%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU820^2 : TPTP v6.1.0. Released v3.7.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n186.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 13:33:14 EDT 2014

% Result   : Timeout 300.02s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU820^2 : TPTP v6.1.0. Released v3.7.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n186.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 11:34:11 CDT 2014
% % CPUTime  : 300.02 
% 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 0x1a66ab8>, <kernel.DependentProduct object at 0x1823cf8>) of role type named in_type
% Using role type
% Declaring in:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0x1a64950>, <kernel.Single object at 0x14f02d8>) of role type named emptyset_type
% Using role type
% Declaring emptyset:fofType
% FOF formula (<kernel.Constant object at 0x1a66ab8>, <kernel.DependentProduct object at 0x1823e18>) of role type named powerset_type
% Using role type
% Declaring powerset:(fofType->fofType)
% FOF formula (<kernel.Constant object at 0x1a66ab8>, <kernel.DependentProduct object at 0x1823d40>) of role type named nonempty_type
% Using role type
% Declaring nonempty:(fofType->Prop)
% FOF formula (((eq (fofType->Prop)) nonempty) (fun (Xx:fofType)=> (not (((eq fofType) Xx) emptyset)))) of role definition named nonempty
% A new definition: (((eq (fofType->Prop)) nonempty) (fun (Xx:fofType)=> (not (((eq fofType) Xx) emptyset))))
% Defined: nonempty:=(fun (Xx:fofType)=> (not (((eq fofType) Xx) emptyset)))
% FOF formula (<kernel.Constant object at 0x14f0518>, <kernel.Sort object at 0x14f7638>) of role type named vacuousDall_type
% Using role type
% Declaring vacuousDall:Prop
% FOF formula (((eq Prop) vacuousDall) (forall (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) emptyset)->(Xphi Xx)))) of role definition named vacuousDall
% A new definition: (((eq Prop) vacuousDall) (forall (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) emptyset)->(Xphi Xx))))
% Defined: vacuousDall:=(forall (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) emptyset)->(Xphi Xx)))
% FOF formula (<kernel.Constant object at 0x14f0518>, <kernel.DependentProduct object at 0x1823830>) of role type named subset_type
% Using role type
% Declaring subset:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0x14f0518>, <kernel.Sort object at 0x14f7638>) of role type named subsetemptysetimpeq_type
% Using role type
% Declaring subsetemptysetimpeq:Prop
% FOF formula (((eq Prop) subsetemptysetimpeq) (forall (A:fofType), (((subset A) emptyset)->(((eq fofType) A) emptyset)))) of role definition named subsetemptysetimpeq
% A new definition: (((eq Prop) subsetemptysetimpeq) (forall (A:fofType), (((subset A) emptyset)->(((eq fofType) A) emptyset))))
% Defined: subsetemptysetimpeq:=(forall (A:fofType), (((subset A) emptyset)->(((eq fofType) A) emptyset)))
% FOF formula (<kernel.Constant object at 0x1823f80>, <kernel.Sort object at 0x14f7638>) of role type named powersetE1_type
% Using role type
% Declaring powersetE1:Prop
% FOF formula (((eq Prop) powersetE1) (forall (A:fofType) (B:fofType), (((in B) (powerset A))->((subset B) A)))) of role definition named powersetE1
% A new definition: (((eq Prop) powersetE1) (forall (A:fofType) (B:fofType), (((in B) (powerset A))->((subset B) A))))
% Defined: powersetE1:=(forall (A:fofType) (B:fofType), (((in B) (powerset A))->((subset B) A)))
% FOF formula (<kernel.Constant object at 0x1823c20>, <kernel.DependentProduct object at 0x1823bd8>) of role type named transitiveset_type
% Using role type
% Declaring transitiveset:(fofType->Prop)
% FOF formula (((eq (fofType->Prop)) transitiveset) (fun (A:fofType)=> (forall (X:fofType), (((in X) A)->((subset X) A))))) of role definition named transitiveset
% A new definition: (((eq (fofType->Prop)) transitiveset) (fun (A:fofType)=> (forall (X:fofType), (((in X) A)->((subset X) A)))))
% Defined: transitiveset:=(fun (A:fofType)=> (forall (X:fofType), (((in X) A)->((subset X) A))))
% FOF formula (<kernel.Constant object at 0x1823bd8>, <kernel.DependentProduct object at 0x1823e60>) of role type named stricttotalorderedByIn_type
% Using role type
% Declaring stricttotalorderedByIn:(fofType->Prop)
% FOF formula (((eq (fofType->Prop)) stricttotalorderedByIn) (fun (A:fofType)=> ((and ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))))) of role definition named stricttotalorderedByIn
% A new definition: (((eq (fofType->Prop)) stricttotalorderedByIn) (fun (A:fofType)=> ((and ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))) (forall (X:fofType), (((in X) A)->(((in X) X)->False))))))
% Defined: stricttotalorderedByIn:=(fun (A:fofType)=> ((and ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))))
% FOF formula (<kernel.Constant object at 0x1823e60>, <kernel.DependentProduct object at 0x1823f80>) of role type named wellorderedByIn_type
% Using role type
% Declaring wellorderedByIn:(fofType->Prop)
% FOF formula (((eq (fofType->Prop)) wellorderedByIn) (fun (A:fofType)=> ((and (stricttotalorderedByIn A)) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))))) of role definition named wellorderedByIn
% A new definition: (((eq (fofType->Prop)) wellorderedByIn) (fun (A:fofType)=> ((and (stricttotalorderedByIn A)) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))))
% Defined: wellorderedByIn:=(fun (A:fofType)=> ((and (stricttotalorderedByIn A)) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))))
% FOF formula (<kernel.Constant object at 0x1823f80>, <kernel.DependentProduct object at 0x1823c20>) of role type named ordinal_type
% Using role type
% Declaring ordinal:(fofType->Prop)
% FOF formula (((eq (fofType->Prop)) ordinal) (fun (Xx:fofType)=> ((and (transitiveset Xx)) (wellorderedByIn Xx)))) of role definition named ordinal
% A new definition: (((eq (fofType->Prop)) ordinal) (fun (Xx:fofType)=> ((and (transitiveset Xx)) (wellorderedByIn Xx))))
% Defined: ordinal:=(fun (Xx:fofType)=> ((and (transitiveset Xx)) (wellorderedByIn Xx)))
% FOF formula (vacuousDall->(subsetemptysetimpeq->(powersetE1->(ordinal emptyset)))) of role conjecture named emptysetOrdinal
% Conjecture to prove = (vacuousDall->(subsetemptysetimpeq->(powersetE1->(ordinal emptyset)))):Prop
% We need to prove ['(vacuousDall->(subsetemptysetimpeq->(powersetE1->(ordinal emptyset))))']
% Parameter fofType:Type.
% Parameter in:(fofType->(fofType->Prop)).
% Parameter emptyset:fofType.
% Parameter powerset:(fofType->fofType).
% Definition nonempty:=(fun (Xx:fofType)=> (not (((eq fofType) Xx) emptyset))):(fofType->Prop).
% Definition vacuousDall:=(forall (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) emptyset)->(Xphi Xx))):Prop.
% Parameter subset:(fofType->(fofType->Prop)).
% Definition subsetemptysetimpeq:=(forall (A:fofType), (((subset A) emptyset)->(((eq fofType) A) emptyset))):Prop.
% Definition powersetE1:=(forall (A:fofType) (B:fofType), (((in B) (powerset A))->((subset B) A))):Prop.
% Definition transitiveset:=(fun (A:fofType)=> (forall (X:fofType), (((in X) A)->((subset X) A)))):(fofType->Prop).
% Definition stricttotalorderedByIn:=(fun (A:fofType)=> ((and ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))) (forall (X:fofType), (((in X) A)->(((in X) X)->False))))):(fofType->Prop).
% Definition wellorderedByIn:=(fun (A:fofType)=> ((and (stricttotalorderedByIn A)) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))):(fofType->Prop).
% Definition ordinal:=(fun (Xx:fofType)=> ((and (transitiveset Xx)) (wellorderedByIn Xx))):(fofType->Prop).
% Trying to prove (vacuousDall->(subsetemptysetimpeq->(powersetE1->(ordinal emptyset))))
% Found x2:=(x (fun (x4:fofType)=> ((subset x4) emptyset))):(forall (Xx:fofType), (((in Xx) emptyset)->((subset Xx) emptyset)))
% Found (x (fun (x4:fofType)=> ((subset x4) emptyset))) as proof of (transitiveset emptyset)
% Found (x (fun (x4:fofType)=> ((subset x4) emptyset))) as proof of (transitiveset emptyset)
% Found x2:=(x (fun (x4:fofType)=> ((subset x4) A))):(forall (Xx:fofType), (((in Xx) emptyset)->((subset Xx) A)))
% Found (x (fun (x4:fofType)=> ((subset x4) A))) as proof of (transitiveset A)
% Found (x (fun (x4:fofType)=> ((subset x4) A))) as proof of (transitiveset A)
% Found (x (fun (x4:fofType)=> ((subset x4) A))) as proof of (transitiveset A)
% Found x2:=(x (fun (x4:fofType)=> ((subset x4) A))):(forall (Xx:fofType), (((in Xx) emptyset)->((subset Xx) A)))
% Found (x (fun (x4:fofType)=> ((subset x4) A))) as proof of (transitiveset A)
% Found (x (fun (x4:fofType)=> ((subset x4) A))) as proof of (transitiveset A)
% Found (x (fun (x4:fofType)=> ((subset x4) A))) as proof of (transitiveset A)
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))):(((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found (eq_ref0 (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) (fun (x:fofType)=> ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eta_expansion_dep0 (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 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 (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found eq_ref00:=(eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))):(((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found (eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) 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: a:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of a
% Found x1:powersetE1
% Instantiate: b:=(forall (A:fofType) (B:fofType), (((in B) (powerset A))->((subset B) A))):Prop
% Found x1 as proof of a
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType
% Found x2 as proof of (P A)
% Found eq_ref00:=(eq_ref0 ((in x4) Y)):(((eq Prop) ((in x4) Y)) ((in x4) Y))
% Found (eq_ref0 ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))):(((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found (eq_ref0 (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))):(((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found (eq_ref0 (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))):(((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found (eq_ref0 (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found x00:((in Y) X)
% Instantiate: A:=X:fofType
% Found x00 as proof of (P A)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found x00:((in Y) X)
% Instantiate: a:=X:fofType;x4:=Y:fofType
% Found x00 as proof of ((in x4) a)
% Found (or_intror00 x00) as proof of (P ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_intror0 ((in x4) a)) x00) as proof of (P ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x00) as proof of (P ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x00) as proof of (P ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of (P ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_introl0 ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> ((eq_substitution00000 x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> (((eq_substitution0000 (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> ((((eq_substitution000 Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((eq_substitution00 a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> (((eq_substitution0 Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x1000:=(x100 x2):((subset A) emptyset)
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found ((x010 (((x1 emptyset) A) x2)) x00) as proof of ((in Y) emptyset)
% Found ((x010 (((x1 emptyset) A) x2)) x00) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset A) emptyset))=> ((x01 x6) (in Y))) (((x1 emptyset) A) x2)) x00) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00) as proof of ((in Y) emptyset)
% Found (x50 (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found ((x5 Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found (or_intror00 (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_intror0 ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (Y:fofType) (x00:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)))) as proof of (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))
% Found (fun (Y:fofType) (x00:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)))) as proof of (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% 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 eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found (eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found (eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found (eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))):(((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found (eq_ref0 (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))):(((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found (eq_ref0 (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% 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 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% 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 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 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found conj00:=(conj0 (wellorderedByIn emptyset)):((transitiveset emptyset)->((wellorderedByIn emptyset)->((and (transitiveset emptyset)) (wellorderedByIn emptyset))))
% Instantiate: a:=((transitiveset emptyset)->((wellorderedByIn emptyset)->((and (transitiveset emptyset)) (wellorderedByIn emptyset)))):Prop
% Found conj00 as proof of a
% Found conj00:=(conj0 (wellorderedByIn emptyset)):((transitiveset emptyset)->((wellorderedByIn emptyset)->((and (transitiveset emptyset)) (wellorderedByIn emptyset))))
% Instantiate: a:=((transitiveset emptyset)->((wellorderedByIn emptyset)->((and (transitiveset emptyset)) (wellorderedByIn emptyset)))):Prop
% Found conj00 as proof of a
% 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 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found (eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found (eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) 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 (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))):(((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found (eq_ref0 (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% 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 (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% 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 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 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 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x5:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x5:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% 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 ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) (fun (x:fofType)=> ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) (fun (x:fofType)=> ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% 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 ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) 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 (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: a:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of a
% Found x:vacuousDall
% Instantiate: b:=(forall (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) emptyset)->(Xphi Xx))):Prop
% Found x as proof of a
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% 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 (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% 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 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: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))):(((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found (eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: a:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of a
% Found x:vacuousDall
% Instantiate: b:=(forall (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) emptyset)->(Xphi Xx))):Prop
% Found x as proof of a
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: a:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of a
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: a:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of a
% Found eq_ref00:=(eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))):(((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found (eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))):(((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found (eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% 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)) (f0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 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)) (f0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: a:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of a
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x8:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x:vacuousDall
% Instantiate: b:=(forall (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) emptyset)->(Xphi Xx))):Prop
% Found x as proof of a
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType;x4:=X:fofType
% Found x2 as proof of (P0 A)
% Found x5:(P2 (f x4))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x5:(P2 (f x4)))=> x5) as proof of (P2 (f0 x4))
% Found (fun (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of ((P2 (f x4))->(P2 (f0 x4)))
% Found (fun (x4:fofType) (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType) (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found x5:(P2 (f x4))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x5:(P2 (f x4)))=> x5) as proof of (P2 (f0 x4))
% Found (fun (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of ((P2 (f x4))->(P2 (f0 x4)))
% Found (fun (x4:fofType) (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType) (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found x2:=(x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))):(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) Xx) Y)) ((in Xx) Y))) ((in Y) Xx))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType;x4:=X:fofType
% Found x2 as proof of (P0 A)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType;x4:=X:fofType
% Found x2 as proof of (P0 A)
% Found x2:((in X) (powerset A))
% Instantiate: A0:=(powerset A):fofType
% Found x2 as proof of (P0 A0)
% Found eq_ref00:=(eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))):(((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found (eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b0)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b0)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b0)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) (fun (x:fofType)=> ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found ((eta_expansion_dep0 (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x101 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% 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 ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% 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 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x2:((in X) (powerset A))
% Instantiate: A0:=(powerset A):fofType
% Found x2 as proof of (P0 A0)
% 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 ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% 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 x2:((in X) (powerset A))
% Instantiate: A0:=(powerset A):fofType
% Found x2 as proof of (P0 A0)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType
% Found x2 as proof of (P0 A)
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: a:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of a
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType
% Found x2 as proof of (P0 A)
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: a:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of a
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: a:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of a
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: a:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of a
% Found eq_ref00:=(eq_ref0 ((in x4) Y)):(((eq Prop) ((in x4) Y)) ((in x4) Y))
% Found (eq_ref0 ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x8:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) emptyset)->(((in X) X)->False)))):(((eq Prop) (forall (X:fofType), (((in X) emptyset)->(((in X) X)->False)))) (forall (X:fofType), (((in X) emptyset)->(((in X) X)->False))))
% Found (eq_ref0 (forall (X:fofType), (((in X) emptyset)->(((in X) X)->False)))) as proof of (((eq Prop) (forall (X:fofType), (((in X) emptyset)->(((in X) X)->False)))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) emptyset)->(((in X) X)->False)))) as proof of (((eq Prop) (forall (X:fofType), (((in X) emptyset)->(((in X) X)->False)))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) emptyset)->(((in X) X)->False)))) as proof of (((eq Prop) (forall (X:fofType), (((in X) emptyset)->(((in X) X)->False)))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) emptyset)->(((in X) X)->False)))) as proof of (((eq Prop) (forall (X:fofType), (((in X) emptyset)->(((in X) X)->False)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))):(((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found (eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))):(((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found (eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found x2:((in X) (powerset A))
% Instantiate: A:=emptyset:fofType;A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found eq_ref00:=(eq_ref0 ((in x4) Y)):(((eq Prop) ((in x4) Y)) ((in x4) Y))
% Found (eq_ref0 ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found eq_ref00:=(eq_ref0 ((in x4) Y)):(((eq Prop) ((in x4) Y)) ((in x4) Y))
% Found (eq_ref0 ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x01:((in Y) X)
% Instantiate: A:=X:fofType
% Found x01 as proof of (P0 A)
% Found x2:((in X) (powerset A))
% Instantiate: A:=emptyset:fofType;A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType
% Found x2 as proof of (P0 A)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found x2:((in X) (powerset A))
% Instantiate: A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x101 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found x01:((in Y) X)
% Instantiate: A:=X:fofType
% Found x01 as proof of (P0 A)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found x600:=(x60 x4):((in Xx) Y)
% Found (x60 x4) as proof of ((in Xx) Y)
% Found ((x6 Y) x4) as proof of ((in Xx) Y)
% Found (((x (in Xx)) Y) x4) as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found (fun (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))
% Found (fun (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))
% Found (fun (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))
% Found (fun (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))
% Found (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))))
% Found (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))
% Found ((conj30 (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found (((conj3 (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found x01:((in Y) X)
% Instantiate: A0:=X:fofType
% Found x01 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) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% 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)) (f0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 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)) (f0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 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)) (f0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 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)) (f0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x5:(P2 (f x4))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x5:(P2 (f x4)))=> x5) as proof of (P2 (f0 x4))
% Found (fun (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of ((P2 (f x4))->(P2 (f0 x4)))
% Found (fun (x4:fofType) (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType) (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found x5:(P2 (f x4))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x5:(P2 (f x4)))=> x5) as proof of (P2 (f0 x4))
% Found (fun (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of ((P2 (f x4))->(P2 (f0 x4)))
% Found (fun (x4:fofType) (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType) (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found x5:(P2 (f x4))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x5:(P2 (f x4)))=> x5) as proof of (P2 (f0 x4))
% Found (fun (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of ((P2 (f x4))->(P2 (f0 x4)))
% Found (fun (x4:fofType) (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType) (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found x2:((in X) (powerset A))
% Instantiate: a:=(powerset A):fofType;x4:=X:fofType
% Found x2 as proof of ((in x4) a)
% Found (or_intror00 x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_intror0 ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found x5:(P2 (f x4))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x5:(P2 (f x4)))=> x5) as proof of (P2 (f0 x4))
% Found (fun (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of ((P2 (f x4))->(P2 (f0 x4)))
% Found (fun (x4:fofType) (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType) (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found x01:((in Y) X)
% Instantiate: A0:=X:fofType
% Found x01 as proof of (P0 A0)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x01:((in Y) X)
% Instantiate: A0:=X:fofType
% Found x01 as proof of (P0 A0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) (fun (x:fofType)=> ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found ((eta_expansion_dep0 (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))):(((eq Prop) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found (eq_ref0 (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found x2:((in X) (powerset A))
% Instantiate: a:=(powerset A):fofType;x4:=X:fofType
% Found x2 as proof of ((in x4) a)
% Found (or_intror00 x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_intror0 ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x02:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x2:((in X) (powerset A))
% Instantiate: a:=(powerset A):fofType;x4:=X:fofType
% Found x2 as proof of ((in x4) a)
% Found (or_intror00 x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_intror0 ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found x2:((in X) (powerset A))
% Instantiate: A0:=(powerset A):fofType
% Found x2 as proof of (P0 A0)
% 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% Found x2:((in X) (powerset A))
% Instantiate: A0:=(powerset A):fofType
% Found x2 as proof of (P0 A0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_introl0 ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> ((eq_substitution00000 x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> (((eq_substitution0000 (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> ((((eq_substitution000 Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((eq_substitution00 a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> (((eq_substitution0 Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x2:((in X) (powerset emptyset))
% Instantiate: A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x101 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found classic0:=(classic (((eq fofType) x4) Y)):((or (((eq fofType) x4) Y)) (not (((eq fofType) x4) Y)))
% Found (classic (((eq fofType) x4) Y)) as proof of ((or (((eq fofType) x4) Y)) b)
% Found (classic (((eq fofType) x4) Y)) as proof of ((or (((eq fofType) x4) Y)) b)
% Found (classic (((eq fofType) x4) Y)) as proof of ((or (((eq fofType) x4) Y)) b)
% Found (classic (((eq fofType) x4) Y)) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))):(((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found (eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b0)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b0)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b0)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b0)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% 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 proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b0
% Found x2:((in X) (powerset A))
% Instantiate: A:=emptyset:fofType;A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found ((x020 (((x1 emptyset) A0) x2)) x01) as proof of ((in Y) emptyset)
% Found ((x020 (((x1 emptyset) A0) x2)) x01) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset A0) emptyset))=> ((x02 x6) (in Y))) (((x1 emptyset) A0) x2)) x01) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01) as proof of ((in Y) emptyset)
% Found (x50 (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found ((x5 Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found (or_intror00 (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_intror0 ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (Y:fofType) (x01:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)))) as proof of (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))
% Found (fun (Y:fofType) (x01:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)))) as proof of (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))
% Found x2:((in X) (powerset A))
% Instantiate: A0:=(powerset A):fofType
% Found x2 as proof of (P0 A0)
% Found x2:((in X) (powerset A))
% Instantiate: A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found x2:((in X) (powerset A))
% Instantiate: A:=emptyset:fofType;A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found x2:((in X) (powerset A))
% Instantiate: A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_introl0 ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> ((eq_substitution00000 x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> (((eq_substitution0000 (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> ((((eq_substitution000 Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((eq_substitution00 a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> (((eq_substitution0 Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))):(((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found (eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b0)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b0)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b0)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b0)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_introl0 ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> ((eq_substitution00000 x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> (((eq_substitution0000 (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> ((((eq_substitution000 Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((eq_substitution00 a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> (((eq_substitution0 Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x1000:=(x100 x2):((subset A0) emptyset)
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found ((x020 (((x1 emptyset) A0) x2)) x01) as proof of ((in Y) emptyset)
% Found ((x020 (((x1 emptyset) A0) x2)) x01) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset A0) emptyset))=> ((x02 x6) (in Y))) (((x1 emptyset) A0) x2)) x01) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01) as proof of ((in Y) emptyset)
% Found (x50 (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found ((x5 Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found (or_intror00 (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_intror0 ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (Y:fofType) (x01:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)))) as proof of (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))
% Found (fun (Y:fofType) (x01:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)))) as proof of (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))
% Found x2:((in X) (powerset A))
% Instantiate: A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found x2:=(x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))):(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) Xx) Y)) ((in Xx) Y))) ((in Y) Xx))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found x2:=(x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))):(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) Xx) Y)) ((in Xx) Y))) ((in Y) Xx))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found classic0:=(classic (((eq fofType) x4) Y)):((or (((eq fofType) x4) Y)) (not (((eq fofType) x4) Y)))
% Found (classic (((eq fofType) x4) Y)) as proof of ((or (((eq fofType) x4) Y)) b)
% Found (classic (((eq fofType) x4) Y)) as proof of ((or (((eq fofType) x4) Y)) b)
% Found (classic (((eq fofType) x4) Y)) as proof of ((or (((eq fofType) x4) Y)) b)
% Found (classic (((eq fofType) x4) Y)) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found x2:=(x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))):(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) Xx) Y)) ((in Xx) Y))) ((in Y) Xx))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))):(((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found (eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))):(((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found (eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eta_expansion_dep0 (fun (x5:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))):(((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found (eq_ref0 (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b0)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b0)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b0)
% Found ((eq_ref Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) as proof of (((eq Prop) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))) b0)
% Found x2:((in X) (powerset A))
% Instantiate: A:=emptyset:fofType;A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found ((x020 (((x1 emptyset) A0) x2)) x01) as proof of ((in Y) emptyset)
% Found ((x020 (((x1 emptyset) A0) x2)) x01) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset A0) emptyset))=> ((x02 x6) (in Y))) (((x1 emptyset) A0) x2)) x01) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01) as proof of ((in Y) emptyset)
% Found (x50 (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found ((x5 Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found (or_intror00 (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_intror0 ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (Y:fofType) (x01:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)))) as proof of (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))
% Found (fun (Y:fofType) (x01:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)))) as proof of (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))
% Found x2:((in X) (powerset A))
% Instantiate: A0:=(powerset A):fofType
% Found x2 as proof of (P0 A0)
% Found x2:((in X) (powerset A))
% Instantiate: A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found x2:((in X) (powerset A))
% Instantiate: A:=emptyset:fofType;A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found x2:((in X) (powerset A))
% Instantiate: A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found eq_ref00:=(eq_ref0 ((in x4) Y)):(((eq Prop) ((in x4) Y)) ((in x4) Y))
% Found (eq_ref0 ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found eq_ref00:=(eq_ref0 ((in x4) Y)):(((eq Prop) ((in x4) Y)) ((in x4) Y))
% Found (eq_ref0 ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found x2:=(x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))):(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) Xx) Y)) ((in Xx) Y))) ((in Y) Xx))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found x2:=(x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))):(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) Xx) Y)) ((in Xx) Y))) ((in Y) Xx))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found x2:=(x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))):(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) Xx) Y)) ((in Xx) Y))) ((in Y) Xx))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) emptyset)->(((in X) X)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) emptyset)->(((in X) X)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) emptyset)->(((in X) X)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:fofType), (((in X) emptyset)->(((in X) X)->False))))
% 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 x2:((in X) (powerset A))
% Instantiate: A:=emptyset:fofType;A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found x2:((in X) (powerset A))
% Instantiate: A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))):(((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found (eq_ref0 (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found x01:((in Y) X)
% Instantiate: A:=X:fofType
% Found x01 as proof of (P0 A)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found ((eta_expansion_dep0 (fun (x5:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x02:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found ((eta_expansion_dep0 (fun (x5:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% 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)) (f0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 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)) (f0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 ((in x4) Y)):(((eq Prop) ((in x4) Y)) ((in x4) Y))
% Found (eq_ref0 ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b0)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b0)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b0)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% 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 x2:=(x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))):(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) Xx) Y)) ((in Xx) Y))) ((in Y) Xx))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% 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 (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found x01:((in Y) X)
% Instantiate: A0:=X:fofType
% Found x01 as proof of (P0 A0)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType
% Found x2 as proof of (P0 A)
% Found x5:(P2 (f x4))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x5:(P2 (f x4)))=> x5) as proof of (P2 (f0 x4))
% Found (fun (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of ((P2 (f x4))->(P2 (f0 x4)))
% Found (fun (x4:fofType) (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType) (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found x5:(P2 (f x4))
% Instantiate: f0:=f:(fofType->Prop)
% Found (fun (x5:(P2 (f x4)))=> x5) as proof of (P2 (f0 x4))
% Found (fun (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of ((P2 (f x4))->(P2 (f0 x4)))
% Found (fun (x4:fofType) (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of (((eq Prop) (f x4)) (f0 x4))
% Found (fun (x4:fofType) (P2:(Prop->Prop)) (x5:(P2 (f x4)))=> x5) as proof of (forall (x:fofType), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found x01:((in Y) X)
% Instantiate: A0:=X:fofType
% Found x01 as proof of (P0 A0)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType
% Found x2 as proof of (P0 A)
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType
% Found x2 as proof of (P0 A)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType
% Found x2 as proof of (P0 A)
% Found x2:((in Xx) A)
% Instantiate: A:=Y:fofType
% Found x2 as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> x2) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> x2) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found x2:((in Xx) A)
% Instantiate: A:=Y:fofType
% Found x2 as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> x2) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> x2) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found x2:((in Xx) A)
% Instantiate: A:=Y:fofType
% Found x2 as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> x2) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> x2) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found x2:((in Xx) A)
% Instantiate: A:=Y:fofType
% Found x2 as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> x2) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> x2) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))):(((eq Prop) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found (eq_ref0 (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A0))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) 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 (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found x2:((in X) (powerset A))
% Instantiate: a:=(powerset A):fofType;x4:=X:fofType
% Found x2 as proof of ((in x4) a)
% Found (or_intror00 x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_intror0 ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% 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 (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x02:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x2:((in X) (powerset A))
% Instantiate: a:=(powerset A):fofType;x4:=X:fofType
% Found x2 as proof of ((in x4) a)
% Found (or_intror00 x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_intror0 ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found eq_ref00:=(eq_ref0 ((in x4) Y)):(((eq Prop) ((in x4) Y)) ((in x4) Y))
% Found (eq_ref0 ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b0)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b0)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b0)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b0)
% Found x600:=(x60 x4):((in Xx) Y)
% Found (x60 x4) as proof of ((in Xx) Y)
% Found ((x6 Y) x4) as proof of ((in Xx) Y)
% Found (((x (in Xx)) Y) x4) as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found (fun (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))
% Found (fun (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))
% Found (fun (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))
% Found (fun (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))
% Found (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))))
% Found (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))
% Found ((conj30 (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found (((conj3 (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found x600:=(x60 x4):((in Xx) Y)
% Found (x60 x4) as proof of ((in Xx) Y)
% Found ((x6 Y) x4) as proof of ((in Xx) Y)
% Found (((x (in Xx)) Y) x4) as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found (fun (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))
% Found (fun (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))
% Found (fun (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))
% Found (fun (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))
% Found (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))))
% Found (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))
% Found ((conj30 (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found (((conj3 (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found x2:((in Xx) A)
% Instantiate: A:=Y:fofType
% Found x2 as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> x2) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> x2) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found x2:((in Xx) A)
% Instantiate: A:=Y:fofType
% Found x2 as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> x2) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> x2) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found (eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found eq_ref00:=(eq_ref0 ((in x4) Y)):(((eq Prop) ((in x4) Y)) ((in x4) Y))
% Found (eq_ref0 ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b0)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b0)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b0)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b0)
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 ((in x4) Y)):(((eq Prop) ((in x4) Y)) ((in x4) Y))
% Found (eq_ref0 ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found eq_ref00:=(eq_ref0 ((in x4) Y)):(((eq Prop) ((in x4) Y)) ((in x4) Y))
% Found (eq_ref0 ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found ((eq_ref Prop) ((in x4) Y)) as proof of (((eq Prop) ((in x4) Y)) b)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_introl0 ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> ((eq_substitution00000 x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> (((eq_substitution0000 (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> ((((eq_substitution000 Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((eq_substitution00 a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> (((eq_substitution0 Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_introl0 ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> ((eq_substitution00000 x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> (((eq_substitution0000 (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> ((((eq_substitution000 Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((eq_substitution00 a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> (((eq_substitution0 Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x600:=(x60 x4):((in Xx) Y)
% Found (x60 x4) as proof of ((in Xx) Y)
% Found ((x6 Y) x4) as proof of ((in Xx) Y)
% Found (((x (in Xx)) Y) x4) as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found (fun (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))
% Found (fun (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))
% Found (fun (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))
% Found (fun (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))
% Found (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))))
% Found (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))
% Found ((conj30 (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found (((conj3 (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found x600:=(x60 x4):((in Xx) Y)
% Found (x60 x4) as proof of ((in Xx) Y)
% Found ((x6 Y) x4) as proof of ((in Xx) Y)
% Found (((x (in Xx)) Y) x4) as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found (fun (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))
% Found (fun (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))
% Found (fun (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))
% Found (fun (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))
% Found (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))))
% Found (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))
% Found ((conj30 (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found (((conj3 (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x1000:=(x100 x2):((subset A0) emptyset)
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found ((x020 (((x1 emptyset) A0) x2)) x01) as proof of ((in Y) emptyset)
% Found ((x020 (((x1 emptyset) A0) x2)) x01) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset A0) emptyset))=> ((x02 x6) (in Y))) (((x1 emptyset) A0) x2)) x01) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01) as proof of ((in Y) emptyset)
% Found (x50 (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found ((x5 Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found (or_intror00 (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_intror0 ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (Y:fofType) (x01:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)))) as proof of (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))
% Found (fun (Y:fofType) (x01:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)))) as proof of (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))
% Found x2:((in X) (powerset A))
% Instantiate: A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found classic0:=(classic (((eq fofType) x4) Y)):((or (((eq fofType) x4) Y)) (not (((eq fofType) x4) Y)))
% Found (classic (((eq fofType) x4) Y)) as proof of ((or (((eq fofType) x4) Y)) b)
% Found (classic (((eq fofType) x4) Y)) as proof of ((or (((eq fofType) x4) Y)) b)
% Found (classic (((eq fofType) x4) Y)) as proof of ((or (((eq fofType) x4) Y)) b)
% Found (classic (((eq fofType) x4) Y)) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found x2:=(x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))):(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) Xx) Y)) ((in Xx) Y))) ((in Y) Xx))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) 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 x00:((in Y) X)
% Instantiate: A:=X:fofType
% Found x00 as proof of (P0 A)
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x5:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (fofType->Prop)) a0) (fun (x:fofType)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found (((eta_expansion fofType) Prop) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType;x4:=X:fofType
% Found x2 as proof of (P0 A)
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) a)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) a)
% 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 x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType;x4:=X:fofType
% Found x2 as proof of (P0 A)
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found (eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b0)
% Found x2:((in X) (powerset A))
% Instantiate: A:=emptyset:fofType;A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found ((x020 (((x1 emptyset) A0) x2)) x01) as proof of ((in Y) emptyset)
% Found ((x020 (((x1 emptyset) A0) x2)) x01) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset A0) emptyset))=> ((x02 x6) (in Y))) (((x1 emptyset) A0) x2)) x01) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01) as proof of ((in Y) emptyset)
% Found (x50 (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found ((x5 Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)) as proof of ((in x4) Y)
% Found (or_intror00 (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_intror0 ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (Y:fofType) (x01:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)))) as proof of (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))
% Found (fun (Y:fofType) (x01:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x01)))) as proof of (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))
% Found x2:((in X) (powerset A))
% Instantiate: A0:=(powerset A):fofType
% Found x2 as proof of (P0 A0)
% Found x2:((in X) (powerset A))
% Instantiate: A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found x2:((in X) (powerset A))
% Instantiate: A:=emptyset:fofType;A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found x2:((in X) (powerset A))
% Instantiate: A0:=X:fofType
% Found x2 as proof of ((in A0) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A0) emptyset)
% Found ((x10 A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found (((x1 emptyset) A0) x2) as proof of ((subset A0) emptyset)
% Found x600:=(x60 x4):((in Xx) Y)
% Found (x60 x4) as proof of ((in Xx) Y)
% Found ((x6 Y) x4) as proof of ((in Xx) Y)
% Found (((x (in Xx)) Y) x4) as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found (fun (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))
% Found (fun (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))
% Found (fun (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))
% Found (fun (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))
% Found (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))))
% Found (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))
% Found ((conj30 (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found (((conj3 (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found x600:=(x60 x4):((in Xx) Y)
% Found (x60 x4) as proof of ((in Xx) Y)
% Found ((x6 Y) x4) as proof of ((in Xx) Y)
% Found (((x (in Xx)) Y) x4) as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found (fun (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))
% Found (fun (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))
% Found (fun (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))
% Found (fun (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))
% Found (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))))
% Found (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))
% Found ((conj30 (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found (((conj3 (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% Found x2:=(x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))):(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) Xx) Y)) ((in Xx) Y))) ((in Y) Xx))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found x00:((in Y) X)
% Instantiate: a:=X:fofType;x4:=Y:fofType
% Found x00 as proof of ((in x4) a)
% Found (or_intror00 x00) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_intror0 ((in x4) a)) x00) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x00) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x00) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found or_introl:(forall (A:Prop) (B:Prop), (A->((or A) B)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (A->((or A) B))):Prop
% Found or_introl as proof of b0
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b0
% Found x2:=(x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))):(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) Xx) Y)) ((in Xx) Y))) ((in Y) Xx))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x2:=(x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))):(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) Xx) Y)) ((in Xx) Y))) ((in Y) Xx))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found x2:=(x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))):(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) Xx) Y)) ((in Xx) Y))) ((in Y) Xx))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Y)
% Found x2:=(x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))):(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) Xx) Y)) ((in Xx) Y))) ((in Y) Xx))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x5:((in Y) X)
% Instantiate: A:=X:fofType
% Found x5 as proof of (P0 A)
% 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 x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x00:((in Y) X)
% Instantiate: A:=X:fofType
% Found x00 as proof of (P0 A)
% Found eq_ref:=(fun (T:Type) (a:T) (P:(T->Prop)) (x:(P a))=> x):(forall (T:Type) (a:T), (((eq T) a) a))
% Instantiate: b:=(forall (T:Type) (a:T), (((eq T) a) a)):Prop
% Found eq_ref as proof of b
% Found functional_extensionality00000:=(fun (x4:(forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y))))))))=> ((functional_extensionality0000 x4) P)):((forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))->((P f)->(P (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))
% Instantiate: b:=((forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))->((P f)->(P (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))):Prop
% Found functional_extensionality00000 as proof of b
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b0
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_introl0 ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> ((eq_substitution00000 x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> (((eq_substitution0000 (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> ((((eq_substitution000 Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((eq_substitution00 a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> (((eq_substitution0 Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x1000:=(x100 x2):((subset A) emptyset)
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found ((x010 (((x1 emptyset) A) x2)) x00) as proof of ((in Y) emptyset)
% Found ((x010 (((x1 emptyset) A) x2)) x00) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset A) emptyset))=> ((x01 x6) (in Y))) (((x1 emptyset) A) x2)) x00) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00) as proof of ((in Y) emptyset)
% Found (x50 (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found ((x5 Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found (or_intror00 (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_intror0 ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (Y:fofType) (x00:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)))) as proof of (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))
% Found (fun (Y:fofType) (x00:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)))) as proof of (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))
% Found (fun (Y:fofType) (x00:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)))) as proof of b
% Found x00:((in Y) X)
% Instantiate: A:=X:fofType
% Found x00 as proof of (P0 A)
% Found x00:((in Y) X)
% Instantiate: A:=X:fofType
% Found x00 as proof of (P0 A)
% Found x600:=(x60 x4):((in Xx) Y)
% Found (x60 x4) as proof of ((in Xx) Y)
% Found ((x6 Y) x4) as proof of ((in Xx) Y)
% Found (((x (in Xx)) Y) x4) as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found (fun (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))
% Found (fun (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))
% Found (fun (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))
% Found (fun (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))
% Found (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))))
% Found (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))
% Found ((conj30 (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found (((conj3 (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% 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: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType
% Found x2 as proof of (P0 A)
% Found x2:((in X) (powerset emptyset))
% Instantiate: a:=(powerset emptyset):fofType;x4:=X:fofType
% Found x2 as proof of ((in x4) a)
% Found (or_intror00 x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_intror0 ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x2) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType
% Found x2 as proof of (P0 A)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType
% Found x2 as proof of (P0 A)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x00:((in Y) X)
% Instantiate: a:=X:fofType;x4:=Y:fofType
% Found x00 as proof of ((in x4) a)
% Found (or_intror00 x00) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_intror0 ((in x4) a)) x00) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x00) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x00) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType
% Found x2 as proof of (P0 A)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x02:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b1)
% Found x00:((in Y) X)
% Instantiate: a:=X:fofType;x4:=Y:fofType
% Found x00 as proof of ((in x4) a)
% Found (or_intror00 x00) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_intror0 ((in x4) a)) x00) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x00) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x00) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found x00:((in Y) X)
% Instantiate: a:=X:fofType;x4:=Y:fofType
% Found x00 as proof of ((in x4) a)
% Found (or_intror00 x00) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_intror0 ((in x4) a)) x00) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x00) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_intror (((eq fofType) x4) a)) ((in x4) a)) x00) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_introl0 ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x6:(((eq fofType) a) Y))=> ((eq_substitution00000 x6) (fun (x8:Prop)=> x8))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x6:(((eq fofType) a) Y))=> (((eq_substitution0000 (fun (x9:fofType)=> ((or (((eq fofType) x4) x9)) ((in x4) x9)))) x6) (fun (x8:Prop)=> x8))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x6:(((eq fofType) Y) Y))=> ((((eq_substitution000 Y) (fun (x9:fofType)=> ((or (((eq fofType) x4) x9)) ((in x4) x9)))) x6) (fun (x8:Prop)=> x8))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x6:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((eq_substitution00 a) Y)) Y) (fun (x9:fofType)=> ((or (((eq fofType) x4) x9)) ((in x4) x9)))) x6) (fun (x8:Prop)=> x8))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x6:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> (((eq_substitution0 Prop) a) Y)) Y) (fun (x9:fofType)=> ((or (((eq fofType) x4) x9)) ((in x4) x9)))) x6) (fun (x8:Prop)=> x8))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x6:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x9:fofType)=> ((or (((eq fofType) x4) x9)) ((in x4) x9)))) x6) (fun (x8:Prop)=> x8))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x5:((in Y) X))=> (((fun (x6:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x9:fofType)=> ((or (((eq fofType) x4) x9)) ((in x4) x9)))) x6) (fun (x8:Prop)=> x8))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x2:=(x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))):(forall (Xx:fofType), (((in Xx) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) Xx) Y)) ((in Xx) Y))) ((in Y) Xx))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4)))))) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_introl0 ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> ((eq_substitution00000 x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> (((eq_substitution0000 (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> ((((eq_substitution000 Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((eq_substitution00 a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> (((eq_substitution0 Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found x0:subsetemptysetimpeq
% Instantiate: b0:=(forall (A:fofType), (((subset A) emptyset)->(((eq fofType) A) emptyset))):Prop
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found x1000:=(x100 x2):((subset A) emptyset)
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found ((x000 (((x1 emptyset) A) x2)) x5) as proof of ((in Y) emptyset)
% Found ((x000 (((x1 emptyset) A) x2)) x5) as proof of ((in Y) emptyset)
% Found (((fun (x7:((subset A) emptyset))=> ((x00 x7) (in Y))) (((x1 emptyset) A) x2)) x5) as proof of ((in Y) emptyset)
% Found (((fun (x7:((subset X) emptyset))=> (((x0 X) x7) (in Y))) (((x1 emptyset) X) x2)) x5) as proof of ((in Y) emptyset)
% Found (((fun (x7:((subset X) emptyset))=> (((x0 X) x7) (in Y))) (((x1 emptyset) X) x2)) x5) as proof of ((in Y) emptyset)
% Found (x60 (((fun (x7:((subset X) emptyset))=> (((x0 X) x7) (in Y))) (((x1 emptyset) X) x2)) x5)) as proof of ((in x4) Y)
% Found ((x6 Y) (((fun (x7:((subset X) emptyset))=> (((x0 X) x7) (in Y))) (((x1 emptyset) X) x2)) x5)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x7:((subset X) emptyset))=> (((x0 X) x7) (in Y))) (((x1 emptyset) X) x2)) x5)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x7:((subset X) emptyset))=> (((x0 X) x7) (in Y))) (((x1 emptyset) X) x2)) x5)) as proof of ((in x4) Y)
% Found (or_intror00 (((x (in x4)) Y) (((fun (x7:((subset X) emptyset))=> (((x0 X) x7) (in Y))) (((x1 emptyset) X) x2)) x5))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_intror0 ((in x4) Y)) (((x (in x4)) Y) (((fun (x7:((subset X) emptyset))=> (((x0 X) x7) (in Y))) (((x1 emptyset) X) x2)) x5))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x7:((subset X) emptyset))=> (((x0 X) x7) (in Y))) (((x1 emptyset) X) x2)) x5))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x5:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x7:((subset X) emptyset))=> (((x0 X) x7) (in Y))) (((x1 emptyset) X) x2)) x5)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (Y:fofType) (x5:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x7:((subset X) emptyset))=> (((x0 X) x7) (in Y))) (((x1 emptyset) X) x2)) x5)))) as proof of (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))
% Found (fun (Y:fofType) (x5:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x7:((subset X) emptyset))=> (((x0 X) x7) (in Y))) (((x1 emptyset) X) x2)) x5)))) as proof of (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))):(((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found (eq_ref0 (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset emptyset))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found x1000:=(x100 x2):((subset A) emptyset)
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found ((x010 (((x1 emptyset) A) x2)) x00) as proof of ((in Y) emptyset)
% Found ((x010 (((x1 emptyset) A) x2)) x00) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset A) emptyset))=> ((x01 x6) (in Y))) (((x1 emptyset) A) x2)) x00) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00) as proof of ((in Y) emptyset)
% Found (x50 (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found ((x5 Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found (or_intror00 (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_intror0 ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (Y:fofType) (x00:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)))) as proof of (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))
% Found (fun (Y:fofType) (x00:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)))) as proof of (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x4)):(((eq Prop) (f1 x4)) (f1 x4))
% Found (eq_ref0 (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found ((eq_ref Prop) (f1 x4)) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (((eq Prop) (f1 x4)) (f0 x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f1 x4))) as proof of (forall (x:fofType), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) a)
% Found ((or_introl0 a) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) a)
% Found (((or_introl (((eq fofType) x4) Y)) a) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) a)
% Found (((or_introl (((eq fofType) x4) Y)) a) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) a)
% Found (((or_introl (((eq fofType) x4) Y)) a) ((eq_ref fofType) x4)) as proof of (P0 a)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_introl0 ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> ((eq_substitution00000 x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> (((eq_substitution0000 (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> ((((eq_substitution000 Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((eq_substitution00 a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> (((eq_substitution0 Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) a)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((or_introl0 ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4)) as proof of (P0 ((or (((eq fofType) x4) a)) ((in x4) a)))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((eq_substitution000000 ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> ((eq_substitution00000 x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) a) Y))=> (((eq_substitution0000 (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) a)) (((or_introl (((eq fofType) x4) a)) ((in x4) a)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> ((((eq_substitution000 Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((eq_substitution00 a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> (((eq_substitution0 Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((fun (x5:(((eq fofType) Y) Y))=> (((((fun (a:fofType)=> ((((eq_substitution fofType) Prop) a) Y)) Y) (fun (x8:fofType)=> ((or (((eq fofType) x4) x8)) ((in x4) x8)))) x5) (fun (x7:Prop)=> x7))) ((eq_ref fofType) Y)) (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x5:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x1000:=(x100 x2):((subset A) emptyset)
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found ((x010 (((x1 emptyset) A) x2)) x00) as proof of ((in Y) emptyset)
% Found ((x010 (((x1 emptyset) A) x2)) x00) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset A) emptyset))=> ((x01 x6) (in Y))) (((x1 emptyset) A) x2)) x00) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00) as proof of ((in Y) emptyset)
% Found (x50 (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found ((x5 Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found (or_intror00 (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_intror0 ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (Y:fofType) (x00:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)))) as proof of (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))
% Found (fun (Y:fofType) (x00:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)))) as proof of (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x1000:=(x100 x2):((subset A) emptyset)
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found ((x010 (((x1 emptyset) A) x2)) x00) as proof of ((in Y) emptyset)
% Found ((x010 (((x1 emptyset) A) x2)) x00) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset A) emptyset))=> ((x01 x6) (in Y))) (((x1 emptyset) A) x2)) x00) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00) as proof of ((in Y) emptyset)
% Found (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00) as proof of ((in Y) emptyset)
% Found (x50 (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found ((x5 Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)) as proof of ((in x4) Y)
% Found (or_intror00 (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_intror0 ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (Y:fofType) (x00:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)))) as proof of (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))
% Found (fun (Y:fofType) (x00:((in Y) X))=> (((or_intror (((eq fofType) x4) Y)) ((in x4) Y)) (((x (in x4)) Y) (((fun (x6:((subset X) emptyset))=> (((x0 X) x6) (in Y))) (((x1 emptyset) X) x2)) x00)))) as proof of (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% 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 (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y)))))
% 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 x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType
% Found x2 as proof of (P0 A)
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))):(((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found (eq_ref0 (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x:vacuousDall
% Instantiate: b:=(forall (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) emptyset)->(Xphi Xx))):Prop
% Found x as proof of a
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found 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 x2:((in Xx) A)
% Instantiate: A:=Y:fofType
% Found x2 as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> x2) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> x2) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x5:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x2:((in Xx) A)
% Instantiate: A:=Y:fofType
% Found x2 as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> x2) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> x2) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found (eq_ref0 (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))) b)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x5:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found x600:=(x60 x4):((in Xx) Y)
% Found (x60 x4) as proof of ((in Xx) Y)
% Found ((x6 Y) x4) as proof of ((in Xx) Y)
% Found (((x (in Xx)) Y) x4) as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found (fun (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))
% Found (fun (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))
% Found (fun (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))
% Found (fun (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))
% Found (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))))
% Found (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))
% Found ((conj30 (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found (((conj3 (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x600:=(x60 x4):((in Xx) Y)
% Found (x60 x4) as proof of ((in Xx) Y)
% Found ((x6 Y) x4) as proof of ((in Xx) Y)
% Found (((x (in Xx)) Y) x4) as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found (fun (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))
% Found (fun (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))
% Found (fun (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))
% Found (fun (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))
% Found (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))))
% Found (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))
% Found ((conj30 (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found (((conj3 (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found x600:=(x60 x4):((in Xx) Y)
% Found (x60 x4) as proof of ((in Xx) Y)
% Found ((x6 Y) x4) as proof of ((in Xx) Y)
% Found (((x (in Xx)) Y) x4) as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found (fun (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))
% Found (fun (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))
% Found (fun (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))
% Found (fun (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))
% Found (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))))
% Found (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))
% Found ((conj30 (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found (((conj3 (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))):(((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found (eq_ref0 (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found x600:=(x60 x4):((in Xx) Y)
% Found (x60 x4) as proof of ((in Xx) Y)
% Found ((x6 Y) x4) as proof of ((in Xx) Y)
% Found (((x (in Xx)) Y) x4) as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found (fun (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))
% Found (fun (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))
% Found (fun (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))
% Found (fun (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))
% Found (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))))
% Found (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))
% Found ((conj30 (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found (((conj3 (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) emptyset)) (X:fofType) (x3:((in X) emptyset)) (Y:fofType) (x4:((in Y) emptyset)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) emptyset)->(forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) emptyset)->(forall (Y:fofType), (((in Y) emptyset)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))):(((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))))))
% Found (eq_ref0 (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) as proof of (((eq Prop) (forall (X:fofType), (((in X) (powerset A))->((nonempty X)->((ex fofType) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y))))))))))) b0)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found x:vacuousDall
% Instantiate: b:=(forall (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) emptyset)->(Xphi Xx))):Prop
% Found x as proof of a
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found x:vacuousDall
% Instantiate: b:=(forall (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) emptyset)->(Xphi Xx))):Prop
% Found x as proof of a
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% 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 eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=(powerset emptyset):fofType;x4:=X:fofType
% Found x2 as proof of (P0 A)
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (f x4))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found x2:((in X) (powerset emptyset))
% Instantiate: A:=X:fofType
% Found x2 as proof of ((in A) (powerset emptyset))
% Found (x100 x2) as proof of ((subset A) emptyset)
% Found ((x10 A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found (((x1 emptyset) A) x2) as proof of ((subset A) emptyset)
% Found x600:=(x60 x4):((in Xx) Y)
% Found (x60 x4) as proof of ((in Xx) Y)
% Found ((x6 Y) x4) as proof of ((in Xx) Y)
% Found (((x (in Xx)) Y) x4) as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found (fun (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))
% Found (fun (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))
% Found (fun (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))
% Found (fun (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))
% Found (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))))
% Found (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))
% Found ((conj30 (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found (((conj3 (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found x600:=(x60 x4):((in Xx) Y)
% Found (x60 x4) as proof of ((in Xx) Y)
% Found ((x6 Y) x4) as proof of ((in Xx) Y)
% Found (((x (in Xx)) Y) x4) as proof of ((in Xx) Y)
% Found (fun (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of ((in Xx) Y)
% Found (fun (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))
% Found (fun (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))
% Found (fun (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))
% Found (fun (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))
% Found (fun (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))
% Found (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y)))))))
% Found (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4)) as proof of (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))
% Found ((conj30 (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found (((conj3 (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X)))))))
% Found ((((conj (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) X))))))) (fun (Xx:fofType) (x2:((in Xx) A)) (X:fofType) (x3:((in X) A)) (Y:fofType) (x4:((in Y) A)) (x5:((and ((in Xx) X)) ((in X) Y)))=> (((x (in Xx)) Y) x4))) (x (fun (x4:fofType)=> (forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) x4) Y)) ((in x4) Y))) ((in Y) x4))))))) as proof of ((and (forall (Xx:fofType), (((in Xx) A)->(forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->(((and ((in Xx) X)) ((in X) Y))->((in Xx) Y))))))))) (forall (X:fofType), (((in X) A)->(forall (Y:fofType), (((in Y) A)->((or ((or (((eq fofType) X) Y)) ((in X) Y))) ((in Y) 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% 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) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x4) Y)) ((in x4) Y))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) x) Y)) ((in x) Y)))))))
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) A)->(((in X) X)->False)))):(((eq Prop) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) (forall (X:fofType), (((in X) A)->(((in X) X)->False))))
% Found (eq_ref0 (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) as proof of (((eq Prop) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) as proof of (((eq Prop) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) as proof of (((eq Prop) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) as proof of (((eq Prop) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) b)
% Found eq_ref00:=(eq_ref0 (forall (X:fofType), (((in X) A)->(((in X) X)->False)))):(((eq Prop) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) (forall (X:fofType), (((in X) A)->(((in X) X)->False))))
% Found (eq_ref0 (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) as proof of (((eq Prop) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) as proof of (((eq Prop) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) as proof of (((eq Prop) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) b)
% Found ((eq_ref Prop) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) as proof of (((eq Prop) (forall (X:fofType), (((in X) A)->(((in X) X)->False)))) b)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_intror00 ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found ((or_intror0 (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)) as proof of ((or ((in x4) Y)) (((eq fofType) x4) Y))
% Found (or_comm_i00 (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_comm_i0 (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x01:((in Y) X))=> (((or_comm_i ((in x4) Y)) (((eq fofType) x4) Y)) (((or_intror ((in x4) Y)) (((eq fofType) x4) Y)) ((eq_ref fofType) x4)))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 a0):(((eq (fofType->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found ((eq_ref (fofType->Prop)) a0) as proof of (((eq (fofType->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and ((in Xx) X)) (forall (Y:fofType), (((in Y) X)->((or (((eq fofType) Xx) Y)) ((in Xx) Y)))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((in x4) Y))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((in x4) Y))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((in x4) Y))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4))) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found eq_ref00:=(eq_ref0 x4):(((eq fofType) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found ((eq_ref fofType) x4) as proof of (((eq fofType) x4) Y)
% Found (or_introl00 ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found ((or_introl0 ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (((or_introl (((eq fofType) x4) Y)) ((in x4) Y)) ((eq_ref fofType) x4)) as proof of ((or (((eq fofType) x4) Y)) ((in x4) Y))
% Found (fun (x00:((in Y) X))=> (((or_introl (((eq fofType) x4) Y)) ((in x
% EOF
%------------------------------------------------------------------------------