (P0 x2)
% Found (((eq_ref fofType) x2) P) as proof of (P0 x2)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref fofType) x2) P) as proof of (P0 x2)
% Found (((eq_ref fofType) x2) P) as proof of (P0 x2)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found ((fun (x3:(((eq fofType) Xy) x2))=> ((eq_sym000 x3) P)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found ((fun (x3:(((eq fofType) Xy) x2))=> (((eq_sym00 x2) x3) P)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found ((fun (x3:(((eq fofType) Xy) x2))=> ((((eq_sym0 Xy) x2) x3) P)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found ((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) P)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) P)) ((eq_ref fofType) Xy))) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) P)) ((eq_ref fofType) Xy))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> ((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) P)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) Xy) x2))=> ((eq_sym000 x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) Xy) x2))=> (((eq_sym00 x2) x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) Xy) x2))=> ((((eq_sym0 Xy) x2) x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P))) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> (((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_trans0000 ((eq_ref fofType) x2)) ((eq_ref fofType) b)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (((eq_trans000 Xy) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((fun (b:fofType)=> ((eq_trans00 b) Xy)) Xy) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((fun (b:fofType)=> (((eq_trans0 x2) b) Xy)) Xy) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_trans00000 ((eq_ref fofType) x2)) ((eq_ref fofType) b)) as proof of ((P x2)->(P Xy))
% Found ((eq_trans00000 ((eq_ref fofType) x2)) ((eq_ref fofType) b)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) x2) b)) (x4:(((eq fofType) b) Xy))=> (((eq_trans0000 x3) x4) P)) ((eq_ref fofType) x2)) ((eq_ref fofType) b)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> ((((eq_trans000 Xy) x3) x4) P)) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((eq_trans00 b) Xy)) Xy) x3) x4) P)) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> (((eq_trans0 x2) b) Xy)) Xy) x3) x4) P)) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) x3) x4) P)) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) x3) x4) P)) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy))) as proof of ((P x2)->(P Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> (((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) x3) x4) P)) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref fofType) x2) P) as proof of (P0 x2)
% Found (((eq_ref fofType) x2) P) as proof of (P0 x2)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found (((eq_trans00000 ((eq_ref fofType) x2)) ((eq_ref fofType) b)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((eq_trans00000 ((eq_ref fofType) x2)) ((eq_ref fofType) b)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:(((eq fofType) x2) b)) (x4:(((eq fofType) b) Xy))=> (((eq_trans0000 x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) ((eq_ref fofType) x2)) ((eq_ref fofType) b)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> ((((eq_trans000 Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((eq_trans00 b) Xy)) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> (((eq_trans0 x2) b) Xy)) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P))) as proof of ((P x2)->(P Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> ((((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) Xy)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found ((x3 Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) Xy)->((in Xx) Xy))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found ((eq_ref0 Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (((eq_ref fofType) Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) Xy)->((in Xx) x2)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x3 x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x0100 (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P x2)->(P Xy))
% Found ((x0100 (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((x010 x3) x4) P)) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> ((((x01 Xy) x3) x4) P)) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) P)) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) P)) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) P)) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) P)) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of (((eq fofType) x2) Xy)
% Found x4:((in Xx) x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x4:((in Xx) x2))=> x4) as proof of ((in Xx) Xy)
% Found (fun (Xx:fofType) (x4:((in Xx) x2))=> x4) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType) (x4:((in Xx) x2))=> x4) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) Xy)))
% Found (x30 (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found ((x3 Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) Xy)->((in Xx) Xy))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found ((eq_ref0 Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (((eq_ref fofType) Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) Xy)->((in Xx) x2)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x3 x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x0100 (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((x0100 (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((x010 x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> ((((x01 Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P))) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found eq_ref00:=(eq_ref0 (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))
% Found (eq_ref0 (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) A)) (((eq fofType) Xx0) (Xf x)))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) A)) (((eq fofType) Xx0) (Xf x)))))
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found eq_ref00:=(eq_ref0 (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))
% Found (eq_ref0 (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) A)) (((eq fofType) Xx0) (Xf x)))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) A)) (((eq fofType) Xx0) (Xf x)))))
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_trans00000 ((eq_ref fofType) x2)) ((eq_ref fofType) b)) as proof of ((P x2)->(P Xy))
% Found ((eq_trans00000 ((eq_ref fofType) x2)) ((eq_ref fofType) b)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) x2) b)) (x4:(((eq fofType) b) Xy))=> (((eq_trans0000 x3) x4) P)) ((eq_ref fofType) x2)) ((eq_ref fofType) b)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> ((((eq_trans000 Xy) x3) x4) P)) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((eq_trans00 b) Xy)) Xy) x3) x4) P)) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> (((eq_trans0 x2) b) Xy)) Xy) x3) x4) P)) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) x3) x4) P)) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) x3) x4) P)) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy))) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) x3) x4) P)) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> (((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) x3) x4) P)) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref fofType) x2) P) as proof of (P0 x2)
% Found (((eq_ref fofType) x2) P) as proof of (P0 x2)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found (((eq_trans00000 ((eq_ref fofType) x2)) ((eq_ref fofType) b)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((eq_trans00000 ((eq_ref fofType) x2)) ((eq_ref fofType) b)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:(((eq fofType) x2) b)) (x4:(((eq fofType) b) Xy))=> (((eq_trans0000 x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) ((eq_ref fofType) x2)) ((eq_ref fofType) b)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> ((((eq_trans000 Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((eq_trans00 b) Xy)) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> (((eq_trans0 x2) b) Xy)) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P))) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> ((((fun (x3:(((eq fofType) x2) Xy)) (x4:(((eq fofType) Xy) Xy))=> (((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P))) as proof of (((eq fofType) x2) Xy)
% Found x3:(P0 x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x3:(P0 x2))=> x3) as proof of (P0 Xy)
% Found (fun (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of (((eq fofType) x2) Xy)
% Found x5:((in Xx) A0)
% Instantiate: x2:=A0:fofType
% Found (fun (x5:((in Xx) A0))=> x5) as proof of ((in Xx) x2)
% Found (fun (Xx:fofType) (x5:((in Xx) A0))=> x5) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType) (x5:((in Xx) A0))=> x5) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) x2)))
% Found (x40 (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found ((x4 x2) (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) A0)))
% Found (x40 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found ((x4 A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x7:(((eq fofType) Xx0) (Xf x4))
% Instantiate: x2:=Xx0:fofType;b:=(Xf x4):fofType
% Found x7 as proof of (((eq fofType) x2) b)
% Found x5:((in Xx) A0)
% Instantiate: x2:=A0:fofType
% Found (fun (x5:((in Xx) A0))=> x5) as proof of ((in Xx) x2)
% Found (fun (Xx:fofType) (x5:((in Xx) A0))=> x5) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType) (x5:((in Xx) A0))=> x5) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) x2)))
% Found (x40 (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found ((x4 x2) (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) A0)))
% Found (x40 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found ((x4 A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found x3:((in Xx0) x2)
% Instantiate: x2:=A:fofType;x4:=Xx0:fofType
% Found x3 as proof of ((in x4) A)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x7:((in Xx) A0)
% Instantiate: x2:=A0:fofType
% Found (fun (x7:((in Xx) A0))=> x7) as proof of ((in Xx) x2)
% Found (fun (Xx:fofType) (x7:((in Xx) A0))=> x7) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType) (x7:((in Xx) A0))=> x7) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) x2)))
% Found (x60 (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) x2)
% Found ((x6 x2) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) x2)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) A0)))
% Found (x60 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found ((x6 A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x7:(((eq fofType) Xx0) (Xf x4))
% Instantiate: x2:=Xx0:fofType;b:=(Xf x4):fofType
% Found x7 as proof of (((eq fofType) x2) b)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) Xy)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found ((x3 Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) Xy)->((in Xx) Xy))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found ((eq_ref0 Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (((eq_ref fofType) Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) Xy)->((in Xx) x2)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x3 x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x010 (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (((eq fofType) x2) Xy)
% Found (((x01 Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (((eq fofType) x2) Xy)
% Found ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (((eq fofType) x2) Xy)
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of (((eq fofType) x2) Xy)
% Found x3:((in Xx0) x2)
% Instantiate: x2:=A:fofType;x4:=Xx0:fofType
% Found x3 as proof of ((in x4) A)
% Found x3:((in Xx0) x2)
% Instantiate: x2:=A:fofType;x4:=Xx0:fofType
% Found x3 as proof of ((in x4) A)
% Found eq_ref00:=(eq_ref0 (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))->((in Xx0) x2))):(((eq Prop) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))->((in Xx0) x2))) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))->((in Xx0) x2)))
% Found (eq_ref0 (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))->((in Xx0) x2))) as proof of (((eq Prop) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))->((in Xx0) x2))) b)
% Found ((eq_ref Prop) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))->((in Xx0) x2))) as proof of (((eq Prop) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))->((in Xx0) x2))) b)
% Found ((eq_ref Prop) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))->((in Xx0) x2))) as proof of (((eq Prop) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))->((in Xx0) x2))) b)
% Found ((eq_ref Prop) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))->((in Xx0) x2))) as proof of (((eq Prop) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))->((in Xx0) x2))) b)
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_ref fofType) x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_ref fofType) x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_ref fofType) x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((eq_ref fofType) x2)) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) Xy)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) Xy)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) Xy)
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((eq_ref fofType) x2)) as proof of (((eq fofType) x2) Xy)
% Found x3:(P0 x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x3:(P0 x2))=> x3) as proof of (P0 Xy)
% Found (fun (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of (((eq fofType) x2) Xy)
% Found eq_ref000:=(eq_ref00 P0):((P0 x2)->(P0 x2))
% Found (eq_ref00 P0) as proof of ((P0 x2)->(P0 Xy))
% Found ((eq_ref0 x2) P0) as proof of ((P0 x2)->(P0 Xy))
% Found (((eq_ref fofType) x2) P0) as proof of ((P0 x2)->(P0 Xy))
% Found (((eq_ref fofType) x2) P0) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (P0:(fofType->Prop))=> (((eq_ref fofType) x2) P0)) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (P0:(fofType->Prop))=> (((eq_ref fofType) x2) P0)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P0:(fofType->Prop))=> (((eq_ref fofType) x2) P0)) as proof of (((eq fofType) x2) Xy)
% Found x7:((in Xx) A0)
% Instantiate: x2:=A0:fofType
% Found (fun (x7:((in Xx) A0))=> x7) as proof of ((in Xx) x2)
% Found (fun (Xx:fofType) (x7:((in Xx) A0))=> x7) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType) (x7:((in Xx) A0))=> x7) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) x2)))
% Found (x60 (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) x2)
% Found ((x6 x2) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) x2)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) A0)))
% Found (x60 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found ((x6 A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found x3:(P x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x3:(P x2))=> x3) as proof of (P Xy)
% Found (fun (P:(fofType->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop)) (x3:(P x2))=> x3) as proof of (((eq fofType) x2) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of ((P x2)->(P Xy))
% Found ((eq_ref0 x2) P) as proof of ((P x2)->(P Xy))
% Found (((eq_ref fofType) x2) P) as proof of ((P x2)->(P Xy))
% Found (((eq_ref fofType) x2) P) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> (((eq_ref fofType) x2) P)) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 (((in Xx0) x2)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))))):(((eq Prop) (((in Xx0) x2)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))))) (((in Xx0) x2)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))
% Found (eq_ref0 (((in Xx0) x2)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))))) as proof of (((eq Prop) (((in Xx0) x2)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))))) b)
% Found ((eq_ref Prop) (((in Xx0) x2)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))))) as proof of (((eq Prop) (((in Xx0) x2)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))))) b)
% Found ((eq_ref Prop) (((in Xx0) x2)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))))) as proof of (((eq Prop) (((in Xx0) x2)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))))) b)
% Found ((eq_ref Prop) (((in Xx0) x2)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))))) as proof of (((eq Prop) (((in Xx0) x2)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))))) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% 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) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) Xy)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) Xy)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) Xy)
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((eq_ref fofType) x2)) as proof of (((eq fofType) x2) Xy)
% Found x9:((in Xx) A0)
% Instantiate: x2:=A0:fofType
% Found (fun (x9:((in Xx) A0))=> x9) as proof of ((in Xx) x2)
% Found (fun (Xx:fofType) (x9:((in Xx) A0))=> x9) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType) (x9:((in Xx) A0))=> x9) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) x2)))
% Found (x80 (fun (Xx:fofType) (x9:((in Xx) A0))=> x9)) as proof of ((subset A0) x2)
% Found ((x8 x2) (fun (Xx:fofType) (x9:((in Xx) A0))=> x9)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x9:((in Xx) A0))=> x9)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x9:((in Xx) A0))=> x9)) as proof of ((subset A0) x2)
% Found x70:=(x7 (fun (x9:fofType)=> ((in Xx) x2))):(((in Xx) x2)->((in Xx) x2))
% Found (x7 (fun (x9:fofType)=> ((in Xx) x2))) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (x7 (fun (x9:fofType)=> ((in Xx) x2)))) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (x7 (fun (x9:fofType)=> ((in Xx) x2)))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) A0)))
% Found (x80 (fun (Xx:fofType)=> (x7 (fun (x9:fofType)=> ((in Xx) x2))))) as proof of ((subset x2) A0)
% Found ((x8 A0) (fun (Xx:fofType)=> (x7 (fun (x9:fofType)=> ((in Xx) x2))))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (x7 (fun (x9:fofType)=> ((in Xx) x2))))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (x7 (fun (x9:fofType)=> ((in Xx) x2))))) as proof of ((subset x2) A0)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: a:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of a
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x1:image1Ex
% Instantiate: b:=(forall (A:fofType) (Xf:(fofType->fofType)), ((ex fofType) (fun (B:fofType)=> (forall (Xx:fofType), ((iff ((in Xx) B)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx) (Xf Xy)))))))))):Prop
% Found x1 as proof of a
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 x2)->(P0 x2))
% Found (eq_ref00 P0) as proof of ((P0 x2)->(P0 Xy))
% Found ((eq_ref0 x2) P0) as proof of ((P0 x2)->(P0 Xy))
% Found (((eq_ref fofType) x2) P0) as proof of ((P0 x2)->(P0 Xy))
% Found (((eq_ref fofType) x2) P0) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (P0:(fofType->Prop))=> (((eq_ref fofType) x2) P0)) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P0:(fofType->Prop))=> (((eq_ref fofType) x2) P0)) as proof of (((eq fofType) x2) Xy)
% Found x3:(P0 x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x3:(P0 x2))=> x3) as proof of (P0 Xy)
% Found (fun (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 (((eq fofType) Xx0) (Xf x4))):(((eq Prop) (((eq fofType) Xx0) (Xf x4))) (((eq fofType) Xx0) (Xf x4)))
% Found (eq_ref0 (((eq fofType) Xx0) (Xf x4))) as proof of (((eq Prop) (((eq fofType) Xx0) (Xf x4))) b)
% Found ((eq_ref Prop) (((eq fofType) Xx0) (Xf x4))) as proof of (((eq Prop) (((eq fofType) Xx0) (Xf x4))) b)
% Found ((eq_ref Prop) (((eq fofType) Xx0) (Xf x4))) as proof of (((eq Prop) (((eq fofType) Xx0) (Xf x4))) b)
% Found ((eq_ref Prop) (((eq fofType) Xx0) (Xf x4))) as proof of (((eq Prop) (((eq fofType) Xx0) (Xf x4))) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found (eq_sym100 ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_sym10 x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (((eq_sym1 Xy) x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found (eq_sym000 ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found ((eq_sym00 x2) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found (((eq_sym0 Xy) x2) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x2) Xy)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy)))))) (fun (x:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) x)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x) Xy))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy)))))) b)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:fofType), ((iff ((in Xx0) x2)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:fofType), ((iff ((in Xx0) x2)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:fofType), ((iff ((in Xx0) x2)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:fofType), ((iff ((in Xx0) x2)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and (forall (Xx0:fofType), ((iff ((in Xx0) x)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x) Xy))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:fofType), ((iff ((in Xx0) x2)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:fofType), ((iff ((in Xx0) x2)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:fofType), ((iff ((in Xx0) x2)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:fofType), ((iff ((in Xx0) x2)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and (forall (Xx0:fofType), ((iff ((in Xx0) x)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x) Xy))))))
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found eq_ref00:=(eq_ref0 (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))
% Found (eq_ref0 (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% 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 x3:(P0 x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x3:(P0 x2))=> x3) as proof of (P0 Xy)
% Found (fun (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of (((eq fofType) x2) Xy)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x9:((in Xx) A0)
% Instantiate: x2:=A0:fofType
% Found (fun (x9:((in Xx) A0))=> x9) as proof of ((in Xx) x2)
% Found (fun (Xx:fofType) (x9:((in Xx) A0))=> x9) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType) (x9:((in Xx) A0))=> x9) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) x2)))
% Found (x80 (fun (Xx:fofType) (x9:((in Xx) A0))=> x9)) as proof of ((subset A0) x2)
% Found ((x8 x2) (fun (Xx:fofType) (x9:((in Xx) A0))=> x9)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x9:((in Xx) A0))=> x9)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x9:((in Xx) A0))=> x9)) as proof of ((subset A0) x2)
% Found x70:=(x7 (fun (x9:fofType)=> ((in Xx) x2))):(((in Xx) x2)->((in Xx) x2))
% Found (x7 (fun (x9:fofType)=> ((in Xx) x2))) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (x7 (fun (x9:fofType)=> ((in Xx) x2)))) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (x7 (fun (x9:fofType)=> ((in Xx) x2)))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) A0)))
% Found (x80 (fun (Xx:fofType)=> (x7 (fun (x9:fofType)=> ((in Xx) x2))))) as proof of ((subset x2) A0)
% Found ((x8 A0) (fun (Xx:fofType)=> (x7 (fun (x9:fofType)=> ((in Xx) x2))))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (x7 (fun (x9:fofType)=> ((in Xx) x2))))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (x7 (fun (x9:fofType)=> ((in Xx) x2))))) as proof of ((subset x2) A0)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found (eq_sym100 ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found ((eq_sym10 x2) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found (((eq_sym1 Xy) x2) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 (((eq fofType) Xx0) (Xf x4))):(((eq Prop) (((eq fofType) Xx0) (Xf x4))) (((eq fofType) Xx0) (Xf x4)))
% Found (eq_ref0 (((eq fofType) Xx0) (Xf x4))) as proof of (((eq Prop) (((eq fofType) Xx0) (Xf x4))) b)
% Found ((eq_ref Prop) (((eq fofType) Xx0) (Xf x4))) as proof of (((eq Prop) (((eq fofType) Xx0) (Xf x4))) b)
% Found ((eq_ref Prop) (((eq fofType) Xx0) (Xf x4))) as proof of (((eq Prop) (((eq fofType) Xx0) (Xf x4))) b)
% Found ((eq_ref Prop) (((eq fofType) Xx0) (Xf x4))) as proof of (((eq Prop) (((eq fofType) Xx0) (Xf x4))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) Xy)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found ((x3 Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) Xy)->((in Xx) Xy))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found ((eq_ref0 Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (((eq_ref fofType) Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) Xy)->((in Xx) x2)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x3 x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x010 (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (((eq fofType) x2) Xy)
% Found (((x01 Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (((eq fofType) x2) Xy)
% Found ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (((eq fofType) x2) Xy)
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of (((eq fofType) x2) Xy)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) A)) (((eq fofType) Xx0) (Xf x)))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) A)) (((eq fofType) Xx0) (Xf x)))))
% Found x4:((in Xx) x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x4:((in Xx) x2))=> x4) as proof of ((in Xx) Xy)
% Found (fun (Xx:fofType) (x4:((in Xx) x2))=> x4) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType) (x4:((in Xx) x2))=> x4) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) Xy)))
% Found (x30 (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found ((x3 Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) Xy)->((in Xx) Xy))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found ((eq_ref0 Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (((eq_ref fofType) Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) Xy)->((in Xx) x2)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x3 x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x010 (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (((x01 Xy) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of (((eq fofType) x2) Xy)
% Found x5:(P Xx0)
% Instantiate: A0:=Xx0:fofType
% Found x5 as proof of (P0 A0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% 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 x5:(P Xx0)
% Instantiate: A0:=Xx0:fofType
% Found x5 as proof of (P0 A0)
% Found x5:(P Xx0)
% Instantiate: b:=Xx0:fofType
% Found x5 as proof of (P0 b)
% Found x5:(P Xx0)
% Instantiate: b:=Xx0:fofType
% Found x5 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found eq_ref00:=(eq_ref0 (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))
% Found (eq_ref0 (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))
% Found (eq_ref0 (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy))))) b)
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_ref fofType) x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_ref fofType) x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_ref fofType) x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((eq_ref fofType) x2)) as proof of (((eq fofType) x2) Xy)
% Found eq_ref000:=(eq_ref00 P0):((P0 x2)->(P0 x2))
% Found (eq_ref00 P0) as proof of (P1 x2)
% Found ((eq_ref0 x2) P0) as proof of (P1 x2)
% Found (((eq_ref fofType) x2) P0) as proof of (P1 x2)
% Found (((eq_ref fofType) x2) P0) as proof of (P1 x2)
% Found x3:(P x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x3:(P x2))=> x3) as proof of (P Xy)
% Found (fun (P:(fofType->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop)) (x3:(P x2))=> x3) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop)) (x3:(P x2))=> x3) as proof of (((eq fofType) x2) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of ((P x2)->(P Xy))
% Found ((eq_ref0 x2) P) as proof of ((P x2)->(P Xy))
% Found (((eq_ref fofType) x2) P) as proof of ((P x2)->(P Xy))
% Found (((eq_ref fofType) x2) P) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x2) P)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> (((eq_ref fofType) x2) P)) as proof of (((eq fofType) x2) Xy)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref fofType) x2) P) as proof of (P0 x2)
% Found (((eq_ref fofType) x2) P) as proof of (P0 x2)
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref fofType) x2) P) as proof of (P0 x2)
% Found (((eq_ref fofType) x2) P) as proof of (P0 x2)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((eq_sym0000 ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) Xy) x2))=> ((eq_sym000 x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) Xy) x2))=> (((eq_sym00 x2) x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) Xy) x2))=> ((((eq_sym0 Xy) x2) x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P))) as proof of ((P x2)->(P Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> (((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) (fun (x5:fofType)=> ((P x2)->(P x5))))) ((eq_ref fofType) Xy)) (((eq_ref fofType) x2) P))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found (eq_sym0000 ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found ((fun (x3:(((eq fofType) Xy) x2))=> ((eq_sym000 x3) P)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found ((fun (x3:(((eq fofType) Xy) x2))=> (((eq_sym00 x2) x3) P)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found ((fun (x3:(((eq fofType) Xy) x2))=> ((((eq_sym0 Xy) x2) x3) P)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found ((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) P)) ((eq_ref fofType) Xy)) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) P)) ((eq_ref fofType) Xy))) as proof of ((P x2)->(P Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> ((fun (x3:(((eq fofType) Xy) x2))=> (((((eq_sym fofType) Xy) x2) x3) P)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) A)) (((eq fofType) Xx0) (Xf x)))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) A)) (((eq fofType) Xx0) (Xf x)))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) A)) (((eq fofType) Xx0) (Xf x)))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) (((eq fofType) Xx0) (Xf x4))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) A)) (((eq fofType) Xx0) (Xf x)))))
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) Xy)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found ((x3 Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) Xy)->((in Xx) Xy))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found ((eq_ref0 Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (((eq_ref fofType) Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) Xy)->((in Xx) x2)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x3 x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x010 (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (((eq fofType) x2) Xy)
% Found (((x01 Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (((eq fofType) x2) Xy)
% Found ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (((eq fofType) x2) Xy)
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found x7:(((eq fofType) Xx0) (Xf x4))
% Instantiate: x2:=(Xf x4):fofType;b:=Xx0:fofType
% Found x7 as proof of (((eq fofType) b) x2)
% Found x7:(((eq fofType) Xx0) (Xf x4))
% Instantiate: a:=Xx0:fofType;b:=(Xf x4):fofType
% Found x7 as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) Xy)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) Xy)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) Xy)
% Found (fun (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2)) as proof of (((eq fofType) x2) Xy)
% Found (fun (x3:(((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2)) as proof of ((((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy))->(((eq fofType) x2) Xy))
% Found (fun (x3:(((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2)) as proof of ((((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))->((((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy))->(((eq fofType) x2) Xy)))
% Found (and_rect00 (fun (x3:(((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2))) as proof of (((eq fofType) x2) Xy)
% Found ((and_rect0 (((eq fofType) x2) Xy)) (fun (x3:(((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2))) as proof of (((eq fofType) x2) Xy)
% Found (((fun (P:Type) (x3:((((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))->((((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy))->P)))=> (((((and_rect (((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy))) P) x3) x000)) (((eq fofType) x2) Xy)) (fun (x3:(((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found x5:(P Xx0)
% Instantiate: A0:=Xx0:fofType
% Found x5 as proof of (P0 A0)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xy)
% Found ((eq_trans0000 ((eq_ref fofType) x2)) ((eq_ref fofType) b)) as proof of (((eq fofType) x2) Xy)
% Found (((eq_trans000 Xy) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found ((((fun (b:fofType)=> ((eq_trans00 b) Xy)) Xy) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found ((((fun (b:fofType)=> (((eq_trans0 x2) b) Xy)) Xy) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found ((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy)) as proof of (((eq fofType) x2) Xy)
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((fun (b:fofType)=> ((((eq_trans fofType) x2) b) Xy)) Xy) ((eq_ref fofType) x2)) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) Xy)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) Xy)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) Xy)
% Found (fun (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2)) as proof of (((eq fofType) x2) Xy)
% Found (fun (x3:(((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2)) as proof of ((((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy))->(((eq fofType) x2) Xy))
% Found (fun (x3:(((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2)) as proof of ((((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))->((((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy))->(((eq fofType) x2) Xy)))
% Found (and_rect00 (fun (x3:(((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2))) as proof of (((eq fofType) x2) Xy)
% Found ((and_rect0 (((eq fofType) x2) Xy)) (fun (x3:(((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2))) as proof of (((eq fofType) x2) Xy)
% Found (((fun (P0:Type) (x3:((((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))->((((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy))->P0)))=> (((((and_rect (((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy))) P0) x3) x000)) (((eq fofType) x2) Xy)) (fun (x3:(((in Xx) Xy)->((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))) (x4:(((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))->((in Xx) Xy)))=> ((eq_ref fofType) x2))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x2)
% 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 x5:(P Xx0)
% Instantiate: A0:=Xx0:fofType
% Found x5 as proof of (P0 A0)
% Found x5:(P Xx0)
% Instantiate: b:=Xx0:fofType
% Found x5 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found x5:(P0 b)
% Instantiate: b:=Xx0:fofType
% Found (fun (x5:(P0 b))=> x5) as proof of (P0 Xx0)
% Found (fun (P0:(fofType->Prop)) (x5:(P0 b))=> x5) as proof of ((P0 b)->(P0 Xx0))
% Found (fun (P0:(fofType->Prop)) (x5:(P0 b))=> x5) as proof of (P b)
% Found x5:(P Xx0)
% Instantiate: b:=Xx0:fofType
% Found x5 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found x5:(P0 b)
% Instantiate: b:=Xx0:fofType
% Found (fun (x5:(P0 b))=> x5) as proof of (P0 Xx0)
% Found (fun (P0:(fofType->Prop)) (x5:(P0 b))=> x5) as proof of ((P0 b)->(P0 Xx0))
% Found (fun (P0:(fofType->Prop)) (x5:(P0 b))=> x5) as proof of (P b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) (Xf x4))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (Xf x4))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (Xf x4))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (Xf x4))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b0)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b0)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b0)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found x3:((in Xx0) x2)
% Instantiate: x2:=A:fofType;x4:=Xx0:fofType
% Found x3 as proof of ((in x4) A)
% Found x5:((in Xx) A0)
% Instantiate: x2:=A0:fofType
% Found (fun (x5:((in Xx) A0))=> x5) as proof of ((in Xx) x2)
% Found (fun (Xx:fofType) (x5:((in Xx) A0))=> x5) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType) (x5:((in Xx) A0))=> x5) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) x2)))
% Found (x40 (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found ((x4 x2) (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) A0)))
% Found (x40 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found ((x4 A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% 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 x3:((in Xx0) x2)
% Instantiate: x2:=A:fofType;x4:=Xx0:fofType
% Found x3 as proof of ((in x4) A)
% Found x5:(P (Xf x4))
% Instantiate: b:=(Xf x4):fofType
% Found x5 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_ref fofType) x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_ref fofType) x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_ref fofType) x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((eq_ref fofType) x2)) as proof of (((eq fofType) x2) Xy)
% Found x5:(P (Xf x4))
% Instantiate: b:=(Xf x4):fofType
% Found x5 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) Xy)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found ((x3 Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) Xy)->((in Xx) Xy))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found ((eq_ref0 Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (((eq_ref fofType) Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) Xy)->((in Xx) x2)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x3 x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x010 (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (((eq fofType) x2) Xy)
% Found (((x01 Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (((eq fofType) x2) Xy)
% Found ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (((eq fofType) x2) Xy)
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) (Xf x4))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (Xf x4))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (Xf x4))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (Xf x4))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b0)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b0)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b0)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b0)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found x5:(P (Xf x4))
% Instantiate: A0:=(Xf x4):fofType
% Found x5 as proof of (P0 A0)
% Found eq_ref000:=(eq_ref00 P0):((P0 x2)->(P0 x2))
% Found (eq_ref00 P0) as proof of ((P0 x2)->(P0 Xy))
% Found ((eq_ref0 x2) P0) as proof of ((P0 x2)->(P0 Xy))
% Found (((eq_ref fofType) x2) P0) as proof of ((P0 x2)->(P0 Xy))
% Found (((eq_ref fofType) x2) P0) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (P0:(fofType->Prop))=> (((eq_ref fofType) x2) P0)) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (P0:(fofType->Prop))=> (((eq_ref fofType) x2) P0)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P0:(fofType->Prop))=> (((eq_ref fofType) x2) P0)) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found (eq_sym000 ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_sym00 x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (((eq_sym0 Xy) x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x2) Xy)
% Found x3:(P0 x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x3:(P0 x2))=> x3) as proof of (P0 Xy)
% Found (fun (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found x5:(P (Xf x4))
% Instantiate: A0:=(Xf x4):fofType
% Found x5 as proof of (P0 A0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) b)
% Found x4:((in Xx) x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x4:((in Xx) x2))=> x4) as proof of ((in Xx) Xy)
% Found (fun (Xx:fofType) (x4:((in Xx) x2))=> x4) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType) (x4:((in Xx) x2))=> x4) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) Xy)))
% Found (x30 (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found ((x3 Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) Xy)->((in Xx) Xy))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found ((eq_ref0 Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (((eq_ref fofType) Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) Xy)->((in Xx) x2)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x3 x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x0100 (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P0)) as proof of ((P0 x2)->(P0 Xy))
% Found (((x0100 (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P0)) as proof of ((P0 x2)->(P0 Xy))
% Found ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((x010 x3) x4) (fun (x6:fofType)=> ((P0 x2)->(P0 x6))))) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P0)) as proof of ((P0 x2)->(P0 Xy))
% Found ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> ((((x01 Xy) x3) x4) (fun (x6:fofType)=> ((P0 x2)->(P0 x6))))) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P0)) as proof of ((P0 x2)->(P0 Xy))
% Found ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) (fun (x6:fofType)=> ((P0 x2)->(P0 x6))))) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P0)) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (P0:(fofType->Prop))=> ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) (fun (x6:fofType)=> ((P0 x2)->(P0 x6))))) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P0))) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P0:(fofType->Prop))=> ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) (fun (x6:fofType)=> ((P0 x2)->(P0 x6))))) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P0))) as proof of (((eq fofType) x2) Xy)
% Found x4:((in Xx) x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x4:((in Xx) x2))=> x4) as proof of ((in Xx) Xy)
% Found (fun (Xx:fofType) (x4:((in Xx) x2))=> x4) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType) (x4:((in Xx) x2))=> x4) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) Xy)))
% Found (x30 (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found ((x3 Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) Xy)->((in Xx) Xy))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found ((eq_ref0 Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (((eq_ref fofType) Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) Xy)->((in Xx) x2)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x3 x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x0100 (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P0 x2)->(P0 Xy))
% Found ((x0100 (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P0 x2)->(P0 Xy))
% Found (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((x010 x3) x4) P0)) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P0 x2)->(P0 Xy))
% Found (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> ((((x01 Xy) x3) x4) P0)) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P0 x2)->(P0 Xy))
% Found (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) P0)) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (P0:(fofType->Prop))=> (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) P0)) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P0:(fofType->Prop))=> (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) P0)) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of (((eq fofType) x2) Xy)
% Found x7:(((eq fofType) Xx0) (Xf x4))
% Instantiate: x2:=(Xf x4):fofType;b:=Xx0:fofType
% Found x7 as proof of (((eq fofType) b) x2)
% Found x7:(((eq fofType) Xx0) (Xf x4))
% Instantiate: a:=Xx0:fofType;b:=(Xf x4):fofType
% Found x7 as proof of (((eq fofType) a) b)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found x4:((in Xx) x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x4:((in Xx) x2))=> x4) as proof of ((in Xx) Xy)
% Found (fun (Xx:fofType) (x4:((in Xx) x2))=> x4) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType) (x4:((in Xx) x2))=> x4) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) Xy)))
% Found (x30 (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found ((x3 Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4)) as proof of ((subset x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) Xy)->((in Xx) Xy))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found ((eq_ref0 Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (((eq_ref fofType) Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) Xy)->((in Xx) x2)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x3 x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x0100 (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P x2)->(P Xy))
% Found ((x0100 (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((x010 x3) x4) P)) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> ((((x01 Xy) x3) x4) P)) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P x2)->(P Xy))
% Found (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) P)) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) P)) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of ((P x2)->(P Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> (((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) P)) (((x x2) Xy) (fun (Xx:fofType) (x4:((in Xx) x2))=> x4))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) Xy)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found ((x3 Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) Xy)->((in Xx) Xy))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found ((eq_ref0 Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (((eq_ref fofType) Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) Xy)->((in Xx) x2)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x3 x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x0100 (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (((x0100 (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((x010 x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> ((((x01 Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P)) as proof of ((P x2)->(P Xy))
% Found (fun (P:(fofType->Prop))=> ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P))) as proof of ((P x2)->(P Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P:(fofType->Prop))=> ((((fun (x3:((subset x2) Xy)) (x4:((subset Xy) x2))=> (((((x0 x2) Xy) x3) x4) (fun (x6:fofType)=> ((P x2)->(P x6))))) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) (((eq_ref fofType) x2) P))) as proof of (((eq fofType) x2) Xy)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x3:((in Xx0) x2)
% Instantiate: x2:=A:fofType;x4:=Xx0:fofType
% Found x3 as proof of ((in x4) A)
% Found x7:((in Xx) A0)
% Instantiate: A0:=(Xf x4):fofType
% Found (fun (x7:((in Xx) A0))=> x7) as proof of ((in Xx) (Xf x4))
% Found (fun (Xx:fofType) (x7:((in Xx) A0))=> x7) as proof of (((in Xx) A0)->((in Xx) (Xf x4)))
% Found (fun (Xx:fofType) (x7:((in Xx) A0))=> x7) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) (Xf x4))))
% Found (x60 (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) (Xf x4))
% Found ((x6 (Xf x4)) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) (Xf x4))
% Found (((x A0) (Xf x4)) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) (Xf x4))
% Found (((x A0) (Xf x4)) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) (Xf x4))
% Found x7:((in Xx) (Xf x4))
% Instantiate: A0:=(Xf x4):fofType
% Found (fun (x7:((in Xx) (Xf x4)))=> x7) as proof of ((in Xx) A0)
% Found (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7) as proof of (((in Xx) (Xf x4))->((in Xx) A0))
% Found (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7) as proof of (forall (Xx:fofType), (((in Xx) (Xf x4))->((in Xx) A0)))
% Found (x60 (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7)) as proof of ((subset (Xf x4)) A0)
% Found ((x6 A0) (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7)) as proof of ((subset (Xf x4)) A0)
% Found (((x (Xf x4)) A0) (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7)) as proof of ((subset (Xf x4)) A0)
% Found (((x (Xf x4)) A0) (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7)) as proof of ((subset (Xf x4)) A0)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_ref fofType) x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_ref fofType) x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_ref fofType) x2) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((eq_ref fofType) x2)) as proof of (((eq fofType) x2) Xy)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x3:((in Xx0) x2)
% Instantiate: x2:=A:fofType;x4:=Xx0:fofType
% Found x3 as proof of ((in x4) A)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy))))))
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (forall (Xx0:fofType), ((iff ((in Xx0) Xx)) ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) Xx) Xy))))))
% Found x7:((in Xx) A0)
% Instantiate: A0:=(Xf x4):fofType
% Found (fun (x7:((in Xx) A0))=> x7) as proof of ((in Xx) (Xf x4))
% Found (fun (Xx:fofType) (x7:((in Xx) A0))=> x7) as proof of (((in Xx) A0)->((in Xx) (Xf x4)))
% Found (fun (Xx:fofType) (x7:((in Xx) A0))=> x7) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) (Xf x4))))
% Found (x60 (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) (Xf x4))
% Found ((x6 (Xf x4)) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) (Xf x4))
% Found (((x A0) (Xf x4)) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) (Xf x4))
% Found (((x A0) (Xf x4)) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) (Xf x4))
% Found x7:(((eq fofType) Xx0) (Xf x4))
% Instantiate: x2:=Xx0:fofType;b:=(Xf x4):fofType
% Found x7 as proof of (((eq fofType) x2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))->((in Xx0) x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))->((in Xx0) x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))->((in Xx0) x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))->((in Xx0) x2)))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found x7:((in Xx) (Xf x4))
% Instantiate: A0:=(Xf x4):fofType
% Found (fun (x7:((in Xx) (Xf x4)))=> x7) as proof of ((in Xx) A0)
% Found (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7) as proof of (((in Xx) (Xf x4))->((in Xx) A0))
% Found (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7) as proof of (forall (Xx:fofType), (((in Xx) (Xf x4))->((in Xx) A0)))
% Found (x60 (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7)) as proof of ((subset (Xf x4)) A0)
% Found ((x6 A0) (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7)) as proof of ((subset (Xf x4)) A0)
% Found (((x (Xf x4)) A0) (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7)) as proof of ((subset (Xf x4)) A0)
% Found (((x (Xf x4)) A0) (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7)) as proof of ((subset (Xf x4)) A0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found x5:(P0 b)
% Instantiate: b:=Xx0:fofType
% Found (fun (x5:(P0 b))=> x5) as proof of (P0 Xx0)
% Found (fun (P0:(fofType->Prop)) (x5:(P0 b))=> x5) as proof of ((P0 b)->(P0 Xx0))
% Found (fun (P0:(fofType->Prop)) (x5:(P0 b))=> x5) as proof of (P b)
% Found x50:(P b)
% Found (fun (x50:(P b))=> x50) as proof of (P b)
% Found (fun (x50:(P b))=> x50) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found x5:(P0 b)
% Instantiate: b:=Xx0:fofType
% Found (fun (x5:(P0 b))=> x5) as proof of (P0 Xx0)
% Found (fun (P0:(fofType->Prop)) (x5:(P0 b))=> x5) as proof of ((P0 b)->(P0 Xx0))
% Found (fun (P0:(fofType->Prop)) (x5:(P0 b))=> x5) as proof of (P b)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b0)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b0)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b0)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) (Xf x4))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (Xf x4))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (Xf x4))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (Xf x4))
% Found x3:(P0 x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x3:(P0 x2))=> x3) as proof of (P0 Xy)
% Found (fun (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of (((eq fofType) x2) Xy)
% Found eq_ref000:=(eq_ref00 P0):((P0 x2)->(P0 x2))
% Found (eq_ref00 P0) as proof of ((P0 x2)->(P0 Xy))
% Found ((eq_ref0 x2) P0) as proof of ((P0 x2)->(P0 Xy))
% Found (((eq_ref fofType) x2) P0) as proof of ((P0 x2)->(P0 Xy))
% Found (((eq_ref fofType) x2) P0) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (P0:(fofType->Prop))=> (((eq_ref fofType) x2) P0)) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (P0:(fofType->Prop))=> (((eq_ref fofType) x2) P0)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P0:(fofType->Prop))=> (((eq_ref fofType) x2) P0)) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))):(((eq Prop) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy))))
% Found (eq_ref0 (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) as proof of (((eq Prop) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) as proof of (((eq Prop) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) as proof of (((eq Prop) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) as proof of (((eq Prop) (forall (Xy:fofType), ((forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))->(((eq fofType) x2) Xy)))) b)
% Found x5:((in Xx) A0)
% Instantiate: x2:=A0:fofType
% Found (fun (x5:((in Xx) A0))=> x5) as proof of ((in Xx) x2)
% Found (fun (Xx:fofType) (x5:((in Xx) A0))=> x5) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType) (x5:((in Xx) A0))=> x5) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) x2)))
% Found (x40 (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found ((x4 x2) (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) A0)))
% Found (x40 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found ((x4 A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found x5:((in Xx) A0)
% Instantiate: x2:=A0:fofType
% Found (fun (x5:((in Xx) A0))=> x5) as proof of ((in Xx) x2)
% Found (fun (Xx:fofType) (x5:((in Xx) A0))=> x5) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType) (x5:((in Xx) A0))=> x5) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) x2)))
% Found (x40 (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found ((x4 x2) (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x5:((in Xx) A0))=> x5)) as proof of ((subset A0) x2)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) A0)))
% Found (x40 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found ((x4 A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found x50:(P b)
% Found (fun (x50:(P b))=> x50) as proof of (P b)
% Found (fun (x50:(P b))=> x50) as proof of (P0 b)
% Found x5:(P (Xf x4))
% Instantiate: b:=(Xf x4):fofType
% Found x5 as proof of (P0 b)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% 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 x50:(P1 (Xf x4))
% Found (fun (x50:(P1 (Xf x4)))=> x50) as proof of (P1 (Xf x4))
% Found (fun (x50:(P1 (Xf x4)))=> x50) as proof of (P2 (Xf x4))
% Found x50:(P1 (Xf x4))
% Found (fun (x50:(P1 (Xf x4)))=> x50) as proof of (P1 (Xf x4))
% Found (fun (x50:(P1 (Xf x4)))=> x50) as proof of (P2 (Xf x4))
% Found x5:(P (Xf x4))
% Instantiate: b:=(Xf x4):fofType
% Found x5 as proof of (P0 b)
% Found x50:(P1 (Xf x4))
% Found (fun (x50:(P1 (Xf x4)))=> x50) as proof of (P1 (Xf x4))
% Found (fun (x50:(P1 (Xf x4)))=> x50) as proof of (P2 (Xf x4))
% Found x50:(P1 (Xf x4))
% Found (fun (x50:(P1 (Xf x4)))=> x50) as proof of (P1 (Xf x4))
% Found (fun (x50:(P1 (Xf x4)))=> x50) as proof of (P2 (Xf x4))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) (Xf x4))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (Xf x4))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (Xf x4))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) (Xf x4))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b0)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b0)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b0)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b0)
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found x3:((in Xx0) x2)
% Instantiate: x2:=A:fofType;x4:=Xx0:fofType
% Found x3 as proof of ((in x4) A)
% Found ((conj20 x3) iff_refl) as proof of ((and ((in x4) A)) b)
% Found (((conj2 b) x3) iff_refl) as proof of ((and ((in x4) A)) b)
% Found ((((conj ((in x4) A)) b) x3) iff_refl) as proof of ((and ((in x4) A)) b)
% Found ((((conj ((in x4) A)) b) x3) iff_refl) as proof of ((and ((in x4) A)) b)
% Found ((((conj ((in x4) A)) b) x3) iff_refl) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found (eq_sym100 ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_sym10 x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (((eq_sym1 Xy) x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x2) Xy)
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found x3:((in Xx0) x2)
% Instantiate: x2:=A:fofType;x4:=Xx0:fofType
% Found x3 as proof of ((in x4) A)
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found x5:(P (Xf x4))
% Instantiate: A0:=(Xf x4):fofType
% Found x5 as proof of (P0 A0)
% Found x3:((in Xx0) x2)
% Instantiate: x2:=A:fofType;x4:=Xx0:fofType
% Found x3 as proof of ((in x4) A)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found x5:(P (Xf x4))
% Instantiate: A0:=(Xf x4):fofType
% Found x5 as proof of (P0 A0)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) Xy))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) Xy)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found ((x3 Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) Xy)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) Xy)->((in Xx) Xy))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found ((eq_ref0 Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (((eq_ref fofType) Xy) (in Xx)) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (((in Xx) Xy)->((in Xx) x2))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) Xy)->((in Xx) x2)))
% Found (x30 (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x3 x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))) as proof of ((subset Xy) x2)
% Found ((x010 (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (((x01 Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx))))) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((x0 x2) Xy) (((x x2) Xy) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))))) (((x Xy) x2) (fun (Xx:fofType)=> (((eq_ref fofType) Xy) (in Xx)))))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x4)
% Found x3:((in Xx0) x2)
% Instantiate: x2:=A:fofType;x4:=Xx0:fofType
% Found x3 as proof of ((in x4) A)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx0)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx0)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx0)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b0)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b0)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b0)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b0)
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq fofType) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found ((eq_ref fofType) Xx0) as proof of (((eq fofType) Xx0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (Xf x4))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) Xx0)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx0)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx0)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b0)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b0)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b0)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b0)
% Found x5:(P Xx0)
% Instantiate: A0:=Xx0:fofType
% Found x5 as proof of (P0 A0)
% Found eq_ref00:=(eq_ref0 x2):(((eq fofType) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq fofType) x2) Xy)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) Xy)
% Found ((eq_ref fofType) x2) as proof of (((eq fofType) x2) Xy)
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((eq_ref fofType) x2)) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((in Xx0) x2)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((in Xx0) x2)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((in Xx0) x2)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((in Xx0) x2)->((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) (((eq fofType) Xx0) (Xf Xy)))))))
% Found x7:((in Xx) A0)
% Instantiate: x2:=A0:fofType
% Found (fun (x7:((in Xx) A0))=> x7) as proof of ((in Xx) x2)
% Found (fun (Xx:fofType) (x7:((in Xx) A0))=> x7) as proof of (((in Xx) A0)->((in Xx) x2))
% Found (fun (Xx:fofType) (x7:((in Xx) A0))=> x7) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) x2)))
% Found (x60 (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) x2)
% Found ((x6 x2) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) x2)
% Found (((x A0) x2) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) x2)
% Found eq_ref000:=(eq_ref00 (in Xx)):(((in Xx) x2)->((in Xx) x2))
% Found (eq_ref00 (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found ((eq_ref0 x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (((eq_ref fofType) x2) (in Xx)) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (((in Xx) x2)->((in Xx) A0))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx))) as proof of (forall (Xx:fofType), (((in Xx) x2)->((in Xx) A0)))
% Found (x60 (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found ((x6 A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found (((x x2) A0) (fun (Xx:fofType)=> (((eq_ref fofType) x2) (in Xx)))) as proof of ((subset x2) A0)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x5:(P Xx0)
% Instantiate: A0:=Xx0:fofType
% Found x5 as proof of (P0 A0)
% Found x5:(P Xx0)
% Instantiate: b:=Xx0:fofType
% Found x5 as proof of (P0 b)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x7:((in Xx) A0)
% Instantiate: A0:=(Xf x4):fofType
% Found (fun (x7:((in Xx) A0))=> x7) as proof of ((in Xx) (Xf x4))
% Found (fun (Xx:fofType) (x7:((in Xx) A0))=> x7) as proof of (((in Xx) A0)->((in Xx) (Xf x4)))
% Found (fun (Xx:fofType) (x7:((in Xx) A0))=> x7) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) (Xf x4))))
% Found (x60 (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) (Xf x4))
% Found ((x6 (Xf x4)) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) (Xf x4))
% Found (((x A0) (Xf x4)) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) (Xf x4))
% Found (((x A0) (Xf x4)) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) (Xf x4))
% Found x7:((in Xx) (Xf x4))
% Instantiate: A0:=(Xf x4):fofType
% Found (fun (x7:((in Xx) (Xf x4)))=> x7) as proof of ((in Xx) A0)
% Found (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7) as proof of (((in Xx) (Xf x4))->((in Xx) A0))
% Found (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7) as proof of (forall (Xx:fofType), (((in Xx) (Xf x4))->((in Xx) A0)))
% Found (x60 (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7)) as proof of ((subset (Xf x4)) A0)
% Found ((x6 A0) (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7)) as proof of ((subset (Xf x4)) A0)
% Found (((x (Xf x4)) A0) (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7)) as proof of ((subset (Xf x4)) A0)
% Found (((x (Xf x4)) A0) (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7)) as proof of ((subset (Xf x4)) A0)
% Found x5:(P Xx0)
% Instantiate: b:=Xx0:fofType
% Found x5 as proof of (P0 b)
% Found x7:(((eq fofType) Xx0) (Xf x4))
% Instantiate: x2:=Xx0:fofType;b:=(Xf x4):fofType
% Found x7 as proof of (((eq fofType) x2) b)
% Found x7:(((eq fofType) Xx0) (Xf x4))
% Instantiate: x2:=Xx0:fofType;b:=(Xf x4):fofType
% Found x7 as proof of (((eq fofType) x2) b)
% Found x7:((in Xx) A0)
% Instantiate: A0:=(Xf x4):fofType
% Found (fun (x7:((in Xx) A0))=> x7) as proof of ((in Xx) (Xf x4))
% Found (fun (Xx:fofType) (x7:((in Xx) A0))=> x7) as proof of (((in Xx) A0)->((in Xx) (Xf x4)))
% Found (fun (Xx:fofType) (x7:((in Xx) A0))=> x7) as proof of (forall (Xx:fofType), (((in Xx) A0)->((in Xx) (Xf x4))))
% Found (x60 (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) (Xf x4))
% Found ((x6 (Xf x4)) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) (Xf x4))
% Found (((x A0) (Xf x4)) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) (Xf x4))
% Found (((x A0) (Xf x4)) (fun (Xx:fofType) (x7:((in Xx) A0))=> x7)) as proof of ((subset A0) (Xf x4))
% Found x7:((in Xx) (Xf x4))
% Instantiate: A0:=(Xf x4):fofType
% Found (fun (x7:((in Xx) (Xf x4)))=> x7) as proof of ((in Xx) A0)
% Found (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7) as proof of (((in Xx) (Xf x4))->((in Xx) A0))
% Found (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7) as proof of (forall (Xx:fofType), (((in Xx) (Xf x4))->((in Xx) A0)))
% Found (x60 (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7)) as proof of ((subset (Xf x4)) A0)
% Found ((x6 A0) (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7)) as proof of ((subset (Xf x4)) A0)
% Found (((x (Xf x4)) A0) (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7)) as proof of ((subset (Xf x4)) A0)
% Found (((x (Xf x4)) A0) (fun (Xx:fofType) (x7:((in Xx) (Xf x4)))=> x7)) as proof of ((subset (Xf x4)) A0)
% Found x3:(P0 x2)
% Instantiate: x2:=Xy:fofType
% Found (fun (x3:(P0 x2))=> x3) as proof of (P0 Xy)
% Found (fun (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P0:(fofType->Prop)) (x3:(P0 x2))=> x3) as proof of (((eq fofType) x2) Xy)
% Found eq_ref000:=(eq_ref00 P0):((P0 x2)->(P0 x2))
% Found (eq_ref00 P0) as proof of ((P0 x2)->(P0 Xy))
% Found ((eq_ref0 x2) P0) as proof of ((P0 x2)->(P0 Xy))
% Found (((eq_ref fofType) x2) P0) as proof of ((P0 x2)->(P0 Xy))
% Found (((eq_ref fofType) x2) P0) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (P0:(fofType->Prop))=> (((eq_ref fofType) x2) P0)) as proof of ((P0 x2)->(P0 Xy))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0)))))))) (P0:(fofType->Prop))=> (((eq_ref fofType) x2) P0)) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq fofType) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found ((eq_ref fofType) Xy) as proof of (((eq fofType) Xy) x2)
% Found (eq_sym000 ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((eq_sym00 x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (((eq_sym0 Xy) x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy)) as proof of (forall (P:(fofType->Prop)), ((P x2)->(P Xy)))
% Found (fun (x00:(forall (Xx:fofType), ((iff ((in Xx) Xy)) ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) (((eq fofType) Xx) (Xf Xy0))))))))=> ((((eq_sym fofType) Xy) x2) ((eq_ref fofType) Xy))) as proof of (((eq fofType) x2) Xy)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found x50:(P b)
% Found (fun (x50:(P b))=> x50) as proof of (P b)
% Found (fun (x50:(P b))=> x50) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found x50:(P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P Xx0)
% Found (fun (x50:(P Xx0))=> x50) as proof of (P0 Xx0)
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found x3:((in Xx0) x2)
% Instantiate: x2:=A:fofType;x4:=Xx0:fofType
% Found x3 as proof of ((in x4) A)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found x3:((in Xx0) x2)
% Instantiate: x2:=A:fofType;x4:=Xx0:fofType
% Found x3 as proof of ((in x4) A)
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found x50:(P b)
% Found (fun (x50:(P b))=> x50) as proof of (P b)
% Found (fun (x50:(P b))=> x50) as proof of (P0 b)
% Found x50:(P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P (Xf x4))
% Found (fun (x50:(P (Xf x4)))=> x50) as proof of (P0 (Xf x4))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx0)
% Found eq_ref00:=(eq_ref0 (Xf x4)):(((eq fofType) (Xf x4)) (Xf x4))
% Found (eq_ref0 (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fofType) (Xf x4)) as proof of (((eq fofType) (Xf x4)) b)
% Found ((eq_ref fof
% EOF
%------------------------------------------------------------------------------