TSTP Solution File: SET669^3 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SET669^3 : TPTP v6.1.0. Released v3.6.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n108.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:30:57 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 : SET669^3 : TPTP v6.1.0. Released v3.6.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n108.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 10:34:06 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 0x15cb638>, <kernel.DependentProduct object at 0x12b7d88>) 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 0x13e9560>, <kernel.DependentProduct object at 0x12b7cb0>) 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 0x13e9560>, <kernel.DependentProduct object at 0x12b7f80>) 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 0x15a8d40>, <kernel.DependentProduct object at 0x12b7e60>) 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 0x15cb638>, <kernel.DependentProduct object at 0x12b7f38>) 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 0x15a8cb0>, <kernel.DependentProduct object at 0x12b7dd0>) 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 0x12b7dd0>, <kernel.DependentProduct object at 0x12b7cb0>) 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 0x12b7e60>, <kernel.DependentProduct object at 0x12b7d40>) 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 0x12b7bd8>, <kernel.DependentProduct object at 0x15ca0e0>) 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 0x12b7bd8>, <kernel.DependentProduct object at 0x15ca440>) 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 0x12b7bd8>, <kernel.DependentProduct object at 0x15caf38>) 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 0x15caf38>, <kernel.DependentProduct object at 0x15ca7a0>) 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 0x15ca7a0>, <kernel.DependentProduct object at 0x15cae18>) 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 0x15cae18>, <kernel.DependentProduct object at 0x15cacf8>) 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/SET008^2.ax, trying next directory
% FOF formula (<kernel.Constant object at 0x15cb488>, <kernel.DependentProduct object at 0x12b7f80>) of role type named cartesian_product_decl
% Using role type
% Declaring cartesian_product:((fofType->Prop)->((fofType->Prop)->(fofType->(fofType->Prop))))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->(fofType->Prop))))) cartesian_product) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType) (V:fofType)=> ((and (X U)) (Y V)))) of role definition named cartesian_product
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->(fofType->Prop))))) cartesian_product) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType) (V:fofType)=> ((and (X U)) (Y V))))
% Defined: cartesian_product:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType) (V:fofType)=> ((and (X U)) (Y V)))
% FOF formula (<kernel.Constant object at 0x15cb638>, <kernel.DependentProduct object at 0x12b7ea8>) of role type named pair_rel_decl
% Using role type
% Declaring pair_rel:(fofType->(fofType->(fofType->(fofType->Prop))))
% FOF formula (((eq (fofType->(fofType->(fofType->(fofType->Prop))))) pair_rel) (fun (X:fofType) (Y:fofType) (U:fofType) (V:fofType)=> ((or (((eq fofType) U) X)) (((eq fofType) V) Y)))) of role definition named pair_rel
% A new definition: (((eq (fofType->(fofType->(fofType->(fofType->Prop))))) pair_rel) (fun (X:fofType) (Y:fofType) (U:fofType) (V:fofType)=> ((or (((eq fofType) U) X)) (((eq fofType) V) Y))))
% Defined: pair_rel:=(fun (X:fofType) (Y:fofType) (U:fofType) (V:fofType)=> ((or (((eq fofType) U) X)) (((eq fofType) V) Y)))
% FOF formula (<kernel.Constant object at 0x15cbbd8>, <kernel.DependentProduct object at 0x12b7cb0>) of role type named id_rel_decl
% Using role type
% Declaring id_rel:((fofType->Prop)->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->(fofType->(fofType->Prop)))) id_rel) (fun (S:(fofType->Prop)) (X:fofType) (Y:fofType)=> ((and (S X)) (((eq fofType) X) Y)))) of role definition named id_rel
% A new definition: (((eq ((fofType->Prop)->(fofType->(fofType->Prop)))) id_rel) (fun (S:(fofType->Prop)) (X:fofType) (Y:fofType)=> ((and (S X)) (((eq fofType) X) Y))))
% Defined: id_rel:=(fun (S:(fofType->Prop)) (X:fofType) (Y:fofType)=> ((and (S X)) (((eq fofType) X) Y)))
% FOF formula (<kernel.Constant object at 0x15cbbd8>, <kernel.DependentProduct object at 0x12b7ea8>) of role type named sub_rel_decl
% Using role type
% Declaring sub_rel:((fofType->(fofType->Prop))->((fofType->(fofType->Prop))->Prop))
% FOF formula (((eq ((fofType->(fofType->Prop))->((fofType->(fofType->Prop))->Prop))) sub_rel) (fun (R1:(fofType->(fofType->Prop))) (R2:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R1 X) Y)->((R2 X) Y))))) of role definition named sub_rel
% A new definition: (((eq ((fofType->(fofType->Prop))->((fofType->(fofType->Prop))->Prop))) sub_rel) (fun (R1:(fofType->(fofType->Prop))) (R2:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R1 X) Y)->((R2 X) Y)))))
% Defined: sub_rel:=(fun (R1:(fofType->(fofType->Prop))) (R2:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R1 X) Y)->((R2 X) Y))))
% FOF formula (<kernel.Constant object at 0x15cb680>, <kernel.DependentProduct object at 0x12b7d88>) of role type named is_rel_on_decl
% Using role type
% Declaring is_rel_on:((fofType->(fofType->Prop))->((fofType->Prop)->((fofType->Prop)->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->((fofType->Prop)->((fofType->Prop)->Prop)))) is_rel_on) (fun (R:(fofType->(fofType->Prop))) (A:(fofType->Prop)) (B:(fofType->Prop))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((and (A X)) (B Y)))))) of role definition named is_rel_on
% A new definition: (((eq ((fofType->(fofType->Prop))->((fofType->Prop)->((fofType->Prop)->Prop)))) is_rel_on) (fun (R:(fofType->(fofType->Prop))) (A:(fofType->Prop)) (B:(fofType->Prop))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((and (A X)) (B Y))))))
% Defined: is_rel_on:=(fun (R:(fofType->(fofType->Prop))) (A:(fofType->Prop)) (B:(fofType->Prop))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((and (A X)) (B Y)))))
% FOF formula (<kernel.Constant object at 0x12b7c68>, <kernel.DependentProduct object at 0x15ca368>) of role type named restrict_rel_domain_decl
% Using role type
% Declaring restrict_rel_domain:((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->(fofType->Prop))))
% FOF formula (((eq ((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->(fofType->Prop))))) restrict_rel_domain) (fun (R:(fofType->(fofType->Prop))) (S:(fofType->Prop)) (X:fofType) (Y:fofType)=> ((and (S X)) ((R X) Y)))) of role definition named restrict_rel_domain
% A new definition: (((eq ((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->(fofType->Prop))))) restrict_rel_domain) (fun (R:(fofType->(fofType->Prop))) (S:(fofType->Prop)) (X:fofType) (Y:fofType)=> ((and (S X)) ((R X) Y))))
% Defined: restrict_rel_domain:=(fun (R:(fofType->(fofType->Prop))) (S:(fofType->Prop)) (X:fofType) (Y:fofType)=> ((and (S X)) ((R X) Y)))
% FOF formula (<kernel.Constant object at 0x12b7248>, <kernel.DependentProduct object at 0x15ca0e0>) of role type named rel_diagonal_decl
% Using role type
% Declaring rel_diagonal:(fofType->(fofType->Prop))
% FOF formula (((eq (fofType->(fofType->Prop))) rel_diagonal) (fun (X:fofType) (Y:fofType)=> (((eq fofType) X) Y))) of role definition named rel_diagonal
% A new definition: (((eq (fofType->(fofType->Prop))) rel_diagonal) (fun (X:fofType) (Y:fofType)=> (((eq fofType) X) Y)))
% Defined: rel_diagonal:=(fun (X:fofType) (Y:fofType)=> (((eq fofType) X) Y))
% FOF formula (<kernel.Constant object at 0x12b7d40>, <kernel.DependentProduct object at 0x15ca170>) of role type named rel_composition_decl
% Using role type
% Declaring rel_composition:((fofType->(fofType->Prop))->((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))
% FOF formula (((eq ((fofType->(fofType->Prop))->((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))) rel_composition) (fun (R1:(fofType->(fofType->Prop))) (R2:(fofType->(fofType->Prop))) (X:fofType) (Z:fofType)=> ((ex fofType) (fun (Y:fofType)=> ((and ((R1 X) Y)) ((R2 Y) Z)))))) of role definition named rel_composition
% A new definition: (((eq ((fofType->(fofType->Prop))->((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))) rel_composition) (fun (R1:(fofType->(fofType->Prop))) (R2:(fofType->(fofType->Prop))) (X:fofType) (Z:fofType)=> ((ex fofType) (fun (Y:fofType)=> ((and ((R1 X) Y)) ((R2 Y) Z))))))
% Defined: rel_composition:=(fun (R1:(fofType->(fofType->Prop))) (R2:(fofType->(fofType->Prop))) (X:fofType) (Z:fofType)=> ((ex fofType) (fun (Y:fofType)=> ((and ((R1 X) Y)) ((R2 Y) Z)))))
% FOF formula (<kernel.Constant object at 0x12b7d40>, <kernel.DependentProduct object at 0x15ca368>) of role type named reflexive_decl
% Using role type
% Declaring reflexive:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) reflexive) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((R X) X)))) of role definition named reflexive
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) reflexive) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((R X) X))))
% Defined: reflexive:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((R X) X)))
% FOF formula (<kernel.Constant object at 0x12b7d40>, <kernel.DependentProduct object at 0x15cacb0>) of role type named irreflexive_decl
% Using role type
% Declaring irreflexive:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) irreflexive) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), (((R X) X)->False)))) of role definition named irreflexive
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) irreflexive) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), (((R X) X)->False))))
% Defined: irreflexive:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), (((R X) X)->False)))
% FOF formula (<kernel.Constant object at 0x15cacb0>, <kernel.DependentProduct object at 0x15ca7a0>) of role type named symmetric_decl
% Using role type
% Declaring symmetric:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) symmetric) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((R Y) X))))) of role definition named symmetric
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) symmetric) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((R Y) X)))))
% Defined: symmetric:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((R Y) X))))
% FOF formula (<kernel.Constant object at 0x15ca7a0>, <kernel.DependentProduct object at 0x15caab8>) of role type named transitive_decl
% Using role type
% Declaring transitive:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) transitive) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Y)) ((R Y) Z))->((R X) Z))))) of role definition named transitive
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) transitive) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Y)) ((R Y) Z))->((R X) Z)))))
% Defined: transitive:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Y)) ((R Y) Z))->((R X) Z))))
% FOF formula (<kernel.Constant object at 0x15caab8>, <kernel.DependentProduct object at 0x15ca368>) of role type named equiv_rel__decl
% Using role type
% Declaring equiv_rel:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) equiv_rel) (fun (R:(fofType->(fofType->Prop)))=> ((and ((and (reflexive R)) (symmetric R))) (transitive R)))) of role definition named equiv_rel
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) equiv_rel) (fun (R:(fofType->(fofType->Prop)))=> ((and ((and (reflexive R)) (symmetric R))) (transitive R))))
% Defined: equiv_rel:=(fun (R:(fofType->(fofType->Prop)))=> ((and ((and (reflexive R)) (symmetric R))) (transitive R)))
% FOF formula (<kernel.Constant object at 0x15ca368>, <kernel.DependentProduct object at 0x15caa28>) of role type named rel_codomain_decl
% Using role type
% Declaring rel_codomain:((fofType->(fofType->Prop))->(fofType->Prop))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->Prop))) rel_codomain) (fun (R:(fofType->(fofType->Prop))) (Y:fofType)=> ((ex fofType) (fun (X:fofType)=> ((R X) Y))))) of role definition named rel_codomain
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->Prop))) rel_codomain) (fun (R:(fofType->(fofType->Prop))) (Y:fofType)=> ((ex fofType) (fun (X:fofType)=> ((R X) Y)))))
% Defined: rel_codomain:=(fun (R:(fofType->(fofType->Prop))) (Y:fofType)=> ((ex fofType) (fun (X:fofType)=> ((R X) Y))))
% FOF formula (<kernel.Constant object at 0x15caa28>, <kernel.DependentProduct object at 0x15ca440>) of role type named rel_domain_decl
% Using role type
% Declaring rel_domain:((fofType->(fofType->Prop))->(fofType->Prop))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->Prop))) rel_domain) (fun (R:(fofType->(fofType->Prop))) (X:fofType)=> ((ex fofType) (fun (Y:fofType)=> ((R X) Y))))) of role definition named rel_domain
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->Prop))) rel_domain) (fun (R:(fofType->(fofType->Prop))) (X:fofType)=> ((ex fofType) (fun (Y:fofType)=> ((R X) Y)))))
% Defined: rel_domain:=(fun (R:(fofType->(fofType->Prop))) (X:fofType)=> ((ex fofType) (fun (Y:fofType)=> ((R X) Y))))
% FOF formula (<kernel.Constant object at 0x15ca440>, <kernel.DependentProduct object at 0x15ca9e0>) of role type named rel_inverse_decl
% Using role type
% Declaring rel_inverse:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) rel_inverse) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((R Y) X))) of role definition named rel_inverse
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) rel_inverse) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((R Y) X)))
% Defined: rel_inverse:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((R Y) X))
% FOF formula (<kernel.Constant object at 0x15ca8c0>, <kernel.DependentProduct object at 0x15c9e18>) of role type named equiv_classes_decl
% Using role type
% Declaring equiv_classes:((fofType->(fofType->Prop))->((fofType->Prop)->Prop))
% FOF formula (((eq ((fofType->(fofType->Prop))->((fofType->Prop)->Prop))) equiv_classes) (fun (R:(fofType->(fofType->Prop))) (S1:(fofType->Prop))=> ((ex fofType) (fun (X:fofType)=> ((and (S1 X)) (forall (Y:fofType), ((iff (S1 Y)) ((R X) Y)))))))) of role definition named equiv_classes
% A new definition: (((eq ((fofType->(fofType->Prop))->((fofType->Prop)->Prop))) equiv_classes) (fun (R:(fofType->(fofType->Prop))) (S1:(fofType->Prop))=> ((ex fofType) (fun (X:fofType)=> ((and (S1 X)) (forall (Y:fofType), ((iff (S1 Y)) ((R X) Y))))))))
% Defined: equiv_classes:=(fun (R:(fofType->(fofType->Prop))) (S1:(fofType->Prop))=> ((ex fofType) (fun (X:fofType)=> ((and (S1 X)) (forall (Y:fofType), ((iff (S1 Y)) ((R X) Y)))))))
% FOF formula (<kernel.Constant object at 0x15ca680>, <kernel.DependentProduct object at 0x15c9c68>) of role type named restrict_rel_codomain_decl
% Using role type
% Declaring restrict_rel_codomain:((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->(fofType->Prop))))
% FOF formula (((eq ((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->(fofType->Prop))))) restrict_rel_codomain) (fun (R:(fofType->(fofType->Prop))) (S:(fofType->Prop)) (X:fofType) (Y:fofType)=> ((and (S Y)) ((R X) Y)))) of role definition named restrict_rel_codomain
% A new definition: (((eq ((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->(fofType->Prop))))) restrict_rel_codomain) (fun (R:(fofType->(fofType->Prop))) (S:(fofType->Prop)) (X:fofType) (Y:fofType)=> ((and (S Y)) ((R X) Y))))
% Defined: restrict_rel_codomain:=(fun (R:(fofType->(fofType->Prop))) (S:(fofType->Prop)) (X:fofType) (Y:fofType)=> ((and (S Y)) ((R X) Y)))
% FOF formula (<kernel.Constant object at 0x15ca680>, <kernel.DependentProduct object at 0x15c9ab8>) of role type named rel_field_decl
% Using role type
% Declaring rel_field:((fofType->(fofType->Prop))->(fofType->Prop))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->Prop))) rel_field) (fun (R:(fofType->(fofType->Prop))) (X:fofType)=> ((or ((rel_domain R) X)) ((rel_codomain R) X)))) of role definition named rel_field
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->Prop))) rel_field) (fun (R:(fofType->(fofType->Prop))) (X:fofType)=> ((or ((rel_domain R) X)) ((rel_codomain R) X))))
% Defined: rel_field:=(fun (R:(fofType->(fofType->Prop))) (X:fofType)=> ((or ((rel_domain R) X)) ((rel_codomain R) X)))
% FOF formula (<kernel.Constant object at 0x15ca680>, <kernel.DependentProduct object at 0x15c98c0>) of role type named well_founded_decl
% Using role type
% Declaring well_founded:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) well_founded) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:(fofType->Prop)) (Z:fofType), ((X Z)->((ex fofType) (fun (Y:fofType)=> ((and (X Y)) (forall (W:fofType), (((R Y) W)->((X W)->False)))))))))) of role definition named well_founded
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) well_founded) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:(fofType->Prop)) (Z:fofType), ((X Z)->((ex fofType) (fun (Y:fofType)=> ((and (X Y)) (forall (W:fofType), (((R Y) W)->((X W)->False))))))))))
% Defined: well_founded:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:(fofType->Prop)) (Z:fofType), ((X Z)->((ex fofType) (fun (Y:fofType)=> ((and (X Y)) (forall (W:fofType), (((R Y) W)->((X W)->False)))))))))
% FOF formula (<kernel.Constant object at 0x15c98c0>, <kernel.DependentProduct object at 0x15c9ab8>) of role type named upwards_well_founded_decl
% Using role type
% Declaring upwards_well_founded:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) upwards_well_founded) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:(fofType->Prop)) (Z:fofType), ((X Z)->((ex fofType) (fun (Y:fofType)=> ((and (X Y)) (forall (W:fofType), (((R Y) Y)->((X W)->False)))))))))) of role definition named upwards_well_founded
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) upwards_well_founded) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:(fofType->Prop)) (Z:fofType), ((X Z)->((ex fofType) (fun (Y:fofType)=> ((and (X Y)) (forall (W:fofType), (((R Y) Y)->((X W)->False))))))))))
% Defined: upwards_well_founded:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:(fofType->Prop)) (Z:fofType), ((X Z)->((ex fofType) (fun (Y:fofType)=> ((and (X Y)) (forall (W:fofType), (((R Y) Y)->((X W)->False)))))))))
% FOF formula (forall (R:(fofType->(fofType->Prop))), (((sub_rel (id_rel (fun (X:fofType)=> True))) R)->((and ((subset (fun (X:fofType)=> True)) (rel_domain R))) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R))))) of role conjecture named thm
% Conjecture to prove = (forall (R:(fofType->(fofType->Prop))), (((sub_rel (id_rel (fun (X:fofType)=> True))) R)->((and ((subset (fun (X:fofType)=> True)) (rel_domain R))) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(forall (R:(fofType->(fofType->Prop))), (((sub_rel (id_rel (fun (X:fofType)=> True))) R)->((and ((subset (fun (X:fofType)=> True)) (rel_domain R))) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R)))))']
% 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 cartesian_product:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType) (V:fofType)=> ((and (X U)) (Y V))):((fofType->Prop)->((fofType->Prop)->(fofType->(fofType->Prop)))).
% Definition pair_rel:=(fun (X:fofType) (Y:fofType) (U:fofType) (V:fofType)=> ((or (((eq fofType) U) X)) (((eq fofType) V) Y))):(fofType->(fofType->(fofType->(fofType->Prop)))).
% Definition id_rel:=(fun (S:(fofType->Prop)) (X:fofType) (Y:fofType)=> ((and (S X)) (((eq fofType) X) Y))):((fofType->Prop)->(fofType->(fofType->Prop))).
% Definition sub_rel:=(fun (R1:(fofType->(fofType->Prop))) (R2:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R1 X) Y)->((R2 X) Y)))):((fofType->(fofType->Prop))->((fofType->(fofType->Prop))->Prop)).
% Definition is_rel_on:=(fun (R:(fofType->(fofType->Prop))) (A:(fofType->Prop)) (B:(fofType->Prop))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((and (A X)) (B Y))))):((fofType->(fofType->Prop))->((fofType->Prop)->((fofType->Prop)->Prop))).
% Definition restrict_rel_domain:=(fun (R:(fofType->(fofType->Prop))) (S:(fofType->Prop)) (X:fofType) (Y:fofType)=> ((and (S X)) ((R X) Y))):((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->(fofType->Prop)))).
% Definition rel_diagonal:=(fun (X:fofType) (Y:fofType)=> (((eq fofType) X) Y)):(fofType->(fofType->Prop)).
% Definition rel_composition:=(fun (R1:(fofType->(fofType->Prop))) (R2:(fofType->(fofType->Prop))) (X:fofType) (Z:fofType)=> ((ex fofType) (fun (Y:fofType)=> ((and ((R1 X) Y)) ((R2 Y) Z))))):((fofType->(fofType->Prop))->((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))).
% Definition reflexive:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((R X) X))):((fofType->(fofType->Prop))->Prop).
% Definition irreflexive:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), (((R X) X)->False))):((fofType->(fofType->Prop))->Prop).
% Definition symmetric:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((R Y) X)))):((fofType->(fofType->Prop))->Prop).
% Definition transitive:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Y)) ((R Y) Z))->((R X) Z)))):((fofType->(fofType->Prop))->Prop).
% Definition equiv_rel:=(fun (R:(fofType->(fofType->Prop)))=> ((and ((and (reflexive R)) (symmetric R))) (transitive R))):((fofType->(fofType->Prop))->Prop).
% Definition rel_codomain:=(fun (R:(fofType->(fofType->Prop))) (Y:fofType)=> ((ex fofType) (fun (X:fofType)=> ((R X) Y)))):((fofType->(fofType->Prop))->(fofType->Prop)).
% Definition rel_domain:=(fun (R:(fofType->(fofType->Prop))) (X:fofType)=> ((ex fofType) (fun (Y:fofType)=> ((R X) Y)))):((fofType->(fofType->Prop))->(fofType->Prop)).
% Definition rel_inverse:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((R Y) X)):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition equiv_classes:=(fun (R:(fofType->(fofType->Prop))) (S1:(fofType->Prop))=> ((ex fofType) (fun (X:fofType)=> ((and (S1 X)) (forall (Y:fofType), ((iff (S1 Y)) ((R X) Y))))))):((fofType->(fofType->Prop))->((fofType->Prop)->Prop)).
% Definition restrict_rel_codomain:=(fun (R:(fofType->(fofType->Prop))) (S:(fofType->Prop)) (X:fofType) (Y:fofType)=> ((and (S Y)) ((R X) Y))):((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->(fofType->Prop)))).
% Definition rel_field:=(fun (R:(fofType->(fofType->Prop))) (X:fofType)=> ((or ((rel_domain R) X)) ((rel_codomain R) X))):((fofType->(fofType->Prop))->(fofType->Prop)).
% Definition well_founded:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:(fofType->Prop)) (Z:fofType), ((X Z)->((ex fofType) (fun (Y:fofType)=> ((and (X Y)) (forall (W:fofType), (((R Y) W)->((X W)->False))))))))):((fofType->(fofType->Prop))->Prop).
% Definition upwards_well_founded:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:(fofType->Prop)) (Z:fofType), ((X Z)->((ex fofType) (fun (Y:fofType)=> ((and (X Y)) (forall (W:fofType), (((R Y) Y)->((X W)->False))))))))):((fofType->(fofType->Prop))->Prop).
% Trying to prove (forall (R:(fofType->(fofType->Prop))), (((sub_rel (id_rel (fun (X:fofType)=> True))) R)->((and ((subset (fun (X:fofType)=> True)) (rel_domain R))) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R)))))
% Found eq_ref00:=(eq_ref0 (fun (X:fofType)=> True)):(((eq (fofType->Prop)) (fun (X:fofType)=> True)) (fun (X:fofType)=> True))
% Found (eq_ref0 (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) 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) (rel_codomain R))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found x00:(P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P0 (fun (X:fofType)=> True))
% Found x00:(P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P0 (fun (X:fofType)=> True))
% Found x00:(P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P0 (fun (X:fofType)=> True))
% Found eq_ref00:=(eq_ref0 (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R))):(((eq Prop) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R))) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R)))
% Found (eq_ref0 (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R))) as proof of (((eq Prop) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R))) b)
% Found ((eq_ref Prop) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R))) as proof of (((eq Prop) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R))) b)
% Found ((eq_ref Prop) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R))) as proof of (((eq Prop) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R))) b)
% Found ((eq_ref Prop) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R))) as proof of (((eq Prop) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R))) b)
% Found eta_expansion000:=(eta_expansion00 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (fun (x:fofType)=> ((rel_codomain R) x)))
% Found (eta_expansion00 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eta_expansion0 Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found x00:(P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P0 (rel_codomain R))
% Found x00:(P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P0 (rel_codomain R))
% Found x10:(P True)
% Found (fun (x10:(P True))=> x10) as proof of (P True)
% Found (fun (x10:(P True))=> x10) as proof of (P0 True)
% Found x10:(P True)
% Found (fun (x10:(P True))=> x10) as proof of (P True)
% Found (fun (x10:(P True))=> x10) as proof of (P0 True)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (fun (x:fofType)=> ((rel_codomain R) x)))
% Found (eta_expansion_dep00 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:fofType)=> True)):(((eq (fofType->Prop)) (fun (X:fofType)=> True)) (fun (x:fofType)=> True))
% Found (eta_expansion_dep00 (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) 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) (rel_codomain R))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found x00:(P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P0 (fun (X:fofType)=> True))
% Found eta_expansion000:=(eta_expansion00 (fun (X:fofType)=> True)):(((eq (fofType->Prop)) (fun (X:fofType)=> True)) (fun (x:fofType)=> True))
% Found (eta_expansion00 (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found ((eta_expansion0 Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) 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) (rel_codomain R))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found eq_ref00:=(eq_ref0 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (rel_codomain R))
% Found (eq_ref0 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eq_ref (fofType->Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eq_ref (fofType->Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eq_ref (fofType->Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((rel_codomain R) x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((rel_codomain R) x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((rel_codomain R) x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((rel_codomain R) x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((rel_codomain R) x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((rel_codomain R) x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((rel_codomain R) x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((rel_codomain R) x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((rel_codomain R) x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((rel_codomain R) x)))
% Found x0:(P (rel_codomain R))
% Instantiate: b:=(rel_codomain R):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 (fun (X:fofType)=> True)):(((eq (fofType->Prop)) (fun (X:fofType)=> True)) (fun (x:fofType)=> True))
% Found (eta_expansion00 (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found ((eta_expansion0 Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found x10:(P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P0 ((rel_codomain R) x0))
% Found x10:(P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P0 ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 (fun (X:fofType)=> True)):(((eq (fofType->Prop)) (fun (X:fofType)=> True)) (fun (X:fofType)=> True))
% Found (eq_ref0 (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b0)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b0)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b0)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) 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) (rel_codomain R))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) (rel_codomain R))
% Found eq_ref00:=(eq_ref0 (fun (X:fofType)=> True)):(((eq (fofType->Prop)) (fun (X:fofType)=> True)) (fun (X:fofType)=> True))
% Found (eq_ref0 (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) 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) (rel_codomain R))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found x0:(P (rel_codomain R))
% Instantiate: f:=(rel_codomain R):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (rel_codomain R))
% Instantiate: f:=(rel_codomain R):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (fun (x:fofType)=> ((rel_codomain R) x)))
% Found (eta_expansion_dep00 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (fun (x:fofType)=> ((rel_codomain R) x)))
% Found (eta_expansion_dep00 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (fun (x:fofType)=> ((rel_codomain R) x)))
% Found (eta_expansion_dep00 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found x1:(P True)
% Instantiate: b:=True:Prop
% Found x1 as proof of (P0 b)
% Found x1:(P True)
% Instantiate: b:=True:Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) True)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) True)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) True)
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) True)
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) True))
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) True)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) True)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) True)
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) True)
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) True))
% Found x00:(P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P0 (fun (X:fofType)=> True))
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P0 (rel_codomain R))
% Found eq_ref00:=(eq_ref0 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (rel_codomain R))
% Found (eq_ref0 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eq_ref (fofType->Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eq_ref (fofType->Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eq_ref (fofType->Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) 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 (X:fofType)=> True))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found x00:(P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P0 (fun (X:fofType)=> True))
% Found x00:(P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P0 (fun (X:fofType)=> True))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found x00:(P1 (rel_codomain R))
% Found (fun (x00:(P1 (rel_codomain R)))=> x00) as proof of (P1 (rel_codomain R))
% Found (fun (x00:(P1 (rel_codomain R)))=> x00) as proof of (P2 (rel_codomain R))
% Found x00:(P1 (rel_codomain R))
% Found (fun (x00:(P1 (rel_codomain R)))=> x00) as proof of (P1 (rel_codomain R))
% Found (fun (x00:(P1 (rel_codomain R)))=> x00) as proof of (P2 (rel_codomain R))
% Found x00:(P1 (rel_codomain R))
% Found (fun (x00:(P1 (rel_codomain R)))=> x00) as proof of (P1 (rel_codomain R))
% Found (fun (x00:(P1 (rel_codomain R)))=> x00) as proof of (P2 (rel_codomain R))
% Found x00:(P1 (rel_codomain R))
% Found (fun (x00:(P1 (rel_codomain R)))=> x00) as proof of (P1 (rel_codomain R))
% Found (fun (x00:(P1 (rel_codomain R)))=> x00) as proof of (P2 (rel_codomain R))
% Found x10:(P True)
% Found (fun (x10:(P True))=> x10) as proof of (P True)
% Found (fun (x10:(P True))=> x10) as proof of (P0 True)
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found x10:(P True)
% Found (fun (x10:(P True))=> x10) as proof of (P True)
% Found (fun (x10:(P True))=> x10) as proof of (P0 True)
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:fofType)=> True)):(((eq (fofType->Prop)) (fun (X:fofType)=> True)) (fun (x:fofType)=> True))
% Found (eta_expansion_dep00 (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (eq_sym010 (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (X:fofType)=> True))) as proof of (P b)
% Found ((eq_sym01 b) (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (X:fofType)=> True))) as proof of (P b)
% Found (((eq_sym0 (fun (X:fofType)=> True)) b) (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (X:fofType)=> True))) as proof of (P b)
% Found (((eq_sym0 (fun (X:fofType)=> True)) b) (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (X:fofType)=> True))) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (fun (x:fofType)=> ((rel_codomain R) x)))
% Found (eta_expansion00 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b0)
% Found ((eta_expansion0 Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b0)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b0)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b0)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (fofType->Prop)) b0) 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 x10:(P1 True)
% Found (fun (x10:(P1 True))=> x10) as proof of (P1 True)
% Found (fun (x10:(P1 True))=> x10) as proof of (P2 True)
% Found x10:(P1 True)
% Found (fun (x10:(P1 True))=> x10) as proof of (P1 True)
% Found (fun (x10:(P1 True))=> x10) as proof of (P2 True)
% Found x10:(P1 True)
% Found (fun (x10:(P1 True))=> x10) as proof of (P1 True)
% Found (fun (x10:(P1 True))=> x10) as proof of (P2 True)
% Found x10:(P1 True)
% Found (fun (x10:(P1 True))=> x10) as proof of (P1 True)
% Found (fun (x10:(P1 True))=> x10) as proof of (P2 True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found x1:(P0 b)
% Instantiate: b:=True:Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 True)
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 True))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found x1:(P0 b)
% Instantiate: b:=True:Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 True)
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 True))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (rel_codomain R))
% Found (eq_ref0 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eq_ref (fofType->Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eq_ref (fofType->Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eq_ref (fofType->Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found eta_expansion000:=(eta_expansion00 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (fun (x:fofType)=> ((rel_codomain R) x)))
% Found (eta_expansion00 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eta_expansion0 Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found or_intror:(forall (A:Prop) (B:Prop), (B->((or A) B)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (B->((or A) B))):Prop
% Found or_intror as proof of b
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_codomain R)))
% 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 (X:fofType)=> True))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found eta_expansion000:=(eta_expansion00 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (fun (x:fofType)=> ((rel_codomain R) x)))
% Found (eta_expansion00 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eta_expansion0 Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) 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 (X:fofType)=> True))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (fun (x:fofType)=> ((rel_codomain R) x)))
% Found (eta_expansion_dep00 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found eq_ref00:=(eq_ref0 (fun (X:fofType)=> True)):(((eq (fofType->Prop)) (fun (X:fofType)=> True)) (fun (X:fofType)=> True))
% Found (eq_ref0 (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) 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) (rel_domain (rel_inverse R)))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (rel_domain (rel_inverse R)))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (rel_domain (rel_inverse R)))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (rel_domain (rel_inverse R)))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (rel_domain (rel_inverse R)))
% Found x1:(P ((rel_codomain R) x0))
% Instantiate: b:=((rel_codomain R) x0):Prop
% Found x1 as proof of (P0 b)
% Found x1:(P ((rel_codomain R) x0))
% Instantiate: b:=((rel_codomain R) x0):Prop
% Found x1 as proof of (P0 b)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (X:fofType)=> True))->(P (fun (x:fofType)=> True)))
% Found (eta_expansion000 P) as proof of (P0 (fun (X:fofType)=> True))
% Found ((eta_expansion00 (fun (X:fofType)=> True)) P) as proof of (P0 (fun (X:fofType)=> True))
% Found (((eta_expansion0 Prop) (fun (X:fofType)=> True)) P) as proof of (P0 (fun (X:fofType)=> True))
% Found ((((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) P) as proof of (P0 (fun (X:fofType)=> True))
% Found ((((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) P) as proof of (P0 (fun (X:fofType)=> True))
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (X:fofType)=> True))->(P (fun (X:fofType)=> True)))
% Found (eq_ref00 P) as proof of (P0 (fun (X:fofType)=> True))
% Found ((eq_ref0 (fun (X:fofType)=> True)) P) as proof of (P0 (fun (X:fofType)=> True))
% Found (((eq_ref (fofType->Prop)) (fun (X:fofType)=> True)) P) as proof of (P0 (fun (X:fofType)=> True))
% Found (((eq_ref (fofType->Prop)) (fun (X:fofType)=> True)) P) as proof of (P0 (fun (X:fofType)=> True))
% Found eq_ref000:=(eq_ref00 P):((P (fun (X:fofType)=> True))->(P (fun (X:fofType)=> True)))
% Found (eq_ref00 P) as proof of (P0 (fun (X:fofType)=> True))
% Found ((eq_ref0 (fun (X:fofType)=> True)) P) as proof of (P0 (fun (X:fofType)=> True))
% Found (((eq_ref (fofType->Prop)) (fun (X:fofType)=> True)) P) as proof of (P0 (fun (X:fofType)=> True))
% Found (((eq_ref (fofType->Prop)) (fun (X:fofType)=> True)) P) as proof of (P0 (fun (X:fofType)=> True))
% Found x1:(P ((rel_codomain R) x0))
% Instantiate: b:=((rel_codomain R) x0):Prop
% Found x1 as proof of (P0 b)
% Found x1:(P ((rel_codomain R) x0))
% Instantiate: b:=((rel_codomain R) x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) (fun (X:fofType)=> True))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (X:fofType)=> True))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (X:fofType)=> True))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (X:fofType)=> True))
% Found eta_expansion000:=(eta_expansion00 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (fun (x:fofType)=> ((rel_codomain R) x)))
% Found (eta_expansion00 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b0)
% Found ((eta_expansion0 Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b0)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b0)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b0)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b0)
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found x00:(P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P0 (rel_codomain R))
% Found x00:(P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P0 (rel_codomain R))
% Found x00:(P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P0 (rel_codomain R))
% Found x00:(P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P0 (rel_codomain R))
% Found x00:(P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P0 (rel_codomain R))
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P (rel_codomain R))
% Found (fun (x00:(P (rel_codomain R)))=> x00) as proof of (P0 (rel_codomain R))
% Found eq_ref00:=(eq_ref0 (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_domain (rel_inverse R)))):(((eq Prop) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_domain (rel_inverse R)))) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_domain (rel_inverse R))))
% Found (eq_ref0 (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_domain (rel_inverse R)))) as proof of (((eq Prop) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_domain (rel_inverse R)))) b)
% Found ((eq_ref Prop) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_domain (rel_inverse R)))) as proof of (((eq Prop) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_domain (rel_inverse R)))) b)
% Found ((eq_ref Prop) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_domain (rel_inverse R)))) as proof of (((eq Prop) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_domain (rel_inverse R)))) b)
% Found ((eq_ref Prop) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_domain (rel_inverse R)))) as proof of (((eq Prop) (((eq (fofType->Prop)) (fun (X:fofType)=> True)) (rel_domain (rel_inverse R)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b0)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b0)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b0)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b0)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b0)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b0)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found x10:(P True)
% Found (fun (x10:(P True))=> x10) as proof of (P True)
% Found (fun (x10:(P True))=> x10) as proof of (P0 True)
% Found x10:(P True)
% Found (fun (x10:(P True))=> x10) as proof of (P True)
% Found (fun (x10:(P True))=> x10) as proof of (P0 True)
% Found x10:(P True)
% Found (fun (x10:(P True))=> x10) as proof of (P True)
% Found (fun (x10:(P True))=> x10) as proof of (P0 True)
% Found x10:(P True)
% Found (fun (x10:(P True))=> x10) as proof of (P True)
% Found (fun (x10:(P True))=> x10) as proof of (P0 True)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P0 (b x0))
% Found x10:(P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P0 (b x0))
% 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 (X:fofType)=> True)):(((eq (fofType->Prop)) (fun (X:fofType)=> True)) (fun (x:fofType)=> True))
% Found (eta_expansion00 (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b0)
% Found ((eta_expansion0 Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b0)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b0)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b0)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b0)
% Found or_intror:(forall (A:Prop) (B:Prop), (B->((or A) B)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (B->((or A) B))):Prop
% Found or_intror as proof of b
% Found x10:(P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P0 ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found x10:(P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P0 ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found x10:(P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P0 ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found x10:(P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P0 ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found x10:(P1 ((rel_codomain R) x0))
% Found (fun (x10:(P1 ((rel_codomain R) x0)))=> x10) as proof of (P1 ((rel_codomain R) x0))
% Found (fun (x10:(P1 ((rel_codomain R) x0)))=> x10) as proof of (P2 ((rel_codomain R) x0))
% Found x10:(P1 ((rel_codomain R) x0))
% Found (fun (x10:(P1 ((rel_codomain R) x0)))=> x10) as proof of (P1 ((rel_codomain R) x0))
% Found (fun (x10:(P1 ((rel_codomain R) x0)))=> x10) as proof of (P2 ((rel_codomain R) x0))
% Found x10:(P1 ((rel_codomain R) x0))
% Found (fun (x10:(P1 ((rel_codomain R) x0)))=> x10) as proof of (P1 ((rel_codomain R) x0))
% Found (fun (x10:(P1 ((rel_codomain R) x0)))=> x10) as proof of (P2 ((rel_codomain R) x0))
% Found x10:(P1 ((rel_codomain R) x0))
% Found (fun (x10:(P1 ((rel_codomain R) x0)))=> x10) as proof of (P1 ((rel_codomain R) x0))
% Found (fun (x10:(P1 ((rel_codomain R) x0)))=> x10) as proof of (P2 ((rel_codomain R) x0))
% Found x10:(P1 ((rel_codomain R) x0))
% Found (fun (x10:(P1 ((rel_codomain R) x0)))=> x10) as proof of (P1 ((rel_codomain R) x0))
% Found (fun (x10:(P1 ((rel_codomain R) x0)))=> x10) as proof of (P2 ((rel_codomain R) x0))
% Found x10:(P1 ((rel_codomain R) x0))
% Found (fun (x10:(P1 ((rel_codomain R) x0)))=> x10) as proof of (P1 ((rel_codomain R) x0))
% Found (fun (x10:(P1 ((rel_codomain R) x0)))=> x10) as proof of (P2 ((rel_codomain R) x0))
% Found x10:(P1 ((rel_codomain R) x0))
% Found (fun (x10:(P1 ((rel_codomain R) x0)))=> x10) as proof of (P1 ((rel_codomain R) x0))
% Found (fun (x10:(P1 ((rel_codomain R) x0)))=> x10) as proof of (P2 ((rel_codomain R) x0))
% Found x10:(P1 ((rel_codomain R) x0))
% Found (fun (x10:(P1 ((rel_codomain R) x0)))=> x10) as proof of (P1 ((rel_codomain R) x0))
% Found (fun (x10:(P1 ((rel_codomain R) x0)))=> x10) as proof of (P2 ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found x1:(P0 b)
% Instantiate: b:=((ex fofType) (fun (X:fofType)=> ((R X) x0))):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 ((rel_codomain R) x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 ((rel_codomain R) x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found x1:(P0 b)
% Instantiate: b:=((ex fofType) (fun (X:fofType)=> ((R X) x0))):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 ((rel_codomain R) x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 ((rel_codomain R) x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (rel_domain (rel_inverse R))):(((eq (fofType->Prop)) (rel_domain (rel_inverse R))) (rel_domain (rel_inverse R)))
% Found (eq_ref0 (rel_domain (rel_inverse R))) as proof of (((eq (fofType->Prop)) (rel_domain (rel_inverse R))) b)
% Found ((eq_ref (fofType->Prop)) (rel_domain (rel_inverse R))) as proof of (((eq (fofType->Prop)) (rel_domain (rel_inverse R))) b)
% Found ((eq_ref (fofType->Prop)) (rel_domain (rel_inverse R))) as proof of (((eq (fofType->Prop)) (rel_domain (rel_inverse R))) b)
% Found ((eq_ref (fofType->Prop)) (rel_domain (rel_inverse R))) as proof of (((eq (fofType->Prop)) (rel_domain (rel_inverse R))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (X:fofType)=> True))
% Found eq_ref00:=(eq_ref0 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (rel_codomain R))
% Found (eq_ref0 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eq_ref (fofType->Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eq_ref (fofType->Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eq_ref (fofType->Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:fofType)=> True)):(((eq (fofType->Prop)) (fun (X:fofType)=> True)) (fun (x:fofType)=> True))
% Found (eta_expansion_dep00 (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) 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) (rel_codomain R))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found eq_ref00:=(eq_ref0 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (rel_codomain R))
% Found (eq_ref0 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eq_ref (fofType->Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eq_ref (fofType->Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eq_ref (fofType->Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (fun (x:fofType)=> ((rel_codomain R) x)))
% Found (eta_expansion_dep00 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found x1:(P True)
% Instantiate: b:=True:Prop
% Found x1 as proof of (P0 b)
% Found x1:(P True)
% Instantiate: b:=True:Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) 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 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) 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 eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (rel_domain (rel_inverse R)))->(P (fun (x:fofType)=> ((rel_domain (rel_inverse R)) x))))
% Found (eta_expansion_dep000 P) as proof of (P0 (rel_domain (rel_inverse R)))
% Found ((eta_expansion_dep00 (rel_domain (rel_inverse R))) P) as proof of (P0 (rel_domain (rel_inverse R)))
% Found (((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (rel_domain (rel_inverse R))) P) as proof of (P0 (rel_domain (rel_inverse R)))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_domain (rel_inverse R))) P) as proof of (P0 (rel_domain (rel_inverse R)))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_domain (rel_inverse R))) P) as proof of (P0 (rel_domain (rel_inverse R)))
% Found eq_ref000:=(eq_ref00 P):((P (rel_domain (rel_inverse R)))->(P (rel_domain (rel_inverse R))))
% Found (eq_ref00 P) as proof of (P0 (rel_domain (rel_inverse R)))
% Found ((eq_ref0 (rel_domain (rel_inverse R))) P) as proof of (P0 (rel_domain (rel_inverse R)))
% Found (((eq_ref (fofType->Prop)) (rel_domain (rel_inverse R))) P) as proof of (P0 (rel_domain (rel_inverse R)))
% Found (((eq_ref (fofType->Prop)) (rel_domain (rel_inverse R))) P) as proof of (P0 (rel_domain (rel_inverse R)))
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_domain (rel_inverse R)) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_domain (rel_inverse R)) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_domain (rel_inverse R)) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_domain (rel_inverse R)) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_domain (rel_inverse R)) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_domain (rel_inverse R)) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_domain (rel_inverse R)) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_domain (rel_inverse R)) x0))
% Found x00:(P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P0 (fun (X:fofType)=> True))
% Found eta_expansion000:=(eta_expansion00 (fun (X:fofType)=> True)):(((eq (fofType->Prop)) (fun (X:fofType)=> True)) (fun (x:fofType)=> True))
% Found (eta_expansion00 (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found ((eta_expansion0 Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) 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) (rel_codomain R))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found eq_ref000:=(eq_ref00 P):((P True)->(P True))
% Found (eq_ref00 P) as proof of (P0 True)
% Found ((eq_ref0 True) P) as proof of (P0 True)
% Found (((eq_ref Prop) True) P) as proof of (P0 True)
% Found (((eq_ref Prop) True) P) as proof of (P0 True)
% Found eq_ref000:=(eq_ref00 P):((P True)->(P True))
% Found (eq_ref00 P) as proof of (P0 True)
% Found ((eq_ref0 True) P) as proof of (P0 True)
% Found (((eq_ref Prop) True) P) as proof of (P0 True)
% Found (((eq_ref Prop) True) P) as proof of (P0 True)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) True)
% Found x00:(P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P0 (fun (X:fofType)=> True))
% Found x00:(P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P (fun (X:fofType)=> True))
% Found (fun (x00:(P (fun (X:fofType)=> True)))=> x00) as proof of (P0 (fun (X:fofType)=> True))
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((rel_codomain R) x0)):(((eq Prop) ((rel_codomain R) x0)) ((rel_codomain R) x0))
% Found (eq_ref0 ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found ((eq_ref Prop) ((rel_codomain R) x0)) as proof of (((eq Prop) ((rel_codomain R) x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) True)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) True)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) True)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) True)
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) True)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) True)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) True)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) True)
% Found eta_expansion000:=(eta_expansion00 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (fun (x:fofType)=> ((rel_codomain R) x)))
% Found (eta_expansion00 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eta_expansion0 Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((rel_codomain R) x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((rel_codomain R) x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((rel_codomain R) x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((rel_codomain R) x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((rel_codomain R) x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((rel_codomain R) x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((rel_codomain R) x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((rel_codomain R) x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((rel_codomain R) x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((rel_codomain R) x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (fun (x:fofType)=> ((rel_codomain R) x)))
% Found (eta_expansion_dep00 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found eta_expansion000:=(eta_expansion00 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (fun (x:fofType)=> ((rel_codomain R) x)))
% Found (eta_expansion00 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eta_expansion0 Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x0:(P (rel_codomain R))
% Instantiate: b:=(rel_codomain R):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found 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) (rel_codomain R))
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (rel_codomain R))
% Found eta_expansion000:=(eta_expansion00 (fun (X:fofType)=> True)):(((eq (fofType->Prop)) (fun (X:fofType)=> True)) (fun (x:fofType)=> True))
% Found (eta_expansion00 (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found ((eta_expansion0 Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b)
% Found x10:(P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P0 ((rel_codomain R) x0))
% Found x10:(P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P0 ((rel_codomain R) x0))
% Found x10:(P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P0 ((rel_codomain R) x0))
% Found x10:(P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P0 ((rel_codomain R) x0))
% Found x10:(P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P0 ((rel_codomain R) x0))
% Found x10:(P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P0 ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P0 ((rel_codomain R) x0))
% Found x10:(P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P ((rel_codomain R) x0))
% Found (fun (x10:(P ((rel_codomain R) x0)))=> x10) as proof of (P0 ((rel_codomain R) x0))
% Found x10:(P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P0 (b x0))
% Found x10:(P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P0 (b x0))
% Found eta_expansion000:=(eta_expansion00 (fun (X:fofType)=> True)):(((eq (fofType->Prop)) (fun (X:fofType)=> True)) (fun (x:fofType)=> True))
% Found (eta_expansion00 (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b0)
% Found ((eta_expansion0 Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b0)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b0)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) b0)
% Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> True)) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> True)) 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) (rel_codomain R))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) (rel_codomain R))
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) (rel_codomain R))
% Found x10:(P True)
% Found (fun (x10:(P True))=> x10) as proof of (P True)
% Found (fun (x10:(P True))=> x10) as proof of (P0 True)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found x10:(P True)
% Found (fun (x10:(P True))=> x10) as proof of (P True)
% Found (fun (x10:(P True))=> x10) as proof of (P0 True)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((rel_codomain R) x0))
% Found eq_ref00:=(eq_ref0 True):(((eq Prop) True) True)
% Found (eq_ref0 True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found ((eq_ref Prop) True) as proof of (((eq Prop) True) b)
% Found eta_expansion000:=(eta_expansion00 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (fun (x:fofType)=> ((rel_codomain R) x)))
% Found (eta_expansion00 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eta_expansion0 Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion fofType) Prop) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (rel_codomain R)):(((eq (fofType->Prop)) (rel_codomain R)) (fun (x:fofType)=> ((rel_codomain R) x)))
% Found (eta_expansion_dep00 (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_codomain R)) as proof of (((eq (fofType->Prop)) (rel_codomain R)) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (rel_codomain R
% EOF
%------------------------------------------------------------------------------