%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SET764^4 : TPTP v6.1.0. Released v3.6.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n101.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 13:31:04 EDT 2014

% Result   : Timeout 300.01s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SET764^4 : TPTP v6.1.0. Released v3.6.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n101.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 10:50:16 CDT 2014
% % CPUTime  : 300.01 
% 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 0x28d13b0>, <kernel.DependentProduct object at 0x28d1cb0>) 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 0x28afcb0>, <kernel.DependentProduct object at 0x28d1f38>) 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 0x25c1638>, <kernel.DependentProduct object at 0x28d13f8>) 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 0x25c1488>, <kernel.DependentProduct object at 0x28d19e0>) 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 0x25c1488>, <kernel.DependentProduct object at 0x28d13f8>) 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 0x25c1cb0>, <kernel.DependentProduct object at 0x28d13b0>) 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 0x28d1908>, <kernel.DependentProduct object at 0x28d19e0>) 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 0x28d13b0>, <kernel.DependentProduct object at 0x28d18c0>) 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 0x28d1878>, <kernel.DependentProduct object at 0x28d0b48>) 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 0x28d1878>, <kernel.DependentProduct object at 0x28d0c68>) 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 0x28d1908>, <kernel.DependentProduct object at 0x28d03f8>) 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 0x28d03f8>, <kernel.DependentProduct object at 0x28d0170>) 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 0x28d0170>, <kernel.DependentProduct object at 0x28d0b48>) 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 0x28d0b48>, <kernel.DependentProduct object at 0x28d05a8>) 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))
% Failed to open /home/cristobal/cocATP/CASC/TPTP/Axioms/SET008^1.ax, trying next directory
% FOF formula (<kernel.Constant object at 0x28d1758>, <kernel.DependentProduct object at 0x28d1cb0>) of role type named fun_image_decl
% Using role type
% Declaring fun_image:((fofType->fofType)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->fofType)->((fofType->Prop)->(fofType->Prop)))) fun_image) (fun (F:(fofType->fofType)) (A:(fofType->Prop)) (Y:fofType)=> ((ex fofType) (fun (X:fofType)=> ((and (A X)) (((eq fofType) Y) (F X))))))) of role definition named fun_image
% A new definition: (((eq ((fofType->fofType)->((fofType->Prop)->(fofType->Prop)))) fun_image) (fun (F:(fofType->fofType)) (A:(fofType->Prop)) (Y:fofType)=> ((ex fofType) (fun (X:fofType)=> ((and (A X)) (((eq fofType) Y) (F X)))))))
% Defined: fun_image:=(fun (F:(fofType->fofType)) (A:(fofType->Prop)) (Y:fofType)=> ((ex fofType) (fun (X:fofType)=> ((and (A X)) (((eq fofType) Y) (F X))))))
% FOF formula (<kernel.Constant object at 0x28d1cb0>, <kernel.DependentProduct object at 0x28d15a8>) of role type named fun_composition_decl
% Using role type
% Declaring fun_composition:((fofType->fofType)->((fofType->fofType)->(fofType->fofType)))
% FOF formula (((eq ((fofType->fofType)->((fofType->fofType)->(fofType->fofType)))) fun_composition) (fun (F:(fofType->fofType)) (G:(fofType->fofType)) (X:fofType)=> (G (F X)))) of role definition named fun_composition
% A new definition: (((eq ((fofType->fofType)->((fofType->fofType)->(fofType->fofType)))) fun_composition) (fun (F:(fofType->fofType)) (G:(fofType->fofType)) (X:fofType)=> (G (F X))))
% Defined: fun_composition:=(fun (F:(fofType->fofType)) (G:(fofType->fofType)) (X:fofType)=> (G (F X)))
% FOF formula (<kernel.Constant object at 0x28d1710>, <kernel.DependentProduct object at 0x28d1908>) of role type named fun_inv_image_decl
% Using role type
% Declaring fun_inv_image:((fofType->fofType)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->fofType)->((fofType->Prop)->(fofType->Prop)))) fun_inv_image) (fun (F:(fofType->fofType)) (B:(fofType->Prop)) (X:fofType)=> ((ex fofType) (fun (Y:fofType)=> ((and (B Y)) (((eq fofType) Y) (F X))))))) of role definition named fun_inv_image
% A new definition: (((eq ((fofType->fofType)->((fofType->Prop)->(fofType->Prop)))) fun_inv_image) (fun (F:(fofType->fofType)) (B:(fofType->Prop)) (X:fofType)=> ((ex fofType) (fun (Y:fofType)=> ((and (B Y)) (((eq fofType) Y) (F X)))))))
% Defined: fun_inv_image:=(fun (F:(fofType->fofType)) (B:(fofType->Prop)) (X:fofType)=> ((ex fofType) (fun (Y:fofType)=> ((and (B Y)) (((eq fofType) Y) (F X))))))
% FOF formula (<kernel.Constant object at 0x28d1908>, <kernel.DependentProduct object at 0x28d13b0>) of role type named fun_injective_decl
% Using role type
% Declaring fun_injective:((fofType->fofType)->Prop)
% FOF formula (((eq ((fofType->fofType)->Prop)) fun_injective) (fun (F:(fofType->fofType))=> (forall (X:fofType) (Y:fofType), ((((eq fofType) (F X)) (F Y))->(((eq fofType) X) Y))))) of role definition named fun_injective
% A new definition: (((eq ((fofType->fofType)->Prop)) fun_injective) (fun (F:(fofType->fofType))=> (forall (X:fofType) (Y:fofType), ((((eq fofType) (F X)) (F Y))->(((eq fofType) X) Y)))))
% Defined: fun_injective:=(fun (F:(fofType->fofType))=> (forall (X:fofType) (Y:fofType), ((((eq fofType) (F X)) (F Y))->(((eq fofType) X) Y))))
% FOF formula (<kernel.Constant object at 0x28d1200>, <kernel.DependentProduct object at 0x28d1f80>) of role type named fun_surjective_decl
% Using role type
% Declaring fun_surjective:((fofType->fofType)->Prop)
% FOF formula (((eq ((fofType->fofType)->Prop)) fun_surjective) (fun (F:(fofType->fofType))=> (forall (Y:fofType), ((ex fofType) (fun (X:fofType)=> (((eq fofType) Y) (F X))))))) of role definition named fun_surjective
% A new definition: (((eq ((fofType->fofType)->Prop)) fun_surjective) (fun (F:(fofType->fofType))=> (forall (Y:fofType), ((ex fofType) (fun (X:fofType)=> (((eq fofType) Y) (F X)))))))
% Defined: fun_surjective:=(fun (F:(fofType->fofType))=> (forall (Y:fofType), ((ex fofType) (fun (X:fofType)=> (((eq fofType) Y) (F X))))))
% FOF formula (<kernel.Constant object at 0x28d1710>, <kernel.DependentProduct object at 0x28d1908>) of role type named fun_bijective_decl
% Using role type
% Declaring fun_bijective:((fofType->fofType)->Prop)
% FOF formula (((eq ((fofType->fofType)->Prop)) fun_bijective) (fun (F:(fofType->fofType))=> ((and (fun_injective F)) (fun_surjective F)))) of role definition named fun_bijective
% A new definition: (((eq ((fofType->fofType)->Prop)) fun_bijective) (fun (F:(fofType->fofType))=> ((and (fun_injective F)) (fun_surjective F))))
% Defined: fun_bijective:=(fun (F:(fofType->fofType))=> ((and (fun_injective F)) (fun_surjective F)))
% FOF formula (<kernel.Constant object at 0x28d1f80>, <kernel.DependentProduct object at 0x28d0368>) of role type named fun_decreasing_decl
% Using role type
% Declaring fun_decreasing:((fofType->fofType)->((fofType->(fofType->Prop))->Prop))
% FOF formula (((eq ((fofType->fofType)->((fofType->(fofType->Prop))->Prop))) fun_decreasing) (fun (F:(fofType->fofType)) (SMALLER:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((SMALLER X) Y)->((SMALLER (F Y)) (F X)))))) of role definition named fun_decreasing
% A new definition: (((eq ((fofType->fofType)->((fofType->(fofType->Prop))->Prop))) fun_decreasing) (fun (F:(fofType->fofType)) (SMALLER:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((SMALLER X) Y)->((SMALLER (F Y)) (F X))))))
% Defined: fun_decreasing:=(fun (F:(fofType->fofType)) (SMALLER:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((SMALLER X) Y)->((SMALLER (F Y)) (F X)))))
% FOF formula (<kernel.Constant object at 0x28d1200>, <kernel.DependentProduct object at 0x28d0c68>) of role type named fun_increasing_decl
% Using role type
% Declaring fun_increasing:((fofType->fofType)->((fofType->(fofType->Prop))->Prop))
% FOF formula (((eq ((fofType->fofType)->((fofType->(fofType->Prop))->Prop))) fun_increasing) (fun (F:(fofType->fofType)) (SMALLER:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((SMALLER X) Y)->((SMALLER (F X)) (F Y)))))) of role definition named fun_increasing
% A new definition: (((eq ((fofType->fofType)->((fofType->(fofType->Prop))->Prop))) fun_increasing) (fun (F:(fofType->fofType)) (SMALLER:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((SMALLER X) Y)->((SMALLER (F X)) (F Y))))))
% Defined: fun_increasing:=(fun (F:(fofType->fofType)) (SMALLER:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((SMALLER X) Y)->((SMALLER (F X)) (F Y)))))
% FOF formula (forall (F:(fofType->fofType)), (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) emptyset)) of role conjecture named thm
% Conjecture to prove = (forall (F:(fofType->fofType)), (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) emptyset)):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(forall (F:(fofType->fofType)), (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) emptyset))']
% 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)).
% Definition fun_image:=(fun (F:(fofType->fofType)) (A:(fofType->Prop)) (Y:fofType)=> ((ex fofType) (fun (X:fofType)=> ((and (A X)) (((eq fofType) Y) (F X)))))):((fofType->fofType)->((fofType->Prop)->(fofType->Prop))).
% Definition fun_composition:=(fun (F:(fofType->fofType)) (G:(fofType->fofType)) (X:fofType)=> (G (F X))):((fofType->fofType)->((fofType->fofType)->(fofType->fofType))).
% Definition fun_inv_image:=(fun (F:(fofType->fofType)) (B:(fofType->Prop)) (X:fofType)=> ((ex fofType) (fun (Y:fofType)=> ((and (B Y)) (((eq fofType) Y) (F X)))))):((fofType->fofType)->((fofType->Prop)->(fofType->Prop))).
% Definition fun_injective:=(fun (F:(fofType->fofType))=> (forall (X:fofType) (Y:fofType), ((((eq fofType) (F X)) (F Y))->(((eq fofType) X) Y)))):((fofType->fofType)->Prop).
% Definition fun_surjective:=(fun (F:(fofType->fofType))=> (forall (Y:fofType), ((ex fofType) (fun (X:fofType)=> (((eq fofType) Y) (F X)))))):((fofType->fofType)->Prop).
% Definition fun_bijective:=(fun (F:(fofType->fofType))=> ((and (fun_injective F)) (fun_surjective F))):((fofType->fofType)->Prop).
% Definition fun_decreasing:=(fun (F:(fofType->fofType)) (SMALLER:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((SMALLER X) Y)->((SMALLER (F Y)) (F X))))):((fofType->fofType)->((fofType->(fofType->Prop))->Prop)).
% Definition fun_increasing:=(fun (F:(fofType->fofType)) (SMALLER:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((SMALLER X) Y)->((SMALLER (F X)) (F Y))))):((fofType->fofType)->((fofType->(fofType->Prop))->Prop)).
% Trying to prove (forall (F:(fofType->fofType)), (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) emptyset))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((fun_inv_image F) emptyset)):(((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) (fun (x:fofType)=> (((fun_inv_image F) emptyset) x)))
% Found (eta_expansion_dep00 ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) 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) 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 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_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found x2:(P emptyset)
% Found (fun (x2:(P emptyset))=> x2) as proof of (P emptyset)
% Found (fun (x2:(P emptyset))=> x2) as proof of (P0 emptyset)
% Found x2:(P emptyset)
% Found (fun (x2:(P emptyset))=> x2) as proof of (P emptyset)
% Found (fun (x2:(P emptyset))=> x2) as proof of (P0 emptyset)
% Found x01:(P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P0 (((fun_inv_image F) emptyset) x))
% Found x01:(P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P0 (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found x01:(P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P0 (emptyset x))
% Found x01:(P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P0 (emptyset x))
% Found x:(P ((fun_inv_image F) emptyset))
% Instantiate: b:=((fun_inv_image F) emptyset):(fofType->Prop)
% Found x as proof of (P0 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 x:(P ((fun_inv_image F) emptyset))
% Instantiate: f:=((fun_inv_image F) emptyset):(fofType->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (emptyset x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (emptyset x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (emptyset x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (emptyset x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (emptyset x)))
% Found x:(P ((fun_inv_image F) emptyset))
% Instantiate: f:=((fun_inv_image F) emptyset):(fofType->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (emptyset x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (emptyset x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (emptyset x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (emptyset x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (emptyset x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 ((fun_inv_image F) emptyset)):(((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) ((fun_inv_image F) emptyset))
% Found (eq_ref0 ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found ((eq_ref (fofType->Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found ((eq_ref (fofType->Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found ((eq_ref (fofType->Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) 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_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 x:(P emptyset)
% Instantiate: b:=emptyset:(fofType->Prop)
% Found x as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 ((fun_inv_image F) emptyset)):(((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) (fun (x:fofType)=> (((fun_inv_image F) emptyset) x)))
% Found (eta_expansion00 ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found ((eta_expansion0 Prop) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found (((eta_expansion fofType) Prop) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found (((eta_expansion fofType) Prop) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found (((eta_expansion fofType) Prop) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) 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 x:(P emptyset)
% Instantiate: f:=emptyset:(fofType->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((fun_inv_image F) emptyset) x)))
% Found x:(P emptyset)
% Instantiate: f:=emptyset:(fofType->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((fun_inv_image F) emptyset) x)))
% Found x0:(P (((fun_inv_image F) emptyset) x))
% Instantiate: b:=(((fun_inv_image F) emptyset) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found x0:(P (((fun_inv_image F) emptyset) x))
% Instantiate: b:=(((fun_inv_image F) emptyset) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) 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 eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% 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) ((fun_inv_image F) emptyset))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P1 emptyset)
% Found (fun (x2:(P1 emptyset))=> x2) as proof of (P1 emptyset)
% Found (fun (x2:(P1 emptyset))=> x2) as proof of (P2 emptyset)
% Found x2:(P1 emptyset)
% Found (fun (x2:(P1 emptyset))=> x2) as proof of (P1 emptyset)
% Found (fun (x2:(P1 emptyset))=> x2) as proof of (P2 emptyset)
% Found eq_ref00:=(eq_ref0 ((fun_inv_image F) emptyset)):(((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) ((fun_inv_image F) emptyset))
% Found (eq_ref0 ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b0)
% Found ((eq_ref (fofType->Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b0)
% Found ((eq_ref (fofType->Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b0)
% Found ((eq_ref (fofType->Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) emptyset)
% Found x01:(P1 (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P1 (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P1 (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P1 (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P2 (((fun_inv_image F) emptyset) x))
% Found x01:(P1 (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P1 (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P1 (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P1 (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P2 (((fun_inv_image F) emptyset) x))
% Found x3:(P emptyset)
% Found (fun (x3:(P emptyset))=> x3) as proof of (P emptyset)
% Found (fun (x3:(P emptyset))=> x3) 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 (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 eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) ((fun_inv_image F) emptyset))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) ((fun_inv_image F) emptyset))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) ((fun_inv_image F) emptyset))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) ((fun_inv_image F) 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) b0)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found x0:(P0 b)
% Instantiate: b:=((ex fofType) (fun (Y:fofType)=> ((and (emptyset Y)) (((eq fofType) Y) (F x))))):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (((fun_inv_image F) emptyset) x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (((fun_inv_image F) emptyset) x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found x0:(P0 b)
% Instantiate: b:=((ex fofType) (fun (Y:fofType)=> ((and (emptyset Y)) (((eq fofType) Y) (F x))))):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (((fun_inv_image F) emptyset) x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (((fun_inv_image F) emptyset) x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x02:(P (((fun_inv_image F) emptyset) x))
% Found (fun (x02:(P (((fun_inv_image F) emptyset) x)))=> x02) as proof of (P (((fun_inv_image F) emptyset) x))
% Found (fun (x02:(P (((fun_inv_image F) emptyset) x)))=> x02) as proof of (P0 (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found x02:(P (((fun_inv_image F) emptyset) x))
% Found (fun (x02:(P (((fun_inv_image F) emptyset) x)))=> x02) as proof of (P (((fun_inv_image F) emptyset) x))
% Found (fun (x02:(P (((fun_inv_image F) emptyset) x)))=> x02) as proof of (P0 (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% 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 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 x0:(P (emptyset x))
% Instantiate: b:=(emptyset x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found x0:(P (emptyset x))
% Instantiate: b:=(emptyset x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) 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) b0)
% Found ((eta_expansion0 Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% Found (((eta_expansion fofType) Prop) emptyset) as proof of (((eq (fofType->Prop)) emptyset) b0)
% 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 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_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:(P (emptyset x))
% Instantiate: b:=(emptyset x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found x0:(P (emptyset x))
% Instantiate: b:=(emptyset x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((fun_inv_image F) emptyset)):(((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) (fun (x:fofType)=> (((fun_inv_image F) emptyset) x)))
% Found (eta_expansion_dep00 ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) 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 eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found x2:(P emptyset)
% Found (fun (x2:(P emptyset))=> x2) as proof of (P emptyset)
% Found (fun (x2:(P emptyset))=> x2) as proof of (P0 emptyset)
% Found x2:(P emptyset)
% Found (fun (x2:(P emptyset))=> x2) as proof of (P emptyset)
% Found (fun (x2:(P emptyset))=> x2) as proof of (P0 emptyset)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found x01:(P1 (emptyset x))
% Found (fun (x01:(P1 (emptyset x)))=> x01) as proof of (P1 (emptyset x))
% Found (fun (x01:(P1 (emptyset x)))=> x01) as proof of (P2 (emptyset x))
% Found x01:(P1 (emptyset x))
% Found (fun (x01:(P1 (emptyset x)))=> x01) as proof of (P1 (emptyset x))
% Found (fun (x01:(P1 (emptyset x)))=> x01) as proof of (P2 (emptyset x))
% Found x01:(P1 (emptyset x))
% Found (fun (x01:(P1 (emptyset x)))=> x01) as proof of (P1 (emptyset x))
% Found (fun (x01:(P1 (emptyset x)))=> x01) as proof of (P2 (emptyset x))
% Found x01:(P1 (emptyset x))
% Found (fun (x01:(P1 (emptyset x)))=> x01) as proof of (P1 (emptyset x))
% Found (fun (x01:(P1 (emptyset x)))=> x01) as proof of (P2 (emptyset x))
% Found x01:(P1 (emptyset x))
% Found (fun (x01:(P1 (emptyset x)))=> x01) as proof of (P1 (emptyset x))
% Found (fun (x01:(P1 (emptyset x)))=> x01) as proof of (P2 (emptyset x))
% Found x01:(P1 (emptyset x))
% Found (fun (x01:(P1 (emptyset x)))=> x01) as proof of (P1 (emptyset x))
% Found (fun (x01:(P1 (emptyset x)))=> x01) as proof of (P2 (emptyset x))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 ((fun_inv_image F) emptyset)):(((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) (fun (x:fofType)=> (((fun_inv_image F) emptyset) x)))
% Found (eta_expansion00 ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b0)
% Found ((eta_expansion0 Prop) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b0)
% Found (((eta_expansion fofType) Prop) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b0)
% Found (((eta_expansion fofType) Prop) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b0)
% Found (((eta_expansion fofType) Prop) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b0)
% Found x02:(P (emptyset x))
% Found (fun (x02:(P (emptyset x)))=> x02) as proof of (P (emptyset x))
% Found (fun (x02:(P (emptyset x)))=> x02) as proof of (P0 (emptyset x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found x02:(P (emptyset x))
% Found (fun (x02:(P (emptyset x)))=> x02) as proof of (P (emptyset x))
% Found (fun (x02:(P (emptyset x)))=> x02) as proof of (P0 (emptyset x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found x02:(P (emptyset x))
% Found (fun (x02:(P (emptyset x)))=> x02) as proof of (P (emptyset x))
% Found (fun (x02:(P (emptyset x)))=> x02) as proof of (P0 (emptyset x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found x02:(P (emptyset x))
% Found (fun (x02:(P (emptyset x)))=> x02) as proof of (P (emptyset x))
% Found (fun (x02:(P (emptyset x)))=> x02) as proof of (P0 (emptyset x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found x0:(P0 b)
% Instantiate: b:=False:Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (emptyset x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (emptyset x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found x0:(P0 b)
% Instantiate: b:=False:Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (emptyset x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (emptyset x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) 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) ((fun_inv_image F) emptyset))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) 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) ((fun_inv_image F) emptyset))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% 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:(P (((fun_inv_image F) emptyset) x))
% Instantiate: b:=(((fun_inv_image F) emptyset) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found x0:(P (((fun_inv_image F) emptyset) x))
% Instantiate: b:=(((fun_inv_image F) emptyset) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found x2:(P emptyset)
% Found (fun (x2:(P emptyset))=> x2) as proof of (P emptyset)
% Found (fun (x2:(P emptyset))=> x2) as proof of (P0 emptyset)
% Found x2:(P emptyset)
% Found (fun (x2:(P emptyset))=> x2) as proof of (P emptyset)
% Found (fun (x2:(P emptyset))=> x2) as proof of (P0 emptyset)
% Found x2:(P emptyset)
% Found (fun (x2:(P emptyset))=> x2) as proof of (P emptyset)
% Found (fun (x2:(P emptyset))=> x2) as proof of (P0 emptyset)
% Found x2:(P emptyset)
% Found (fun (x2:(P emptyset))=> x2) as proof of (P emptyset)
% Found (fun (x2:(P emptyset))=> x2) as proof of (P0 emptyset)
% Found x01:(P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P0 (((fun_inv_image F) emptyset) x))
% Found x01:(P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P0 (((fun_inv_image F) emptyset) x))
% Found x0:(P (((fun_inv_image F) emptyset) x))
% Instantiate: b:=((fun_inv_image F) emptyset):(fofType->Prop)
% Found x0 as proof of (P0 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:(P (((fun_inv_image F) emptyset) x))
% Instantiate: b:=((fun_inv_image F) emptyset):(fofType->Prop)
% Found x0 as proof of (P0 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 x01:(P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P0 (((fun_inv_image F) emptyset) x))
% Found x01:(P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P0 (((fun_inv_image F) emptyset) x))
% Found x2:(P ((fun_inv_image F) emptyset))
% Found (fun (x2:(P ((fun_inv_image F) emptyset)))=> x2) as proof of (P ((fun_inv_image F) emptyset))
% Found (fun (x2:(P ((fun_inv_image F) emptyset)))=> x2) as proof of (P0 ((fun_inv_image F) emptyset))
% Found x2:(P ((fun_inv_image F) emptyset))
% Found (fun (x2:(P ((fun_inv_image F) emptyset)))=> x2) as proof of (P ((fun_inv_image F) emptyset))
% Found (fun (x2:(P ((fun_inv_image F) emptyset)))=> x2) as proof of (P0 ((fun_inv_image F) emptyset))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% 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_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 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found x01:(P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P0 (emptyset x))
% Found x01:(P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P0 (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P0 (emptyset x))
% Found x01:(P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P0 (emptyset x))
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x02:(P (((fun_inv_image F) emptyset) x))
% Found (fun (x02:(P (((fun_inv_image F) emptyset) x)))=> x02) as proof of (P (((fun_inv_image F) emptyset) x))
% Found (fun (x02:(P (((fun_inv_image F) emptyset) x)))=> x02) as proof of (P0 (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found x02:(P (((fun_inv_image F) emptyset) x))
% Found (fun (x02:(P (((fun_inv_image F) emptyset) x)))=> x02) as proof of (P (((fun_inv_image F) emptyset) x))
% Found (fun (x02:(P (((fun_inv_image F) emptyset) x)))=> x02) as proof of (P0 (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found x01:(P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P0 (emptyset x))
% Found x01:(P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P0 (emptyset x))
% Found x01:(P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P0 (emptyset x))
% Found x01:(P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P0 (emptyset x))
% Found x01:(P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P0 (emptyset x))
% Found x01:(P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P (emptyset x))
% Found (fun (x01:(P (emptyset x)))=> x01) as proof of (P0 (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P0 (((fun_inv_image F) emptyset) x))
% Found x01:(P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P0 (((fun_inv_image F) emptyset) x))
% Found x01:(P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P (((fun_inv_image F) emptyset) x))
% Found (fun (x01:(P (((fun_inv_image F) emptyset) x)))=> x01) as proof of (P0 (((fun_inv_image F) emptyset) x))
% Found x:(P ((fun_inv_image F) emptyset))
% Instantiate: b:=((fun_inv_image F) emptyset):(fofType->Prop)
% Found x as proof of (P0 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 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b0)
% Found x:(P ((fun_inv_image F) emptyset))
% Instantiate: f:=((fun_inv_image F) emptyset):(fofType->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (emptyset x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (emptyset x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (emptyset x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (emptyset x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (emptyset x)))
% Found x:(P ((fun_inv_image F) emptyset))
% Instantiate: f:=((fun_inv_image F) emptyset):(fofType->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (emptyset x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (emptyset x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (emptyset x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (emptyset x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (emptyset x)))
% Found eq_ref00:=(eq_ref0 ((fun_inv_image F) emptyset)):(((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) ((fun_inv_image F) emptyset))
% Found (eq_ref0 ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found ((eq_ref (fofType->Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found ((eq_ref (fofType->Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found ((eq_ref (fofType->Prop)) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) 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 eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% 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 x:(P emptyset)
% Instantiate: b:=emptyset:(fofType->Prop)
% Found x as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 ((fun_inv_image F) emptyset)):(((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) (fun (x:fofType)=> (((fun_inv_image F) emptyset) x)))
% Found (eta_expansion00 ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found ((eta_expansion0 Prop) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found (((eta_expansion fofType) Prop) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found (((eta_expansion fofType) Prop) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found (((eta_expansion fofType) Prop) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found x:(P ((fun_inv_image F) emptyset))
% Instantiate: a:=((fun_inv_image F) emptyset):(fofType->Prop)
% Found x as proof of (P0 a)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) 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 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 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 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((fun_inv_image F) emptyset) x))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found x:(P emptyset)
% Instantiate: f:=emptyset:(fofType->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((fun_inv_image F) emptyset) x)))
% Found x:(P emptyset)
% Instantiate: f:=emptyset:(fofType->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((fun_inv_image F) emptyset) x)))
% Found x:(P emptyset)
% Instantiate: f:=emptyset:(fofType->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((fun_inv_image F) emptyset) x)))
% Found x:(P emptyset)
% Instantiate: f:=emptyset:(fofType->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((fun_inv_image F) emptyset) x0))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((fun_inv_image F) emptyset) x)))
% Found x0:(P (((fun_inv_image F) emptyset) x))
% Instantiate: b:=(((fun_inv_image F) emptyset) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found x0:(P (((fun_inv_image F) emptyset) x))
% Instantiate: b:=(((fun_inv_image F) emptyset) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found x0:(P (((fun_inv_image F) emptyset) x))
% Instantiate: b:=(((fun_inv_image F) emptyset) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found x0:(P (((fun_inv_image F) emptyset) x))
% Instantiate: b:=(((fun_inv_image F) emptyset) x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (emptyset x)):(((eq Prop) (emptyset x)) (emptyset x))
% Found (eq_ref0 (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found ((eq_ref Prop) (emptyset x)) as proof of (((eq Prop) (emptyset x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% 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) ((fun_inv_image F) emptyset))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% 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) ((fun_inv_image F) emptyset))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% 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_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) ((fun_inv_image F) emptyset))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((fun_inv_image F) emptyset))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 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 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 x:(P1 emptyset)
% Instantiate: b:=emptyset:(fofType->Prop)
% Found x as proof of (P2 b)
% Found eta_expansion000:=(eta_expansion00 ((fun_inv_image F) emptyset)):(((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) (fun (x:fofType)=> (((fun_inv_image F) emptyset) x)))
% Found (eta_expansion00 ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found ((eta_expansion0 Prop) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found (((eta_expansion fofType) Prop) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found (((eta_expansion fofType) Prop) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found (((eta_expansion fofType) Prop) ((fun_inv_image F) emptyset)) as proof of (((eq (fofType->Prop)) ((fun_inv_image F) emptyset)) b)
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) emptyset) x))
% Found (eq_ref0 (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found ((eq_ref Prop) (((fun_inv_image F) emptyset) x)) as proof of (((eq Prop) (((fun_inv_image F) emptyset) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (emptyset x))
% Found eq_ref00:=(eq_ref0 (((fun_inv_image F) emptyset) x)):(((eq Prop) (((fun_inv_image F) emptyset) x)) (((fun_inv_image F) e
% EOF
%------------------------------------------------------------------------------