TSTP Solution File: SET611^3 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SET611^3 : TPTP v6.1.0. Released v3.6.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n098.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 13:30:50 EDT 2014
% Result : Timeout 300.06s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SET611^3 : TPTP v6.1.0. Released v3.6.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n098.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 10:19:21 CDT 2014
% % CPUTime : 300.06
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% Failed to open /home/cristobal/cocATP/CASC/TPTP/Axioms/SET008^0.ax, trying next directory
% FOF formula (<kernel.Constant object at 0xe0f560>, <kernel.DependentProduct object at 0xb39dd0>) of role type named in_decl
% Using role type
% Declaring in:(fofType->((fofType->Prop)->Prop))
% FOF formula (((eq (fofType->((fofType->Prop)->Prop))) in) (fun (X:fofType) (M:(fofType->Prop))=> (M X))) of role definition named in
% A new definition: (((eq (fofType->((fofType->Prop)->Prop))) in) (fun (X:fofType) (M:(fofType->Prop))=> (M X)))
% Defined: in:=(fun (X:fofType) (M:(fofType->Prop))=> (M X))
% FOF formula (<kernel.Constant object at 0xe0f560>, <kernel.DependentProduct object at 0xb39cb0>) of role type named is_a_decl
% Using role type
% Declaring is_a:(fofType->((fofType->Prop)->Prop))
% FOF formula (((eq (fofType->((fofType->Prop)->Prop))) is_a) (fun (X:fofType) (M:(fofType->Prop))=> (M X))) of role definition named is_a
% A new definition: (((eq (fofType->((fofType->Prop)->Prop))) is_a) (fun (X:fofType) (M:(fofType->Prop))=> (M X)))
% Defined: is_a:=(fun (X:fofType) (M:(fofType->Prop))=> (M X))
% FOF formula (<kernel.Constant object at 0xf5bdd0>, <kernel.DependentProduct object at 0xb395a8>) of role type named emptyset_decl
% Using role type
% Declaring emptyset:(fofType->Prop)
% FOF formula (((eq (fofType->Prop)) emptyset) (fun (X:fofType)=> False)) of role definition named emptyset
% A new definition: (((eq (fofType->Prop)) emptyset) (fun (X:fofType)=> False))
% Defined: emptyset:=(fun (X:fofType)=> False)
% FOF formula (<kernel.Constant object at 0xf5bdd0>, <kernel.DependentProduct object at 0xb39cb0>) of role type named unord_pair_decl
% Using role type
% Declaring unord_pair:(fofType->(fofType->(fofType->Prop)))
% FOF formula (((eq (fofType->(fofType->(fofType->Prop)))) unord_pair) (fun (X:fofType) (Y:fofType) (U:fofType)=> ((or (((eq fofType) U) X)) (((eq fofType) U) Y)))) of role definition named unord_pair
% A new definition: (((eq (fofType->(fofType->(fofType->Prop)))) unord_pair) (fun (X:fofType) (Y:fofType) (U:fofType)=> ((or (((eq fofType) U) X)) (((eq fofType) U) Y))))
% Defined: unord_pair:=(fun (X:fofType) (Y:fofType) (U:fofType)=> ((or (((eq fofType) U) X)) (((eq fofType) U) Y)))
% FOF formula (<kernel.Constant object at 0xf5bdd0>, <kernel.DependentProduct object at 0xb395a8>) of role type named singleton_decl
% Using role type
% Declaring singleton:(fofType->(fofType->Prop))
% FOF formula (((eq (fofType->(fofType->Prop))) singleton) (fun (X:fofType) (U:fofType)=> (((eq fofType) U) X))) of role definition named singleton
% A new definition: (((eq (fofType->(fofType->Prop))) singleton) (fun (X:fofType) (U:fofType)=> (((eq fofType) U) X)))
% Defined: singleton:=(fun (X:fofType) (U:fofType)=> (((eq fofType) U) X))
% FOF formula (<kernel.Constant object at 0xb395a8>, <kernel.DependentProduct object at 0xb39b00>) of role type named union_decl
% Using role type
% Declaring union:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) union) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or (X U)) (Y U)))) of role definition named union
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) union) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or (X U)) (Y U))))
% Defined: union:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or (X U)) (Y U)))
% FOF formula (<kernel.Constant object at 0xb39b00>, <kernel.DependentProduct object at 0xb39dd0>) of role type named excl_union_decl
% Using role type
% Declaring excl_union:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) excl_union) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or ((and (X U)) ((Y U)->False))) ((and ((X U)->False)) (Y U))))) of role definition named excl_union
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) excl_union) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or ((and (X U)) ((Y U)->False))) ((and ((X U)->False)) (Y U)))))
% Defined: excl_union:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or ((and (X U)) ((Y U)->False))) ((and ((X U)->False)) (Y U))))
% FOF formula (<kernel.Constant object at 0xb39dd0>, <kernel.DependentProduct object at 0xb39f80>) of role type named intersection_decl
% Using role type
% Declaring intersection:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) intersection) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) (Y U)))) of role definition named intersection
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) intersection) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) (Y U))))
% Defined: intersection:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) (Y U)))
% FOF formula (<kernel.Constant object at 0xb39f80>, <kernel.DependentProduct object at 0xb39440>) of role type named setminus_decl
% Using role type
% Declaring setminus:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) setminus) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) ((Y U)->False)))) of role definition named setminus
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) setminus) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) ((Y U)->False))))
% Defined: setminus:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) ((Y U)->False)))
% FOF formula (<kernel.Constant object at 0xb39440>, <kernel.DependentProduct object at 0xb395a8>) of role type named complement_decl
% Using role type
% Declaring complement:((fofType->Prop)->(fofType->Prop))
% FOF formula (((eq ((fofType->Prop)->(fofType->Prop))) complement) (fun (X:(fofType->Prop)) (U:fofType)=> ((X U)->False))) of role definition named complement
% A new definition: (((eq ((fofType->Prop)->(fofType->Prop))) complement) (fun (X:(fofType->Prop)) (U:fofType)=> ((X U)->False)))
% Defined: complement:=(fun (X:(fofType->Prop)) (U:fofType)=> ((X U)->False))
% FOF formula (<kernel.Constant object at 0xb395a8>, <kernel.DependentProduct object at 0xb39b90>) of role type named disjoint_decl
% Using role type
% Declaring disjoint:((fofType->Prop)->((fofType->Prop)->Prop))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) disjoint) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((eq (fofType->Prop)) ((intersection X) Y)) emptyset))) of role definition named disjoint
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) disjoint) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((eq (fofType->Prop)) ((intersection X) Y)) emptyset)))
% Defined: disjoint:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((eq (fofType->Prop)) ((intersection X) Y)) emptyset))
% FOF formula (<kernel.Constant object at 0xe101b8>, <kernel.DependentProduct object at 0xb39830>) of role type named subset_decl
% Using role type
% Declaring subset:((fofType->Prop)->((fofType->Prop)->Prop))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) subset) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (forall (U:fofType), ((X U)->(Y U))))) of role definition named subset
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) subset) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (forall (U:fofType), ((X U)->(Y U)))))
% Defined: subset:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (forall (U:fofType), ((X U)->(Y U))))
% FOF formula (<kernel.Constant object at 0xe102d8>, <kernel.DependentProduct object at 0xb397a0>) of role type named meets_decl
% Using role type
% Declaring meets:((fofType->Prop)->((fofType->Prop)->Prop))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) meets) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))))) of role definition named meets
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) meets) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))))
% Defined: meets:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))))
% FOF formula (<kernel.Constant object at 0xe10200>, <kernel.DependentProduct object at 0xb39440>) of role type named misses_decl
% Using role type
% Declaring misses:((fofType->Prop)->((fofType->Prop)->Prop))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) misses) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))->False))) of role definition named misses
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) misses) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))->False)))
% Defined: misses:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))->False))
% FOF formula (forall (A:(fofType->Prop)) (B:(fofType->Prop)), ((iff (((eq (fofType->Prop)) ((intersection A) B)) emptyset)) (((eq (fofType->Prop)) ((setminus A) B)) A))) of role conjecture named thm
% Conjecture to prove = (forall (A:(fofType->Prop)) (B:(fofType->Prop)), ((iff (((eq (fofType->Prop)) ((intersection A) B)) emptyset)) (((eq (fofType->Prop)) ((setminus A) B)) A))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(forall (A:(fofType->Prop)) (B:(fofType->Prop)), ((iff (((eq (fofType->Prop)) ((intersection A) B)) emptyset)) (((eq (fofType->Prop)) ((setminus A) B)) A)))']
% Parameter fofType:Type.
% Definition in:=(fun (X:fofType) (M:(fofType->Prop))=> (M X)):(fofType->((fofType->Prop)->Prop)).
% Definition is_a:=(fun (X:fofType) (M:(fofType->Prop))=> (M X)):(fofType->((fofType->Prop)->Prop)).
% Definition emptyset:=(fun (X:fofType)=> False):(fofType->Prop).
% Definition unord_pair:=(fun (X:fofType) (Y:fofType) (U:fofType)=> ((or (((eq fofType) U) X)) (((eq fofType) U) Y))):(fofType->(fofType->(fofType->Prop))).
% Definition singleton:=(fun (X:fofType) (U:fofType)=> (((eq fofType) U) X)):(fofType->(fofType->Prop)).
% Definition union:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or (X U)) (Y U))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition excl_union:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or ((and (X U)) ((Y U)->False))) ((and ((X U)->False)) (Y U)))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition intersection:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) (Y U))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition setminus:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) ((Y U)->False))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition complement:=(fun (X:(fofType->Prop)) (U:fofType)=> ((X U)->False)):((fofType->Prop)->(fofType->Prop)).
% Definition disjoint:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((eq (fofType->Prop)) ((intersection X) Y)) emptyset)):((fofType->Prop)->((fofType->Prop)->Prop)).
% Definition subset:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (forall (U:fofType), ((X U)->(Y U)))):((fofType->Prop)->((fofType->Prop)->Prop)).
% Definition meets:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))):((fofType->Prop)->((fofType->Prop)->Prop)).
% Definition misses:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))->False)):((fofType->Prop)->((fofType->Prop)->Prop)).
% Trying to prove (forall (A:(fofType->Prop)) (B:(fofType->Prop)), ((iff (((eq (fofType->Prop)) ((intersection A) B)) emptyset)) (((eq (fofType->Prop)) ((setminus A) B)) A)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found eta_expansion000:=(eta_expansion00 ((setminus A) B)):(((eq (fofType->Prop)) ((setminus A) B)) (fun (x:fofType)=> (((setminus A) B) x)))
% Found (eta_expansion00 ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found ((eta_expansion0 Prop) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found (((eta_expansion fofType) Prop) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found (((eta_expansion fofType) Prop) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found (((eta_expansion fofType) Prop) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found eq_ref00:=(eq_ref0 ((intersection A) B)):(((eq (fofType->Prop)) ((intersection A) B)) ((intersection A) B))
% Found (eq_ref0 ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) 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) emptyset)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))):((P ((intersection A) B))->(P ((intersection A) B)))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))):((P ((intersection A) B))->(P ((intersection A) B)))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))):((P ((intersection A) B))->(P ((intersection A) B)))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found eq_ref00:=(eq_ref0 ((((eq (fofType->Prop)) ((setminus A) B)) A)->((disjoint A) B))):(((eq Prop) ((((eq (fofType->Prop)) ((setminus A) B)) A)->((disjoint A) B))) ((((eq (fofType->Prop)) ((setminus A) B)) A)->((disjoint A) B)))
% Found (eq_ref0 ((((eq (fofType->Prop)) ((setminus A) B)) A)->((disjoint A) B))) as proof of (((eq Prop) ((((eq (fofType->Prop)) ((setminus A) B)) A)->((disjoint A) B))) b)
% Found ((eq_ref Prop) ((((eq (fofType->Prop)) ((setminus A) B)) A)->((disjoint A) B))) as proof of (((eq Prop) ((((eq (fofType->Prop)) ((setminus A) B)) A)->((disjoint A) B))) b)
% Found ((eq_ref Prop) ((((eq (fofType->Prop)) ((setminus A) B)) A)->((disjoint A) B))) as proof of (((eq Prop) ((((eq (fofType->Prop)) ((setminus A) B)) A)->((disjoint A) B))) b)
% Found ((eq_ref Prop) ((((eq (fofType->Prop)) ((setminus A) B)) A)->((disjoint A) B))) as proof of (((eq Prop) ((((eq (fofType->Prop)) ((setminus A) B)) A)->((disjoint A) B))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found eq_ref00:=(eq_ref0 ((setminus A) B)):(((eq (fofType->Prop)) ((setminus A) B)) ((setminus A) B))
% Found (eq_ref0 ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found eta_expansion000:=(eta_expansion00 ((intersection A) B)):(((eq (fofType->Prop)) ((intersection A) B)) (fun (x:fofType)=> (((intersection A) B) x)))
% Found (eta_expansion00 ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found ((eta_expansion0 Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found eta_expansion000:=(eta_expansion00 ((intersection ((setminus A) B)) B)):(((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) (fun (x:fofType)=> (((intersection ((setminus A) B)) B) x)))
% Found (eta_expansion00 ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found ((eta_expansion0 Prop) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))):((P ((intersection A) B))->(P ((intersection A) B)))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((intersection ((setminus A) B)) B))->(P (fun (x:fofType)=> (((intersection ((setminus A) B)) B) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((eta_expansion_dep00 ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))):((P ((intersection A) B))->(P ((intersection A) B)))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))):((P ((intersection A) B))->(P ((intersection A) B)))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found eq_ref000:=(eq_ref00 P):((P ((intersection ((setminus A) B)) B))->(P ((intersection ((setminus A) B)) B)))
% Found (eq_ref00 P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((eq_ref0 ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found eq_ref000:=(eq_ref00 P):((P ((intersection ((setminus A) B)) B))->(P ((intersection ((setminus A) B)) B)))
% Found (eq_ref00 P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((eq_ref0 ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found eq_ref00:=(eq_ref0 (((disjoint A) B)->(((eq (fofType->Prop)) ((setminus A) B)) A))):(((eq Prop) (((disjoint A) B)->(((eq (fofType->Prop)) ((setminus A) B)) A))) (((disjoint A) B)->(((eq (fofType->Prop)) ((setminus A) B)) A)))
% Found (eq_ref0 (((disjoint A) B)->(((eq (fofType->Prop)) ((setminus A) B)) A))) as proof of (((eq Prop) (((disjoint A) B)->(((eq (fofType->Prop)) ((setminus A) B)) A))) b)
% Found ((eq_ref Prop) (((disjoint A) B)->(((eq (fofType->Prop)) ((setminus A) B)) A))) as proof of (((eq Prop) (((disjoint A) B)->(((eq (fofType->Prop)) ((setminus A) B)) A))) b)
% Found ((eq_ref Prop) (((disjoint A) B)->(((eq (fofType->Prop)) ((setminus A) B)) A))) as proof of (((eq Prop) (((disjoint A) B)->(((eq (fofType->Prop)) ((setminus A) B)) A))) b)
% Found ((eq_ref Prop) (((disjoint A) B)->(((eq (fofType->Prop)) ((setminus A) B)) A))) as proof of (((eq Prop) (((disjoint A) B)->(((eq (fofType->Prop)) ((setminus A) B)) A))) b)
% Found eq_ref00:=(eq_ref0 ((intersection ((setminus A) B)) B)):(((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) ((intersection ((setminus A) B)) B))
% Found (eq_ref0 ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((setminus A) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus A) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus A) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus 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) ((intersection A) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection A) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection A) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection A) B))
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((intersection ((setminus A) B)) B))->(P (fun (x:fofType)=> (((intersection ((setminus A) B)) B) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((eta_expansion_dep00 ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((intersection ((setminus A) B)) B))->(P (fun (x:fofType)=> (((intersection ((setminus A) B)) B) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((eta_expansion_dep00 ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((intersection ((setminus A) B)) B))->(P (fun (x:fofType)=> (((intersection ((setminus A) B)) B) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((eta_expansion_dep00 ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P A))):((P A)->(P A))
% Found (x (fun (x0:(fofType->Prop))=> (P A))) as proof of (P0 A)
% Found (x (fun (x0:(fofType->Prop))=> (P A))) as proof of (P0 A)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P A))):((P A)->(P A))
% Found (x (fun (x0:(fofType->Prop))=> (P A))) as proof of (P0 A)
% Found (x (fun (x0:(fofType->Prop))=> (P A))) as proof of (P0 A)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P emptyset))):((P emptyset)->(P emptyset))
% Found (x (fun (x0:(fofType->Prop))=> (P emptyset))) as proof of (P0 emptyset)
% Found (x (fun (x0:(fofType->Prop))=> (P emptyset))) as proof of (P0 emptyset)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P emptyset))):((P emptyset)->(P emptyset))
% Found (x (fun (x0:(fofType->Prop))=> (P emptyset))) as proof of (P0 emptyset)
% Found (x (fun (x0:(fofType->Prop))=> (P emptyset))) as proof of (P0 emptyset)
% Found eq_ref00:=(eq_ref0 (((intersection A) B) x0)):(((eq Prop) (((intersection A) B) x0)) (((intersection A) B) x0))
% Found (eq_ref0 (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (((intersection A) B) x0)):(((eq Prop) (((intersection A) B) x0)) (((intersection A) B) x0))
% Found (eq_ref0 (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (((setminus A) B) x0)):(((eq Prop) (((setminus A) B) x0)) (((setminus A) B) x0))
% Found (eq_ref0 (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found eq_ref00:=(eq_ref0 (((setminus A) B) x0)):(((eq Prop) (((setminus A) B) x0)) (((setminus A) B) x0))
% Found (eq_ref0 (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found x1:=(x (fun (x1:(fofType->Prop))=> (P (((intersection A) B) x0)))):((P (((intersection A) B) x0))->(P (((intersection A) B) x0)))
% Found (x (fun (x1:(fofType->Prop))=> (P (((intersection A) B) x0)))) as proof of (P0 (((intersection A) B) x0))
% Found (x (fun (x1:(fofType->Prop))=> (P (((intersection A) B) x0)))) as proof of (P0 (((intersection A) B) x0))
% Found x1:=(x (fun (x1:(fofType->Prop))=> (P (((intersection A) B) x0)))):((P (((intersection A) B) x0))->(P (((intersection A) B) x0)))
% Found (x (fun (x1:(fofType->Prop))=> (P (((intersection A) B) x0)))) as proof of (P0 (((intersection A) B) x0))
% Found (x (fun (x1:(fofType->Prop))=> (P (((intersection A) B) x0)))) as proof of (P0 (((intersection A) B) x0))
% Found x1:=(x (fun (x1:(fofType->Prop))=> (P (((setminus A) B) x0)))):((P (((setminus A) B) x0))->(P (((setminus A) B) x0)))
% Found (x (fun (x1:(fofType->Prop))=> (P (((setminus A) B) x0)))) as proof of (P0 (((setminus A) B) x0))
% Found (x (fun (x1:(fofType->Prop))=> (P (((setminus A) B) x0)))) as proof of (P0 (((setminus A) B) x0))
% Found x1:=(x (fun (x1:(fofType->Prop))=> (P (((setminus A) B) x0)))):((P (((setminus A) B) x0))->(P (((setminus A) B) x0)))
% Found (x (fun (x1:(fofType->Prop))=> (P (((setminus A) B) x0)))) as proof of (P0 (((setminus A) B) x0))
% Found (x (fun (x1:(fofType->Prop))=> (P (((setminus A) B) x0)))) as proof of (P0 (((setminus A) B) x0))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P emptyset)->(P (fun (x:fofType)=> (emptyset x))))
% Found (eta_expansion_dep000 P) as proof of (P0 emptyset)
% Found ((eta_expansion_dep00 emptyset) P) as proof of (P0 emptyset)
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) P) as proof of (P0 emptyset)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((setminus A) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus A) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus A) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus A) B))
% Found eta_expansion_dep000:=(eta_expansion_dep00 A):(((eq (fofType->Prop)) A) (fun (x:fofType)=> (A x)))
% Found (eta_expansion_dep00 A) as proof of (((eq (fofType->Prop)) A) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) A) as proof of (((eq (fofType->Prop)) A) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) A) as proof of (((eq (fofType->Prop)) A) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) A) as proof of (((eq (fofType->Prop)) A) b)
% Found (((eta_expansion_dep fofType) (fun (x1: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) ((intersection ((setminus A) B)) B))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) ((intersection A) B))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection A) B))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection A) B))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection A) B))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection A) B))
% Found eta_expansion_dep000:=(eta_expansion_dep00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion_dep00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found eq_ref00:=(eq_ref0 ((setminus A) B)):(((eq (fofType->Prop)) ((setminus A) B)) ((setminus A) B))
% Found (eq_ref0 ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found eta_expansion000:=(eta_expansion00 ((intersection A) B)):(((eq (fofType->Prop)) ((intersection A) B)) (fun (x:fofType)=> (((intersection A) B) x)))
% Found (eta_expansion00 ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found ((eta_expansion0 Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) 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) emptyset)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))):((P ((intersection A) B))->(P ((intersection A) B)))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found eta_expansion0000:=(eta_expansion000 P):((P emptyset)->(P (fun (x:fofType)=> (emptyset x))))
% Found (eta_expansion000 P) as proof of (P0 emptyset)
% Found ((eta_expansion00 emptyset) P) as proof of (P0 emptyset)
% Found (((eta_expansion0 Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found eta_expansion0000:=(eta_expansion000 P):((P emptyset)->(P (fun (x:fofType)=> (emptyset x))))
% Found (eta_expansion000 P) as proof of (P0 emptyset)
% Found ((eta_expansion00 emptyset) P) as proof of (P0 emptyset)
% Found (((eta_expansion0 Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found eta_expansion0000:=(eta_expansion000 P):((P emptyset)->(P (fun (x:fofType)=> (emptyset x))))
% Found (eta_expansion000 P) as proof of (P0 emptyset)
% Found ((eta_expansion00 emptyset) P) as proof of (P0 emptyset)
% Found (((eta_expansion0 Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found eta_expansion0000:=(eta_expansion000 P):((P emptyset)->(P (fun (x:fofType)=> (emptyset x))))
% Found (eta_expansion000 P) as proof of (P0 emptyset)
% Found ((eta_expansion00 emptyset) P) as proof of (P0 emptyset)
% Found (((eta_expansion0 Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found eta_expansion0000:=(eta_expansion000 P):((P emptyset)->(P (fun (x:fofType)=> (emptyset x))))
% Found (eta_expansion000 P) as proof of (P0 emptyset)
% Found ((eta_expansion00 emptyset) P) as proof of (P0 emptyset)
% Found (((eta_expansion0 Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P A))):((P A)->(P A))
% Found (x (fun (x0:(fofType->Prop))=> (P A))) as proof of (P0 A)
% Found (x (fun (x0:(fofType->Prop))=> (P A))) as proof of (P0 A)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P A))):((P A)->(P A))
% Found (x (fun (x0:(fofType->Prop))=> (P A))) as proof of (P0 A)
% Found (x (fun (x0:(fofType->Prop))=> (P A))) as proof of (P0 A)
% Found eta_expansion0000:=(eta_expansion000 P):((P emptyset)->(P (fun (x:fofType)=> (emptyset x))))
% Found (eta_expansion000 P) as proof of (P0 emptyset)
% Found ((eta_expansion00 emptyset) P) as proof of (P0 emptyset)
% Found (((eta_expansion0 Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P emptyset))):((P emptyset)->(P emptyset))
% Found (x (fun (x0:(fofType->Prop))=> (P emptyset))) as proof of (P0 emptyset)
% Found (x (fun (x0:(fofType->Prop))=> (P emptyset))) as proof of (P0 emptyset)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P emptyset))):((P emptyset)->(P emptyset))
% Found (x (fun (x0:(fofType->Prop))=> (P emptyset))) as proof of (P0 emptyset)
% Found (x (fun (x0:(fofType->Prop))=> (P emptyset))) as proof of (P0 emptyset)
% Found x0:(P ((setminus A) B))
% Instantiate: b:=((setminus A) B):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found x0:(P ((intersection A) B))
% Instantiate: b:=((intersection A) B):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((intersection ((setminus A) B)) B) x00)):(((eq Prop) (((intersection ((setminus A) B)) B) x00)) (((intersection ((setminus A) B)) B) x00))
% Found (eq_ref0 (((intersection ((setminus A) B)) B) x00)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x00)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x00)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x00)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x00)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x00))
% Found eq_ref00:=(eq_ref0 (((intersection ((setminus A) B)) B) x00)):(((eq Prop) (((intersection ((setminus A) B)) B) x00)) (((intersection ((setminus A) B)) B) x00))
% Found (eq_ref0 (((intersection ((setminus A) B)) B) x00)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x00)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x00)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x00)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x00)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x00))
% Found eq_ref00:=(eq_ref0 A):(((eq (fofType->Prop)) A) A)
% Found (eq_ref0 A) as proof of (((eq (fofType->Prop)) A) b)
% Found ((eq_ref (fofType->Prop)) A) as proof of (((eq (fofType->Prop)) A) b)
% Found ((eq_ref (fofType->Prop)) A) as proof of (((eq (fofType->Prop)) A) b)
% Found ((eq_ref (fofType->Prop)) A) as proof of (((eq (fofType->Prop)) A) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion_dep00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eq_ref00:=(eq_ref0 (((intersection ((setminus A) B)) B) x0)):(((eq Prop) (((intersection ((setminus A) B)) B) x0)) (((intersection ((setminus A) B)) B) x0))
% Found (eq_ref0 (((intersection ((setminus A) B)) B) x0)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x0)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x0)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x0)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (((intersection ((setminus A) B)) B) x1)):(((eq Prop) (((intersection ((setminus A) B)) B) x1)) (((intersection ((setminus A) B)) B) x1))
% Found (eq_ref0 (((intersection ((setminus A) B)) B) x1)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x1)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x1)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x1)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x1))
% Found eq_ref00:=(eq_ref0 (((intersection ((setminus A) B)) B) x1)):(((eq Prop) (((intersection ((setminus A) B)) B) x1)) (((intersection ((setminus A) B)) B) x1))
% Found (eq_ref0 (((intersection ((setminus A) B)) B) x1)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x1)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x1)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x1)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x1))
% Found eq_ref00:=(eq_ref0 (((intersection ((setminus A) B)) B) x0)):(((eq Prop) (((intersection ((setminus A) B)) B) x0)) (((intersection ((setminus A) B)) B) x0))
% Found (eq_ref0 (((intersection ((setminus A) B)) B) x0)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x0)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x0)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x0)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (((intersection A) B) x0)):(((eq Prop) (((intersection A) B) x0)) (((intersection A) B) x0))
% Found (eq_ref0 (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (((intersection A) B) x0)):(((eq Prop) (((intersection A) B) x0)) (((intersection A) B) x0))
% Found (eq_ref0 (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))):((P ((intersection A) B))->(P ((intersection A) B)))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found eq_ref00:=(eq_ref0 (((setminus A) B) x0)):(((eq Prop) (((setminus A) B) x0)) (((setminus A) B) x0))
% Found (eq_ref0 (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found eq_ref00:=(eq_ref0 (((setminus A) B) x0)):(((eq Prop) (((setminus A) B) x0)) (((setminus A) B) x0))
% Found (eq_ref0 (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))):((P ((intersection A) B))->(P ((intersection A) B)))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found eq_ref000:=(eq_ref00 P):((P (((intersection ((setminus A) B)) B) x00))->(P (((intersection ((setminus A) B)) B) x00)))
% Found (eq_ref00 P) as proof of (P0 (((intersection ((setminus A) B)) B) x00))
% Found ((eq_ref0 (((intersection ((setminus A) B)) B) x00)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x00))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x00))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x00))
% Found eq_ref000:=(eq_ref00 P):((P (((intersection ((setminus A) B)) B) x00))->(P (((intersection ((setminus A) B)) B) x00)))
% Found (eq_ref00 P) as proof of (P0 (((intersection ((setminus A) B)) B) x00))
% Found ((eq_ref0 (((intersection ((setminus A) B)) B) x00)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x00))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x00))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x00))
% Found eq_ref000:=(eq_ref00 P):((P (((intersection ((setminus A) B)) B) x1))->(P (((intersection ((setminus A) B)) B) x1)))
% Found (eq_ref00 P) as proof of (P0 (((intersection ((setminus A) B)) B) x1))
% Found ((eq_ref0 (((intersection ((setminus A) B)) B) x1)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x1))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x1))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x1))
% Found eq_ref000:=(eq_ref00 P):((P (((intersection ((setminus A) B)) B) x0))->(P (((intersection ((setminus A) B)) B) x0)))
% Found (eq_ref00 P) as proof of (P0 (((intersection ((setminus A) B)) B) x0))
% Found ((eq_ref0 (((intersection ((setminus A) B)) B) x0)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x0))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x0))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x0))
% Found eq_ref000:=(eq_ref00 P):((P (((intersection ((setminus A) B)) B) x0))->(P (((intersection ((setminus A) B)) B) x0)))
% Found (eq_ref00 P) as proof of (P0 (((intersection ((setminus A) B)) B) x0))
% Found ((eq_ref0 (((intersection ((setminus A) B)) B) x0)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x0))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x0))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x0))
% Found eq_ref000:=(eq_ref00 P):((P (((intersection ((setminus A) B)) B) x1))->(P (((intersection ((setminus A) B)) B) x1)))
% Found (eq_ref00 P) as proof of (P0 (((intersection ((setminus A) B)) B) x1))
% Found ((eq_ref0 (((intersection ((setminus A) B)) B) x1)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x1))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x1))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x1))
% Found x1:=(x (fun (x1:(fofType->Prop))=> (P (((setminus A) B) x0)))):((P (((setminus A) B) x0))->(P (((setminus A) B) x0)))
% Found (x (fun (x1:(fofType->Prop))=> (P (((setminus A) B) x0)))) as proof of (P0 (((setminus A) B) x0))
% Found (x (fun (x1:(fofType->Prop))=> (P (((setminus A) B) x0)))) as proof of (P0 (((setminus A) B) x0))
% Found x1:=(x (fun (x1:(fofType->Prop))=> (P (((setminus A) B) x0)))):((P (((setminus A) B) x0))->(P (((setminus A) B) x0)))
% Found (x (fun (x1:(fofType->Prop))=> (P (((setminus A) B) x0)))) as proof of (P0 (((setminus A) B) x0))
% Found (x (fun (x1:(fofType->Prop))=> (P (((setminus A) B) x0)))) as proof of (P0 (((setminus A) B) x0))
% Found x1:=(x (fun (x1:(fofType->Prop))=> (P (((intersection A) B) x0)))):((P (((intersection A) B) x0))->(P (((intersection A) B) x0)))
% Found (x (fun (x1:(fofType->Prop))=> (P (((intersection A) B) x0)))) as proof of (P0 (((intersection A) B) x0))
% Found (x (fun (x1:(fofType->Prop))=> (P (((intersection A) B) x0)))) as proof of (P0 (((intersection A) B) x0))
% Found x1:=(x (fun (x1:(fofType->Prop))=> (P (((intersection A) B) x0)))):((P (((intersection A) B) x0))->(P (((intersection A) B) x0)))
% Found (x (fun (x1:(fofType->Prop))=> (P (((intersection A) B) x0)))) as proof of (P0 (((intersection A) B) x0))
% Found (x (fun (x1:(fofType->Prop))=> (P (((intersection A) B) x0)))) as proof of (P0 (((intersection A) B) x0))
% Found eta_expansion0000:=(eta_expansion000 P):((P emptyset)->(P (fun (x:fofType)=> (emptyset x))))
% Found (eta_expansion000 P) as proof of (P0 emptyset)
% Found ((eta_expansion00 emptyset) P) as proof of (P0 emptyset)
% Found (((eta_expansion0 Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found x0:(P ((intersection A) B))
% Instantiate: f:=((intersection A) B):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P ((setminus A) B))
% Instantiate: f:=((setminus A) B):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P ((intersection A) B))
% Instantiate: f:=((intersection A) B):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P ((setminus A) B))
% Instantiate: f:=((setminus A) B):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 ((((eq (fofType->Prop)) ((setminus A) B)) A)->(((eq (fofType->Prop)) ((intersection A) B)) emptyset))):(((eq Prop) ((((eq (fofType->Prop)) ((setminus A) B)) A)->(((eq (fofType->Prop)) ((intersection A) B)) emptyset))) ((((eq (fofType->Prop)) ((setminus A) B)) A)->(((eq (fofType->Prop)) ((intersection A) B)) emptyset)))
% Found (eq_ref0 ((((eq (fofType->Prop)) ((setminus A) B)) A)->(((eq (fofType->Prop)) ((intersection A) B)) emptyset))) as proof of (((eq Prop) ((((eq (fofType->Prop)) ((setminus A) B)) A)->(((eq (fofType->Prop)) ((intersection A) B)) emptyset))) b)
% Found ((eq_ref Prop) ((((eq (fofType->Prop)) ((setminus A) B)) A)->(((eq (fofType->Prop)) ((intersection A) B)) emptyset))) as proof of (((eq Prop) ((((eq (fofType->Prop)) ((setminus A) B)) A)->(((eq (fofType->Prop)) ((intersection A) B)) emptyset))) b)
% Found ((eq_ref Prop) ((((eq (fofType->Prop)) ((setminus A) B)) A)->(((eq (fofType->Prop)) ((intersection A) B)) emptyset))) as proof of (((eq Prop) ((((eq (fofType->Prop)) ((setminus A) B)) A)->(((eq (fofType->Prop)) ((intersection A) B)) emptyset))) b)
% Found ((eq_ref Prop) ((((eq (fofType->Prop)) ((setminus A) B)) A)->(((eq (fofType->Prop)) ((intersection A) B)) emptyset))) as proof of (((eq Prop) ((((eq (fofType->Prop)) ((setminus A) B)) A)->(((eq (fofType->Prop)) ((intersection A) B)) emptyset))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P emptyset)->(P (fun (x:fofType)=> (emptyset x))))
% Found (eta_expansion_dep000 P) as proof of (P0 emptyset)
% Found ((eta_expansion_dep00 emptyset) P) as proof of (P0 emptyset)
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) P) as proof of (P0 emptyset)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found eta_expansion000:=(eta_expansion00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion0 Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection ((setminus A) B)) B))
% Found eta_expansion000:=(eta_expansion00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion0 Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))):((P ((intersection A) B))->(P ((intersection A) B)))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found eq_ref00:=(eq_ref0 ((setminus A) B)):(((eq (fofType->Prop)) ((setminus A) B)) ((setminus A) B))
% Found (eq_ref0 ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found eq_ref00:=(eq_ref0 ((setminus A) B)):(((eq (fofType->Prop)) ((setminus A) B)) ((setminus A) B))
% Found (eq_ref0 ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found eta_expansion000:=(eta_expansion00 ((intersection A) B)):(((eq (fofType->Prop)) ((intersection A) B)) (fun (x:fofType)=> (((intersection A) B) x)))
% Found (eta_expansion00 ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found ((eta_expansion0 Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found eq_ref00:=(eq_ref0 ((intersection A) B)):(((eq (fofType->Prop)) ((intersection A) B)) ((intersection A) B))
% Found (eq_ref0 ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found ((eq_ref (fofType->Prop)) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found eta_expansion000:=(eta_expansion00 ((intersection A) B)):(((eq (fofType->Prop)) ((intersection A) B)) (fun (x:fofType)=> (((intersection A) B) x)))
% Found (eta_expansion00 ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found ((eta_expansion0 Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found eta_expansion000:=(eta_expansion00 ((intersection ((setminus A) B)) B)):(((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) (fun (x:fofType)=> (((intersection ((setminus A) B)) B) x)))
% Found (eta_expansion00 ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found ((eta_expansion0 Prop) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (emptyset x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (emptyset x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (A x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (A x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (A x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (A x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (A x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (A x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (A x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (A x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (A x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (A x)))
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))):((P ((intersection A) B))->(P ((intersection A) B)))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((intersection ((setminus A) B)) B))->(P (fun (x:fofType)=> (((intersection ((setminus A) B)) B) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((eta_expansion_dep00 ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found x0:(P ((setminus A) B))
% Instantiate: b:=((setminus A) B):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found x0:(P ((intersection A) B))
% Instantiate: b:=((intersection A) B):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found x1:(P ((intersection ((setminus A) B)) B))
% Instantiate: b:=((intersection ((setminus A) B)) B):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))):((P ((intersection A) B))->(P ((intersection A) B)))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))):((P ((intersection A) B))->(P ((intersection A) B)))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found eta_expansion000:=(eta_expansion00 A):(((eq (fofType->Prop)) A) (fun (x:fofType)=> (A x)))
% Found (eta_expansion00 A) as proof of (((eq (fofType->Prop)) A) b)
% Found ((eta_expansion0 Prop) A) as proof of (((eq (fofType->Prop)) A) b)
% Found (((eta_expansion fofType) Prop) A) as proof of (((eq (fofType->Prop)) A) b)
% Found (((eta_expansion fofType) Prop) A) as proof of (((eq (fofType->Prop)) A) b)
% Found (((eta_expansion fofType) Prop) A) as proof of (((eq (fofType->Prop)) A) b)
% Found eta_expansion000:=(eta_expansion00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion0 Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P emptyset)->(P (fun (x:fofType)=> (emptyset x))))
% Found (eta_expansion_dep000 P) as proof of (P0 emptyset)
% Found ((eta_expansion_dep00 emptyset) P) as proof of (P0 emptyset)
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) P) as proof of (P0 emptyset)
% Found eta_expansion0000:=(eta_expansion000 P):((P emptyset)->(P (fun (x:fofType)=> (emptyset x))))
% Found (eta_expansion000 P) as proof of (P0 emptyset)
% Found ((eta_expansion00 emptyset) P) as proof of (P0 emptyset)
% Found (((eta_expansion0 Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found eta_expansion_dep000:=(eta_expansion_dep00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion_dep00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eq_ref000:=(eq_ref00 P):((P ((intersection ((setminus A) B)) B))->(P ((intersection ((setminus A) B)) B)))
% Found (eq_ref00 P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((eq_ref0 ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found eq_ref000:=(eq_ref00 P):((P emptyset)->(P emptyset))
% Found (eq_ref00 P) as proof of (P0 emptyset)
% Found ((eq_ref0 emptyset) P) as proof of (P0 emptyset)
% Found (((eq_ref (fofType->Prop)) emptyset) P) as proof of (P0 emptyset)
% Found (((eq_ref (fofType->Prop)) emptyset) P) as proof of (P0 emptyset)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P emptyset)->(P (fun (x:fofType)=> (emptyset x))))
% Found (eta_expansion_dep000 P) as proof of (P0 emptyset)
% Found ((eta_expansion_dep00 emptyset) P) as proof of (P0 emptyset)
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) P) as proof of (P0 emptyset)
% Found eq_ref000:=(eq_ref00 P):((P ((intersection ((setminus A) B)) B))->(P ((intersection ((setminus A) B)) B)))
% Found (eq_ref00 P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((eq_ref0 ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P emptyset)->(P (fun (x:fofType)=> (emptyset x))))
% Found (eta_expansion_dep000 P) as proof of (P0 emptyset)
% Found ((eta_expansion_dep00 emptyset) P) as proof of (P0 emptyset)
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) P) as proof of (P0 emptyset)
% Found eq_ref000:=(eq_ref00 P):((P emptyset)->(P emptyset))
% Found (eq_ref00 P) as proof of (P0 emptyset)
% Found ((eq_ref0 emptyset) P) as proof of (P0 emptyset)
% Found (((eq_ref (fofType->Prop)) emptyset) P) as proof of (P0 emptyset)
% Found (((eq_ref (fofType->Prop)) emptyset) P) as proof of (P0 emptyset)
% Found eq_ref00:=(eq_ref0 (((intersection ((setminus A) B)) B) x00)):(((eq Prop) (((intersection ((setminus A) B)) B) x00)) (((intersection ((setminus A) B)) B) x00))
% Found (eq_ref0 (((intersection ((setminus A) B)) B) x00)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x00)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x00)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x00)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x00)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x00))
% Found eq_ref00:=(eq_ref0 (((intersection ((setminus A) B)) B) x00)):(((eq Prop) (((intersection ((setminus A) B)) B) x00)) (((intersection ((setminus A) B)) B) x00))
% Found (eq_ref0 (((intersection ((setminus A) B)) B) x00)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x00)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x00)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x00)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x00)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x00))
% Found eq_ref00:=(eq_ref0 (((intersection ((setminus A) B)) B) x1)):(((eq Prop) (((intersection ((setminus A) B)) B) x1)) (((intersection ((setminus A) B)) B) x1))
% Found (eq_ref0 (((intersection ((setminus A) B)) B) x1)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x1)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x1)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x1)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x1))
% Found eq_ref00:=(eq_ref0 (((intersection ((setminus A) B)) B) x1)):(((eq Prop) (((intersection ((setminus A) B)) B) x1)) (((intersection ((setminus A) B)) B) x1))
% Found (eq_ref0 (((intersection ((setminus A) B)) B) x1)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x1)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x1)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x1)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x1))
% Found eq_ref00:=(eq_ref0 (((intersection ((setminus A) B)) B) x0)):(((eq Prop) (((intersection ((setminus A) B)) B) x0)) (((intersection ((setminus A) B)) B) x0))
% Found (eq_ref0 (((intersection ((setminus A) B)) B) x0)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x0)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x0)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x0)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (((intersection ((setminus A) B)) B) x0)):(((eq Prop) (((intersection ((setminus A) B)) B) x0)) (((intersection ((setminus A) B)) B) x0))
% Found (eq_ref0 (((intersection ((setminus A) B)) B) x0)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x0)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x0)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x0)) b)
% Found ((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) as proof of (((eq Prop) (((intersection ((setminus A) B)) B) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection A) B) x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection A) B) x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus A) B) x0))
% Found eq_ref00:=(eq_ref0 (A x0)):(((eq Prop) (A x0)) (A x0))
% Found (eq_ref0 (A x0)) as proof of (((eq Prop) (A x0)) b)
% Found ((eq_ref Prop) (A x0)) as proof of (((eq Prop) (A x0)) b)
% Found ((eq_ref Prop) (A x0)) as proof of (((eq Prop) (A x0)) b)
% Found ((eq_ref Prop) (A x0)) as proof of (((eq Prop) (A x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus A) B) x0))
% Found eq_ref00:=(eq_ref0 (A x0)):(((eq Prop) (A x0)) (A x0))
% Found (eq_ref0 (A x0)) as proof of (((eq Prop) (A x0)) b)
% Found ((eq_ref Prop) (A x0)) as proof of (((eq Prop) (A x0)) b)
% Found ((eq_ref Prop) (A x0)) as proof of (((eq Prop) (A x0)) b)
% Found ((eq_ref Prop) (A x0)) as proof of (((eq Prop) (A x0)) b)
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection A) B) x0))
% Found eq_ref00:=(eq_ref0 (emptyset x0)):(((eq Prop) (emptyset x0)) (emptyset x0))
% Found (eq_ref0 (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found ((eq_ref Prop) (emptyset x0)) as proof of (((eq Prop) (emptyset x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection A) B) x0))
% Found eq_ref00:=(eq_ref0 (A x0)):(((eq Prop) (A x0)) (A x0))
% Found (eq_ref0 (A x0)) as proof of (((eq Prop) (A x0)) b)
% Found ((eq_ref Prop) (A x0)) as proof of (((eq Prop) (A x0)) b)
% Found ((eq_ref Prop) (A x0)) as proof of (((eq Prop) (A x0)) b)
% Found ((eq_ref Prop) (A x0)) as proof of (((eq Prop) (A x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus A) B) x0))
% Found eq_ref00:=(eq_ref0 (A x0)):(((eq Prop) (A x0)) (A x0))
% Found (eq_ref0 (A x0)) as proof of (((eq Prop) (A x0)) b)
% Found ((eq_ref Prop) (A x0)) as proof of (((eq Prop) (A x0)) b)
% Found ((eq_ref Prop) (A x0)) as proof of (((eq Prop) (A x0)) b)
% Found ((eq_ref Prop) (A x0)) as proof of (((eq Prop) (A x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((setminus A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus A) B) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((setminus A) B) x0))
% Found eq_ref00:=(eq_ref0 ((((eq (fofType->Prop)) ((intersection A) B)) emptyset)->(((eq (fofType->Prop)) ((setminus A) B)) A))):(((eq Prop) ((((eq (fofType->Prop)) ((intersection A) B)) emptyset)->(((eq (fofType->Prop)) ((setminus A) B)) A))) ((((eq (fofType->Prop)) ((intersection A) B)) emptyset)->(((eq (fofType->Prop)) ((setminus A) B)) A)))
% Found (eq_ref0 ((((eq (fofType->Prop)) ((intersection A) B)) emptyset)->(((eq (fofType->Prop)) ((setminus A) B)) A))) as proof of (((eq Prop) ((((eq (fofType->Prop)) ((intersection A) B)) emptyset)->(((eq (fofType->Prop)) ((setminus A) B)) A))) b)
% Found ((eq_ref Prop) ((((eq (fofType->Prop)) ((intersection A) B)) emptyset)->(((eq (fofType->Prop)) ((setminus A) B)) A))) as proof of (((eq Prop) ((((eq (fofType->Prop)) ((intersection A) B)) emptyset)->(((eq (fofType->Prop)) ((setminus A) B)) A))) b)
% Found ((eq_ref Prop) ((((eq (fofType->Prop)) ((intersection A) B)) emptyset)->(((eq (fofType->Prop)) ((setminus A) B)) A))) as proof of (((eq Prop) ((((eq (fofType->Prop)) ((intersection A) B)) emptyset)->(((eq (fofType->Prop)) ((setminus A) B)) A))) b)
% Found ((eq_ref Prop) ((((eq (fofType->Prop)) ((intersection A) B)) emptyset)->(((eq (fofType->Prop)) ((setminus A) B)) A))) as proof of (((eq Prop) ((((eq (fofType->Prop)) ((intersection A) B)) emptyset)->(((eq (fofType->Prop)) ((setminus A) B)) A))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (P b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (P b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (P b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (P b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) 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) emptyset)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found eta_expansion000:=(eta_expansion00 ((intersection ((setminus A) B)) B)):(((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) (fun (x:fofType)=> (((intersection ((setminus A) B)) B) x)))
% Found (eta_expansion00 ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found ((eta_expansion0 Prop) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 A):(((eq (fofType->Prop)) A) (fun (x:fofType)=> (A x)))
% Found (eta_expansion_dep00 A) as proof of (((eq (fofType->Prop)) A) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) A) as proof of (((eq (fofType->Prop)) A) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) A) as proof of (((eq (fofType->Prop)) A) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) A) as proof of (((eq (fofType->Prop)) A) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) A) as proof of (((eq (fofType->Prop)) A) b)
% Found eq_ref000:=(eq_ref00 P):((P (((intersection ((setminus A) B)) B) x00))->(P (((intersection ((setminus A) B)) B) x00)))
% Found (eq_ref00 P) as proof of (P0 (((intersection ((setminus A) B)) B) x00))
% Found ((eq_ref0 (((intersection ((setminus A) B)) B) x00)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x00))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x00))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x00))
% Found eq_ref000:=(eq_ref00 P):((P (((intersection ((setminus A) B)) B) x00))->(P (((intersection ((setminus A) B)) B) x00)))
% Found (eq_ref00 P) as proof of (P0 (((intersection ((setminus A) B)) B) x00))
% Found ((eq_ref0 (((intersection ((setminus A) B)) B) x00)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x00))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x00))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x00)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x00))
% Found x0:(P ((setminus A) B))
% Instantiate: f:=((setminus A) B):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P ((setminus A) B))
% Instantiate: f:=((setminus A) B):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P ((intersection A) B))
% Instantiate: f:=((intersection A) B):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x1:=(x (fun (x1:(fofType->Prop))=> (P (A x0)))):((P (A x0))->(P (A x0)))
% Found (x (fun (x1:(fofType->Prop))=> (P (A x0)))) as proof of (P0 (A x0))
% Found (x (fun (x1:(fofType->Prop))=> (P (A x0)))) as proof of (P0 (A x0))
% Found x1:=(x (fun (x1:(fofType->Prop))=> (P (A x0)))):((P (A x0))->(P (A x0)))
% Found (x (fun (x1:(fofType->Prop))=> (P (A x0)))) as proof of (P0 (A x0))
% Found (x (fun (x1:(fofType->Prop))=> (P (A x0)))) as proof of (P0 (A x0))
% Found x0:(P ((intersection A) B))
% Instantiate: f:=((intersection A) B):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x1:(P ((intersection ((setminus A) B)) B))
% Instantiate: f:=((intersection ((setminus A) B)) B):(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found x1:(P ((intersection ((setminus A) B)) B))
% Instantiate: f:=((intersection ((setminus A) B)) B):(fofType->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref000:=(eq_ref00 P):((P (((intersection ((setminus A) B)) B) x0))->(P (((intersection ((setminus A) B)) B) x0)))
% Found (eq_ref00 P) as proof of (P0 (((intersection ((setminus A) B)) B) x0))
% Found ((eq_ref0 (((intersection ((setminus A) B)) B) x0)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x0))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x0))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x0))
% Found eq_ref000:=(eq_ref00 P):((P (((intersection ((setminus A) B)) B) x0))->(P (((intersection ((setminus A) B)) B) x0)))
% Found (eq_ref00 P) as proof of (P0 (((intersection ((setminus A) B)) B) x0))
% Found ((eq_ref0 (((intersection ((setminus A) B)) B) x0)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x0))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x0))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x0)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x0))
% Found eq_ref000:=(eq_ref00 P):((P (((intersection ((setminus A) B)) B) x1))->(P (((intersection ((setminus A) B)) B) x1)))
% Found (eq_ref00 P) as proof of (P0 (((intersection ((setminus A) B)) B) x1))
% Found ((eq_ref0 (((intersection ((setminus A) B)) B) x1)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x1))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x1))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x1))
% Found eq_ref000:=(eq_ref00 P):((P (((intersection ((setminus A) B)) B) x1))->(P (((intersection ((setminus A) B)) B) x1)))
% Found (eq_ref00 P) as proof of (P0 (((intersection ((setminus A) B)) B) x1))
% Found ((eq_ref0 (((intersection ((setminus A) B)) B) x1)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x1))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x1))
% Found (((eq_ref Prop) (((intersection ((setminus A) B)) B) x1)) P) as proof of (P0 (((intersection ((setminus A) B)) B) x1))
% Found x1:=(x (fun (x1:(fofType->Prop))=> (P (emptyset x0)))):((P (emptyset x0))->(P (emptyset x0)))
% Found (x (fun (x1:(fofType->Prop))=> (P (emptyset x0)))) as proof of (P0 (emptyset x0))
% Found (x (fun (x1:(fofType->Prop))=> (P (emptyset x0)))) as proof of (P0 (emptyset x0))
% Found x1:=(x (fun (x1:(fofType->Prop))=> (P (emptyset x0)))):((P (emptyset x0))->(P (emptyset x0)))
% Found (x (fun (x1:(fofType->Prop))=> (P (emptyset x0)))) as proof of (P0 (emptyset x0))
% Found (x (fun (x1:(fofType->Prop))=> (P (emptyset x0)))) as proof of (P0 (emptyset x0))
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 ((setminus A) B)):(((eq (fofType->Prop)) ((setminus A) B)) ((setminus A) B))
% Found (eq_ref0 ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b0)
% Found ((eq_ref (fofType->Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b0)
% Found ((eq_ref (fofType->Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b0)
% Found ((eq_ref (fofType->Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b0)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))):((P ((intersection A) B))->(P ((intersection A) B)))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found (x (fun (x0:(fofType->Prop))=> (P ((intersection A) B)))) as proof of (P0 ((intersection A) B))
% Found eta_expansion0000:=(eta_expansion000 P):((P emptyset)->(P (fun (x:fofType)=> (emptyset x))))
% Found (eta_expansion000 P) as proof of (P0 emptyset)
% Found ((eta_expansion00 emptyset) P) as proof of (P0 emptyset)
% Found (((eta_expansion0 Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found ((((eta_expansion fofType) Prop) emptyset) P) as proof of (P0 emptyset)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found eta_expansion000:=(eta_expansion00 ((setminus A) B)):(((eq (fofType->Prop)) ((setminus A) B)) (fun (x:fofType)=> (((setminus A) B) x)))
% Found (eta_expansion00 ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found ((eta_expansion0 Prop) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found (((eta_expansion fofType) Prop) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found (((eta_expansion fofType) Prop) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found (((eta_expansion fofType) Prop) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found eq_ref00:=(eq_ref0 ((intersection ((setminus A) B)) B)):(((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) ((intersection ((setminus A) B)) B))
% Found (eq_ref0 ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((intersection A) B)):(((eq (fofType->Prop)) ((intersection A) B)) (fun (x:fofType)=> (((intersection A) B) x)))
% Found (eta_expansion_dep00 ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found eta_expansion000:=(eta_expansion00 ((intersection A) B)):(((eq (fofType->Prop)) ((intersection A) B)) (fun (x:fofType)=> (((intersection A) B) x)))
% Found (eta_expansion00 ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found ((eta_expansion0 Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion fofType) Prop) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found eq_ref00:=(eq_ref0 ((intersection ((setminus A) B)) B)):(((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) ((intersection ((setminus A) B)) B))
% Found (eq_ref0 ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found ((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (emptyset x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (emptyset x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (emptyset x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (emptyset x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (emptyset x)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((intersection ((setminus A) B)) B))->(P (fun (x:fofType)=> (((intersection ((setminus A) B)) B) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((eta_expansion_dep00 ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((setminus A) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus A) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus A) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((setminus 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 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (emptyset x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (emptyset x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (emptyset x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (emptyset x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (emptyset x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (emptyset x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (emptyset x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (emptyset x)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((intersection A) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection A) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection A) B))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((intersection A) B))
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (A x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (A x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (A x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (A x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (A x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (A x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (A x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (A x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (A x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (A x)))
% Found eq_ref000:=(eq_ref00 P):((P ((intersection ((setminus A) B)) B))->(P ((intersection ((setminus A) B)) B)))
% Found (eq_ref00 P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((eq_ref0 ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((intersection ((setminus A) B)) B)):(((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) (fun (x:fofType)=> (((intersection ((setminus A) B)) B) x)))
% Found (eta_expansion_dep00 ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) as proof of (((eq (fofType->Prop)) ((intersection ((setminus A) B)) B)) 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) emptyset)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) emptyset)
% Found x0:(P A)
% Instantiate: b:=A:(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found x0:(P emptyset)
% Instantiate: b:=emptyset:(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((setminus A) B)):(((eq (fofType->Prop)) ((setminus A) B)) (fun (x:fofType)=> (((setminus A) B) x)))
% Found (eta_expansion_dep00 ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((intersection A) B)):(((eq (fofType->Prop)) ((intersection A) B)) (fun (x:fofType)=> (((intersection A) B) x)))
% Found (eta_expansion_dep00 ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection A) B)) as proof of (((eq (fofType->Prop)) ((intersection A) B)) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((intersection ((setminus A) B)) B))->(P (fun (x:fofType)=> (((intersection ((setminus A) B)) B) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((eta_expansion_dep00 ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found eq_ref000:=(eq_ref00 P):((P ((intersection ((setminus A) B)) B))->(P ((intersection ((setminus A) B)) B)))
% Found (eq_ref00 P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((eq_ref0 ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eq_ref (fofType->Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) 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 x1:(P ((intersection ((setminus A) B)) B))
% Instantiate: b:=((intersection ((setminus A) B)) B):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((intersection ((setminus A) B)) B))->(P (fun (x:fofType)=> (((intersection ((setminus A) B)) B) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((eta_expansion_dep00 ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P ((intersection ((setminus A) B)) B))->(P (fun (x:fofType)=> (((intersection ((setminus A) B)) B) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((eta_expansion_dep00 ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) ((intersection ((setminus A) B)) B)) P) as proof of (P0 ((intersection ((setminus A) B)) B))
% Found eq_ref000:=(eq_ref00 P):((P (emptyset x0))->(P (emptyset x0)))
% Found (eq_ref00 P) as proof of (P0 (emptyset x0))
% Found ((eq_ref0 (emptyset x0)) P) as proof of (P0 (emptyset x0))
% Found (((eq_ref Prop) (emptyset x0)) P) as proof of (P0 (emptyset x0))
% Found (((eq_ref Prop) (emptyset x0)) P) as proof of (P0 (emptyset x0))
% Found eq_ref000:=(eq_ref00 P):((P (emptyset x0))->(P (emptyset x0)))
% Found (eq_ref00 P) as proof of (P0 (emptyset x0))
% Found ((eq_ref0 (emptyset x0)) P) as proof of (P0 (emptyset x0))
% Found (((eq_ref Prop) (emptyset x0)) P) as proof of (P0 (emptyset x0))
% Found (((eq_ref Prop) (emptyset x0)) P) as proof of (P0 (emptyset x0))
% Found eta_expansion000:=(eta_expansion00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion0 Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eq_ref000:=(eq_ref00 P):((P (emptyset x0))->(P (emptyset x0)))
% Found (eq_ref00 P) as proof of (P0 (emptyset x0))
% Found ((eq_ref0 (emptyset x0)) P) as proof of (P0 (emptyset x0))
% Found (((eq_ref Prop) (emptyset x0)) P) as proof of (P0 (emptyset x0))
% Found (((eq_ref Prop) (emptyset x0)) P) as proof of (P0 (emptyset x0))
% Found eq_ref000:=(eq_ref00 P):((P (emptyset x0))->(P (emptyset x0)))
% Found (eq_ref00 P) as proof of (P0 (emptyset x0))
% Found ((eq_ref0 (emptyset x0)) P) as proof of (P0 (emptyset x0))
% Found (((eq_ref Prop) (emptyset x0)) P) as proof of (P0 (emptyset x0))
% Found (((eq_ref Prop) (emptyset x0)) P) as proof of (P0 (emptyset x0))
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P A))):((P A)->(P A))
% Found (x (fun (x0:(fofType->Prop))=> (P A))) as proof of (P0 A)
% Found (x (fun (x0:(fofType->Prop))=> (P A))) as proof of (P0 A)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P A))):((P A)->(P A))
% Found (x (fun (x0:(fofType->Prop))=> (P A))) as proof of (P0 A)
% Found (x (fun (x0:(fofType->Prop))=> (P A))) as proof of (P0 A)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P emptyset))):((P emptyset)->(P emptyset))
% Found (x (fun (x0:(fofType->Prop))=> (P emptyset))) as proof of (P0 emptyset)
% Found (x (fun (x0:(fofType->Prop))=> (P emptyset))) as proof of (P0 emptyset)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P emptyset))):((P emptyset)->(P emptyset))
% Found (x (fun (x0:(fofType->Prop))=> (P emptyset))) as proof of (P0 emptyset)
% Found (x (fun (x0:(fofType->Prop))=> (P emptyset))) as proof of (P0 emptyset)
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P ((setminus A) B)))):((P ((setminus A) B))->(P ((setminus A) B)))
% Found (x (fun (x0:(fofType->Prop))=> (P ((setminus A) B)))) as proof of (P0 ((setminus A) B))
% Found (x (fun (x0:(fofType->Prop))=> (P ((setminus A) B)))) as proof of (P0 ((setminus A) B))
% Found x0:=(x (fun (x0:(fofType->Prop))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:(fofType->Prop))=> (P b))) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P (emptyset x0))->(P (emptyset x0)))
% Found (eq_ref00 P) as proof of (P0 emptyset)
% Found ((eq_ref0 (emptyset x0)) P) as proof of (P0 emptyset)
% Found (((eq_ref Prop) (emptyset x0)) P) as proof of (P0 emptyset)
% Found (((eq_ref Prop) (emptyset x0)) P) as proof of (P0 emptyset)
% Found eq_ref000:=(eq_ref00 P):((P (emptyset x0))->(P (emptyset x0)))
% Found (eq_ref00 P) as proof of (P0 emptyset)
% Found ((eq_ref0 (emptyset x0)) P) as proof of (P0 emptyset)
% Found (((eq_ref Prop) (emptyset x0)) P) as proof of (P0 emptyset)
% Found (((eq_ref Prop) (emptyset x0)) P) as proof of (P0 emptyset)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref (fofType->Prop)) b) as proof of (P b)
% Found ((eq_ref (fofType->Prop)) b) as proof of (P b)
% Found ((eq_ref (fofType->Prop)) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 emptyset):(((eq (fofType->Prop)) emptyset) emptyset)
% Found (eq_ref0 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eq_ref (fofType->Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (P b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (P b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (P b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (P b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion0 Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) A)
% Found eta_expansion000:=(eta_expansion00 ((setminus A) B)):(((eq (fofType->Prop)) ((setminus A) B)) (fun (x:fofType)=> (((setminus A) B) x)))
% Found (eta_expansion00 ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found ((eta_expansion0 Prop) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found (((eta_expansion fofType) Prop) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found (((eta_expansion fofType) Prop) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found (((eta_expansion fofType) Prop) ((setminus A) B)) as proof of (((eq (fofType->Prop)) ((setminus A) B)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (P b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b) as proof of (P b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (P b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (P b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 emptyset):(((eq (fofType->Prop)) emptyset) (fun (x:fofType)=> (emptyset x)))
% Found (eta_expansion_dep00 emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) 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 (((setminus A) B) x0)):(((eq Prop) (((setminus A) B) x0)) (((setminus A) B) x0))
% Found (eq_ref0 (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found eq_ref00:=(eq_ref0 (((intersection A) B) x0)):(((eq Prop) (((intersection A) B) x0)) (((intersection A) B) x0))
% Found (eq_ref0 (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (((intersection A) B) x0)):(((eq Prop) (((intersection A) B) x0)) (((intersection A) B) x0))
% Found (eq_ref0 (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found ((eq_ref Prop) (((intersection A) B) x0)) as proof of (((eq Prop) (((intersection A) B) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x0))
% Found eq_ref00:=(eq_ref0 (((setminus A) B) x0)):(((eq Prop) (((setminus A) B) x0)) (((setminus A) B) x0))
% Found (eq_ref0 (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found ((eq_ref Prop) (((setminus A) B) x0)) as proof of (((eq Prop) (((setminus A) B) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (A x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((setminus A) B)) B) x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((setminus A) B)) B) x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((setminus A) B)) B) x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((setminus A) B)) B) x00))
% Found eq_ref00:=(eq_ref0 (emptyset x00)):(((eq Prop) (emptyset x00)) (emptyset x00))
% Found (eq_ref0 (emptyset x00)) as proof of (((eq Prop) (emptyset x00)) b)
% Found ((eq_ref Prop) (emptyset x00)) as proof of (((eq Prop) (emptyset x00)) b)
% Found ((eq_ref Prop) (emptyset x00)) as proof of (((eq Prop) (emptyset x00)) b)
% Found ((eq_ref Prop) (emptyset x00)) as proof of (((eq Prop) (emptyset x00)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((intersection ((setminus A) B)) B) x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((setminus A) B)) B) x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((setminus A) B)) B) x00))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((intersection ((setminus A) B)) B) x00))
% Found eq_ref00:=(eq_ref0 (emptyset x00)):(((eq Prop) (emptyset x00)) (emptyset x00))
% Found (eq_ref0 (emptyset x00)) as proof of (((eq Prop) (emptyset x00
% EOF
%------------------------------------------------------------------------------