%------------------------------------------------------------------------------ % File : cocATP---0.2.0 % Problem : SWW473^2 : TPTP v6.1.0. Released v5.3.0. % Transfm : none % Format : tptp:raw % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % Computer : n096.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 13:37:23 EDT 2014 % Result : Timeout 300.08s % Output : None % Verified : % SZS Type : None (Parsing solution fails) % Syntax : Number of formulae : 0 % Comments : %------------------------------------------------------------------------------ %----NO SOLUTION OUTPUT BY SYSTEM %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % % Problem : SWW473^2 : TPTP v6.1.0. Released v5.3.0. % % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % % Computer : n096.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 09:20:16 CDT 2014 % % CPUTime : 300.08 % Python 2.7.5 % Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/'] % FOF formula (<kernel.Constant object at 0x21f3638>, <kernel.Type object at 0x21f3bd8>) of role type named ty_ty_t__a % Using role type % Declaring x_a:Type % FOF formula (<kernel.Constant object at 0x2339fc8>, <kernel.Type object at 0x21f3ab8>) of role type named ty_ty_tc__Com__Ocom % Using role type % Declaring com:Type % FOF formula (<kernel.Constant object at 0x21f3fc8>, <kernel.Type object at 0x21f3368>) of role type named ty_ty_tc__Com__Opname % Using role type % Declaring pname:Type % FOF formula (<kernel.Constant object at 0x21f3bd8>, <kernel.Type object at 0x21f3ab8>) of role type named ty_ty_tc__Nat__Onat % Using role type % Declaring nat:Type % FOF formula (<kernel.Constant object at 0x21f3440>, <kernel.Type object at 0x25cecb0>) of role type named ty_ty_tc__Option__Ooption_Itc__Com__Ocom_J % Using role type % Declaring option_com:Type % FOF formula (<kernel.Constant object at 0x21f36c8>, <kernel.DependentProduct object at 0x25ce998>) of role type named sy_c_Com_Obody % Using role type % Declaring body:(pname->option_com) % FOF formula (<kernel.Constant object at 0x21f3950>, <kernel.DependentProduct object at 0x25ceb00>) of role type named sy_c_Finite__Set_Ocard_000_062_It__a_M_Eo_J % Using role type % Declaring finite_card_a_o:(((x_a->Prop)->Prop)->nat) % FOF formula (<kernel.Constant object at 0x21f36c8>, <kernel.DependentProduct object at 0x25ce878>) of role type named sy_c_Finite__Set_Ocard_000_062_Itc__Com__Opname_M_Eo_J % Using role type % Declaring finite_card_pname_o:(((pname->Prop)->Prop)->nat) % FOF formula (<kernel.Constant object at 0x21f3bd8>, <kernel.DependentProduct object at 0x25ce9e0>) of role type named sy_c_Finite__Set_Ocard_000_062_Itc__Nat__Onat_M_Eo_J % Using role type % Declaring finite_card_nat_o:(((nat->Prop)->Prop)->nat) % FOF formula (<kernel.Constant object at 0x21f3bd8>, <kernel.DependentProduct object at 0x25ce8c0>) of role type named sy_c_Finite__Set_Ocard_000t__a % Using role type % Declaring finite_card_a:((x_a->Prop)->nat) % FOF formula (<kernel.Constant object at 0x21f3bd8>, <kernel.DependentProduct object at 0x25ceb90>) of role type named sy_c_Finite__Set_Ocard_000tc__Com__Opname % Using role type % Declaring finite_card_pname:((pname->Prop)->nat) % FOF formula (<kernel.Constant object at 0x25ce9e0>, <kernel.DependentProduct object at 0x25ce6c8>) of role type named sy_c_Finite__Set_Ocard_000tc__Nat__Onat % Using role type % Declaring finite_card_nat:((nat->Prop)->nat) % FOF formula (<kernel.Constant object at 0x25ceab8>, <kernel.DependentProduct object at 0x25ce878>) of role type named sy_c_Finite__Set_Ofinite_000_062_I_062_It__a_M_Eo_J_M_Eo_J % Using role type % Declaring finite_finite_a_o_o:((((x_a->Prop)->Prop)->Prop)->Prop) % FOF formula (<kernel.Constant object at 0x25ce998>, <kernel.DependentProduct object at 0x25ce7a0>) of role type named sy_c_Finite__Set_Ofinite_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_J % Using role type % Declaring finite1066544169me_o_o:((((pname->Prop)->Prop)->Prop)->Prop) % FOF formula (<kernel.Constant object at 0x25ce6c8>, <kernel.DependentProduct object at 0x25ce638>) of role type named sy_c_Finite__Set_Ofinite_000_062_I_062_Itc__Nat__Onat_M_Eo_J_M_Eo_J % Using role type % Declaring finite1676163439at_o_o:((((nat->Prop)->Prop)->Prop)->Prop) % FOF formula (<kernel.Constant object at 0x25ce878>, <kernel.DependentProduct object at 0x25ce638>) of role type named sy_c_Finite__Set_Ofinite_000_062_It__a_M_Eo_J % Using role type % Declaring finite_finite_a_o:(((x_a->Prop)->Prop)->Prop) % FOF formula (<kernel.Constant object at 0x25ce7a0>, <kernel.DependentProduct object at 0x25ce638>) of role type named sy_c_Finite__Set_Ofinite_000_062_Itc__Com__Opname_M_Eo_J % Using role type % Declaring finite297249702name_o:(((pname->Prop)->Prop)->Prop) % FOF formula (<kernel.Constant object at 0x25ce758>, <kernel.DependentProduct object at 0x25ce638>) of role type named sy_c_Finite__Set_Ofinite_000_062_Itc__Nat__Onat_M_Eo_J % Using role type % Declaring finite_finite_nat_o:(((nat->Prop)->Prop)->Prop) % FOF formula (<kernel.Constant object at 0x25ceb00>, <kernel.DependentProduct object at 0x25ce3f8>) of role type named sy_c_Finite__Set_Ofinite_000t__a % Using role type % Declaring finite_finite_a:((x_a->Prop)->Prop) % FOF formula (<kernel.Constant object at 0x25ce560>, <kernel.DependentProduct object at 0x25ce4d0>) of role type named sy_c_Finite__Set_Ofinite_000tc__Com__Opname % Using role type % Declaring finite_finite_pname:((pname->Prop)->Prop) % FOF formula (<kernel.Constant object at 0x25ce638>, <kernel.DependentProduct object at 0x25ceb00>) of role type named sy_c_Finite__Set_Ofinite_000tc__Nat__Onat % Using role type % Declaring finite_finite_nat:((nat->Prop)->Prop) % FOF formula (<kernel.Constant object at 0x25ce7a0>, <kernel.DependentProduct object at 0x25cec20>) of role type named sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat % Using role type % Declaring minus_minus_nat:(nat->(nat->nat)) % FOF formula (<kernel.Constant object at 0x25ce368>, <kernel.Constant object at 0x25cec20>) of role type named sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat % Using role type % Declaring one_one_nat:nat % FOF formula (<kernel.Constant object at 0x25ce638>, <kernel.DependentProduct object at 0x25ceb00>) of role type named sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat % Using role type % Declaring plus_plus_nat:(nat->(nat->nat)) % FOF formula (<kernel.Constant object at 0x25ce4d0>, <kernel.Constant object at 0x25ceb00>) of role type named sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat % Using role type % Declaring zero_zero_nat:nat % FOF formula (<kernel.Constant object at 0x25ce368>, <kernel.DependentProduct object at 0x25ce7a0>) of role type named sy_c_Nat_OSuc % Using role type % Declaring suc:(nat->nat) % FOF formula (<kernel.Constant object at 0x25ce710>, <kernel.DependentProduct object at 0x25ce050>) of role type named sy_c_Nat_Onat_Onat__case_000_Eo % Using role type % Declaring nat_case_o:(Prop->((nat->Prop)->(nat->Prop))) % FOF formula (<kernel.Constant object at 0x25ce248>, <kernel.DependentProduct object at 0x25ced40>) of role type named sy_c_Option_Othe_000tc__Com__Ocom % Using role type % Declaring the_com:(option_com->com) % FOF formula (<kernel.Constant object at 0x25ce4d0>, <kernel.DependentProduct object at 0x25ce050>) of role type named sy_c_Orderings_Obot__class_Obot_000_062_I_062_It__a_M_Eo_J_M_Eo_J % Using role type % Declaring bot_bot_a_o_o:((x_a->Prop)->Prop) % FOF formula (<kernel.Constant object at 0x25ce710>, <kernel.DependentProduct object at 0x25ce248>) of role type named sy_c_Orderings_Obot__class_Obot_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_J % Using role type % Declaring bot_bot_pname_o_o:((pname->Prop)->Prop) % FOF formula (<kernel.Constant object at 0x25ce320>, <kernel.DependentProduct object at 0x25ce050>) of role type named sy_c_Orderings_Obot__class_Obot_000_062_I_062_Itc__Nat__Onat_M_Eo_J_M_Eo_J % Using role type % Declaring bot_bot_nat_o_o:((nat->Prop)->Prop) % FOF formula (<kernel.Constant object at 0x25ce290>, <kernel.DependentProduct object at 0x25cecf8>) of role type named sy_c_Orderings_Obot__class_Obot_000_062_It__a_M_Eo_J % Using role type % Declaring bot_bot_a_o:(x_a->Prop) % FOF formula (<kernel.Constant object at 0x25ce200>, <kernel.DependentProduct object at 0x25ce7e8>) of role type named sy_c_Orderings_Obot__class_Obot_000_062_Itc__Com__Opname_M_Eo_J % Using role type % Declaring bot_bot_pname_o:(pname->Prop) % FOF formula (<kernel.Constant object at 0x25ce050>, <kernel.DependentProduct object at 0x25cefc8>) of role type named sy_c_Orderings_Obot__class_Obot_000_062_Itc__Nat__Onat_M_Eo_J % Using role type % Declaring bot_bot_nat_o:(nat->Prop) % FOF formula (<kernel.Constant object at 0x25cecf8>, <kernel.Sort object at 0x20bd6c8>) of role type named sy_c_Orderings_Obot__class_Obot_000_Eo % Using role type % Declaring bot_bot_o:Prop % FOF formula (<kernel.Constant object at 0x25ce638>, <kernel.Constant object at 0x25ce050>) of role type named sy_c_Orderings_Obot__class_Obot_000tc__Nat__Onat % Using role type % Declaring bot_bot_nat:nat % FOF formula (<kernel.Constant object at 0x25ce7e8>, <kernel.DependentProduct object at 0x25ce248>) of role type named sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat % Using role type % Declaring ord_less_nat:(nat->(nat->Prop)) % FOF formula (<kernel.Constant object at 0x25ce098>, <kernel.DependentProduct object at 0x25ce0e0>) of role type named sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_It__a_M_Eo_J_M_Eo_J % Using role type % Declaring ord_less_eq_a_o_o:(((x_a->Prop)->Prop)->(((x_a->Prop)->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x25ce050>, <kernel.DependentProduct object at 0x25ce5f0>) of role type named sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_ % Using role type % Declaring ord_le1205211808me_o_o:(((pname->Prop)->Prop)->(((pname->Prop)->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x25cecf8>, <kernel.DependentProduct object at 0x25ce128>) of role type named sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_Itc__Nat__Onat_M_Eo_J_M_Eo_J % Using role type % Declaring ord_less_eq_nat_o_o:(((nat->Prop)->Prop)->(((nat->Prop)->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x25ce0e0>, <kernel.DependentProduct object at 0x25ce170>) of role type named sy_c_Orderings_Oord__class_Oless__eq_000_062_It__a_M_Eo_J % Using role type % Declaring ord_less_eq_a_o:((x_a->Prop)->((x_a->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x25ce7e8>, <kernel.DependentProduct object at 0x25ceef0>) of role type named sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Com__Opname_M_Eo_J % Using role type % Declaring ord_less_eq_pname_o:((pname->Prop)->((pname->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x25ce830>, <kernel.DependentProduct object at 0x25ce5a8>) of role type named sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Nat__Onat_M_Eo_J % Using role type % Declaring ord_less_eq_nat_o:((nat->Prop)->((nat->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x25cea70>, <kernel.DependentProduct object at 0x25ce0e0>) of role type named sy_c_Orderings_Oord__class_Oless__eq_000_Eo % Using role type % Declaring ord_less_eq_o:(Prop->(Prop->Prop)) % FOF formula (<kernel.Constant object at 0x25ce7e8>, <kernel.DependentProduct object at 0x25ceef0>) of role type named sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat % Using role type % Declaring ord_less_eq_nat:(nat->(nat->Prop)) % FOF formula (<kernel.Constant object at 0x25ce5a8>, <kernel.DependentProduct object at 0x25ce950>) of role type named sy_c_Set_OCollect_000_062_I_062_It__a_M_Eo_J_M_Eo_J % Using role type % Declaring collect_a_o_o:((((x_a->Prop)->Prop)->Prop)->(((x_a->Prop)->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x25ce0e0>, <kernel.DependentProduct object at 0x25ce830>) of role type named sy_c_Set_OCollect_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_J % Using role type % Declaring collect_pname_o_o:((((pname->Prop)->Prop)->Prop)->(((pname->Prop)->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x25ceef0>, <kernel.DependentProduct object at 0x25cea70>) of role type named sy_c_Set_OCollect_000_062_I_062_Itc__Nat__Onat_M_Eo_J_M_Eo_J % Using role type % Declaring collect_nat_o_o:((((nat->Prop)->Prop)->Prop)->(((nat->Prop)->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x25ce128>, <kernel.DependentProduct object at 0x2624e18>) of role type named sy_c_Set_OCollect_000_062_It__a_M_Eo_J % Using role type % Declaring collect_a_o:(((x_a->Prop)->Prop)->((x_a->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x25ce5a8>, <kernel.DependentProduct object at 0x2624758>) of role type named sy_c_Set_OCollect_000_062_Itc__Com__Opname_M_Eo_J % Using role type % Declaring collect_pname_o:(((pname->Prop)->Prop)->((pname->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x25ceef0>, <kernel.DependentProduct object at 0x2624170>) of role type named sy_c_Set_OCollect_000_062_Itc__Nat__Onat_M_Eo_J % Using role type % Declaring collect_nat_o:(((nat->Prop)->Prop)->((nat->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x25ce128>, <kernel.DependentProduct object at 0x2624758>) of role type named sy_c_Set_OCollect_000t__a % Using role type % Declaring collect_a:((x_a->Prop)->(x_a->Prop)) % FOF formula (<kernel.Constant object at 0x25ceef0>, <kernel.DependentProduct object at 0x2624170>) of role type named sy_c_Set_OCollect_000tc__Com__Opname % Using role type % Declaring collect_pname:((pname->Prop)->(pname->Prop)) % FOF formula (<kernel.Constant object at 0x25ceef0>, <kernel.DependentProduct object at 0x2624e18>) of role type named sy_c_Set_OCollect_000tc__Nat__Onat % Using role type % Declaring collect_nat:((nat->Prop)->(nat->Prop)) % FOF formula (<kernel.Constant object at 0x25ce128>, <kernel.DependentProduct object at 0x2624758>) of role type named sy_c_Set_Oimage_000_062_It__a_M_Eo_J_000t__a % Using role type % Declaring image_a_o_a:(((x_a->Prop)->x_a)->(((x_a->Prop)->Prop)->(x_a->Prop))) % FOF formula (<kernel.Constant object at 0x25ce128>, <kernel.DependentProduct object at 0x2624fc8>) of role type named sy_c_Set_Oimage_000_062_It__a_M_Eo_J_000tc__Com__Opname % Using role type % Declaring image_a_o_pname:(((x_a->Prop)->pname)->(((x_a->Prop)->Prop)->(pname->Prop))) % FOF formula (<kernel.Constant object at 0x26243f8>, <kernel.DependentProduct object at 0x2624200>) of role type named sy_c_Set_Oimage_000_062_It__a_M_Eo_J_000tc__Nat__Onat % Using role type % Declaring image_a_o_nat:(((x_a->Prop)->nat)->(((x_a->Prop)->Prop)->(nat->Prop))) % FOF formula (<kernel.Constant object at 0x26242d8>, <kernel.DependentProduct object at 0x2624e18>) of role type named sy_c_Set_Oimage_000_062_Itc__Com__Opname_M_Eo_J_000t__a % Using role type % Declaring image_pname_o_a:(((pname->Prop)->x_a)->(((pname->Prop)->Prop)->(x_a->Prop))) % FOF formula (<kernel.Constant object at 0x2624050>, <kernel.DependentProduct object at 0x2624e60>) of role type named sy_c_Set_Oimage_000_062_Itc__Com__Opname_M_Eo_J_000tc__Com__Opname % Using role type % Declaring image_pname_o_pname:(((pname->Prop)->pname)->(((pname->Prop)->Prop)->(pname->Prop))) % FOF formula (<kernel.Constant object at 0x2624758>, <kernel.DependentProduct object at 0x26243f8>) of role type named sy_c_Set_Oimage_000_062_Itc__Com__Opname_M_Eo_J_000tc__Nat__Onat % Using role type % Declaring image_pname_o_nat:(((pname->Prop)->nat)->(((pname->Prop)->Prop)->(nat->Prop))) % FOF formula (<kernel.Constant object at 0x2624fc8>, <kernel.DependentProduct object at 0x26242d8>) of role type named sy_c_Set_Oimage_000_062_Itc__Nat__Onat_M_Eo_J_000t__a % Using role type % Declaring image_nat_o_a:(((nat->Prop)->x_a)->(((nat->Prop)->Prop)->(x_a->Prop))) % FOF formula (<kernel.Constant object at 0x2624200>, <kernel.DependentProduct object at 0x26243f8>) of role type named sy_c_Set_Oimage_000_062_Itc__Nat__Onat_M_Eo_J_000tc__Com__Opname % Using role type % Declaring image_nat_o_pname:(((nat->Prop)->pname)->(((nat->Prop)->Prop)->(pname->Prop))) % FOF formula (<kernel.Constant object at 0x26247a0>, <kernel.DependentProduct object at 0x2624e60>) of role type named sy_c_Set_Oimage_000_062_Itc__Nat__Onat_M_Eo_J_000tc__Nat__Onat % Using role type % Declaring image_nat_o_nat:(((nat->Prop)->nat)->(((nat->Prop)->Prop)->(nat->Prop))) % FOF formula (<kernel.Constant object at 0x2624758>, <kernel.DependentProduct object at 0x2624200>) of role type named sy_c_Set_Oimage_000t__a_000_062_It__a_M_Eo_J % Using role type % Declaring image_a_a_o:((x_a->(x_a->Prop))->((x_a->Prop)->((x_a->Prop)->Prop))) % FOF formula (<kernel.Constant object at 0x26242d8>, <kernel.DependentProduct object at 0x25cf680>) of role type named sy_c_Set_Oimage_000t__a_000_062_Itc__Com__Opname_M_Eo_J % Using role type % Declaring image_a_pname_o:((x_a->(pname->Prop))->((x_a->Prop)->((pname->Prop)->Prop))) % FOF formula (<kernel.Constant object at 0x2624200>, <kernel.DependentProduct object at 0x25cfc20>) of role type named sy_c_Set_Oimage_000t__a_000_062_Itc__Nat__Onat_M_Eo_J % Using role type % Declaring image_a_nat_o:((x_a->(nat->Prop))->((x_a->Prop)->((nat->Prop)->Prop))) % FOF formula (<kernel.Constant object at 0x2624cb0>, <kernel.DependentProduct object at 0x25cfcb0>) of role type named sy_c_Set_Oimage_000t__a_000t__a % Using role type % Declaring image_a_a:((x_a->x_a)->((x_a->Prop)->(x_a->Prop))) % FOF formula (<kernel.Constant object at 0x26242d8>, <kernel.DependentProduct object at 0x25cfb48>) of role type named sy_c_Set_Oimage_000t__a_000tc__Com__Opname % Using role type % Declaring image_a_pname:((x_a->pname)->((x_a->Prop)->(pname->Prop))) % FOF formula (<kernel.Constant object at 0x2624cb0>, <kernel.DependentProduct object at 0x25cfe18>) of role type named sy_c_Set_Oimage_000t__a_000tc__Nat__Onat % Using role type % Declaring image_a_nat:((x_a->nat)->((x_a->Prop)->(nat->Prop))) % FOF formula (<kernel.Constant object at 0x26242d8>, <kernel.DependentProduct object at 0x25cfb48>) of role type named sy_c_Set_Oimage_000tc__Com__Opname_000_062_It__a_M_Eo_J % Using role type % Declaring image_pname_a_o:((pname->(x_a->Prop))->((pname->Prop)->((x_a->Prop)->Prop))) % FOF formula (<kernel.Constant object at 0x26242d8>, <kernel.DependentProduct object at 0x25cfe18>) of role type named sy_c_Set_Oimage_000tc__Com__Opname_000_062_Itc__Com__Opname_M_Eo_J % Using role type % Declaring image_pname_pname_o:((pname->(pname->Prop))->((pname->Prop)->((pname->Prop)->Prop))) % FOF formula (<kernel.Constant object at 0x25cff38>, <kernel.DependentProduct object at 0x25cfb48>) of role type named sy_c_Set_Oimage_000tc__Com__Opname_000_062_Itc__Nat__Onat_M_Eo_J % Using role type % Declaring image_pname_nat_o:((pname->(nat->Prop))->((pname->Prop)->((nat->Prop)->Prop))) % FOF formula (<kernel.Constant object at 0x25cf290>, <kernel.DependentProduct object at 0x25cf170>) of role type named sy_c_Set_Oimage_000tc__Com__Opname_000t__a % Using role type % Declaring image_pname_a:((pname->x_a)->((pname->Prop)->(x_a->Prop))) % FOF formula (<kernel.Constant object at 0x25cfc68>, <kernel.DependentProduct object at 0x25cfbd8>) of role type named sy_c_Set_Oimage_000tc__Com__Opname_000tc__Com__Opname % Using role type % Declaring image_pname_pname:((pname->pname)->((pname->Prop)->(pname->Prop))) % FOF formula (<kernel.Constant object at 0x25cfab8>, <kernel.DependentProduct object at 0x25cfbd8>) of role type named sy_c_Set_Oimage_000tc__Com__Opname_000tc__Nat__Onat % Using role type % Declaring image_pname_nat:((pname->nat)->((pname->Prop)->(nat->Prop))) % FOF formula (<kernel.Constant object at 0x25cfc20>, <kernel.DependentProduct object at 0x25cfbd8>) of role type named sy_c_Set_Oimage_000tc__Nat__Onat_000_062_It__a_M_Eo_J % Using role type % Declaring image_nat_a_o:((nat->(x_a->Prop))->((nat->Prop)->((x_a->Prop)->Prop))) % FOF formula (<kernel.Constant object at 0x25cfdd0>, <kernel.DependentProduct object at 0x25cfbd8>) of role type named sy_c_Set_Oimage_000tc__Nat__Onat_000_062_Itc__Com__Opname_M_Eo_J % Using role type % Declaring image_nat_pname_o:((nat->(pname->Prop))->((nat->Prop)->((pname->Prop)->Prop))) % FOF formula (<kernel.Constant object at 0x25cfc68>, <kernel.DependentProduct object at 0x25cfc20>) of role type named sy_c_Set_Oimage_000tc__Nat__Onat_000_062_Itc__Nat__Onat_M_Eo_J % Using role type % Declaring image_nat_nat_o:((nat->(nat->Prop))->((nat->Prop)->((nat->Prop)->Prop))) % FOF formula (<kernel.Constant object at 0x25cf290>, <kernel.DependentProduct object at 0x21d8908>) of role type named sy_c_Set_Oimage_000tc__Nat__Onat_000t__a % Using role type % Declaring image_nat_a:((nat->x_a)->((nat->Prop)->(x_a->Prop))) % FOF formula (<kernel.Constant object at 0x25cfc20>, <kernel.DependentProduct object at 0x21d8758>) of role type named sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Com__Opname % Using role type % Declaring image_nat_pname:((nat->pname)->((nat->Prop)->(pname->Prop))) % FOF formula (<kernel.Constant object at 0x25cfc68>, <kernel.DependentProduct object at 0x21d87a0>) of role type named sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Nat__Onat % Using role type % Declaring image_nat_nat:((nat->nat)->((nat->Prop)->(nat->Prop))) % FOF formula (<kernel.Constant object at 0x25cfc20>, <kernel.DependentProduct object at 0x21d8830>) of role type named sy_c_Set_Oinsert_000_062_It__a_M_Eo_J % Using role type % Declaring insert_a_o:((x_a->Prop)->(((x_a->Prop)->Prop)->((x_a->Prop)->Prop))) % FOF formula (<kernel.Constant object at 0x25cfc68>, <kernel.DependentProduct object at 0x21d87e8>) of role type named sy_c_Set_Oinsert_000_062_Itc__Com__Opname_M_Eo_J % Using role type % Declaring insert_pname_o:((pname->Prop)->(((pname->Prop)->Prop)->((pname->Prop)->Prop))) % FOF formula (<kernel.Constant object at 0x25cf290>, <kernel.DependentProduct object at 0x21d8758>) of role type named sy_c_Set_Oinsert_000_062_Itc__Nat__Onat_M_Eo_J % Using role type % Declaring insert_nat_o:((nat->Prop)->(((nat->Prop)->Prop)->((nat->Prop)->Prop))) % FOF formula (<kernel.Constant object at 0x25cf290>, <kernel.DependentProduct object at 0x21d8830>) of role type named sy_c_Set_Oinsert_000t__a % Using role type % Declaring insert_a:(x_a->((x_a->Prop)->(x_a->Prop))) % FOF formula (<kernel.Constant object at 0x21d8878>, <kernel.DependentProduct object at 0x21d8908>) of role type named sy_c_Set_Oinsert_000tc__Com__Opname % Using role type % Declaring insert_pname:(pname->((pname->Prop)->(pname->Prop))) % FOF formula (<kernel.Constant object at 0x21d8638>, <kernel.DependentProduct object at 0x21d8518>) of role type named sy_c_Set_Oinsert_000tc__Nat__Onat % Using role type % Declaring insert_nat:(nat->((nat->Prop)->(nat->Prop))) % FOF formula (<kernel.Constant object at 0x21d86c8>, <kernel.DependentProduct object at 0x21d8560>) of role type named sy_c_fequal_000_062_It__a_M_Eo_J % Using role type % Declaring fequal_a_o:((x_a->Prop)->((x_a->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x21d8908>, <kernel.DependentProduct object at 0x21d88c0>) of role type named sy_c_fequal_000_062_Itc__Com__Opname_M_Eo_J % Using role type % Declaring fequal_pname_o:((pname->Prop)->((pname->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x21d8638>, <kernel.DependentProduct object at 0x21d85f0>) of role type named sy_c_fequal_000_062_Itc__Nat__Onat_M_Eo_J % Using role type % Declaring fequal_nat_o:((nat->Prop)->((nat->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x21d8878>, <kernel.DependentProduct object at 0x21d88c0>) of role type named sy_c_fequal_000t__a % Using role type % Declaring fequal_a:(x_a->(x_a->Prop)) % FOF formula (<kernel.Constant object at 0x21d87a0>, <kernel.DependentProduct object at 0x21d8878>) of role type named sy_c_fequal_000tc__Com__Opname % Using role type % Declaring fequal_pname:(pname->(pname->Prop)) % FOF formula (<kernel.Constant object at 0x21d87e8>, <kernel.DependentProduct object at 0x21d8488>) of role type named sy_c_fequal_000tc__Nat__Onat % Using role type % Declaring fequal_nat:(nat->(nat->Prop)) % FOF formula (<kernel.Constant object at 0x21d8908>, <kernel.DependentProduct object at 0x21d84d0>) of role type named sy_c_member_000_062_It__a_M_Eo_J % Using role type % Declaring member_a_o:((x_a->Prop)->(((x_a->Prop)->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x21d8878>, <kernel.DependentProduct object at 0x21d85f0>) of role type named sy_c_member_000_062_Itc__Com__Opname_M_Eo_J % Using role type % Declaring member_pname_o:((pname->Prop)->(((pname->Prop)->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x21d87e8>, <kernel.DependentProduct object at 0x21d89e0>) of role type named sy_c_member_000_062_Itc__Nat__Onat_M_Eo_J % Using role type % Declaring member_nat_o:((nat->Prop)->(((nat->Prop)->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x21d8440>, <kernel.DependentProduct object at 0x21d8a70>) of role type named sy_c_member_000t__a % Using role type % Declaring member_a:(x_a->((x_a->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x21d8638>, <kernel.DependentProduct object at 0x21d8b00>) of role type named sy_c_member_000tc__Com__Opname % Using role type % Declaring member_pname:(pname->((pname->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x21d8518>, <kernel.DependentProduct object at 0x21d8b48>) of role type named sy_c_member_000tc__Nat__Onat % Using role type % Declaring member_nat:(nat->((nat->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x21d87e8>, <kernel.DependentProduct object at 0x21d8a70>) of role type named sy_v_G % Using role type % Declaring g:(x_a->Prop) % FOF formula (<kernel.Constant object at 0x21d8638>, <kernel.DependentProduct object at 0x21d8b90>) of role type named sy_v_P % Using role type % Declaring p:((x_a->Prop)->((x_a->Prop)->Prop)) % FOF formula (<kernel.Constant object at 0x21d8518>, <kernel.DependentProduct object at 0x21d8a28>) of role type named sy_v_U % Using role type % Declaring u:(pname->Prop) % FOF formula (<kernel.Constant object at 0x21d8440>, <kernel.DependentProduct object at 0x21d8bd8>) of role type named sy_v_mgt % Using role type % Declaring mgt:(com->x_a) % FOF formula (<kernel.Constant object at 0x21d8b90>, <kernel.DependentProduct object at 0x21d8c20>) of role type named sy_v_mgt__call % Using role type % Declaring mgt_call:(pname->x_a) % FOF formula (<kernel.Constant object at 0x21d8a28>, <kernel.Constant object at 0x21d8c20>) of role type named sy_v_na % Using role type % Declaring na:nat % FOF formula (<kernel.Constant object at 0x21d8440>, <kernel.Constant object at 0x21d8c20>) of role type named sy_v_pn % Using role type % Declaring pn:pname % FOF formula (<kernel.Constant object at 0x21d8b90>, <kernel.DependentProduct object at 0x21d8cb0>) of role type named sy_v_wt % Using role type % Declaring wt:(com->Prop) % FOF formula (forall (Ts:(x_a->Prop)) (G:(x_a->Prop)), (((ord_less_eq_a_o Ts) G)->((p G) Ts))) of role axiom named fact_0_assms_I1_J % A new axiom: (forall (Ts:(x_a->Prop)) (G:(x_a->Prop)), (((ord_less_eq_a_o Ts) G)->((p G) Ts))) % FOF formula (forall (A_92:((nat->Prop)->Prop)), ((finite_finite_nat_o A_92)->(finite1676163439at_o_o (collect_nat_o_o (fun (B_47:((nat->Prop)->Prop))=> ((ord_less_eq_nat_o_o B_47) A_92)))))) of role axiom named fact_1_finite__Collect__subsets % A new axiom: (forall (A_92:((nat->Prop)->Prop)), ((finite_finite_nat_o A_92)->(finite1676163439at_o_o (collect_nat_o_o (fun (B_47:((nat->Prop)->Prop))=> ((ord_less_eq_nat_o_o B_47) A_92)))))) % FOF formula (forall (A_92:((pname->Prop)->Prop)), ((finite297249702name_o A_92)->(finite1066544169me_o_o (collect_pname_o_o (fun (B_47:((pname->Prop)->Prop))=> ((ord_le1205211808me_o_o B_47) A_92)))))) of role axiom named fact_2_finite__Collect__subsets % A new axiom: (forall (A_92:((pname->Prop)->Prop)), ((finite297249702name_o A_92)->(finite1066544169me_o_o (collect_pname_o_o (fun (B_47:((pname->Prop)->Prop))=> ((ord_le1205211808me_o_o B_47) A_92)))))) % FOF formula (forall (A_92:((x_a->Prop)->Prop)), ((finite_finite_a_o A_92)->(finite_finite_a_o_o (collect_a_o_o (fun (B_47:((x_a->Prop)->Prop))=> ((ord_less_eq_a_o_o B_47) A_92)))))) of role axiom named fact_3_finite__Collect__subsets % A new axiom: (forall (A_92:((x_a->Prop)->Prop)), ((finite_finite_a_o A_92)->(finite_finite_a_o_o (collect_a_o_o (fun (B_47:((x_a->Prop)->Prop))=> ((ord_less_eq_a_o_o B_47) A_92)))))) % FOF formula (forall (A_92:(x_a->Prop)), ((finite_finite_a A_92)->(finite_finite_a_o (collect_a_o (fun (B_47:(x_a->Prop))=> ((ord_less_eq_a_o B_47) A_92)))))) of role axiom named fact_4_finite__Collect__subsets % A new axiom: (forall (A_92:(x_a->Prop)), ((finite_finite_a A_92)->(finite_finite_a_o (collect_a_o (fun (B_47:(x_a->Prop))=> ((ord_less_eq_a_o B_47) A_92)))))) % FOF formula (forall (A_92:(pname->Prop)), ((finite_finite_pname A_92)->(finite297249702name_o (collect_pname_o (fun (B_47:(pname->Prop))=> ((ord_less_eq_pname_o B_47) A_92)))))) of role axiom named fact_5_finite__Collect__subsets % A new axiom: (forall (A_92:(pname->Prop)), ((finite_finite_pname A_92)->(finite297249702name_o (collect_pname_o (fun (B_47:(pname->Prop))=> ((ord_less_eq_pname_o B_47) A_92)))))) % FOF formula (forall (A_92:(nat->Prop)), ((finite_finite_nat A_92)->(finite_finite_nat_o (collect_nat_o (fun (B_47:(nat->Prop))=> ((ord_less_eq_nat_o B_47) A_92)))))) of role axiom named fact_6_finite__Collect__subsets % A new axiom: (forall (A_92:(nat->Prop)), ((finite_finite_nat A_92)->(finite_finite_nat_o (collect_nat_o (fun (B_47:(nat->Prop))=> ((ord_less_eq_nat_o B_47) A_92)))))) % FOF formula (forall (H:(pname->(nat->Prop))) (F_25:(pname->Prop)), ((finite_finite_pname F_25)->(finite_finite_nat_o ((image_pname_nat_o H) F_25)))) of role axiom named fact_7_finite__imageI % A new axiom: (forall (H:(pname->(nat->Prop))) (F_25:(pname->Prop)), ((finite_finite_pname F_25)->(finite_finite_nat_o ((image_pname_nat_o H) F_25)))) % FOF formula (forall (H:(pname->(pname->Prop))) (F_25:(pname->Prop)), ((finite_finite_pname F_25)->(finite297249702name_o ((image_pname_pname_o H) F_25)))) of role axiom named fact_8_finite__imageI % A new axiom: (forall (H:(pname->(pname->Prop))) (F_25:(pname->Prop)), ((finite_finite_pname F_25)->(finite297249702name_o ((image_pname_pname_o H) F_25)))) % FOF formula (forall (H:(pname->(x_a->Prop))) (F_25:(pname->Prop)), ((finite_finite_pname F_25)->(finite_finite_a_o ((image_pname_a_o H) F_25)))) of role axiom named fact_9_finite__imageI % A new axiom: (forall (H:(pname->(x_a->Prop))) (F_25:(pname->Prop)), ((finite_finite_pname F_25)->(finite_finite_a_o ((image_pname_a_o H) F_25)))) % FOF formula (forall (H:(nat->x_a)) (F_25:(nat->Prop)), ((finite_finite_nat F_25)->(finite_finite_a ((image_nat_a H) F_25)))) of role axiom named fact_10_finite__imageI % A new axiom: (forall (H:(nat->x_a)) (F_25:(nat->Prop)), ((finite_finite_nat F_25)->(finite_finite_a ((image_nat_a H) F_25)))) % FOF formula (forall (H:(nat->(nat->Prop))) (F_25:(nat->Prop)), ((finite_finite_nat F_25)->(finite_finite_nat_o ((image_nat_nat_o H) F_25)))) of role axiom named fact_11_finite__imageI % A new axiom: (forall (H:(nat->(nat->Prop))) (F_25:(nat->Prop)), ((finite_finite_nat F_25)->(finite_finite_nat_o ((image_nat_nat_o H) F_25)))) % FOF formula (forall (H:(nat->(pname->Prop))) (F_25:(nat->Prop)), ((finite_finite_nat F_25)->(finite297249702name_o ((image_nat_pname_o H) F_25)))) of role axiom named fact_12_finite__imageI % A new axiom: (forall (H:(nat->(pname->Prop))) (F_25:(nat->Prop)), ((finite_finite_nat F_25)->(finite297249702name_o ((image_nat_pname_o H) F_25)))) % FOF formula (forall (H:(nat->(x_a->Prop))) (F_25:(nat->Prop)), ((finite_finite_nat F_25)->(finite_finite_a_o ((image_nat_a_o H) F_25)))) of role axiom named fact_13_finite__imageI % A new axiom: (forall (H:(nat->(x_a->Prop))) (F_25:(nat->Prop)), ((finite_finite_nat F_25)->(finite_finite_a_o ((image_nat_a_o H) F_25)))) % FOF formula (forall (H:(x_a->pname)) (F_25:(x_a->Prop)), ((finite_finite_a F_25)->(finite_finite_pname ((image_a_pname H) F_25)))) of role axiom named fact_14_finite__imageI % A new axiom: (forall (H:(x_a->pname)) (F_25:(x_a->Prop)), ((finite_finite_a F_25)->(finite_finite_pname ((image_a_pname H) F_25)))) % FOF formula (forall (H:((nat->Prop)->pname)) (F_25:((nat->Prop)->Prop)), ((finite_finite_nat_o F_25)->(finite_finite_pname ((image_nat_o_pname H) F_25)))) of role axiom named fact_15_finite__imageI % A new axiom: (forall (H:((nat->Prop)->pname)) (F_25:((nat->Prop)->Prop)), ((finite_finite_nat_o F_25)->(finite_finite_pname ((image_nat_o_pname H) F_25)))) % FOF formula (forall (H:((pname->Prop)->pname)) (F_25:((pname->Prop)->Prop)), ((finite297249702name_o F_25)->(finite_finite_pname ((image_pname_o_pname H) F_25)))) of role axiom named fact_16_finite__imageI % A new axiom: (forall (H:((pname->Prop)->pname)) (F_25:((pname->Prop)->Prop)), ((finite297249702name_o F_25)->(finite_finite_pname ((image_pname_o_pname H) F_25)))) % FOF formula (forall (H:((x_a->Prop)->pname)) (F_25:((x_a->Prop)->Prop)), ((finite_finite_a_o F_25)->(finite_finite_pname ((image_a_o_pname H) F_25)))) of role axiom named fact_17_finite__imageI % A new axiom: (forall (H:((x_a->Prop)->pname)) (F_25:((x_a->Prop)->Prop)), ((finite_finite_a_o F_25)->(finite_finite_pname ((image_a_o_pname H) F_25)))) % FOF formula (forall (H:(x_a->nat)) (F_25:(x_a->Prop)), ((finite_finite_a F_25)->(finite_finite_nat ((image_a_nat H) F_25)))) of role axiom named fact_18_finite__imageI % A new axiom: (forall (H:(x_a->nat)) (F_25:(x_a->Prop)), ((finite_finite_a F_25)->(finite_finite_nat ((image_a_nat H) F_25)))) % FOF formula (forall (H:((nat->Prop)->nat)) (F_25:((nat->Prop)->Prop)), ((finite_finite_nat_o F_25)->(finite_finite_nat ((image_nat_o_nat H) F_25)))) of role axiom named fact_19_finite__imageI % A new axiom: (forall (H:((nat->Prop)->nat)) (F_25:((nat->Prop)->Prop)), ((finite_finite_nat_o F_25)->(finite_finite_nat ((image_nat_o_nat H) F_25)))) % FOF formula (forall (H:((pname->Prop)->nat)) (F_25:((pname->Prop)->Prop)), ((finite297249702name_o F_25)->(finite_finite_nat ((image_pname_o_nat H) F_25)))) of role axiom named fact_20_finite__imageI % A new axiom: (forall (H:((pname->Prop)->nat)) (F_25:((pname->Prop)->Prop)), ((finite297249702name_o F_25)->(finite_finite_nat ((image_pname_o_nat H) F_25)))) % FOF formula (forall (H:((x_a->Prop)->nat)) (F_25:((x_a->Prop)->Prop)), ((finite_finite_a_o F_25)->(finite_finite_nat ((image_a_o_nat H) F_25)))) of role axiom named fact_21_finite__imageI % A new axiom: (forall (H:((x_a->Prop)->nat)) (F_25:((x_a->Prop)->Prop)), ((finite_finite_a_o F_25)->(finite_finite_nat ((image_a_o_nat H) F_25)))) % FOF formula (forall (H:(pname->x_a)) (F_25:(pname->Prop)), ((finite_finite_pname F_25)->(finite_finite_a ((image_pname_a H) F_25)))) of role axiom named fact_22_finite__imageI % A new axiom: (forall (H:(pname->x_a)) (F_25:(pname->Prop)), ((finite_finite_pname F_25)->(finite_finite_a ((image_pname_a H) F_25)))) % FOF formula (forall (A_91:(nat->Prop)) (A_90:((nat->Prop)->Prop)), ((finite_finite_nat_o A_90)->(finite_finite_nat_o ((insert_nat_o A_91) A_90)))) of role axiom named fact_23_finite_OinsertI % A new axiom: (forall (A_91:(nat->Prop)) (A_90:((nat->Prop)->Prop)), ((finite_finite_nat_o A_90)->(finite_finite_nat_o ((insert_nat_o A_91) A_90)))) % FOF formula (forall (A_91:(pname->Prop)) (A_90:((pname->Prop)->Prop)), ((finite297249702name_o A_90)->(finite297249702name_o ((insert_pname_o A_91) A_90)))) of role axiom named fact_24_finite_OinsertI % A new axiom: (forall (A_91:(pname->Prop)) (A_90:((pname->Prop)->Prop)), ((finite297249702name_o A_90)->(finite297249702name_o ((insert_pname_o A_91) A_90)))) % FOF formula (forall (A_91:(x_a->Prop)) (A_90:((x_a->Prop)->Prop)), ((finite_finite_a_o A_90)->(finite_finite_a_o ((insert_a_o A_91) A_90)))) of role axiom named fact_25_finite_OinsertI % A new axiom: (forall (A_91:(x_a->Prop)) (A_90:((x_a->Prop)->Prop)), ((finite_finite_a_o A_90)->(finite_finite_a_o ((insert_a_o A_91) A_90)))) % FOF formula (forall (A_91:pname) (A_90:(pname->Prop)), ((finite_finite_pname A_90)->(finite_finite_pname ((insert_pname A_91) A_90)))) of role axiom named fact_26_finite_OinsertI % A new axiom: (forall (A_91:pname) (A_90:(pname->Prop)), ((finite_finite_pname A_90)->(finite_finite_pname ((insert_pname A_91) A_90)))) % FOF formula (forall (A_91:nat) (A_90:(nat->Prop)), ((finite_finite_nat A_90)->(finite_finite_nat ((insert_nat A_91) A_90)))) of role axiom named fact_27_finite_OinsertI % A new axiom: (forall (A_91:nat) (A_90:(nat->Prop)), ((finite_finite_nat A_90)->(finite_finite_nat ((insert_nat A_91) A_90)))) % FOF formula (forall (A_91:x_a) (A_90:(x_a->Prop)), ((finite_finite_a A_90)->(finite_finite_a ((insert_a A_91) A_90)))) of role axiom named fact_28_finite_OinsertI % A new axiom: (forall (A_91:x_a) (A_90:(x_a->Prop)), ((finite_finite_a A_90)->(finite_finite_a ((insert_a A_91) A_90)))) % FOF formula (forall (F_24:(pname->pname)) (A_89:(pname->Prop)), ((finite_finite_pname A_89)->((ord_less_eq_nat (finite_card_pname ((image_pname_pname F_24) A_89))) (finite_card_pname A_89)))) of role axiom named fact_29_card__image__le % A new axiom: (forall (F_24:(pname->pname)) (A_89:(pname->Prop)), ((finite_finite_pname A_89)->((ord_less_eq_nat (finite_card_pname ((image_pname_pname F_24) A_89))) (finite_card_pname A_89)))) % FOF formula (forall (F_24:(x_a->x_a)) (A_89:(x_a->Prop)), ((finite_finite_a A_89)->((ord_less_eq_nat (finite_card_a ((image_a_a F_24) A_89))) (finite_card_a A_89)))) of role axiom named fact_30_card__image__le % A new axiom: (forall (F_24:(x_a->x_a)) (A_89:(x_a->Prop)), ((finite_finite_a A_89)->((ord_less_eq_nat (finite_card_a ((image_a_a F_24) A_89))) (finite_card_a A_89)))) % FOF formula (forall (F_24:((nat->Prop)->x_a)) (A_89:((nat->Prop)->Prop)), ((finite_finite_nat_o A_89)->((ord_less_eq_nat (finite_card_a ((image_nat_o_a F_24) A_89))) (finite_card_nat_o A_89)))) of role axiom named fact_31_card__image__le % A new axiom: (forall (F_24:((nat->Prop)->x_a)) (A_89:((nat->Prop)->Prop)), ((finite_finite_nat_o A_89)->((ord_less_eq_nat (finite_card_a ((image_nat_o_a F_24) A_89))) (finite_card_nat_o A_89)))) % FOF formula (forall (F_24:((pname->Prop)->x_a)) (A_89:((pname->Prop)->Prop)), ((finite297249702name_o A_89)->((ord_less_eq_nat (finite_card_a ((image_pname_o_a F_24) A_89))) (finite_card_pname_o A_89)))) of role axiom named fact_32_card__image__le % A new axiom: (forall (F_24:((pname->Prop)->x_a)) (A_89:((pname->Prop)->Prop)), ((finite297249702name_o A_89)->((ord_less_eq_nat (finite_card_a ((image_pname_o_a F_24) A_89))) (finite_card_pname_o A_89)))) % FOF formula (forall (F_24:((x_a->Prop)->x_a)) (A_89:((x_a->Prop)->Prop)), ((finite_finite_a_o A_89)->((ord_less_eq_nat (finite_card_a ((image_a_o_a F_24) A_89))) (finite_card_a_o A_89)))) of role axiom named fact_33_card__image__le % A new axiom: (forall (F_24:((x_a->Prop)->x_a)) (A_89:((x_a->Prop)->Prop)), ((finite_finite_a_o A_89)->((ord_less_eq_nat (finite_card_a ((image_a_o_a F_24) A_89))) (finite_card_a_o A_89)))) % FOF formula (forall (F_24:(pname->nat)) (A_89:(pname->Prop)), ((finite_finite_pname A_89)->((ord_less_eq_nat (finite_card_nat ((image_pname_nat F_24) A_89))) (finite_card_pname A_89)))) of role axiom named fact_34_card__image__le % A new axiom: (forall (F_24:(pname->nat)) (A_89:(pname->Prop)), ((finite_finite_pname A_89)->((ord_less_eq_nat (finite_card_nat ((image_pname_nat F_24) A_89))) (finite_card_pname A_89)))) % FOF formula (forall (F_24:(x_a->nat)) (A_89:(x_a->Prop)), ((finite_finite_a A_89)->((ord_less_eq_nat (finite_card_nat ((image_a_nat F_24) A_89))) (finite_card_a A_89)))) of role axiom named fact_35_card__image__le % A new axiom: (forall (F_24:(x_a->nat)) (A_89:(x_a->Prop)), ((finite_finite_a A_89)->((ord_less_eq_nat (finite_card_nat ((image_a_nat F_24) A_89))) (finite_card_a A_89)))) % FOF formula (forall (F_24:((nat->Prop)->nat)) (A_89:((nat->Prop)->Prop)), ((finite_finite_nat_o A_89)->((ord_less_eq_nat (finite_card_nat ((image_nat_o_nat F_24) A_89))) (finite_card_nat_o A_89)))) of role axiom named fact_36_card__image__le % A new axiom: (forall (F_24:((nat->Prop)->nat)) (A_89:((nat->Prop)->Prop)), ((finite_finite_nat_o A_89)->((ord_less_eq_nat (finite_card_nat ((image_nat_o_nat F_24) A_89))) (finite_card_nat_o A_89)))) % FOF formula (forall (F_24:((pname->Prop)->nat)) (A_89:((pname->Prop)->Prop)), ((finite297249702name_o A_89)->((ord_less_eq_nat (finite_card_nat ((image_pname_o_nat F_24) A_89))) (finite_card_pname_o A_89)))) of role axiom named fact_37_card__image__le % A new axiom: (forall (F_24:((pname->Prop)->nat)) (A_89:((pname->Prop)->Prop)), ((finite297249702name_o A_89)->((ord_less_eq_nat (finite_card_nat ((image_pname_o_nat F_24) A_89))) (finite_card_pname_o A_89)))) % FOF formula (forall (F_24:((x_a->Prop)->nat)) (A_89:((x_a->Prop)->Prop)), ((finite_finite_a_o A_89)->((ord_less_eq_nat (finite_card_nat ((image_a_o_nat F_24) A_89))) (finite_card_a_o A_89)))) of role axiom named fact_38_card__image__le % A new axiom: (forall (F_24:((x_a->Prop)->nat)) (A_89:((x_a->Prop)->Prop)), ((finite_finite_a_o A_89)->((ord_less_eq_nat (finite_card_nat ((image_a_o_nat F_24) A_89))) (finite_card_a_o A_89)))) % FOF formula (forall (F_24:(x_a->pname)) (A_89:(x_a->Prop)), ((finite_finite_a A_89)->((ord_less_eq_nat (finite_card_pname ((image_a_pname F_24) A_89))) (finite_card_a A_89)))) of role axiom named fact_39_card__image__le % A new axiom: (forall (F_24:(x_a->pname)) (A_89:(x_a->Prop)), ((finite_finite_a A_89)->((ord_less_eq_nat (finite_card_pname ((image_a_pname F_24) A_89))) (finite_card_a A_89)))) % FOF formula (forall (F_24:(nat->pname)) (A_89:(nat->Prop)), ((finite_finite_nat A_89)->((ord_less_eq_nat (finite_card_pname ((image_nat_pname F_24) A_89))) (finite_card_nat A_89)))) of role axiom named fact_40_card__image__le % A new axiom: (forall (F_24:(nat->pname)) (A_89:(nat->Prop)), ((finite_finite_nat A_89)->((ord_less_eq_nat (finite_card_pname ((image_nat_pname F_24) A_89))) (finite_card_nat A_89)))) % FOF formula (forall (F_24:(pname->x_a)) (A_89:(pname->Prop)), ((finite_finite_pname A_89)->((ord_less_eq_nat (finite_card_a ((image_pname_a F_24) A_89))) (finite_card_pname A_89)))) of role axiom named fact_41_card__image__le % A new axiom: (forall (F_24:(pname->x_a)) (A_89:(pname->Prop)), ((finite_finite_pname A_89)->((ord_less_eq_nat (finite_card_a ((image_pname_a F_24) A_89))) (finite_card_pname A_89)))) % FOF formula (forall (A_88:((nat->Prop)->Prop)) (B_46:((nat->Prop)->Prop)), ((finite_finite_nat_o B_46)->(((ord_less_eq_nat_o_o A_88) B_46)->((ord_less_eq_nat (finite_card_nat_o A_88)) (finite_card_nat_o B_46))))) of role axiom named fact_42_card__mono % A new axiom: (forall (A_88:((nat->Prop)->Prop)) (B_46:((nat->Prop)->Prop)), ((finite_finite_nat_o B_46)->(((ord_less_eq_nat_o_o A_88) B_46)->((ord_less_eq_nat (finite_card_nat_o A_88)) (finite_card_nat_o B_46))))) % FOF formula (forall (A_88:((pname->Prop)->Prop)) (B_46:((pname->Prop)->Prop)), ((finite297249702name_o B_46)->(((ord_le1205211808me_o_o A_88) B_46)->((ord_less_eq_nat (finite_card_pname_o A_88)) (finite_card_pname_o B_46))))) of role axiom named fact_43_card__mono % A new axiom: (forall (A_88:((pname->Prop)->Prop)) (B_46:((pname->Prop)->Prop)), ((finite297249702name_o B_46)->(((ord_le1205211808me_o_o A_88) B_46)->((ord_less_eq_nat (finite_card_pname_o A_88)) (finite_card_pname_o B_46))))) % FOF formula (forall (A_88:((x_a->Prop)->Prop)) (B_46:((x_a->Prop)->Prop)), ((finite_finite_a_o B_46)->(((ord_less_eq_a_o_o A_88) B_46)->((ord_less_eq_nat (finite_card_a_o A_88)) (finite_card_a_o B_46))))) of role axiom named fact_44_card__mono % A new axiom: (forall (A_88:((x_a->Prop)->Prop)) (B_46:((x_a->Prop)->Prop)), ((finite_finite_a_o B_46)->(((ord_less_eq_a_o_o A_88) B_46)->((ord_less_eq_nat (finite_card_a_o A_88)) (finite_card_a_o B_46))))) % FOF formula (forall (A_88:(pname->Prop)) (B_46:(pname->Prop)), ((finite_finite_pname B_46)->(((ord_less_eq_pname_o A_88) B_46)->((ord_less_eq_nat (finite_card_pname A_88)) (finite_card_pname B_46))))) of role axiom named fact_45_card__mono % A new axiom: (forall (A_88:(pname->Prop)) (B_46:(pname->Prop)), ((finite_finite_pname B_46)->(((ord_less_eq_pname_o A_88) B_46)->((ord_less_eq_nat (finite_card_pname A_88)) (finite_card_pname B_46))))) % FOF formula (forall (A_88:(x_a->Prop)) (B_46:(x_a->Prop)), ((finite_finite_a B_46)->(((ord_less_eq_a_o A_88) B_46)->((ord_less_eq_nat (finite_card_a A_88)) (finite_card_a B_46))))) of role axiom named fact_46_card__mono % A new axiom: (forall (A_88:(x_a->Prop)) (B_46:(x_a->Prop)), ((finite_finite_a B_46)->(((ord_less_eq_a_o A_88) B_46)->((ord_less_eq_nat (finite_card_a A_88)) (finite_card_a B_46))))) % FOF formula (forall (A_88:(nat->Prop)) (B_46:(nat->Prop)), ((finite_finite_nat B_46)->(((ord_less_eq_nat_o A_88) B_46)->((ord_less_eq_nat (finite_card_nat A_88)) (finite_card_nat B_46))))) of role axiom named fact_47_card__mono % A new axiom: (forall (A_88:(nat->Prop)) (B_46:(nat->Prop)), ((finite_finite_nat B_46)->(((ord_less_eq_nat_o A_88) B_46)->((ord_less_eq_nat (finite_card_nat A_88)) (finite_card_nat B_46))))) % FOF formula (forall (A_87:((nat->Prop)->Prop)) (B_45:((nat->Prop)->Prop)), ((finite_finite_nat_o B_45)->(((ord_less_eq_nat_o_o A_87) B_45)->(((ord_less_eq_nat (finite_card_nat_o B_45)) (finite_card_nat_o A_87))->(((eq ((nat->Prop)->Prop)) A_87) B_45))))) of role axiom named fact_48_card__seteq % A new axiom: (forall (A_87:((nat->Prop)->Prop)) (B_45:((nat->Prop)->Prop)), ((finite_finite_nat_o B_45)->(((ord_less_eq_nat_o_o A_87) B_45)->(((ord_less_eq_nat (finite_card_nat_o B_45)) (finite_card_nat_o A_87))->(((eq ((nat->Prop)->Prop)) A_87) B_45))))) % FOF formula (forall (A_87:((pname->Prop)->Prop)) (B_45:((pname->Prop)->Prop)), ((finite297249702name_o B_45)->(((ord_le1205211808me_o_o A_87) B_45)->(((ord_less_eq_nat (finite_card_pname_o B_45)) (finite_card_pname_o A_87))->(((eq ((pname->Prop)->Prop)) A_87) B_45))))) of role axiom named fact_49_card__seteq % A new axiom: (forall (A_87:((pname->Prop)->Prop)) (B_45:((pname->Prop)->Prop)), ((finite297249702name_o B_45)->(((ord_le1205211808me_o_o A_87) B_45)->(((ord_less_eq_nat (finite_card_pname_o B_45)) (finite_card_pname_o A_87))->(((eq ((pname->Prop)->Prop)) A_87) B_45))))) % FOF formula (forall (A_87:((x_a->Prop)->Prop)) (B_45:((x_a->Prop)->Prop)), ((finite_finite_a_o B_45)->(((ord_less_eq_a_o_o A_87) B_45)->(((ord_less_eq_nat (finite_card_a_o B_45)) (finite_card_a_o A_87))->(((eq ((x_a->Prop)->Prop)) A_87) B_45))))) of role axiom named fact_50_card__seteq % A new axiom: (forall (A_87:((x_a->Prop)->Prop)) (B_45:((x_a->Prop)->Prop)), ((finite_finite_a_o B_45)->(((ord_less_eq_a_o_o A_87) B_45)->(((ord_less_eq_nat (finite_card_a_o B_45)) (finite_card_a_o A_87))->(((eq ((x_a->Prop)->Prop)) A_87) B_45))))) % FOF formula (forall (A_87:(pname->Prop)) (B_45:(pname->Prop)), ((finite_finite_pname B_45)->(((ord_less_eq_pname_o A_87) B_45)->(((ord_less_eq_nat (finite_card_pname B_45)) (finite_card_pname A_87))->(((eq (pname->Prop)) A_87) B_45))))) of role axiom named fact_51_card__seteq % A new axiom: (forall (A_87:(pname->Prop)) (B_45:(pname->Prop)), ((finite_finite_pname B_45)->(((ord_less_eq_pname_o A_87) B_45)->(((ord_less_eq_nat (finite_card_pname B_45)) (finite_card_pname A_87))->(((eq (pname->Prop)) A_87) B_45))))) % FOF formula (forall (A_87:(x_a->Prop)) (B_45:(x_a->Prop)), ((finite_finite_a B_45)->(((ord_less_eq_a_o A_87) B_45)->(((ord_less_eq_nat (finite_card_a B_45)) (finite_card_a A_87))->(((eq (x_a->Prop)) A_87) B_45))))) of role axiom named fact_52_card__seteq % A new axiom: (forall (A_87:(x_a->Prop)) (B_45:(x_a->Prop)), ((finite_finite_a B_45)->(((ord_less_eq_a_o A_87) B_45)->(((ord_less_eq_nat (finite_card_a B_45)) (finite_card_a A_87))->(((eq (x_a->Prop)) A_87) B_45))))) % FOF formula (forall (A_87:(nat->Prop)) (B_45:(nat->Prop)), ((finite_finite_nat B_45)->(((ord_less_eq_nat_o A_87) B_45)->(((ord_less_eq_nat (finite_card_nat B_45)) (finite_card_nat A_87))->(((eq (nat->Prop)) A_87) B_45))))) of role axiom named fact_53_card__seteq % A new axiom: (forall (A_87:(nat->Prop)) (B_45:(nat->Prop)), ((finite_finite_nat B_45)->(((ord_less_eq_nat_o A_87) B_45)->(((ord_less_eq_nat (finite_card_nat B_45)) (finite_card_nat A_87))->(((eq (nat->Prop)) A_87) B_45))))) % FOF formula (forall (X_33:(nat->Prop)) (A_86:((nat->Prop)->Prop)), ((finite_finite_nat_o A_86)->((ord_less_eq_nat (finite_card_nat_o A_86)) (finite_card_nat_o ((insert_nat_o X_33) A_86))))) of role axiom named fact_54_card__insert__le % A new axiom: (forall (X_33:(nat->Prop)) (A_86:((nat->Prop)->Prop)), ((finite_finite_nat_o A_86)->((ord_less_eq_nat (finite_card_nat_o A_86)) (finite_card_nat_o ((insert_nat_o X_33) A_86))))) % FOF formula (forall (X_33:(pname->Prop)) (A_86:((pname->Prop)->Prop)), ((finite297249702name_o A_86)->((ord_less_eq_nat (finite_card_pname_o A_86)) (finite_card_pname_o ((insert_pname_o X_33) A_86))))) of role axiom named fact_55_card__insert__le % A new axiom: (forall (X_33:(pname->Prop)) (A_86:((pname->Prop)->Prop)), ((finite297249702name_o A_86)->((ord_less_eq_nat (finite_card_pname_o A_86)) (finite_card_pname_o ((insert_pname_o X_33) A_86))))) % FOF formula (forall (X_33:(x_a->Prop)) (A_86:((x_a->Prop)->Prop)), ((finite_finite_a_o A_86)->((ord_less_eq_nat (finite_card_a_o A_86)) (finite_card_a_o ((insert_a_o X_33) A_86))))) of role axiom named fact_56_card__insert__le % A new axiom: (forall (X_33:(x_a->Prop)) (A_86:((x_a->Prop)->Prop)), ((finite_finite_a_o A_86)->((ord_less_eq_nat (finite_card_a_o A_86)) (finite_card_a_o ((insert_a_o X_33) A_86))))) % FOF formula (forall (X_33:pname) (A_86:(pname->Prop)), ((finite_finite_pname A_86)->((ord_less_eq_nat (finite_card_pname A_86)) (finite_card_pname ((insert_pname X_33) A_86))))) of role axiom named fact_57_card__insert__le % A new axiom: (forall (X_33:pname) (A_86:(pname->Prop)), ((finite_finite_pname A_86)->((ord_less_eq_nat (finite_card_pname A_86)) (finite_card_pname ((insert_pname X_33) A_86))))) % FOF formula (forall (X_33:nat) (A_86:(nat->Prop)), ((finite_finite_nat A_86)->((ord_less_eq_nat (finite_card_nat A_86)) (finite_card_nat ((insert_nat X_33) A_86))))) of role axiom named fact_58_card__insert__le % A new axiom: (forall (X_33:nat) (A_86:(nat->Prop)), ((finite_finite_nat A_86)->((ord_less_eq_nat (finite_card_nat A_86)) (finite_card_nat ((insert_nat X_33) A_86))))) % FOF formula (forall (X_33:x_a) (A_86:(x_a->Prop)), ((finite_finite_a A_86)->((ord_less_eq_nat (finite_card_a A_86)) (finite_card_a ((insert_a X_33) A_86))))) of role axiom named fact_59_card__insert__le % A new axiom: (forall (X_33:x_a) (A_86:(x_a->Prop)), ((finite_finite_a A_86)->((ord_less_eq_nat (finite_card_a A_86)) (finite_card_a ((insert_a X_33) A_86))))) % FOF formula (forall (X_32:(nat->Prop)) (A_85:((nat->Prop)->Prop)), ((finite_finite_nat_o A_85)->((and (((member_nat_o X_32) A_85)->(((eq nat) (finite_card_nat_o ((insert_nat_o X_32) A_85))) (finite_card_nat_o A_85)))) ((((member_nat_o X_32) A_85)->False)->(((eq nat) (finite_card_nat_o ((insert_nat_o X_32) A_85))) (suc (finite_card_nat_o A_85))))))) of role axiom named fact_60_card__insert__if % A new axiom: (forall (X_32:(nat->Prop)) (A_85:((nat->Prop)->Prop)), ((finite_finite_nat_o A_85)->((and (((member_nat_o X_32) A_85)->(((eq nat) (finite_card_nat_o ((insert_nat_o X_32) A_85))) (finite_card_nat_o A_85)))) ((((member_nat_o X_32) A_85)->False)->(((eq nat) (finite_card_nat_o ((insert_nat_o X_32) A_85))) (suc (finite_card_nat_o A_85))))))) % FOF formula (forall (X_32:(pname->Prop)) (A_85:((pname->Prop)->Prop)), ((finite297249702name_o A_85)->((and (((member_pname_o X_32) A_85)->(((eq nat) (finite_card_pname_o ((insert_pname_o X_32) A_85))) (finite_card_pname_o A_85)))) ((((member_pname_o X_32) A_85)->False)->(((eq nat) (finite_card_pname_o ((insert_pname_o X_32) A_85))) (suc (finite_card_pname_o A_85))))))) of role axiom named fact_61_card__insert__if % A new axiom: (forall (X_32:(pname->Prop)) (A_85:((pname->Prop)->Prop)), ((finite297249702name_o A_85)->((and (((member_pname_o X_32) A_85)->(((eq nat) (finite_card_pname_o ((insert_pname_o X_32) A_85))) (finite_card_pname_o A_85)))) ((((member_pname_o X_32) A_85)->False)->(((eq nat) (finite_card_pname_o ((insert_pname_o X_32) A_85))) (suc (finite_card_pname_o A_85))))))) % FOF formula (forall (X_32:(x_a->Prop)) (A_85:((x_a->Prop)->Prop)), ((finite_finite_a_o A_85)->((and (((member_a_o X_32) A_85)->(((eq nat) (finite_card_a_o ((insert_a_o X_32) A_85))) (finite_card_a_o A_85)))) ((((member_a_o X_32) A_85)->False)->(((eq nat) (finite_card_a_o ((insert_a_o X_32) A_85))) (suc (finite_card_a_o A_85))))))) of role axiom named fact_62_card__insert__if % A new axiom: (forall (X_32:(x_a->Prop)) (A_85:((x_a->Prop)->Prop)), ((finite_finite_a_o A_85)->((and (((member_a_o X_32) A_85)->(((eq nat) (finite_card_a_o ((insert_a_o X_32) A_85))) (finite_card_a_o A_85)))) ((((member_a_o X_32) A_85)->False)->(((eq nat) (finite_card_a_o ((insert_a_o X_32) A_85))) (suc (finite_card_a_o A_85))))))) % FOF formula (forall (X_32:nat) (A_85:(nat->Prop)), ((finite_finite_nat A_85)->((and (((member_nat X_32) A_85)->(((eq nat) (finite_card_nat ((insert_nat X_32) A_85))) (finite_card_nat A_85)))) ((((member_nat X_32) A_85)->False)->(((eq nat) (finite_card_nat ((insert_nat X_32) A_85))) (suc (finite_card_nat A_85))))))) of role axiom named fact_63_card__insert__if % A new axiom: (forall (X_32:nat) (A_85:(nat->Prop)), ((finite_finite_nat A_85)->((and (((member_nat X_32) A_85)->(((eq nat) (finite_card_nat ((insert_nat X_32) A_85))) (finite_card_nat A_85)))) ((((member_nat X_32) A_85)->False)->(((eq nat) (finite_card_nat ((insert_nat X_32) A_85))) (suc (finite_card_nat A_85))))))) % FOF formula (forall (X_32:pname) (A_85:(pname->Prop)), ((finite_finite_pname A_85)->((and (((member_pname X_32) A_85)->(((eq nat) (finite_card_pname ((insert_pname X_32) A_85))) (finite_card_pname A_85)))) ((((member_pname X_32) A_85)->False)->(((eq nat) (finite_card_pname ((insert_pname X_32) A_85))) (suc (finite_card_pname A_85))))))) of role axiom named fact_64_card__insert__if % A new axiom: (forall (X_32:pname) (A_85:(pname->Prop)), ((finite_finite_pname A_85)->((and (((member_pname X_32) A_85)->(((eq nat) (finite_card_pname ((insert_pname X_32) A_85))) (finite_card_pname A_85)))) ((((member_pname X_32) A_85)->False)->(((eq nat) (finite_card_pname ((insert_pname X_32) A_85))) (suc (finite_card_pname A_85))))))) % FOF formula (forall (X_32:x_a) (A_85:(x_a->Prop)), ((finite_finite_a A_85)->((and (((member_a X_32) A_85)->(((eq nat) (finite_card_a ((insert_a X_32) A_85))) (finite_card_a A_85)))) ((((member_a X_32) A_85)->False)->(((eq nat) (finite_card_a ((insert_a X_32) A_85))) (suc (finite_card_a A_85))))))) of role axiom named fact_65_card__insert__if % A new axiom: (forall (X_32:x_a) (A_85:(x_a->Prop)), ((finite_finite_a A_85)->((and (((member_a X_32) A_85)->(((eq nat) (finite_card_a ((insert_a X_32) A_85))) (finite_card_a A_85)))) ((((member_a X_32) A_85)->False)->(((eq nat) (finite_card_a ((insert_a X_32) A_85))) (suc (finite_card_a A_85))))))) % FOF formula (forall (X_31:(nat->Prop)) (A_84:((nat->Prop)->Prop)), ((finite_finite_nat_o A_84)->((((member_nat_o X_31) A_84)->False)->(((eq nat) (finite_card_nat_o ((insert_nat_o X_31) A_84))) (suc (finite_card_nat_o A_84)))))) of role axiom named fact_66_card__insert__disjoint % A new axiom: (forall (X_31:(nat->Prop)) (A_84:((nat->Prop)->Prop)), ((finite_finite_nat_o A_84)->((((member_nat_o X_31) A_84)->False)->(((eq nat) (finite_card_nat_o ((insert_nat_o X_31) A_84))) (suc (finite_card_nat_o A_84)))))) % FOF formula (forall (X_31:(pname->Prop)) (A_84:((pname->Prop)->Prop)), ((finite297249702name_o A_84)->((((member_pname_o X_31) A_84)->False)->(((eq nat) (finite_card_pname_o ((insert_pname_o X_31) A_84))) (suc (finite_card_pname_o A_84)))))) of role axiom named fact_67_card__insert__disjoint % A new axiom: (forall (X_31:(pname->Prop)) (A_84:((pname->Prop)->Prop)), ((finite297249702name_o A_84)->((((member_pname_o X_31) A_84)->False)->(((eq nat) (finite_card_pname_o ((insert_pname_o X_31) A_84))) (suc (finite_card_pname_o A_84)))))) % FOF formula (forall (X_31:(x_a->Prop)) (A_84:((x_a->Prop)->Prop)), ((finite_finite_a_o A_84)->((((member_a_o X_31) A_84)->False)->(((eq nat) (finite_card_a_o ((insert_a_o X_31) A_84))) (suc (finite_card_a_o A_84)))))) of role axiom named fact_68_card__insert__disjoint % A new axiom: (forall (X_31:(x_a->Prop)) (A_84:((x_a->Prop)->Prop)), ((finite_finite_a_o A_84)->((((member_a_o X_31) A_84)->False)->(((eq nat) (finite_card_a_o ((insert_a_o X_31) A_84))) (suc (finite_card_a_o A_84)))))) % FOF formula (forall (X_31:nat) (A_84:(nat->Prop)), ((finite_finite_nat A_84)->((((member_nat X_31) A_84)->False)->(((eq nat) (finite_card_nat ((insert_nat X_31) A_84))) (suc (finite_card_nat A_84)))))) of role axiom named fact_69_card__insert__disjoint % A new axiom: (forall (X_31:nat) (A_84:(nat->Prop)), ((finite_finite_nat A_84)->((((member_nat X_31) A_84)->False)->(((eq nat) (finite_card_nat ((insert_nat X_31) A_84))) (suc (finite_card_nat A_84)))))) % FOF formula (forall (X_31:pname) (A_84:(pname->Prop)), ((finite_finite_pname A_84)->((((member_pname X_31) A_84)->False)->(((eq nat) (finite_card_pname ((insert_pname X_31) A_84))) (suc (finite_card_pname A_84)))))) of role axiom named fact_70_card__insert__disjoint % A new axiom: (forall (X_31:pname) (A_84:(pname->Prop)), ((finite_finite_pname A_84)->((((member_pname X_31) A_84)->False)->(((eq nat) (finite_card_pname ((insert_pname X_31) A_84))) (suc (finite_card_pname A_84)))))) % FOF formula (forall (X_31:x_a) (A_84:(x_a->Prop)), ((finite_finite_a A_84)->((((member_a X_31) A_84)->False)->(((eq nat) (finite_card_a ((insert_a X_31) A_84))) (suc (finite_card_a A_84)))))) of role axiom named fact_71_card__insert__disjoint % A new axiom: (forall (X_31:x_a) (A_84:(x_a->Prop)), ((finite_finite_a A_84)->((((member_a X_31) A_84)->False)->(((eq nat) (finite_card_a ((insert_a X_31) A_84))) (suc (finite_card_a A_84)))))) % FOF formula (forall (Q_1:(x_a->Prop)) (P_10:(x_a->Prop)), (((or (finite_finite_a (collect_a P_10))) (finite_finite_a (collect_a Q_1)))->(finite_finite_a (collect_a (fun (X_1:x_a)=> ((and (P_10 X_1)) (Q_1 X_1))))))) of role axiom named fact_72_finite__Collect__conjI % A new axiom: (forall (Q_1:(x_a->Prop)) (P_10:(x_a->Prop)), (((or (finite_finite_a (collect_a P_10))) (finite_finite_a (collect_a Q_1)))->(finite_finite_a (collect_a (fun (X_1:x_a)=> ((and (P_10 X_1)) (Q_1 X_1))))))) % FOF formula (forall (Q_1:((nat->Prop)->Prop)) (P_10:((nat->Prop)->Prop)), (((or (finite_finite_nat_o (collect_nat_o P_10))) (finite_finite_nat_o (collect_nat_o Q_1)))->(finite_finite_nat_o (collect_nat_o (fun (X_1:(nat->Prop))=> ((and (P_10 X_1)) (Q_1 X_1))))))) of role axiom named fact_73_finite__Collect__conjI % A new axiom: (forall (Q_1:((nat->Prop)->Prop)) (P_10:((nat->Prop)->Prop)), (((or (finite_finite_nat_o (collect_nat_o P_10))) (finite_finite_nat_o (collect_nat_o Q_1)))->(finite_finite_nat_o (collect_nat_o (fun (X_1:(nat->Prop))=> ((and (P_10 X_1)) (Q_1 X_1))))))) % FOF formula (forall (Q_1:((pname->Prop)->Prop)) (P_10:((pname->Prop)->Prop)), (((or (finite297249702name_o (collect_pname_o P_10))) (finite297249702name_o (collect_pname_o Q_1)))->(finite297249702name_o (collect_pname_o (fun (X_1:(pname->Prop))=> ((and (P_10 X_1)) (Q_1 X_1))))))) of role axiom named fact_74_finite__Collect__conjI % A new axiom: (forall (Q_1:((pname->Prop)->Prop)) (P_10:((pname->Prop)->Prop)), (((or (finite297249702name_o (collect_pname_o P_10))) (finite297249702name_o (collect_pname_o Q_1)))->(finite297249702name_o (collect_pname_o (fun (X_1:(pname->Prop))=> ((and (P_10 X_1)) (Q_1 X_1))))))) % FOF formula (forall (Q_1:((x_a->Prop)->Prop)) (P_10:((x_a->Prop)->Prop)), (((or (finite_finite_a_o (collect_a_o P_10))) (finite_finite_a_o (collect_a_o Q_1)))->(finite_finite_a_o (collect_a_o (fun (X_1:(x_a->Prop))=> ((and (P_10 X_1)) (Q_1 X_1))))))) of role axiom named fact_75_finite__Collect__conjI % A new axiom: (forall (Q_1:((x_a->Prop)->Prop)) (P_10:((x_a->Prop)->Prop)), (((or (finite_finite_a_o (collect_a_o P_10))) (finite_finite_a_o (collect_a_o Q_1)))->(finite_finite_a_o (collect_a_o (fun (X_1:(x_a->Prop))=> ((and (P_10 X_1)) (Q_1 X_1))))))) % FOF formula (forall (Q_1:(pname->Prop)) (P_10:(pname->Prop)), (((or (finite_finite_pname (collect_pname P_10))) (finite_finite_pname (collect_pname Q_1)))->(finite_finite_pname (collect_pname (fun (X_1:pname)=> ((and (P_10 X_1)) (Q_1 X_1))))))) of role axiom named fact_76_finite__Collect__conjI % A new axiom: (forall (Q_1:(pname->Prop)) (P_10:(pname->Prop)), (((or (finite_finite_pname (collect_pname P_10))) (finite_finite_pname (collect_pname Q_1)))->(finite_finite_pname (collect_pname (fun (X_1:pname)=> ((and (P_10 X_1)) (Q_1 X_1))))))) % FOF formula (forall (Q_1:(nat->Prop)) (P_10:(nat->Prop)), (((or (finite_finite_nat (collect_nat P_10))) (finite_finite_nat (collect_nat Q_1)))->(finite_finite_nat (collect_nat (fun (X_1:nat)=> ((and (P_10 X_1)) (Q_1 X_1))))))) of role axiom named fact_77_finite__Collect__conjI % A new axiom: (forall (Q_1:(nat->Prop)) (P_10:(nat->Prop)), (((or (finite_finite_nat (collect_nat P_10))) (finite_finite_nat (collect_nat Q_1)))->(finite_finite_nat (collect_nat (fun (X_1:nat)=> ((and (P_10 X_1)) (Q_1 X_1))))))) % FOF formula (forall (N:nat) (M:nat), (((ord_less_eq_nat N) M)->(((eq nat) ((minus_minus_nat (suc M)) N)) (suc ((minus_minus_nat M) N))))) of role axiom named fact_78_Suc__diff__le % A new axiom: (forall (N:nat) (M:nat), (((ord_less_eq_nat N) M)->(((eq nat) ((minus_minus_nat (suc M)) N)) (suc ((minus_minus_nat M) N))))) % FOF formula (forall (K:nat), (finite_finite_nat (collect_nat (fun (N_2:nat)=> ((ord_less_eq_nat N_2) K))))) of role axiom named fact_79_finite__Collect__le__nat % A new axiom: (forall (K:nat), (finite_finite_nat (collect_nat (fun (N_2:nat)=> ((ord_less_eq_nat N_2) K))))) % FOF formula (forall (N:nat), (((eq nat) (finite_card_nat (collect_nat (fun (I_1:nat)=> ((ord_less_eq_nat I_1) N))))) (suc N))) of role axiom named fact_80_card__Collect__le__nat % A new axiom: (forall (N:nat), (((eq nat) (finite_card_nat (collect_nat (fun (I_1:nat)=> ((ord_less_eq_nat I_1) N))))) (suc N))) % FOF formula (forall (X:nat) (Y:nat), ((((eq nat) (suc X)) (suc Y))->(((eq nat) X) Y))) of role axiom named fact_81_Suc__inject % A new axiom: (forall (X:nat) (Y:nat), ((((eq nat) (suc X)) (suc Y))->(((eq nat) X) Y))) % FOF formula (forall (Nat_2:nat) (Nat:nat), ((iff (((eq nat) (suc Nat_2)) (suc Nat))) (((eq nat) Nat_2) Nat))) of role axiom named fact_82_nat_Oinject % A new axiom: (forall (Nat_2:nat) (Nat:nat), ((iff (((eq nat) (suc Nat_2)) (suc Nat))) (((eq nat) Nat_2) Nat))) % FOF formula (forall (N:nat), (not (((eq nat) (suc N)) N))) of role axiom named fact_83_Suc__n__not__n % A new axiom: (forall (N:nat), (not (((eq nat) (suc N)) N))) % FOF formula (forall (N:nat), (not (((eq nat) N) (suc N)))) of role axiom named fact_84_n__not__Suc__n % A new axiom: (forall (N:nat), (not (((eq nat) N) (suc N)))) % FOF formula (forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->(((ord_less_eq_nat N) M)->(((eq nat) M) N)))) of role axiom named fact_85_le__antisym % A new axiom: (forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->(((ord_less_eq_nat N) M)->(((eq nat) M) N)))) % FOF formula (forall (K:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->(((ord_less_eq_nat J_1) K)->((ord_less_eq_nat _TPTP_I) K)))) of role axiom named fact_86_le__trans % A new axiom: (forall (K:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->(((ord_less_eq_nat J_1) K)->((ord_less_eq_nat _TPTP_I) K)))) % FOF formula (forall (M:nat) (N:nat), ((((eq nat) M) N)->((ord_less_eq_nat M) N))) of role axiom named fact_87_eq__imp__le % A new axiom: (forall (M:nat) (N:nat), ((((eq nat) M) N)->((ord_less_eq_nat M) N))) % FOF formula (forall (M:nat) (N:nat), ((or ((ord_less_eq_nat M) N)) ((ord_less_eq_nat N) M))) of role axiom named fact_88_nat__le__linear % A new axiom: (forall (M:nat) (N:nat), ((or ((ord_less_eq_nat M) N)) ((ord_less_eq_nat N) M))) % FOF formula (forall (N:nat), ((ord_less_eq_nat N) N)) of role axiom named fact_89_le__refl % A new axiom: (forall (N:nat), ((ord_less_eq_nat N) N)) % FOF formula (forall (_TPTP_I:nat) (J_1:nat) (K:nat), (((eq nat) ((minus_minus_nat ((minus_minus_nat _TPTP_I) J_1)) K)) ((minus_minus_nat ((minus_minus_nat _TPTP_I) K)) J_1))) of role axiom named fact_90_diff__commute % A new axiom: (forall (_TPTP_I:nat) (J_1:nat) (K:nat), (((eq nat) ((minus_minus_nat ((minus_minus_nat _TPTP_I) J_1)) K)) ((minus_minus_nat ((minus_minus_nat _TPTP_I) K)) J_1))) % FOF formula (forall (P_9:(x_a->Prop)) (Q:(x_a->Prop)), ((iff (finite_finite_a (collect_a (fun (X_1:x_a)=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite_finite_a (collect_a P_9))) (finite_finite_a (collect_a Q))))) of role axiom named fact_91_finite__Collect__disjI % A new axiom: (forall (P_9:(x_a->Prop)) (Q:(x_a->Prop)), ((iff (finite_finite_a (collect_a (fun (X_1:x_a)=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite_finite_a (collect_a P_9))) (finite_finite_a (collect_a Q))))) % FOF formula (forall (P_9:((nat->Prop)->Prop)) (Q:((nat->Prop)->Prop)), ((iff (finite_finite_nat_o (collect_nat_o (fun (X_1:(nat->Prop))=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite_finite_nat_o (collect_nat_o P_9))) (finite_finite_nat_o (collect_nat_o Q))))) of role axiom named fact_92_finite__Collect__disjI % A new axiom: (forall (P_9:((nat->Prop)->Prop)) (Q:((nat->Prop)->Prop)), ((iff (finite_finite_nat_o (collect_nat_o (fun (X_1:(nat->Prop))=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite_finite_nat_o (collect_nat_o P_9))) (finite_finite_nat_o (collect_nat_o Q))))) % FOF formula (forall (P_9:((pname->Prop)->Prop)) (Q:((pname->Prop)->Prop)), ((iff (finite297249702name_o (collect_pname_o (fun (X_1:(pname->Prop))=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite297249702name_o (collect_pname_o P_9))) (finite297249702name_o (collect_pname_o Q))))) of role axiom named fact_93_finite__Collect__disjI % A new axiom: (forall (P_9:((pname->Prop)->Prop)) (Q:((pname->Prop)->Prop)), ((iff (finite297249702name_o (collect_pname_o (fun (X_1:(pname->Prop))=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite297249702name_o (collect_pname_o P_9))) (finite297249702name_o (collect_pname_o Q))))) % FOF formula (forall (P_9:((x_a->Prop)->Prop)) (Q:((x_a->Prop)->Prop)), ((iff (finite_finite_a_o (collect_a_o (fun (X_1:(x_a->Prop))=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite_finite_a_o (collect_a_o P_9))) (finite_finite_a_o (collect_a_o Q))))) of role axiom named fact_94_finite__Collect__disjI % A new axiom: (forall (P_9:((x_a->Prop)->Prop)) (Q:((x_a->Prop)->Prop)), ((iff (finite_finite_a_o (collect_a_o (fun (X_1:(x_a->Prop))=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite_finite_a_o (collect_a_o P_9))) (finite_finite_a_o (collect_a_o Q))))) % FOF formula (forall (P_9:(pname->Prop)) (Q:(pname->Prop)), ((iff (finite_finite_pname (collect_pname (fun (X_1:pname)=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite_finite_pname (collect_pname P_9))) (finite_finite_pname (collect_pname Q))))) of role axiom named fact_95_finite__Collect__disjI % A new axiom: (forall (P_9:(pname->Prop)) (Q:(pname->Prop)), ((iff (finite_finite_pname (collect_pname (fun (X_1:pname)=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite_finite_pname (collect_pname P_9))) (finite_finite_pname (collect_pname Q))))) % FOF formula (forall (P_9:(nat->Prop)) (Q:(nat->Prop)), ((iff (finite_finite_nat (collect_nat (fun (X_1:nat)=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite_finite_nat (collect_nat P_9))) (finite_finite_nat (collect_nat Q))))) of role axiom named fact_96_finite__Collect__disjI % A new axiom: (forall (P_9:(nat->Prop)) (Q:(nat->Prop)), ((iff (finite_finite_nat (collect_nat (fun (X_1:nat)=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite_finite_nat (collect_nat P_9))) (finite_finite_nat (collect_nat Q))))) % FOF formula (forall (A_83:(nat->Prop)) (A_82:((nat->Prop)->Prop)), ((iff (finite_finite_nat_o ((insert_nat_o A_83) A_82))) (finite_finite_nat_o A_82))) of role axiom named fact_97_finite__insert % A new axiom: (forall (A_83:(nat->Prop)) (A_82:((nat->Prop)->Prop)), ((iff (finite_finite_nat_o ((insert_nat_o A_83) A_82))) (finite_finite_nat_o A_82))) % FOF formula (forall (A_83:(pname->Prop)) (A_82:((pname->Prop)->Prop)), ((iff (finite297249702name_o ((insert_pname_o A_83) A_82))) (finite297249702name_o A_82))) of role axiom named fact_98_finite__insert % A new axiom: (forall (A_83:(pname->Prop)) (A_82:((pname->Prop)->Prop)), ((iff (finite297249702name_o ((insert_pname_o A_83) A_82))) (finite297249702name_o A_82))) % FOF formula (forall (A_83:(x_a->Prop)) (A_82:((x_a->Prop)->Prop)), ((iff (finite_finite_a_o ((insert_a_o A_83) A_82))) (finite_finite_a_o A_82))) of role axiom named fact_99_finite__insert % A new axiom: (forall (A_83:(x_a->Prop)) (A_82:((x_a->Prop)->Prop)), ((iff (finite_finite_a_o ((insert_a_o A_83) A_82))) (finite_finite_a_o A_82))) % FOF formula (forall (A_83:pname) (A_82:(pname->Prop)), ((iff (finite_finite_pname ((insert_pname A_83) A_82))) (finite_finite_pname A_82))) of role axiom named fact_100_finite__insert % A new axiom: (forall (A_83:pname) (A_82:(pname->Prop)), ((iff (finite_finite_pname ((insert_pname A_83) A_82))) (finite_finite_pname A_82))) % FOF formula (forall (A_83:nat) (A_82:(nat->Prop)), ((iff (finite_finite_nat ((insert_nat A_83) A_82))) (finite_finite_nat A_82))) of role axiom named fact_101_finite__insert % A new axiom: (forall (A_83:nat) (A_82:(nat->Prop)), ((iff (finite_finite_nat ((insert_nat A_83) A_82))) (finite_finite_nat A_82))) % FOF formula (forall (A_83:x_a) (A_82:(x_a->Prop)), ((iff (finite_finite_a ((insert_a A_83) A_82))) (finite_finite_a A_82))) of role axiom named fact_102_finite__insert % A new axiom: (forall (A_83:x_a) (A_82:(x_a->Prop)), ((iff (finite_finite_a ((insert_a A_83) A_82))) (finite_finite_a A_82))) % FOF formula (forall (A_81:((nat->Prop)->Prop)) (B_44:((nat->Prop)->Prop)), (((ord_less_eq_nat_o_o A_81) B_44)->((finite_finite_nat_o B_44)->(finite_finite_nat_o A_81)))) of role axiom named fact_103_finite__subset % A new axiom: (forall (A_81:((nat->Prop)->Prop)) (B_44:((nat->Prop)->Prop)), (((ord_less_eq_nat_o_o A_81) B_44)->((finite_finite_nat_o B_44)->(finite_finite_nat_o A_81)))) % FOF formula (forall (A_81:((pname->Prop)->Prop)) (B_44:((pname->Prop)->Prop)), (((ord_le1205211808me_o_o A_81) B_44)->((finite297249702name_o B_44)->(finite297249702name_o A_81)))) of role axiom named fact_104_finite__subset % A new axiom: (forall (A_81:((pname->Prop)->Prop)) (B_44:((pname->Prop)->Prop)), (((ord_le1205211808me_o_o A_81) B_44)->((finite297249702name_o B_44)->(finite297249702name_o A_81)))) % FOF formula (forall (A_81:((x_a->Prop)->Prop)) (B_44:((x_a->Prop)->Prop)), (((ord_less_eq_a_o_o A_81) B_44)->((finite_finite_a_o B_44)->(finite_finite_a_o A_81)))) of role axiom named fact_105_finite__subset % A new axiom: (forall (A_81:((x_a->Prop)->Prop)) (B_44:((x_a->Prop)->Prop)), (((ord_less_eq_a_o_o A_81) B_44)->((finite_finite_a_o B_44)->(finite_finite_a_o A_81)))) % FOF formula (forall (A_81:(x_a->Prop)) (B_44:(x_a->Prop)), (((ord_less_eq_a_o A_81) B_44)->((finite_finite_a B_44)->(finite_finite_a A_81)))) of role axiom named fact_106_finite__subset % A new axiom: (forall (A_81:(x_a->Prop)) (B_44:(x_a->Prop)), (((ord_less_eq_a_o A_81) B_44)->((finite_finite_a B_44)->(finite_finite_a A_81)))) % FOF formula (forall (A_81:(pname->Prop)) (B_44:(pname->Prop)), (((ord_less_eq_pname_o A_81) B_44)->((finite_finite_pname B_44)->(finite_finite_pname A_81)))) of role axiom named fact_107_finite__subset % A new axiom: (forall (A_81:(pname->Prop)) (B_44:(pname->Prop)), (((ord_less_eq_pname_o A_81) B_44)->((finite_finite_pname B_44)->(finite_finite_pname A_81)))) % FOF formula (forall (A_81:(nat->Prop)) (B_44:(nat->Prop)), (((ord_less_eq_nat_o A_81) B_44)->((finite_finite_nat B_44)->(finite_finite_nat A_81)))) of role axiom named fact_108_finite__subset % A new axiom: (forall (A_81:(nat->Prop)) (B_44:(nat->Prop)), (((ord_less_eq_nat_o A_81) B_44)->((finite_finite_nat B_44)->(finite_finite_nat A_81)))) % FOF formula (forall (A_80:((nat->Prop)->Prop)) (B_43:((nat->Prop)->Prop)), ((finite_finite_nat_o B_43)->(((ord_less_eq_nat_o_o A_80) B_43)->(finite_finite_nat_o A_80)))) of role axiom named fact_109_rev__finite__subset % A new axiom: (forall (A_80:((nat->Prop)->Prop)) (B_43:((nat->Prop)->Prop)), ((finite_finite_nat_o B_43)->(((ord_less_eq_nat_o_o A_80) B_43)->(finite_finite_nat_o A_80)))) % FOF formula (forall (A_80:((pname->Prop)->Prop)) (B_43:((pname->Prop)->Prop)), ((finite297249702name_o B_43)->(((ord_le1205211808me_o_o A_80) B_43)->(finite297249702name_o A_80)))) of role axiom named fact_110_rev__finite__subset % A new axiom: (forall (A_80:((pname->Prop)->Prop)) (B_43:((pname->Prop)->Prop)), ((finite297249702name_o B_43)->(((ord_le1205211808me_o_o A_80) B_43)->(finite297249702name_o A_80)))) % FOF formula (forall (A_80:((x_a->Prop)->Prop)) (B_43:((x_a->Prop)->Prop)), ((finite_finite_a_o B_43)->(((ord_less_eq_a_o_o A_80) B_43)->(finite_finite_a_o A_80)))) of role axiom named fact_111_rev__finite__subset % A new axiom: (forall (A_80:((x_a->Prop)->Prop)) (B_43:((x_a->Prop)->Prop)), ((finite_finite_a_o B_43)->(((ord_less_eq_a_o_o A_80) B_43)->(finite_finite_a_o A_80)))) % FOF formula (forall (A_80:(x_a->Prop)) (B_43:(x_a->Prop)), ((finite_finite_a B_43)->(((ord_less_eq_a_o A_80) B_43)->(finite_finite_a A_80)))) of role axiom named fact_112_rev__finite__subset % A new axiom: (forall (A_80:(x_a->Prop)) (B_43:(x_a->Prop)), ((finite_finite_a B_43)->(((ord_less_eq_a_o A_80) B_43)->(finite_finite_a A_80)))) % FOF formula (forall (A_80:(pname->Prop)) (B_43:(pname->Prop)), ((finite_finite_pname B_43)->(((ord_less_eq_pname_o A_80) B_43)->(finite_finite_pname A_80)))) of role axiom named fact_113_rev__finite__subset % A new axiom: (forall (A_80:(pname->Prop)) (B_43:(pname->Prop)), ((finite_finite_pname B_43)->(((ord_less_eq_pname_o A_80) B_43)->(finite_finite_pname A_80)))) % FOF formula (forall (A_80:(nat->Prop)) (B_43:(nat->Prop)), ((finite_finite_nat B_43)->(((ord_less_eq_nat_o A_80) B_43)->(finite_finite_nat A_80)))) of role axiom named fact_114_rev__finite__subset % A new axiom: (forall (A_80:(nat->Prop)) (B_43:(nat->Prop)), ((finite_finite_nat B_43)->(((ord_less_eq_nat_o A_80) B_43)->(finite_finite_nat A_80)))) % FOF formula (forall (M:nat) (N:nat), (((ord_less_eq_nat (suc M)) N)->((ord_less_eq_nat M) N))) of role axiom named fact_115_Suc__leD % A new axiom: (forall (M:nat) (N:nat), (((ord_less_eq_nat (suc M)) N)->((ord_less_eq_nat M) N))) % FOF formula (forall (M:nat) (N:nat), (((ord_less_eq_nat M) (suc N))->((((ord_less_eq_nat M) N)->False)->(((eq nat) M) (suc N))))) of role axiom named fact_116_le__SucE % A new axiom: (forall (M:nat) (N:nat), (((ord_less_eq_nat M) (suc N))->((((ord_less_eq_nat M) N)->False)->(((eq nat) M) (suc N))))) % FOF formula (forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((ord_less_eq_nat M) (suc N)))) of role axiom named fact_117_le__SucI % A new axiom: (forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((ord_less_eq_nat M) (suc N)))) % FOF formula (forall (N:nat) (M:nat), ((iff ((ord_less_eq_nat (suc N)) (suc M))) ((ord_less_eq_nat N) M))) of role axiom named fact_118_Suc__le__mono % A new axiom: (forall (N:nat) (M:nat), ((iff ((ord_less_eq_nat (suc N)) (suc M))) ((ord_less_eq_nat N) M))) % FOF formula (forall (M:nat) (N:nat), ((iff ((ord_less_eq_nat M) (suc N))) ((or ((ord_less_eq_nat M) N)) (((eq nat) M) (suc N))))) of role axiom named fact_119_le__Suc__eq % A new axiom: (forall (M:nat) (N:nat), ((iff ((ord_less_eq_nat M) (suc N))) ((or ((ord_less_eq_nat M) N)) (((eq nat) M) (suc N))))) % FOF formula (forall (M:nat) (N:nat), ((iff (((ord_less_eq_nat M) N)->False)) ((ord_less_eq_nat (suc N)) M))) of role axiom named fact_120_not__less__eq__eq % A new axiom: (forall (M:nat) (N:nat), ((iff (((ord_less_eq_nat M) N)->False)) ((ord_less_eq_nat (suc N)) M))) % FOF formula (forall (N:nat), (((ord_less_eq_nat (suc N)) N)->False)) of role axiom named fact_121_Suc__n__not__le__n % A new axiom: (forall (N:nat), (((ord_less_eq_nat (suc N)) N)->False)) % FOF formula (forall (M:nat) (N:nat) (K:nat), (((eq nat) ((minus_minus_nat ((minus_minus_nat (suc M)) N)) (suc K))) ((minus_minus_nat ((minus_minus_nat M) N)) K))) of role axiom named fact_122_Suc__diff__diff % A new axiom: (forall (M:nat) (N:nat) (K:nat), (((eq nat) ((minus_minus_nat ((minus_minus_nat (suc M)) N)) (suc K))) ((minus_minus_nat ((minus_minus_nat M) N)) K))) % FOF formula (forall (M:nat) (N:nat), (((eq nat) ((minus_minus_nat (suc M)) (suc N))) ((minus_minus_nat M) N))) of role axiom named fact_123_diff__Suc__Suc % A new axiom: (forall (M:nat) (N:nat), (((eq nat) ((minus_minus_nat (suc M)) (suc N))) ((minus_minus_nat M) N))) % FOF formula (forall (N:nat) (K:nat) (M:nat), (((ord_less_eq_nat K) M)->(((ord_less_eq_nat K) N)->((iff ((ord_less_eq_nat ((minus_minus_nat M) K)) ((minus_minus_nat N) K))) ((ord_less_eq_nat M) N))))) of role axiom named fact_124_le__diff__iff % A new axiom: (forall (N:nat) (K:nat) (M:nat), (((ord_less_eq_nat K) M)->(((ord_less_eq_nat K) N)->((iff ((ord_less_eq_nat ((minus_minus_nat M) K)) ((minus_minus_nat N) K))) ((ord_less_eq_nat M) N))))) % FOF formula (forall (N:nat) (K:nat) (M:nat), (((ord_less_eq_nat K) M)->(((ord_less_eq_nat K) N)->(((eq nat) ((minus_minus_nat ((minus_minus_nat M) K)) ((minus_minus_nat N) K))) ((minus_minus_nat M) N))))) of role axiom named fact_125_Nat_Odiff__diff__eq % A new axiom: (forall (N:nat) (K:nat) (M:nat), (((ord_less_eq_nat K) M)->(((ord_less_eq_nat K) N)->(((eq nat) ((minus_minus_nat ((minus_minus_nat M) K)) ((minus_minus_nat N) K))) ((minus_minus_nat M) N))))) % FOF formula (forall (N:nat) (K:nat) (M:nat), (((ord_less_eq_nat K) M)->(((ord_less_eq_nat K) N)->((iff (((eq nat) ((minus_minus_nat M) K)) ((minus_minus_nat N) K))) (((eq nat) M) N))))) of role axiom named fact_126_eq__diff__iff % A new axiom: (forall (N:nat) (K:nat) (M:nat), (((ord_less_eq_nat K) M)->(((ord_less_eq_nat K) N)->((iff (((eq nat) ((minus_minus_nat M) K)) ((minus_minus_nat N) K))) (((eq nat) M) N))))) % FOF formula (forall (_TPTP_I:nat) (N:nat), (((ord_less_eq_nat _TPTP_I) N)->(((eq nat) ((minus_minus_nat N) ((minus_minus_nat N) _TPTP_I))) _TPTP_I))) of role axiom named fact_127_diff__diff__cancel % A new axiom: (forall (_TPTP_I:nat) (N:nat), (((ord_less_eq_nat _TPTP_I) N)->(((eq nat) ((minus_minus_nat N) ((minus_minus_nat N) _TPTP_I))) _TPTP_I))) % FOF formula (forall (L:nat) (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((ord_less_eq_nat ((minus_minus_nat M) L)) ((minus_minus_nat N) L)))) of role axiom named fact_128_diff__le__mono % A new axiom: (forall (L:nat) (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((ord_less_eq_nat ((minus_minus_nat M) L)) ((minus_minus_nat N) L)))) % FOF formula (forall (L:nat) (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((ord_less_eq_nat ((minus_minus_nat L) N)) ((minus_minus_nat L) M)))) of role axiom named fact_129_diff__le__mono2 % A new axiom: (forall (L:nat) (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((ord_less_eq_nat ((minus_minus_nat L) N)) ((minus_minus_nat L) M)))) % FOF formula (forall (M:nat) (N:nat), ((ord_less_eq_nat ((minus_minus_nat M) N)) M)) of role axiom named fact_130_diff__le__self % A new axiom: (forall (M:nat) (N:nat), ((ord_less_eq_nat ((minus_minus_nat M) N)) M)) % FOF formula (forall (B_42:(x_a->Prop)) (F_23:(x_a->x_a)) (A_79:(x_a->Prop)), ((finite_finite_a A_79)->(((ord_less_eq_a_o B_42) ((image_a_a F_23) A_79))->(finite_finite_a B_42)))) of role axiom named fact_131_finite__surj % A new axiom: (forall (B_42:(x_a->Prop)) (F_23:(x_a->x_a)) (A_79:(x_a->Prop)), ((finite_finite_a A_79)->(((ord_less_eq_a_o B_42) ((image_a_a F_23) A_79))->(finite_finite_a B_42)))) % FOF formula (forall (B_42:(x_a->Prop)) (F_23:((nat->Prop)->x_a)) (A_79:((nat->Prop)->Prop)), ((finite_finite_nat_o A_79)->(((ord_less_eq_a_o B_42) ((image_nat_o_a F_23) A_79))->(finite_finite_a B_42)))) of role axiom named fact_132_finite__surj % A new axiom: (forall (B_42:(x_a->Prop)) (F_23:((nat->Prop)->x_a)) (A_79:((nat->Prop)->Prop)), ((finite_finite_nat_o A_79)->(((ord_less_eq_a_o B_42) ((image_nat_o_a F_23) A_79))->(finite_finite_a B_42)))) % FOF formula (forall (B_42:(x_a->Prop)) (F_23:((pname->Prop)->x_a)) (A_79:((pname->Prop)->Prop)), ((finite297249702name_o A_79)->(((ord_less_eq_a_o B_42) ((image_pname_o_a F_23) A_79))->(finite_finite_a B_42)))) of role axiom named fact_133_finite__surj % A new axiom: (forall (B_42:(x_a->Prop)) (F_23:((pname->Prop)->x_a)) (A_79:((pname->Prop)->Prop)), ((finite297249702name_o A_79)->(((ord_less_eq_a_o B_42) ((image_pname_o_a F_23) A_79))->(finite_finite_a B_42)))) % FOF formula (forall (B_42:(x_a->Prop)) (F_23:((x_a->Prop)->x_a)) (A_79:((x_a->Prop)->Prop)), ((finite_finite_a_o A_79)->(((ord_less_eq_a_o B_42) ((image_a_o_a F_23) A_79))->(finite_finite_a B_42)))) of role axiom named fact_134_finite__surj % A new axiom: (forall (B_42:(x_a->Prop)) (F_23:((x_a->Prop)->x_a)) (A_79:((x_a->Prop)->Prop)), ((finite_finite_a_o A_79)->(((ord_less_eq_a_o B_42) ((image_a_o_a F_23) A_79))->(finite_finite_a B_42)))) % FOF formula (forall (B_42:((nat->Prop)->Prop)) (F_23:(pname->(nat->Prop))) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_less_eq_nat_o_o B_42) ((image_pname_nat_o F_23) A_79))->(finite_finite_nat_o B_42)))) of role axiom named fact_135_finite__surj % A new axiom: (forall (B_42:((nat->Prop)->Prop)) (F_23:(pname->(nat->Prop))) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_less_eq_nat_o_o B_42) ((image_pname_nat_o F_23) A_79))->(finite_finite_nat_o B_42)))) % FOF formula (forall (B_42:((pname->Prop)->Prop)) (F_23:(pname->(pname->Prop))) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_le1205211808me_o_o B_42) ((image_pname_pname_o F_23) A_79))->(finite297249702name_o B_42)))) of role axiom named fact_136_finite__surj % A new axiom: (forall (B_42:((pname->Prop)->Prop)) (F_23:(pname->(pname->Prop))) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_le1205211808me_o_o B_42) ((image_pname_pname_o F_23) A_79))->(finite297249702name_o B_42)))) % FOF formula (forall (B_42:((x_a->Prop)->Prop)) (F_23:(pname->(x_a->Prop))) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_less_eq_a_o_o B_42) ((image_pname_a_o F_23) A_79))->(finite_finite_a_o B_42)))) of role axiom named fact_137_finite__surj % A new axiom: (forall (B_42:((x_a->Prop)->Prop)) (F_23:(pname->(x_a->Prop))) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_less_eq_a_o_o B_42) ((image_pname_a_o F_23) A_79))->(finite_finite_a_o B_42)))) % FOF formula (forall (B_42:(pname->Prop)) (F_23:(pname->pname)) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_less_eq_pname_o B_42) ((image_pname_pname F_23) A_79))->(finite_finite_pname B_42)))) of role axiom named fact_138_finite__surj % A new axiom: (forall (B_42:(pname->Prop)) (F_23:(pname->pname)) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_less_eq_pname_o B_42) ((image_pname_pname F_23) A_79))->(finite_finite_pname B_42)))) % FOF formula (forall (B_42:(nat->Prop)) (F_23:(pname->nat)) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_less_eq_nat_o B_42) ((image_pname_nat F_23) A_79))->(finite_finite_nat B_42)))) of role axiom named fact_139_finite__surj % A new axiom: (forall (B_42:(nat->Prop)) (F_23:(pname->nat)) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_less_eq_nat_o B_42) ((image_pname_nat F_23) A_79))->(finite_finite_nat B_42)))) % FOF formula (forall (B_42:(x_a->Prop)) (F_23:(nat->x_a)) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_less_eq_a_o B_42) ((image_nat_a F_23) A_79))->(finite_finite_a B_42)))) of role axiom named fact_140_finite__surj % A new axiom: (forall (B_42:(x_a->Prop)) (F_23:(nat->x_a)) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_less_eq_a_o B_42) ((image_nat_a F_23) A_79))->(finite_finite_a B_42)))) % FOF formula (forall (B_42:((nat->Prop)->Prop)) (F_23:(nat->(nat->Prop))) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_less_eq_nat_o_o B_42) ((image_nat_nat_o F_23) A_79))->(finite_finite_nat_o B_42)))) of role axiom named fact_141_finite__surj % A new axiom: (forall (B_42:((nat->Prop)->Prop)) (F_23:(nat->(nat->Prop))) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_less_eq_nat_o_o B_42) ((image_nat_nat_o F_23) A_79))->(finite_finite_nat_o B_42)))) % FOF formula (forall (B_42:((pname->Prop)->Prop)) (F_23:(nat->(pname->Prop))) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_le1205211808me_o_o B_42) ((image_nat_pname_o F_23) A_79))->(finite297249702name_o B_42)))) of role axiom named fact_142_finite__surj % A new axiom: (forall (B_42:((pname->Prop)->Prop)) (F_23:(nat->(pname->Prop))) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_le1205211808me_o_o B_42) ((image_nat_pname_o F_23) A_79))->(finite297249702name_o B_42)))) % FOF formula (forall (B_42:((x_a->Prop)->Prop)) (F_23:(nat->(x_a->Prop))) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_less_eq_a_o_o B_42) ((image_nat_a_o F_23) A_79))->(finite_finite_a_o B_42)))) of role axiom named fact_143_finite__surj % A new axiom: (forall (B_42:((x_a->Prop)->Prop)) (F_23:(nat->(x_a->Prop))) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_less_eq_a_o_o B_42) ((image_nat_a_o F_23) A_79))->(finite_finite_a_o B_42)))) % FOF formula (forall (B_42:(pname->Prop)) (F_23:(nat->pname)) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_less_eq_pname_o B_42) ((image_nat_pname F_23) A_79))->(finite_finite_pname B_42)))) of role axiom named fact_144_finite__surj % A new axiom: (forall (B_42:(pname->Prop)) (F_23:(nat->pname)) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_less_eq_pname_o B_42) ((image_nat_pname F_23) A_79))->(finite_finite_pname B_42)))) % FOF formula (forall (B_42:(nat->Prop)) (F_23:(nat->nat)) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_less_eq_nat_o B_42) ((image_nat_nat F_23) A_79))->(finite_finite_nat B_42)))) of role axiom named fact_145_finite__surj % A new axiom: (forall (B_42:(nat->Prop)) (F_23:(nat->nat)) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_less_eq_nat_o B_42) ((image_nat_nat F_23) A_79))->(finite_finite_nat B_42)))) % FOF formula (forall (B_42:(pname->Prop)) (F_23:(x_a->pname)) (A_79:(x_a->Prop)), ((finite_finite_a A_79)->(((ord_less_eq_pname_o B_42) ((image_a_pname F_23) A_79))->(finite_finite_pname B_42)))) of role axiom named fact_146_finite__surj % A new axiom: (forall (B_42:(pname->Prop)) (F_23:(x_a->pname)) (A_79:(x_a->Prop)), ((finite_finite_a A_79)->(((ord_less_eq_pname_o B_42) ((image_a_pname F_23) A_79))->(finite_finite_pname B_42)))) % FOF formula (forall (B_42:(pname->Prop)) (F_23:((nat->Prop)->pname)) (A_79:((nat->Prop)->Prop)), ((finite_finite_nat_o A_79)->(((ord_less_eq_pname_o B_42) ((image_nat_o_pname F_23) A_79))->(finite_finite_pname B_42)))) of role axiom named fact_147_finite__surj % A new axiom: (forall (B_42:(pname->Prop)) (F_23:((nat->Prop)->pname)) (A_79:((nat->Prop)->Prop)), ((finite_finite_nat_o A_79)->(((ord_less_eq_pname_o B_42) ((image_nat_o_pname F_23) A_79))->(finite_finite_pname B_42)))) % FOF formula (forall (B_42:(pname->Prop)) (F_23:((pname->Prop)->pname)) (A_79:((pname->Prop)->Prop)), ((finite297249702name_o A_79)->(((ord_less_eq_pname_o B_42) ((image_pname_o_pname F_23) A_79))->(finite_finite_pname B_42)))) of role axiom named fact_148_finite__surj % A new axiom: (forall (B_42:(pname->Prop)) (F_23:((pname->Prop)->pname)) (A_79:((pname->Prop)->Prop)), ((finite297249702name_o A_79)->(((ord_less_eq_pname_o B_42) ((image_pname_o_pname F_23) A_79))->(finite_finite_pname B_42)))) % FOF formula (forall (B_42:(pname->Prop)) (F_23:((x_a->Prop)->pname)) (A_79:((x_a->Prop)->Prop)), ((finite_finite_a_o A_79)->(((ord_less_eq_pname_o B_42) ((image_a_o_pname F_23) A_79))->(finite_finite_pname B_42)))) of role axiom named fact_149_finite__surj % A new axiom: (forall (B_42:(pname->Prop)) (F_23:((x_a->Prop)->pname)) (A_79:((x_a->Prop)->Prop)), ((finite_finite_a_o A_79)->(((ord_less_eq_pname_o B_42) ((image_a_o_pname F_23) A_79))->(finite_finite_pname B_42)))) % FOF formula (forall (B_42:(nat->Prop)) (F_23:(x_a->nat)) (A_79:(x_a->Prop)), ((finite_finite_a A_79)->(((ord_less_eq_nat_o B_42) ((image_a_nat F_23) A_79))->(finite_finite_nat B_42)))) of role axiom named fact_150_finite__surj % A new axiom: (forall (B_42:(nat->Prop)) (F_23:(x_a->nat)) (A_79:(x_a->Prop)), ((finite_finite_a A_79)->(((ord_less_eq_nat_o B_42) ((image_a_nat F_23) A_79))->(finite_finite_nat B_42)))) % FOF formula (forall (B_42:(nat->Prop)) (F_23:((nat->Prop)->nat)) (A_79:((nat->Prop)->Prop)), ((finite_finite_nat_o A_79)->(((ord_less_eq_nat_o B_42) ((image_nat_o_nat F_23) A_79))->(finite_finite_nat B_42)))) of role axiom named fact_151_finite__surj % A new axiom: (forall (B_42:(nat->Prop)) (F_23:((nat->Prop)->nat)) (A_79:((nat->Prop)->Prop)), ((finite_finite_nat_o A_79)->(((ord_less_eq_nat_o B_42) ((image_nat_o_nat F_23) A_79))->(finite_finite_nat B_42)))) % FOF formula (forall (B_42:(nat->Prop)) (F_23:((pname->Prop)->nat)) (A_79:((pname->Prop)->Prop)), ((finite297249702name_o A_79)->(((ord_less_eq_nat_o B_42) ((image_pname_o_nat F_23) A_79))->(finite_finite_nat B_42)))) of role axiom named fact_152_finite__surj % A new axiom: (forall (B_42:(nat->Prop)) (F_23:((pname->Prop)->nat)) (A_79:((pname->Prop)->Prop)), ((finite297249702name_o A_79)->(((ord_less_eq_nat_o B_42) ((image_pname_o_nat F_23) A_79))->(finite_finite_nat B_42)))) % FOF formula (forall (B_42:(nat->Prop)) (F_23:((x_a->Prop)->nat)) (A_79:((x_a->Prop)->Prop)), ((finite_finite_a_o A_79)->(((ord_less_eq_nat_o B_42) ((image_a_o_nat F_23) A_79))->(finite_finite_nat B_42)))) of role axiom named fact_153_finite__surj % A new axiom: (forall (B_42:(nat->Prop)) (F_23:((x_a->Prop)->nat)) (A_79:((x_a->Prop)->Prop)), ((finite_finite_a_o A_79)->(((ord_less_eq_nat_o B_42) ((image_a_o_nat F_23) A_79))->(finite_finite_nat B_42)))) % FOF formula (forall (B_42:(x_a->Prop)) (F_23:(pname->x_a)) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_less_eq_a_o B_42) ((image_pname_a F_23) A_79))->(finite_finite_a B_42)))) of role axiom named fact_154_finite__surj % A new axiom: (forall (B_42:(x_a->Prop)) (F_23:(pname->x_a)) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_less_eq_a_o B_42) ((image_pname_a F_23) A_79))->(finite_finite_a B_42)))) % FOF formula (forall (F_22:((nat->Prop)->x_a)) (A_78:((nat->Prop)->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_nat_o_a F_22) A_78))->((ex ((nat->Prop)->Prop)) (fun (C_12:((nat->Prop)->Prop))=> ((and ((and ((ord_less_eq_nat_o_o C_12) A_78)) (finite_finite_nat_o C_12))) (((eq (x_a->Prop)) B_41) ((image_nat_o_a F_22) C_12)))))))) of role axiom named fact_155_finite__subset__image % A new axiom: (forall (F_22:((nat->Prop)->x_a)) (A_78:((nat->Prop)->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_nat_o_a F_22) A_78))->((ex ((nat->Prop)->Prop)) (fun (C_12:((nat->Prop)->Prop))=> ((and ((and ((ord_less_eq_nat_o_o C_12) A_78)) (finite_finite_nat_o C_12))) (((eq (x_a->Prop)) B_41) ((image_nat_o_a F_22) C_12)))))))) % FOF formula (forall (F_22:((pname->Prop)->x_a)) (A_78:((pname->Prop)->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_pname_o_a F_22) A_78))->((ex ((pname->Prop)->Prop)) (fun (C_12:((pname->Prop)->Prop))=> ((and ((and ((ord_le1205211808me_o_o C_12) A_78)) (finite297249702name_o C_12))) (((eq (x_a->Prop)) B_41) ((image_pname_o_a F_22) C_12)))))))) of role axiom named fact_156_finite__subset__image % A new axiom: (forall (F_22:((pname->Prop)->x_a)) (A_78:((pname->Prop)->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_pname_o_a F_22) A_78))->((ex ((pname->Prop)->Prop)) (fun (C_12:((pname->Prop)->Prop))=> ((and ((and ((ord_le1205211808me_o_o C_12) A_78)) (finite297249702name_o C_12))) (((eq (x_a->Prop)) B_41) ((image_pname_o_a F_22) C_12)))))))) % FOF formula (forall (F_22:((x_a->Prop)->x_a)) (A_78:((x_a->Prop)->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_a_o_a F_22) A_78))->((ex ((x_a->Prop)->Prop)) (fun (C_12:((x_a->Prop)->Prop))=> ((and ((and ((ord_less_eq_a_o_o C_12) A_78)) (finite_finite_a_o C_12))) (((eq (x_a->Prop)) B_41) ((image_a_o_a F_22) C_12)))))))) of role axiom named fact_157_finite__subset__image % A new axiom: (forall (F_22:((x_a->Prop)->x_a)) (A_78:((x_a->Prop)->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_a_o_a F_22) A_78))->((ex ((x_a->Prop)->Prop)) (fun (C_12:((x_a->Prop)->Prop))=> ((and ((and ((ord_less_eq_a_o_o C_12) A_78)) (finite_finite_a_o C_12))) (((eq (x_a->Prop)) B_41) ((image_a_o_a F_22) C_12)))))))) % FOF formula (forall (F_22:(x_a->x_a)) (A_78:(x_a->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_a_a F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq (x_a->Prop)) B_41) ((image_a_a F_22) C_12)))))))) of role axiom named fact_158_finite__subset__image % A new axiom: (forall (F_22:(x_a->x_a)) (A_78:(x_a->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_a_a F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq (x_a->Prop)) B_41) ((image_a_a F_22) C_12)))))))) % FOF formula (forall (F_22:(x_a->(nat->Prop))) (A_78:(x_a->Prop)) (B_41:((nat->Prop)->Prop)), ((finite_finite_nat_o B_41)->(((ord_less_eq_nat_o_o B_41) ((image_a_nat_o F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq ((nat->Prop)->Prop)) B_41) ((image_a_nat_o F_22) C_12)))))))) of role axiom named fact_159_finite__subset__image % A new axiom: (forall (F_22:(x_a->(nat->Prop))) (A_78:(x_a->Prop)) (B_41:((nat->Prop)->Prop)), ((finite_finite_nat_o B_41)->(((ord_less_eq_nat_o_o B_41) ((image_a_nat_o F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq ((nat->Prop)->Prop)) B_41) ((image_a_nat_o F_22) C_12)))))))) % FOF formula (forall (F_22:(x_a->(pname->Prop))) (A_78:(x_a->Prop)) (B_41:((pname->Prop)->Prop)), ((finite297249702name_o B_41)->(((ord_le1205211808me_o_o B_41) ((image_a_pname_o F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq ((pname->Prop)->Prop)) B_41) ((image_a_pname_o F_22) C_12)))))))) of role axiom named fact_160_finite__subset__image % A new axiom: (forall (F_22:(x_a->(pname->Prop))) (A_78:(x_a->Prop)) (B_41:((pname->Prop)->Prop)), ((finite297249702name_o B_41)->(((ord_le1205211808me_o_o B_41) ((image_a_pname_o F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq ((pname->Prop)->Prop)) B_41) ((image_a_pname_o F_22) C_12)))))))) % FOF formula (forall (F_22:(x_a->(x_a->Prop))) (A_78:(x_a->Prop)) (B_41:((x_a->Prop)->Prop)), ((finite_finite_a_o B_41)->(((ord_less_eq_a_o_o B_41) ((image_a_a_o F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq ((x_a->Prop)->Prop)) B_41) ((image_a_a_o F_22) C_12)))))))) of role axiom named fact_161_finite__subset__image % A new axiom: (forall (F_22:(x_a->(x_a->Prop))) (A_78:(x_a->Prop)) (B_41:((x_a->Prop)->Prop)), ((finite_finite_a_o B_41)->(((ord_less_eq_a_o_o B_41) ((image_a_a_o F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq ((x_a->Prop)->Prop)) B_41) ((image_a_a_o F_22) C_12)))))))) % FOF formula (forall (F_22:(x_a->pname)) (A_78:(x_a->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_a_pname F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq (pname->Prop)) B_41) ((image_a_pname F_22) C_12)))))))) of role axiom named fact_162_finite__subset__image % A new axiom: (forall (F_22:(x_a->pname)) (A_78:(x_a->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_a_pname F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq (pname->Prop)) B_41) ((image_a_pname F_22) C_12)))))))) % FOF formula (forall (F_22:((nat->Prop)->pname)) (A_78:((nat->Prop)->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_nat_o_pname F_22) A_78))->((ex ((nat->Prop)->Prop)) (fun (C_12:((nat->Prop)->Prop))=> ((and ((and ((ord_less_eq_nat_o_o C_12) A_78)) (finite_finite_nat_o C_12))) (((eq (pname->Prop)) B_41) ((image_nat_o_pname F_22) C_12)))))))) of role axiom named fact_163_finite__subset__image % A new axiom: (forall (F_22:((nat->Prop)->pname)) (A_78:((nat->Prop)->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_nat_o_pname F_22) A_78))->((ex ((nat->Prop)->Prop)) (fun (C_12:((nat->Prop)->Prop))=> ((and ((and ((ord_less_eq_nat_o_o C_12) A_78)) (finite_finite_nat_o C_12))) (((eq (pname->Prop)) B_41) ((image_nat_o_pname F_22) C_12)))))))) % FOF formula (forall (F_22:((pname->Prop)->pname)) (A_78:((pname->Prop)->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_pname_o_pname F_22) A_78))->((ex ((pname->Prop)->Prop)) (fun (C_12:((pname->Prop)->Prop))=> ((and ((and ((ord_le1205211808me_o_o C_12) A_78)) (finite297249702name_o C_12))) (((eq (pname->Prop)) B_41) ((image_pname_o_pname F_22) C_12)))))))) of role axiom named fact_164_finite__subset__image % A new axiom: (forall (F_22:((pname->Prop)->pname)) (A_78:((pname->Prop)->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_pname_o_pname F_22) A_78))->((ex ((pname->Prop)->Prop)) (fun (C_12:((pname->Prop)->Prop))=> ((and ((and ((ord_le1205211808me_o_o C_12) A_78)) (finite297249702name_o C_12))) (((eq (pname->Prop)) B_41) ((image_pname_o_pname F_22) C_12)))))))) % FOF formula (forall (F_22:((x_a->Prop)->pname)) (A_78:((x_a->Prop)->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_a_o_pname F_22) A_78))->((ex ((x_a->Prop)->Prop)) (fun (C_12:((x_a->Prop)->Prop))=> ((and ((and ((ord_less_eq_a_o_o C_12) A_78)) (finite_finite_a_o C_12))) (((eq (pname->Prop)) B_41) ((image_a_o_pname F_22) C_12)))))))) of role axiom named fact_165_finite__subset__image % A new axiom: (forall (F_22:((x_a->Prop)->pname)) (A_78:((x_a->Prop)->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_a_o_pname F_22) A_78))->((ex ((x_a->Prop)->Prop)) (fun (C_12:((x_a->Prop)->Prop))=> ((and ((and ((ord_less_eq_a_o_o C_12) A_78)) (finite_finite_a_o C_12))) (((eq (pname->Prop)) B_41) ((image_a_o_pname F_22) C_12)))))))) % FOF formula (forall (F_22:(x_a->nat)) (A_78:(x_a->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_a_nat F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq (nat->Prop)) B_41) ((image_a_nat F_22) C_12)))))))) of role axiom named fact_166_finite__subset__image % A new axiom: (forall (F_22:(x_a->nat)) (A_78:(x_a->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_a_nat F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq (nat->Prop)) B_41) ((image_a_nat F_22) C_12)))))))) % FOF formula (forall (F_22:((nat->Prop)->nat)) (A_78:((nat->Prop)->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_nat_o_nat F_22) A_78))->((ex ((nat->Prop)->Prop)) (fun (C_12:((nat->Prop)->Prop))=> ((and ((and ((ord_less_eq_nat_o_o C_12) A_78)) (finite_finite_nat_o C_12))) (((eq (nat->Prop)) B_41) ((image_nat_o_nat F_22) C_12)))))))) of role axiom named fact_167_finite__subset__image % A new axiom: (forall (F_22:((nat->Prop)->nat)) (A_78:((nat->Prop)->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_nat_o_nat F_22) A_78))->((ex ((nat->Prop)->Prop)) (fun (C_12:((nat->Prop)->Prop))=> ((and ((and ((ord_less_eq_nat_o_o C_12) A_78)) (finite_finite_nat_o C_12))) (((eq (nat->Prop)) B_41) ((image_nat_o_nat F_22) C_12)))))))) % FOF formula (forall (F_22:((pname->Prop)->nat)) (A_78:((pname->Prop)->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_pname_o_nat F_22) A_78))->((ex ((pname->Prop)->Prop)) (fun (C_12:((pname->Prop)->Prop))=> ((and ((and ((ord_le1205211808me_o_o C_12) A_78)) (finite297249702name_o C_12))) (((eq (nat->Prop)) B_41) ((image_pname_o_nat F_22) C_12)))))))) of role axiom named fact_168_finite__subset__image % A new axiom: (forall (F_22:((pname->Prop)->nat)) (A_78:((pname->Prop)->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_pname_o_nat F_22) A_78))->((ex ((pname->Prop)->Prop)) (fun (C_12:((pname->Prop)->Prop))=> ((and ((and ((ord_le1205211808me_o_o C_12) A_78)) (finite297249702name_o C_12))) (((eq (nat->Prop)) B_41) ((image_pname_o_nat F_22) C_12)))))))) % FOF formula (forall (F_22:((x_a->Prop)->nat)) (A_78:((x_a->Prop)->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_a_o_nat F_22) A_78))->((ex ((x_a->Prop)->Prop)) (fun (C_12:((x_a->Prop)->Prop))=> ((and ((and ((ord_less_eq_a_o_o C_12) A_78)) (finite_finite_a_o C_12))) (((eq (nat->Prop)) B_41) ((image_a_o_nat F_22) C_12)))))))) of role axiom named fact_169_finite__subset__image % A new axiom: (forall (F_22:((x_a->Prop)->nat)) (A_78:((x_a->Prop)->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_a_o_nat F_22) A_78))->((ex ((x_a->Prop)->Prop)) (fun (C_12:((x_a->Prop)->Prop))=> ((and ((and ((ord_less_eq_a_o_o C_12) A_78)) (finite_finite_a_o C_12))) (((eq (nat->Prop)) B_41) ((image_a_o_nat F_22) C_12)))))))) % FOF formula (forall (F_22:(pname->(nat->Prop))) (A_78:(pname->Prop)) (B_41:((nat->Prop)->Prop)), ((finite_finite_nat_o B_41)->(((ord_less_eq_nat_o_o B_41) ((image_pname_nat_o F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq ((nat->Prop)->Prop)) B_41) ((image_pname_nat_o F_22) C_12)))))))) of role axiom named fact_170_finite__subset__image % A new axiom: (forall (F_22:(pname->(nat->Prop))) (A_78:(pname->Prop)) (B_41:((nat->Prop)->Prop)), ((finite_finite_nat_o B_41)->(((ord_less_eq_nat_o_o B_41) ((image_pname_nat_o F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq ((nat->Prop)->Prop)) B_41) ((image_pname_nat_o F_22) C_12)))))))) % FOF formula (forall (F_22:(pname->(pname->Prop))) (A_78:(pname->Prop)) (B_41:((pname->Prop)->Prop)), ((finite297249702name_o B_41)->(((ord_le1205211808me_o_o B_41) ((image_pname_pname_o F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq ((pname->Prop)->Prop)) B_41) ((image_pname_pname_o F_22) C_12)))))))) of role axiom named fact_171_finite__subset__image % A new axiom: (forall (F_22:(pname->(pname->Prop))) (A_78:(pname->Prop)) (B_41:((pname->Prop)->Prop)), ((finite297249702name_o B_41)->(((ord_le1205211808me_o_o B_41) ((image_pname_pname_o F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq ((pname->Prop)->Prop)) B_41) ((image_pname_pname_o F_22) C_12)))))))) % FOF formula (forall (F_22:(pname->(x_a->Prop))) (A_78:(pname->Prop)) (B_41:((x_a->Prop)->Prop)), ((finite_finite_a_o B_41)->(((ord_less_eq_a_o_o B_41) ((image_pname_a_o F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq ((x_a->Prop)->Prop)) B_41) ((image_pname_a_o F_22) C_12)))))))) of role axiom named fact_172_finite__subset__image % A new axiom: (forall (F_22:(pname->(x_a->Prop))) (A_78:(pname->Prop)) (B_41:((x_a->Prop)->Prop)), ((finite_finite_a_o B_41)->(((ord_less_eq_a_o_o B_41) ((image_pname_a_o F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq ((x_a->Prop)->Prop)) B_41) ((image_pname_a_o F_22) C_12)))))))) % FOF formula (forall (F_22:(pname->pname)) (A_78:(pname->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_pname_pname F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq (pname->Prop)) B_41) ((image_pname_pname F_22) C_12)))))))) of role axiom named fact_173_finite__subset__image % A new axiom: (forall (F_22:(pname->pname)) (A_78:(pname->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_pname_pname F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq (pname->Prop)) B_41) ((image_pname_pname F_22) C_12)))))))) % FOF formula (forall (F_22:(pname->nat)) (A_78:(pname->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_pname_nat F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq (nat->Prop)) B_41) ((image_pname_nat F_22) C_12)))))))) of role axiom named fact_174_finite__subset__image % A new axiom: (forall (F_22:(pname->nat)) (A_78:(pname->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_pname_nat F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq (nat->Prop)) B_41) ((image_pname_nat F_22) C_12)))))))) % FOF formula (forall (F_22:(nat->x_a)) (A_78:(nat->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_nat_a F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq (x_a->Prop)) B_41) ((image_nat_a F_22) C_12)))))))) of role axiom named fact_175_finite__subset__image % A new axiom: (forall (F_22:(nat->x_a)) (A_78:(nat->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_nat_a F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq (x_a->Prop)) B_41) ((image_nat_a F_22) C_12)))))))) % FOF formula (forall (F_22:(nat->(nat->Prop))) (A_78:(nat->Prop)) (B_41:((nat->Prop)->Prop)), ((finite_finite_nat_o B_41)->(((ord_less_eq_nat_o_o B_41) ((image_nat_nat_o F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq ((nat->Prop)->Prop)) B_41) ((image_nat_nat_o F_22) C_12)))))))) of role axiom named fact_176_finite__subset__image % A new axiom: (forall (F_22:(nat->(nat->Prop))) (A_78:(nat->Prop)) (B_41:((nat->Prop)->Prop)), ((finite_finite_nat_o B_41)->(((ord_less_eq_nat_o_o B_41) ((image_nat_nat_o F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq ((nat->Prop)->Prop)) B_41) ((image_nat_nat_o F_22) C_12)))))))) % FOF formula (forall (F_22:(nat->(pname->Prop))) (A_78:(nat->Prop)) (B_41:((pname->Prop)->Prop)), ((finite297249702name_o B_41)->(((ord_le1205211808me_o_o B_41) ((image_nat_pname_o F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq ((pname->Prop)->Prop)) B_41) ((image_nat_pname_o F_22) C_12)))))))) of role axiom named fact_177_finite__subset__image % A new axiom: (forall (F_22:(nat->(pname->Prop))) (A_78:(nat->Prop)) (B_41:((pname->Prop)->Prop)), ((finite297249702name_o B_41)->(((ord_le1205211808me_o_o B_41) ((image_nat_pname_o F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq ((pname->Prop)->Prop)) B_41) ((image_nat_pname_o F_22) C_12)))))))) % FOF formula (forall (F_22:(nat->(x_a->Prop))) (A_78:(nat->Prop)) (B_41:((x_a->Prop)->Prop)), ((finite_finite_a_o B_41)->(((ord_less_eq_a_o_o B_41) ((image_nat_a_o F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq ((x_a->Prop)->Prop)) B_41) ((image_nat_a_o F_22) C_12)))))))) of role axiom named fact_178_finite__subset__image % A new axiom: (forall (F_22:(nat->(x_a->Prop))) (A_78:(nat->Prop)) (B_41:((x_a->Prop)->Prop)), ((finite_finite_a_o B_41)->(((ord_less_eq_a_o_o B_41) ((image_nat_a_o F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq ((x_a->Prop)->Prop)) B_41) ((image_nat_a_o F_22) C_12)))))))) % FOF formula (forall (F_22:(nat->pname)) (A_78:(nat->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_nat_pname F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq (pname->Prop)) B_41) ((image_nat_pname F_22) C_12)))))))) of role axiom named fact_179_finite__subset__image % A new axiom: (forall (F_22:(nat->pname)) (A_78:(nat->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_nat_pname F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq (pname->Prop)) B_41) ((image_nat_pname F_22) C_12)))))))) % FOF formula (forall (F_22:(nat->nat)) (A_78:(nat->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_nat_nat F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq (nat->Prop)) B_41) ((image_nat_nat F_22) C_12)))))))) of role axiom named fact_180_finite__subset__image % A new axiom: (forall (F_22:(nat->nat)) (A_78:(nat->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_nat_nat F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq (nat->Prop)) B_41) ((image_nat_nat F_22) C_12)))))))) % FOF formula (forall (F_22:(pname->x_a)) (A_78:(pname->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_pname_a F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq (x_a->Prop)) B_41) ((image_pname_a F_22) C_12)))))))) of role axiom named fact_181_finite__subset__image % A new axiom: (forall (F_22:(pname->x_a)) (A_78:(pname->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_pname_a F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq (x_a->Prop)) B_41) ((image_pname_a F_22) C_12)))))))) % FOF formula (forall (N_4:nat) (N_3:nat) (F_21:(nat->Prop)), ((forall (N_2:nat), ((ord_less_eq_o (F_21 N_2)) (F_21 (suc N_2))))->(((ord_less_eq_nat N_4) N_3)->((ord_less_eq_o (F_21 N_4)) (F_21 N_3))))) of role axiom named fact_182_lift__Suc__mono__le % A new axiom: (forall (N_4:nat) (N_3:nat) (F_21:(nat->Prop)), ((forall (N_2:nat), ((ord_less_eq_o (F_21 N_2)) (F_21 (suc N_2))))->(((ord_less_eq_nat N_4) N_3)->((ord_less_eq_o (F_21 N_4)) (F_21 N_3))))) % FOF formula (forall (N_4:nat) (N_3:nat) (F_21:(nat->(pname->Prop))), ((forall (N_2:nat), ((ord_less_eq_pname_o (F_21 N_2)) (F_21 (suc N_2))))->(((ord_less_eq_nat N_4) N_3)->((ord_less_eq_pname_o (F_21 N_4)) (F_21 N_3))))) of role axiom named fact_183_lift__Suc__mono__le % A new axiom: (forall (N_4:nat) (N_3:nat) (F_21:(nat->(pname->Prop))), ((forall (N_2:nat), ((ord_less_eq_pname_o (F_21 N_2)) (F_21 (suc N_2))))->(((ord_less_eq_nat N_4) N_3)->((ord_less_eq_pname_o (F_21 N_4)) (F_21 N_3))))) % FOF formula (forall (N_4:nat) (N_3:nat) (F_21:(nat->(nat->Prop))), ((forall (N_2:nat), ((ord_less_eq_nat_o (F_21 N_2)) (F_21 (suc N_2))))->(((ord_less_eq_nat N_4) N_3)->((ord_less_eq_nat_o (F_21 N_4)) (F_21 N_3))))) of role axiom named fact_184_lift__Suc__mono__le % A new axiom: (forall (N_4:nat) (N_3:nat) (F_21:(nat->(nat->Prop))), ((forall (N_2:nat), ((ord_less_eq_nat_o (F_21 N_2)) (F_21 (suc N_2))))->(((ord_less_eq_nat N_4) N_3)->((ord_less_eq_nat_o (F_21 N_4)) (F_21 N_3))))) % FOF formula (forall (N_4:nat) (N_3:nat) (F_21:(nat->(x_a->Prop))), ((forall (N_2:nat), ((ord_less_eq_a_o (F_21 N_2)) (F_21 (suc N_2))))->(((ord_less_eq_nat N_4) N_3)->((ord_less_eq_a_o (F_21 N_4)) (F_21 N_3))))) of role axiom named fact_185_lift__Suc__mono__le % A new axiom: (forall (N_4:nat) (N_3:nat) (F_21:(nat->(x_a->Prop))), ((forall (N_2:nat), ((ord_less_eq_a_o (F_21 N_2)) (F_21 (suc N_2))))->(((ord_less_eq_nat N_4) N_3)->((ord_less_eq_a_o (F_21 N_4)) (F_21 N_3))))) % FOF formula (forall (N_4:nat) (N_3:nat) (F_21:(nat->nat)), ((forall (N_2:nat), ((ord_less_eq_nat (F_21 N_2)) (F_21 (suc N_2))))->(((ord_less_eq_nat N_4) N_3)->((ord_less_eq_nat (F_21 N_4)) (F_21 N_3))))) of role axiom named fact_186_lift__Suc__mono__le % A new axiom: (forall (N_4:nat) (N_3:nat) (F_21:(nat->nat)), ((forall (N_2:nat), ((ord_less_eq_nat (F_21 N_2)) (F_21 (suc N_2))))->(((ord_less_eq_nat N_4) N_3)->((ord_less_eq_nat (F_21 N_4)) (F_21 N_3))))) % FOF formula (forall (F_20:(nat->pname)) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite_finite_pname ((image_nat_pname F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_187_pigeonhole__infinite % A new axiom: (forall (F_20:(nat->pname)) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite_finite_pname ((image_nat_pname F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(x_a->pname)) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite_finite_pname ((image_a_pname F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_188_pigeonhole__infinite % A new axiom: (forall (F_20:(x_a->pname)) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite_finite_pname ((image_a_pname F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(pname->pname)) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite_finite_pname ((image_pname_pname F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_189_pigeonhole__infinite % A new axiom: (forall (F_20:(pname->pname)) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite_finite_pname ((image_pname_pname F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:((nat->Prop)->pname)) (A_77:((nat->Prop)->Prop)), (((finite_finite_nat_o A_77)->False)->((finite_finite_pname ((image_nat_o_pname F_20) A_77))->((ex (nat->Prop)) (fun (X_1:(nat->Prop))=> ((and ((member_nat_o X_1) A_77)) ((finite_finite_nat_o (collect_nat_o (fun (A_2:(nat->Prop))=> ((and ((member_nat_o A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_190_pigeonhole__infinite % A new axiom: (forall (F_20:((nat->Prop)->pname)) (A_77:((nat->Prop)->Prop)), (((finite_finite_nat_o A_77)->False)->((finite_finite_pname ((image_nat_o_pname F_20) A_77))->((ex (nat->Prop)) (fun (X_1:(nat->Prop))=> ((and ((member_nat_o X_1) A_77)) ((finite_finite_nat_o (collect_nat_o (fun (A_2:(nat->Prop))=> ((and ((member_nat_o A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:((pname->Prop)->pname)) (A_77:((pname->Prop)->Prop)), (((finite297249702name_o A_77)->False)->((finite_finite_pname ((image_pname_o_pname F_20) A_77))->((ex (pname->Prop)) (fun (X_1:(pname->Prop))=> ((and ((member_pname_o X_1) A_77)) ((finite297249702name_o (collect_pname_o (fun (A_2:(pname->Prop))=> ((and ((member_pname_o A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_191_pigeonhole__infinite % A new axiom: (forall (F_20:((pname->Prop)->pname)) (A_77:((pname->Prop)->Prop)), (((finite297249702name_o A_77)->False)->((finite_finite_pname ((image_pname_o_pname F_20) A_77))->((ex (pname->Prop)) (fun (X_1:(pname->Prop))=> ((and ((member_pname_o X_1) A_77)) ((finite297249702name_o (collect_pname_o (fun (A_2:(pname->Prop))=> ((and ((member_pname_o A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:((x_a->Prop)->pname)) (A_77:((x_a->Prop)->Prop)), (((finite_finite_a_o A_77)->False)->((finite_finite_pname ((image_a_o_pname F_20) A_77))->((ex (x_a->Prop)) (fun (X_1:(x_a->Prop))=> ((and ((member_a_o X_1) A_77)) ((finite_finite_a_o (collect_a_o (fun (A_2:(x_a->Prop))=> ((and ((member_a_o A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_192_pigeonhole__infinite % A new axiom: (forall (F_20:((x_a->Prop)->pname)) (A_77:((x_a->Prop)->Prop)), (((finite_finite_a_o A_77)->False)->((finite_finite_pname ((image_a_o_pname F_20) A_77))->((ex (x_a->Prop)) (fun (X_1:(x_a->Prop))=> ((and ((member_a_o X_1) A_77)) ((finite_finite_a_o (collect_a_o (fun (A_2:(x_a->Prop))=> ((and ((member_a_o A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(nat->nat)) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite_finite_nat ((image_nat_nat F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_193_pigeonhole__infinite % A new axiom: (forall (F_20:(nat->nat)) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite_finite_nat ((image_nat_nat F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(x_a->nat)) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite_finite_nat ((image_a_nat F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_194_pigeonhole__infinite % A new axiom: (forall (F_20:(x_a->nat)) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite_finite_nat ((image_a_nat F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(pname->nat)) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite_finite_nat ((image_pname_nat F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_195_pigeonhole__infinite % A new axiom: (forall (F_20:(pname->nat)) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite_finite_nat ((image_pname_nat F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:((nat->Prop)->nat)) (A_77:((nat->Prop)->Prop)), (((finite_finite_nat_o A_77)->False)->((finite_finite_nat ((image_nat_o_nat F_20) A_77))->((ex (nat->Prop)) (fun (X_1:(nat->Prop))=> ((and ((member_nat_o X_1) A_77)) ((finite_finite_nat_o (collect_nat_o (fun (A_2:(nat->Prop))=> ((and ((member_nat_o A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_196_pigeonhole__infinite % A new axiom: (forall (F_20:((nat->Prop)->nat)) (A_77:((nat->Prop)->Prop)), (((finite_finite_nat_o A_77)->False)->((finite_finite_nat ((image_nat_o_nat F_20) A_77))->((ex (nat->Prop)) (fun (X_1:(nat->Prop))=> ((and ((member_nat_o X_1) A_77)) ((finite_finite_nat_o (collect_nat_o (fun (A_2:(nat->Prop))=> ((and ((member_nat_o A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:((pname->Prop)->nat)) (A_77:((pname->Prop)->Prop)), (((finite297249702name_o A_77)->False)->((finite_finite_nat ((image_pname_o_nat F_20) A_77))->((ex (pname->Prop)) (fun (X_1:(pname->Prop))=> ((and ((member_pname_o X_1) A_77)) ((finite297249702name_o (collect_pname_o (fun (A_2:(pname->Prop))=> ((and ((member_pname_o A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_197_pigeonhole__infinite % A new axiom: (forall (F_20:((pname->Prop)->nat)) (A_77:((pname->Prop)->Prop)), (((finite297249702name_o A_77)->False)->((finite_finite_nat ((image_pname_o_nat F_20) A_77))->((ex (pname->Prop)) (fun (X_1:(pname->Prop))=> ((and ((member_pname_o X_1) A_77)) ((finite297249702name_o (collect_pname_o (fun (A_2:(pname->Prop))=> ((and ((member_pname_o A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:((x_a->Prop)->nat)) (A_77:((x_a->Prop)->Prop)), (((finite_finite_a_o A_77)->False)->((finite_finite_nat ((image_a_o_nat F_20) A_77))->((ex (x_a->Prop)) (fun (X_1:(x_a->Prop))=> ((and ((member_a_o X_1) A_77)) ((finite_finite_a_o (collect_a_o (fun (A_2:(x_a->Prop))=> ((and ((member_a_o A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_198_pigeonhole__infinite % A new axiom: (forall (F_20:((x_a->Prop)->nat)) (A_77:((x_a->Prop)->Prop)), (((finite_finite_a_o A_77)->False)->((finite_finite_nat ((image_a_o_nat F_20) A_77))->((ex (x_a->Prop)) (fun (X_1:(x_a->Prop))=> ((and ((member_a_o X_1) A_77)) ((finite_finite_a_o (collect_a_o (fun (A_2:(x_a->Prop))=> ((and ((member_a_o A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(nat->x_a)) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite_finite_a ((image_nat_a F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq x_a) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_199_pigeonhole__infinite % A new axiom: (forall (F_20:(nat->x_a)) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite_finite_a ((image_nat_a F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq x_a) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(nat->(nat->Prop))) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite_finite_nat_o ((image_nat_nat_o F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq (nat->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_200_pigeonhole__infinite % A new axiom: (forall (F_20:(nat->(nat->Prop))) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite_finite_nat_o ((image_nat_nat_o F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq (nat->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(nat->(pname->Prop))) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite297249702name_o ((image_nat_pname_o F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq (pname->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_201_pigeonhole__infinite % A new axiom: (forall (F_20:(nat->(pname->Prop))) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite297249702name_o ((image_nat_pname_o F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq (pname->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(nat->(x_a->Prop))) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite_finite_a_o ((image_nat_a_o F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq (x_a->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_202_pigeonhole__infinite % A new axiom: (forall (F_20:(nat->(x_a->Prop))) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite_finite_a_o ((image_nat_a_o F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq (x_a->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(pname->(nat->Prop))) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite_finite_nat_o ((image_pname_nat_o F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq (nat->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_203_pigeonhole__infinite % A new axiom: (forall (F_20:(pname->(nat->Prop))) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite_finite_nat_o ((image_pname_nat_o F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq (nat->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(pname->(pname->Prop))) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite297249702name_o ((image_pname_pname_o F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq (pname->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_204_pigeonhole__infinite % A new axiom: (forall (F_20:(pname->(pname->Prop))) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite297249702name_o ((image_pname_pname_o F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq (pname->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(pname->(x_a->Prop))) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite_finite_a_o ((image_pname_a_o F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq (x_a->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_205_pigeonhole__infinite % A new axiom: (forall (F_20:(pname->(x_a->Prop))) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite_finite_a_o ((image_pname_a_o F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq (x_a->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(x_a->x_a)) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite_finite_a ((image_a_a F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq x_a) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_206_pigeonhole__infinite % A new axiom: (forall (F_20:(x_a->x_a)) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite_finite_a ((image_a_a F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq x_a) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(x_a->(nat->Prop))) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite_finite_nat_o ((image_a_nat_o F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq (nat->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_207_pigeonhole__infinite % A new axiom: (forall (F_20:(x_a->(nat->Prop))) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite_finite_nat_o ((image_a_nat_o F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq (nat->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(x_a->(pname->Prop))) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite297249702name_o ((image_a_pname_o F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq (pname->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_208_pigeonhole__infinite % A new axiom: (forall (F_20:(x_a->(pname->Prop))) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite297249702name_o ((image_a_pname_o F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq (pname->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(x_a->(x_a->Prop))) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite_finite_a_o ((image_a_a_o F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq (x_a->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_209_pigeonhole__infinite % A new axiom: (forall (F_20:(x_a->(x_a->Prop))) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite_finite_a_o ((image_a_a_o F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq (x_a->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (F_20:(pname->x_a)) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite_finite_a ((image_pname_a F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq x_a) (F_20 A_2)) (F_20 X_1))))))->False))))))) of role axiom named fact_210_pigeonhole__infinite % A new axiom: (forall (F_20:(pname->x_a)) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite_finite_a ((image_pname_a F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq x_a) (F_20 A_2)) (F_20 X_1))))))->False))))))) % FOF formula (forall (A_76:(pname->Prop)) (B_40:nat) (F_19:(pname->nat)) (X_30:pname), ((((eq nat) B_40) (F_19 X_30))->(((member_pname X_30) A_76)->((member_nat B_40) ((image_pname_nat F_19) A_76))))) of role axiom named fact_211_image__eqI % A new axiom: (forall (A_76:(pname->Prop)) (B_40:nat) (F_19:(pname->nat)) (X_30:pname), ((((eq nat) B_40) (F_19 X_30))->(((member_pname X_30) A_76)->((member_nat B_40) ((image_pname_nat F_19) A_76))))) % FOF formula (forall (A_76:(x_a->Prop)) (B_40:nat) (F_19:(x_a->nat)) (X_30:x_a), ((((eq nat) B_40) (F_19 X_30))->(((member_a X_30) A_76)->((member_nat B_40) ((image_a_nat F_19) A_76))))) of role axiom named fact_212_image__eqI % A new axiom: (forall (A_76:(x_a->Prop)) (B_40:nat) (F_19:(x_a->nat)) (X_30:x_a), ((((eq nat) B_40) (F_19 X_30))->(((member_a X_30) A_76)->((member_nat B_40) ((image_a_nat F_19) A_76))))) % FOF formula (forall (A_76:(nat->Prop)) (B_40:pname) (F_19:(nat->pname)) (X_30:nat), ((((eq pname) B_40) (F_19 X_30))->(((member_nat X_30) A_76)->((member_pname B_40) ((image_nat_pname F_19) A_76))))) of role axiom named fact_213_image__eqI % A new axiom: (forall (A_76:(nat->Prop)) (B_40:pname) (F_19:(nat->pname)) (X_30:nat), ((((eq pname) B_40) (F_19 X_30))->(((member_nat X_30) A_76)->((member_pname B_40) ((image_nat_pname F_19) A_76))))) % FOF formula (forall (A_76:(nat->Prop)) (B_40:x_a) (F_19:(nat->x_a)) (X_30:nat), ((((eq x_a) B_40) (F_19 X_30))->(((member_nat X_30) A_76)->((member_a B_40) ((image_nat_a F_19) A_76))))) of role axiom named fact_214_image__eqI % A new axiom: (forall (A_76:(nat->Prop)) (B_40:x_a) (F_19:(nat->x_a)) (X_30:nat), ((((eq x_a) B_40) (F_19 X_30))->(((member_nat X_30) A_76)->((member_a B_40) ((image_nat_a F_19) A_76))))) % FOF formula (forall (A_76:(pname->Prop)) (B_40:x_a) (F_19:(pname->x_a)) (X_30:pname), ((((eq x_a) B_40) (F_19 X_30))->(((member_pname X_30) A_76)->((member_a B_40) ((image_pname_a F_19) A_76))))) of role axiom named fact_215_image__eqI % A new axiom: (forall (A_76:(pname->Prop)) (B_40:x_a) (F_19:(pname->x_a)) (X_30:pname), ((((eq x_a) B_40) (F_19 X_30))->(((member_pname X_30) A_76)->((member_a B_40) ((image_pname_a F_19) A_76))))) % FOF formula (forall (A_75:(pname->Prop)) (B_39:(pname->Prop)), (((ord_less_eq_pname_o A_75) B_39)->(((ord_less_eq_pname_o B_39) A_75)->(((eq (pname->Prop)) A_75) B_39)))) of role axiom named fact_216_equalityI % A new axiom: (forall (A_75:(pname->Prop)) (B_39:(pname->Prop)), (((ord_less_eq_pname_o A_75) B_39)->(((ord_less_eq_pname_o B_39) A_75)->(((eq (pname->Prop)) A_75) B_39)))) % FOF formula (forall (A_75:(nat->Prop)) (B_39:(nat->Prop)), (((ord_less_eq_nat_o A_75) B_39)->(((ord_less_eq_nat_o B_39) A_75)->(((eq (nat->Prop)) A_75) B_39)))) of role axiom named fact_217_equalityI % A new axiom: (forall (A_75:(nat->Prop)) (B_39:(nat->Prop)), (((ord_less_eq_nat_o A_75) B_39)->(((ord_less_eq_nat_o B_39) A_75)->(((eq (nat->Prop)) A_75) B_39)))) % FOF formula (forall (A_75:(x_a->Prop)) (B_39:(x_a->Prop)), (((ord_less_eq_a_o A_75) B_39)->(((ord_less_eq_a_o B_39) A_75)->(((eq (x_a->Prop)) A_75) B_39)))) of role axiom named fact_218_equalityI % A new axiom: (forall (A_75:(x_a->Prop)) (B_39:(x_a->Prop)), (((ord_less_eq_a_o A_75) B_39)->(((ord_less_eq_a_o B_39) A_75)->(((eq (x_a->Prop)) A_75) B_39)))) % FOF formula (forall (C_11:nat) (A_74:(nat->Prop)) (B_38:(nat->Prop)), (((ord_less_eq_nat_o A_74) B_38)->(((member_nat C_11) A_74)->((member_nat C_11) B_38)))) of role axiom named fact_219_subsetD % A new axiom: (forall (C_11:nat) (A_74:(nat->Prop)) (B_38:(nat->Prop)), (((ord_less_eq_nat_o A_74) B_38)->(((member_nat C_11) A_74)->((member_nat C_11) B_38)))) % FOF formula (forall (C_11:pname) (A_74:(pname->Prop)) (B_38:(pname->Prop)), (((ord_less_eq_pname_o A_74) B_38)->(((member_pname C_11) A_74)->((member_pname C_11) B_38)))) of role axiom named fact_220_subsetD % A new axiom: (forall (C_11:pname) (A_74:(pname->Prop)) (B_38:(pname->Prop)), (((ord_less_eq_pname_o A_74) B_38)->(((member_pname C_11) A_74)->((member_pname C_11) B_38)))) % FOF formula (forall (C_11:x_a) (A_74:(x_a->Prop)) (B_38:(x_a->Prop)), (((ord_less_eq_a_o A_74) B_38)->(((member_a C_11) A_74)->((member_a C_11) B_38)))) of role axiom named fact_221_subsetD % A new axiom: (forall (C_11:x_a) (A_74:(x_a->Prop)) (B_38:(x_a->Prop)), (((ord_less_eq_a_o A_74) B_38)->(((member_a C_11) A_74)->((member_a C_11) B_38)))) % FOF formula (forall (B_37:nat) (A_73:nat) (B_36:(nat->Prop)), (((((member_nat A_73) B_36)->False)->(((eq nat) A_73) B_37))->((member_nat A_73) ((insert_nat B_37) B_36)))) of role axiom named fact_222_insertCI % A new axiom: (forall (B_37:nat) (A_73:nat) (B_36:(nat->Prop)), (((((member_nat A_73) B_36)->False)->(((eq nat) A_73) B_37))->((member_nat A_73) ((insert_nat B_37) B_36)))) % FOF formula (forall (B_37:pname) (A_73:pname) (B_36:(pname->Prop)), (((((member_pname A_73) B_36)->False)->(((eq pname) A_73) B_37))->((member_pname A_73) ((insert_pname B_37) B_36)))) of role axiom named fact_223_insertCI % A new axiom: (forall (B_37:pname) (A_73:pname) (B_36:(pname->Prop)), (((((member_pname A_73) B_36)->False)->(((eq pname) A_73) B_37))->((member_pname A_73) ((insert_pname B_37) B_36)))) % FOF formula (forall (B_37:x_a) (A_73:x_a) (B_36:(x_a->Prop)), (((((member_a A_73) B_36)->False)->(((eq x_a) A_73) B_37))->((member_a A_73) ((insert_a B_37) B_36)))) of role axiom named fact_224_insertCI % A new axiom: (forall (B_37:x_a) (A_73:x_a) (B_36:(x_a->Prop)), (((((member_a A_73) B_36)->False)->(((eq x_a) A_73) B_37))->((member_a A_73) ((insert_a B_37) B_36)))) % FOF formula (forall (A_72:nat) (B_35:nat) (A_71:(nat->Prop)), (((member_nat A_72) ((insert_nat B_35) A_71))->((not (((eq nat) A_72) B_35))->((member_nat A_72) A_71)))) of role axiom named fact_225_insertE % A new axiom: (forall (A_72:nat) (B_35:nat) (A_71:(nat->Prop)), (((member_nat A_72) ((insert_nat B_35) A_71))->((not (((eq nat) A_72) B_35))->((member_nat A_72) A_71)))) % FOF formula (forall (A_72:pname) (B_35:pname) (A_71:(pname->Prop)), (((member_pname A_72) ((insert_pname B_35) A_71))->((not (((eq pname) A_72) B_35))->((member_pname A_72) A_71)))) of role axiom named fact_226_insertE % A new axiom: (forall (A_72:pname) (B_35:pname) (A_71:(pname->Prop)), (((member_pname A_72) ((insert_pname B_35) A_71))->((not (((eq pname) A_72) B_35))->((member_pname A_72) A_71)))) % FOF formula (forall (A_72:x_a) (B_35:x_a) (A_71:(x_a->Prop)), (((member_a A_72) ((insert_a B_35) A_71))->((not (((eq x_a) A_72) B_35))->((member_a A_72) A_71)))) of role axiom named fact_227_insertE % A new axiom: (forall (A_72:x_a) (B_35:x_a) (A_71:(x_a->Prop)), (((member_a A_72) ((insert_a B_35) A_71))->((not (((eq x_a) A_72) B_35))->((member_a A_72) A_71)))) % FOF formula (forall (_TPTP_I:nat) (P:(nat->Prop)) (K:nat), ((P K)->((forall (N_2:nat), ((P (suc N_2))->(P N_2)))->(P ((minus_minus_nat K) _TPTP_I))))) of role axiom named fact_228_zero__induct__lemma % A new axiom: (forall (_TPTP_I:nat) (P:(nat->Prop)) (K:nat), ((P K)->((forall (N_2:nat), ((P (suc N_2))->(P N_2)))->(P ((minus_minus_nat K) _TPTP_I))))) % FOF formula (forall (N:nat) (M_2:nat), (((ord_less_eq_nat (suc N)) M_2)->((ex nat) (fun (M_1:nat)=> (((eq nat) M_2) (suc M_1)))))) of role axiom named fact_229_Suc__le__D % A new axiom: (forall (N:nat) (M_2:nat), (((ord_less_eq_nat (suc N)) M_2)->((ex nat) (fun (M_1:nat)=> (((eq nat) M_2) (suc M_1)))))) % FOF formula (forall (A_70:nat) (B_34:(nat->Prop)), ((member_nat A_70) ((insert_nat A_70) B_34))) of role axiom named fact_230_insertI1 % A new axiom: (forall (A_70:nat) (B_34:(nat->Prop)), ((member_nat A_70) ((insert_nat A_70) B_34))) % FOF formula (forall (A_70:pname) (B_34:(pname->Prop)), ((member_pname A_70) ((insert_pname A_70) B_34))) of role axiom named fact_231_insertI1 % A new axiom: (forall (A_70:pname) (B_34:(pname->Prop)), ((member_pname A_70) ((insert_pname A_70) B_34))) % FOF formula (forall (A_70:x_a) (B_34:(x_a->Prop)), ((member_a A_70) ((insert_a A_70) B_34))) of role axiom named fact_232_insertI1 % A new axiom: (forall (A_70:x_a) (B_34:(x_a->Prop)), ((member_a A_70) ((insert_a A_70) B_34))) % FOF formula (forall (A_69:(nat->Prop)) (B_33:((nat->Prop)->Prop)), (((eq ((nat->Prop)->Prop)) ((insert_nat_o A_69) B_33)) (collect_nat_o (fun (X_1:(nat->Prop))=> ((or (((eq (nat->Prop)) X_1) A_69)) ((member_nat_o X_1) B_33)))))) of role axiom named fact_233_insert__compr % A new axiom: (forall (A_69:(nat->Prop)) (B_33:((nat->Prop)->Prop)), (((eq ((nat->Prop)->Prop)) ((insert_nat_o A_69) B_33)) (collect_nat_o (fun (X_1:(nat->Prop))=> ((or (((eq (nat->Prop)) X_1) A_69)) ((member_nat_o X_1) B_33)))))) % FOF formula (forall (A_69:(pname->Prop)) (B_33:((pname->Prop)->Prop)), (((eq ((pname->Prop)->Prop)) ((insert_pname_o A_69) B_33)) (collect_pname_o (fun (X_1:(pname->Prop))=> ((or (((eq (pname->Prop)) X_1) A_69)) ((member_pname_o X_1) B_33)))))) of role axiom named fact_234_insert__compr % A new axiom: (forall (A_69:(pname->Prop)) (B_33:((pname->Prop)->Prop)), (((eq ((pname->Prop)->Prop)) ((insert_pname_o A_69) B_33)) (collect_pname_o (fun (X_1:(pname->Prop))=> ((or (((eq (pname->Prop)) X_1) A_69)) ((member_pname_o X_1) B_33)))))) % FOF formula (forall (A_69:(x_a->Prop)) (B_33:((x_a->Prop)->Prop)), (((eq ((x_a->Prop)->Prop)) ((insert_a_o A_69) B_33)) (collect_a_o (fun (X_1:(x_a->Prop))=> ((or (((eq (x_a->Prop)) X_1) A_69)) ((member_a_o X_1) B_33)))))) of role axiom named fact_235_insert__compr % A new axiom: (forall (A_69:(x_a->Prop)) (B_33:((x_a->Prop)->Prop)), (((eq ((x_a->Prop)->Prop)) ((insert_a_o A_69) B_33)) (collect_a_o (fun (X_1:(x_a->Prop))=> ((or (((eq (x_a->Prop)) X_1) A_69)) ((member_a_o X_1) B_33)))))) % FOF formula (forall (A_69:nat) (B_33:(nat->Prop)), (((eq (nat->Prop)) ((insert_nat A_69) B_33)) (collect_nat (fun (X_1:nat)=> ((or (((eq nat) X_1) A_69)) ((member_nat X_1) B_33)))))) of role axiom named fact_236_insert__compr % A new axiom: (forall (A_69:nat) (B_33:(nat->Prop)), (((eq (nat->Prop)) ((insert_nat A_69) B_33)) (collect_nat (fun (X_1:nat)=> ((or (((eq nat) X_1) A_69)) ((member_nat X_1) B_33)))))) % FOF formula (forall (A_69:pname) (B_33:(pname->Prop)), (((eq (pname->Prop)) ((insert_pname A_69) B_33)) (collect_pname (fun (X_1:pname)=> ((or (((eq pname) X_1) A_69)) ((member_pname X_1) B_33)))))) of role axiom named fact_237_insert__compr % A new axiom: (forall (A_69:pname) (B_33:(pname->Prop)), (((eq (pname->Prop)) ((insert_pname A_69) B_33)) (collect_pname (fun (X_1:pname)=> ((or (((eq pname) X_1) A_69)) ((member_pname X_1) B_33)))))) % FOF formula (forall (A_69:x_a) (B_33:(x_a->Prop)), (((eq (x_a->Prop)) ((insert_a A_69) B_33)) (collect_a (fun (X_1:x_a)=> ((or (((eq x_a) X_1) A_69)) ((member_a X_1) B_33)))))) of role axiom named fact_238_insert__compr % A new axiom: (forall (A_69:x_a) (B_33:(x_a->Prop)), (((eq (x_a->Prop)) ((insert_a A_69) B_33)) (collect_a (fun (X_1:x_a)=> ((or (((eq x_a) X_1) A_69)) ((member_a X_1) B_33)))))) % FOF formula (forall (A_68:pname) (P_8:(pname->Prop)), (((eq (pname->Prop)) ((insert_pname A_68) (collect_pname P_8))) (collect_pname (fun (U_1:pname)=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq pname) U_1) A_68))) (P_8 U_1)))))) of role axiom named fact_239_insert__Collect % A new axiom: (forall (A_68:pname) (P_8:(pname->Prop)), (((eq (pname->Prop)) ((insert_pname A_68) (collect_pname P_8))) (collect_pname (fun (U_1:pname)=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq pname) U_1) A_68))) (P_8 U_1)))))) % FOF formula (forall (A_68:(nat->Prop)) (P_8:((nat->Prop)->Prop)), (((eq ((nat->Prop)->Prop)) ((insert_nat_o A_68) (collect_nat_o P_8))) (collect_nat_o (fun (U_1:(nat->Prop))=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq (nat->Prop)) U_1) A_68))) (P_8 U_1)))))) of role axiom named fact_240_insert__Collect % A new axiom: (forall (A_68:(nat->Prop)) (P_8:((nat->Prop)->Prop)), (((eq ((nat->Prop)->Prop)) ((insert_nat_o A_68) (collect_nat_o P_8))) (collect_nat_o (fun (U_1:(nat->Prop))=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq (nat->Prop)) U_1) A_68))) (P_8 U_1)))))) % FOF formula (forall (A_68:(pname->Prop)) (P_8:((pname->Prop)->Prop)), (((eq ((pname->Prop)->Prop)) ((insert_pname_o A_68) (collect_pname_o P_8))) (collect_pname_o (fun (U_1:(pname->Prop))=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq (pname->Prop)) U_1) A_68))) (P_8 U_1)))))) of role axiom named fact_241_insert__Collect % A new axiom: (forall (A_68:(pname->Prop)) (P_8:((pname->Prop)->Prop)), (((eq ((pname->Prop)->Prop)) ((insert_pname_o A_68) (collect_pname_o P_8))) (collect_pname_o (fun (U_1:(pname->Prop))=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq (pname->Prop)) U_1) A_68))) (P_8 U_1)))))) % FOF formula (forall (A_68:(x_a->Prop)) (P_8:((x_a->Prop)->Prop)), (((eq ((x_a->Prop)->Prop)) ((insert_a_o A_68) (collect_a_o P_8))) (collect_a_o (fun (U_1:(x_a->Prop))=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq (x_a->Prop)) U_1) A_68))) (P_8 U_1)))))) of role axiom named fact_242_insert__Collect % A new axiom: (forall (A_68:(x_a->Prop)) (P_8:((x_a->Prop)->Prop)), (((eq ((x_a->Prop)->Prop)) ((insert_a_o A_68) (collect_a_o P_8))) (collect_a_o (fun (U_1:(x_a->Prop))=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq (x_a->Prop)) U_1) A_68))) (P_8 U_1)))))) % FOF formula (forall (A_68:nat) (P_8:(nat->Prop)), (((eq (nat->Prop)) ((insert_nat A_68) (collect_nat P_8))) (collect_nat (fun (U_1:nat)=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq nat) U_1) A_68))) (P_8 U_1)))))) of role axiom named fact_243_insert__Collect % A new axiom: (forall (A_68:nat) (P_8:(nat->Prop)), (((eq (nat->Prop)) ((insert_nat A_68) (collect_nat P_8))) (collect_nat (fun (U_1:nat)=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq nat) U_1) A_68))) (P_8 U_1)))))) % FOF formula (forall (A_68:x_a) (P_8:(x_a->Prop)), (((eq (x_a->Prop)) ((insert_a A_68) (collect_a P_8))) (collect_a (fun (U_1:x_a)=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq x_a) U_1) A_68))) (P_8 U_1)))))) of role axiom named fact_244_insert__Collect % A new axiom: (forall (A_68:x_a) (P_8:(x_a->Prop)), (((eq (x_a->Prop)) ((insert_a A_68) (collect_a P_8))) (collect_a (fun (U_1:x_a)=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq x_a) U_1) A_68))) (P_8 U_1)))))) % FOF formula (forall (X_29:pname) (A_67:(pname->Prop)), (((eq (pname->Prop)) ((insert_pname X_29) ((insert_pname X_29) A_67))) ((insert_pname X_29) A_67))) of role axiom named fact_245_insert__absorb2 % A new axiom: (forall (X_29:pname) (A_67:(pname->Prop)), (((eq (pname->Prop)) ((insert_pname X_29) ((insert_pname X_29) A_67))) ((insert_pname X_29) A_67))) % FOF formula (forall (X_29:nat) (A_67:(nat->Prop)), (((eq (nat->Prop)) ((insert_nat X_29) ((insert_nat X_29) A_67))) ((insert_nat X_29) A_67))) of role axiom named fact_246_insert__absorb2 % A new axiom: (forall (X_29:nat) (A_67:(nat->Prop)), (((eq (nat->Prop)) ((insert_nat X_29) ((insert_nat X_29) A_67))) ((insert_nat X_29) A_67))) % FOF formula (forall (X_29:x_a) (A_67:(x_a->Prop)), (((eq (x_a->Prop)) ((insert_a X_29) ((insert_a X_29) A_67))) ((insert_a X_29) A_67))) of role axiom named fact_247_insert__absorb2 % A new axiom: (forall (X_29:x_a) (A_67:(x_a->Prop)), (((eq (x_a->Prop)) ((insert_a X_29) ((insert_a X_29) A_67))) ((insert_a X_29) A_67))) % FOF formula (forall (X_28:pname) (Y_12:pname) (A_66:(pname->Prop)), (((eq (pname->Prop)) ((insert_pname X_28) ((insert_pname Y_12) A_66))) ((insert_pname Y_12) ((insert_pname X_28) A_66)))) of role axiom named fact_248_insert__commute % A new axiom: (forall (X_28:pname) (Y_12:pname) (A_66:(pname->Prop)), (((eq (pname->Prop)) ((insert_pname X_28) ((insert_pname Y_12) A_66))) ((insert_pname Y_12) ((insert_pname X_28) A_66)))) % FOF formula (forall (X_28:nat) (Y_12:nat) (A_66:(nat->Prop)), (((eq (nat->Prop)) ((insert_nat X_28) ((insert_nat Y_12) A_66))) ((insert_nat Y_12) ((insert_nat X_28) A_66)))) of role axiom named fact_249_insert__commute % A new axiom: (forall (X_28:nat) (Y_12:nat) (A_66:(nat->Prop)), (((eq (nat->Prop)) ((insert_nat X_28) ((insert_nat Y_12) A_66))) ((insert_nat Y_12) ((insert_nat X_28) A_66)))) % FOF formula (forall (X_28:x_a) (Y_12:x_a) (A_66:(x_a->Prop)), (((eq (x_a->Prop)) ((insert_a X_28) ((insert_a Y_12) A_66))) ((insert_a Y_12) ((insert_a X_28) A_66)))) of role axiom named fact_250_insert__commute % A new axiom: (forall (X_28:x_a) (Y_12:x_a) (A_66:(x_a->Prop)), (((eq (x_a->Prop)) ((insert_a X_28) ((insert_a Y_12) A_66))) ((insert_a Y_12) ((insert_a X_28) A_66)))) % FOF formula (forall (A_65:nat) (B_32:nat) (A_64:(nat->Prop)), ((iff ((member_nat A_65) ((insert_nat B_32) A_64))) ((or (((eq nat) A_65) B_32)) ((member_nat A_65) A_64)))) of role axiom named fact_251_insert__iff % A new axiom: (forall (A_65:nat) (B_32:nat) (A_64:(nat->Prop)), ((iff ((member_nat A_65) ((insert_nat B_32) A_64))) ((or (((eq nat) A_65) B_32)) ((member_nat A_65) A_64)))) % FOF formula (forall (A_65:pname) (B_32:pname) (A_64:(pname->Prop)), ((iff ((member_pname A_65) ((insert_pname B_32) A_64))) ((or (((eq pname) A_65) B_32)) ((member_pname A_65) A_64)))) of role axiom named fact_252_insert__iff % A new axiom: (forall (A_65:pname) (B_32:pname) (A_64:(pname->Prop)), ((iff ((member_pname A_65) ((insert_pname B_32) A_64))) ((or (((eq pname) A_65) B_32)) ((member_pname A_65) A_64)))) % FOF formula (forall (A_65:x_a) (B_32:x_a) (A_64:(x_a->Prop)), ((iff ((member_a A_65) ((insert_a B_32) A_64))) ((or (((eq x_a) A_65) B_32)) ((member_a A_65) A_64)))) of role axiom named fact_253_insert__iff % A new axiom: (forall (A_65:x_a) (B_32:x_a) (A_64:(x_a->Prop)), ((iff ((member_a A_65) ((insert_a B_32) A_64))) ((or (((eq x_a) A_65) B_32)) ((member_a A_65) A_64)))) % FOF formula (forall (Y_11:pname) (A_63:(pname->Prop)) (X_27:pname), ((iff (((insert_pname Y_11) A_63) X_27)) ((or (((eq pname) Y_11) X_27)) (A_63 X_27)))) of role axiom named fact_254_insert__code % A new axiom: (forall (Y_11:pname) (A_63:(pname->Prop)) (X_27:pname), ((iff (((insert_pname Y_11) A_63) X_27)) ((or (((eq pname) Y_11) X_27)) (A_63 X_27)))) % FOF formula (forall (Y_11:nat) (A_63:(nat->Prop)) (X_27:nat), ((iff (((insert_nat Y_11) A_63) X_27)) ((or (((eq nat) Y_11) X_27)) (A_63 X_27)))) of role axiom named fact_255_insert__code % A new axiom: (forall (Y_11:nat) (A_63:(nat->Prop)) (X_27:nat), ((iff (((insert_nat Y_11) A_63) X_27)) ((or (((eq nat) Y_11) X_27)) (A_63 X_27)))) % FOF formula (forall (Y_11:x_a) (A_63:(x_a->Prop)) (X_27:x_a), ((iff (((insert_a Y_11) A_63) X_27)) ((or (((eq x_a) Y_11) X_27)) (A_63 X_27)))) of role axiom named fact_256_insert__code % A new axiom: (forall (Y_11:x_a) (A_63:(x_a->Prop)) (X_27:x_a), ((iff (((insert_a Y_11) A_63) X_27)) ((or (((eq x_a) Y_11) X_27)) (A_63 X_27)))) % FOF formula (forall (B_31:(nat->Prop)) (X_26:nat) (A_62:(nat->Prop)), ((((member_nat X_26) A_62)->False)->((((member_nat X_26) B_31)->False)->((iff (((eq (nat->Prop)) ((insert_nat X_26) A_62)) ((insert_nat X_26) B_31))) (((eq (nat->Prop)) A_62) B_31))))) of role axiom named fact_257_insert__ident % A new axiom: (forall (B_31:(nat->Prop)) (X_26:nat) (A_62:(nat->Prop)), ((((member_nat X_26) A_62)->False)->((((member_nat X_26) B_31)->False)->((iff (((eq (nat->Prop)) ((insert_nat X_26) A_62)) ((insert_nat X_26) B_31))) (((eq (nat->Prop)) A_62) B_31))))) % FOF formula (forall (B_31:(pname->Prop)) (X_26:pname) (A_62:(pname->Prop)), ((((member_pname X_26) A_62)->False)->((((member_pname X_26) B_31)->False)->((iff (((eq (pname->Prop)) ((insert_pname X_26) A_62)) ((insert_pname X_26) B_31))) (((eq (pname->Prop)) A_62) B_31))))) of role axiom named fact_258_insert__ident % A new axiom: (forall (B_31:(pname->Prop)) (X_26:pname) (A_62:(pname->Prop)), ((((member_pname X_26) A_62)->False)->((((member_pname X_26) B_31)->False)->((iff (((eq (pname->Prop)) ((insert_pname X_26) A_62)) ((insert_pname X_26) B_31))) (((eq (pname->Prop)) A_62) B_31))))) % FOF formula (forall (B_31:(x_a->Prop)) (X_26:x_a) (A_62:(x_a->Prop)), ((((member_a X_26) A_62)->False)->((((member_a X_26) B_31)->False)->((iff (((eq (x_a->Prop)) ((insert_a X_26) A_62)) ((insert_a X_26) B_31))) (((eq (x_a->Prop)) A_62) B_31))))) of role axiom named fact_259_insert__ident % A new axiom: (forall (B_31:(x_a->Prop)) (X_26:x_a) (A_62:(x_a->Prop)), ((((member_a X_26) A_62)->False)->((((member_a X_26) B_31)->False)->((iff (((eq (x_a->Prop)) ((insert_a X_26) A_62)) ((insert_a X_26) B_31))) (((eq (x_a->Prop)) A_62) B_31))))) % FOF formula (forall (B_30:nat) (A_61:nat) (B_29:(nat->Prop)), (((member_nat A_61) B_29)->((member_nat A_61) ((insert_nat B_30) B_29)))) of role axiom named fact_260_insertI2 % A new axiom: (forall (B_30:nat) (A_61:nat) (B_29:(nat->Prop)), (((member_nat A_61) B_29)->((member_nat A_61) ((insert_nat B_30) B_29)))) % FOF formula (forall (B_30:pname) (A_61:pname) (B_29:(pname->Prop)), (((member_pname A_61) B_29)->((member_pname A_61) ((insert_pname B_30) B_29)))) of role axiom named fact_261_insertI2 % A new axiom: (forall (B_30:pname) (A_61:pname) (B_29:(pname->Prop)), (((member_pname A_61) B_29)->((member_pname A_61) ((insert_pname B_30) B_29)))) % FOF formula (forall (B_30:x_a) (A_61:x_a) (B_29:(x_a->Prop)), (((member_a A_61) B_29)->((member_a A_61) ((insert_a B_30) B_29)))) of role axiom named fact_262_insertI2 % A new axiom: (forall (B_30:x_a) (A_61:x_a) (B_29:(x_a->Prop)), (((member_a A_61) B_29)->((member_a A_61) ((insert_a B_30) B_29)))) % FOF formula (forall (A_60:nat) (A_59:(nat->Prop)), (((member_nat A_60) A_59)->(((eq (nat->Prop)) ((insert_nat A_60) A_59)) A_59))) of role axiom named fact_263_insert__absorb % A new axiom: (forall (A_60:nat) (A_59:(nat->Prop)), (((member_nat A_60) A_59)->(((eq (nat->Prop)) ((insert_nat A_60) A_59)) A_59))) % FOF formula (forall (A_60:pname) (A_59:(pname->Prop)), (((member_pname A_60) A_59)->(((eq (pname->Prop)) ((insert_pname A_60) A_59)) A_59))) of role axiom named fact_264_insert__absorb % A new axiom: (forall (A_60:pname) (A_59:(pname->Prop)), (((member_pname A_60) A_59)->(((eq (pname->Prop)) ((insert_pname A_60) A_59)) A_59))) % FOF formula (forall (A_60:x_a) (A_59:(x_a->Prop)), (((member_a A_60) A_59)->(((eq (x_a->Prop)) ((insert_a A_60) A_59)) A_59))) of role axiom named fact_265_insert__absorb % A new axiom: (forall (A_60:x_a) (A_59:(x_a->Prop)), (((member_a A_60) A_59)->(((eq (x_a->Prop)) ((insert_a A_60) A_59)) A_59))) % FOF formula (forall (A_58:(pname->Prop)), ((ord_less_eq_pname_o A_58) A_58)) of role axiom named fact_266_subset__refl % A new axiom: (forall (A_58:(pname->Prop)), ((ord_less_eq_pname_o A_58) A_58)) % FOF formula (forall (A_58:(nat->Prop)), ((ord_less_eq_nat_o A_58) A_58)) of role axiom named fact_267_subset__refl % A new axiom: (forall (A_58:(nat->Prop)), ((ord_less_eq_nat_o A_58) A_58)) % FOF formula (forall (A_58:(x_a->Prop)), ((ord_less_eq_a_o A_58) A_58)) of role axiom named fact_268_subset__refl % A new axiom: (forall (A_58:(x_a->Prop)), ((ord_less_eq_a_o A_58) A_58)) % FOF formula (forall (A_57:(pname->Prop)) (B_28:(pname->Prop)), ((iff (((eq (pname->Prop)) A_57) B_28)) ((and ((ord_less_eq_pname_o A_57) B_28)) ((ord_less_eq_pname_o B_28) A_57)))) of role axiom named fact_269_set__eq__subset % A new axiom: (forall (A_57:(pname->Prop)) (B_28:(pname->Prop)), ((iff (((eq (pname->Prop)) A_57) B_28)) ((and ((ord_less_eq_pname_o A_57) B_28)) ((ord_less_eq_pname_o B_28) A_57)))) % FOF formula (forall (A_57:(nat->Prop)) (B_28:(nat->Prop)), ((iff (((eq (nat->Prop)) A_57) B_28)) ((and ((ord_less_eq_nat_o A_57) B_28)) ((ord_less_eq_nat_o B_28) A_57)))) of role axiom named fact_270_set__eq__subset % A new axiom: (forall (A_57:(nat->Prop)) (B_28:(nat->Prop)), ((iff (((eq (nat->Prop)) A_57) B_28)) ((and ((ord_less_eq_nat_o A_57) B_28)) ((ord_less_eq_nat_o B_28) A_57)))) % FOF formula (forall (A_57:(x_a->Prop)) (B_28:(x_a->Prop)), ((iff (((eq (x_a->Prop)) A_57) B_28)) ((and ((ord_less_eq_a_o A_57) B_28)) ((ord_less_eq_a_o B_28) A_57)))) of role axiom named fact_271_set__eq__subset % A new axiom: (forall (A_57:(x_a->Prop)) (B_28:(x_a->Prop)), ((iff (((eq (x_a->Prop)) A_57) B_28)) ((and ((ord_less_eq_a_o A_57) B_28)) ((ord_less_eq_a_o B_28) A_57)))) % FOF formula (forall (A_56:(pname->Prop)) (B_27:(pname->Prop)), ((((eq (pname->Prop)) A_56) B_27)->((ord_less_eq_pname_o A_56) B_27))) of role axiom named fact_272_equalityD1 % A new axiom: (forall (A_56:(pname->Prop)) (B_27:(pname->Prop)), ((((eq (pname->Prop)) A_56) B_27)->((ord_less_eq_pname_o A_56) B_27))) % FOF formula (forall (A_56:(nat->Prop)) (B_27:(nat->Prop)), ((((eq (nat->Prop)) A_56) B_27)->((ord_less_eq_nat_o A_56) B_27))) of role axiom named fact_273_equalityD1 % A new axiom: (forall (A_56:(nat->Prop)) (B_27:(nat->Prop)), ((((eq (nat->Prop)) A_56) B_27)->((ord_less_eq_nat_o A_56) B_27))) % FOF formula (forall (A_56:(x_a->Prop)) (B_27:(x_a->Prop)), ((((eq (x_a->Prop)) A_56) B_27)->((ord_less_eq_a_o A_56) B_27))) of role axiom named fact_274_equalityD1 % A new axiom: (forall (A_56:(x_a->Prop)) (B_27:(x_a->Prop)), ((((eq (x_a->Prop)) A_56) B_27)->((ord_less_eq_a_o A_56) B_27))) % FOF formula (forall (A_55:(pname->Prop)) (B_26:(pname->Prop)), ((((eq (pname->Prop)) A_55) B_26)->((ord_less_eq_pname_o B_26) A_55))) of role axiom named fact_275_equalityD2 % A new axiom: (forall (A_55:(pname->Prop)) (B_26:(pname->Prop)), ((((eq (pname->Prop)) A_55) B_26)->((ord_less_eq_pname_o B_26) A_55))) % FOF formula (forall (A_55:(nat->Prop)) (B_26:(nat->Prop)), ((((eq (nat->Prop)) A_55) B_26)->((ord_less_eq_nat_o B_26) A_55))) of role axiom named fact_276_equalityD2 % A new axiom: (forall (A_55:(nat->Prop)) (B_26:(nat->Prop)), ((((eq (nat->Prop)) A_55) B_26)->((ord_less_eq_nat_o B_26) A_55))) % FOF formula (forall (A_55:(x_a->Prop)) (B_26:(x_a->Prop)), ((((eq (x_a->Prop)) A_55) B_26)->((ord_less_eq_a_o B_26) A_55))) of role axiom named fact_277_equalityD2 % A new axiom: (forall (A_55:(x_a->Prop)) (B_26:(x_a->Prop)), ((((eq (x_a->Prop)) A_55) B_26)->((ord_less_eq_a_o B_26) A_55))) % FOF formula (forall (X_25:nat) (A_54:(nat->Prop)) (B_25:(nat->Prop)), (((ord_less_eq_nat_o A_54) B_25)->(((member_nat X_25) A_54)->((member_nat X_25) B_25)))) of role axiom named fact_278_in__mono % A new axiom: (forall (X_25:nat) (A_54:(nat->Prop)) (B_25:(nat->Prop)), (((ord_less_eq_nat_o A_54) B_25)->(((member_nat X_25) A_54)->((member_nat X_25) B_25)))) % FOF formula (forall (X_25:pname) (A_54:(pname->Prop)) (B_25:(pname->Prop)), (((ord_less_eq_pname_o A_54) B_25)->(((member_pname X_25) A_54)->((member_pname X_25) B_25)))) of role axiom named fact_279_in__mono % A new axiom: (forall (X_25:pname) (A_54:(pname->Prop)) (B_25:(pname->Prop)), (((ord_less_eq_pname_o A_54) B_25)->(((member_pname X_25) A_54)->((member_pname X_25) B_25)))) % FOF formula (forall (X_25:x_a) (A_54:(x_a->Prop)) (B_25:(x_a->Prop)), (((ord_less_eq_a_o A_54) B_25)->(((member_a X_25) A_54)->((member_a X_25) B_25)))) of role axiom named fact_280_in__mono % A new axiom: (forall (X_25:x_a) (A_54:(x_a->Prop)) (B_25:(x_a->Prop)), (((ord_less_eq_a_o A_54) B_25)->(((member_a X_25) A_54)->((member_a X_25) B_25)))) % FOF formula (forall (B_24:(nat->Prop)) (X_24:nat) (A_53:(nat->Prop)), (((member_nat X_24) A_53)->(((ord_less_eq_nat_o A_53) B_24)->((member_nat X_24) B_24)))) of role axiom named fact_281_set__rev__mp % A new axiom: (forall (B_24:(nat->Prop)) (X_24:nat) (A_53:(nat->Prop)), (((member_nat X_24) A_53)->(((ord_less_eq_nat_o A_53) B_24)->((member_nat X_24) B_24)))) % FOF formula (forall (B_24:(pname->Prop)) (X_24:pname) (A_53:(pname->Prop)), (((member_pname X_24) A_53)->(((ord_less_eq_pname_o A_53) B_24)->((member_pname X_24) B_24)))) of role axiom named fact_282_set__rev__mp % A new axiom: (forall (B_24:(pname->Prop)) (X_24:pname) (A_53:(pname->Prop)), (((member_pname X_24) A_53)->(((ord_less_eq_pname_o A_53) B_24)->((member_pname X_24) B_24)))) % FOF formula (forall (B_24:(x_a->Prop)) (X_24:x_a) (A_53:(x_a->Prop)), (((member_a X_24) A_53)->(((ord_less_eq_a_o A_53) B_24)->((member_a X_24) B_24)))) of role axiom named fact_283_set__rev__mp % A new axiom: (forall (B_24:(x_a->Prop)) (X_24:x_a) (A_53:(x_a->Prop)), (((member_a X_24) A_53)->(((ord_less_eq_a_o A_53) B_24)->((member_a X_24) B_24)))) % FOF formula (forall (X_23:nat) (A_52:(nat->Prop)) (B_23:(nat->Prop)), (((ord_less_eq_nat_o A_52) B_23)->(((member_nat X_23) A_52)->((member_nat X_23) B_23)))) of role axiom named fact_284_set__mp % A new axiom: (forall (X_23:nat) (A_52:(nat->Prop)) (B_23:(nat->Prop)), (((ord_less_eq_nat_o A_52) B_23)->(((member_nat X_23) A_52)->((member_nat X_23) B_23)))) % FOF formula (forall (X_23:pname) (A_52:(pname->Prop)) (B_23:(pname->Prop)), (((ord_less_eq_pname_o A_52) B_23)->(((member_pname X_23) A_52)->((member_pname X_23) B_23)))) of role axiom named fact_285_set__mp % A new axiom: (forall (X_23:pname) (A_52:(pname->Prop)) (B_23:(pname->Prop)), (((ord_less_eq_pname_o A_52) B_23)->(((member_pname X_23) A_52)->((member_pname X_23) B_23)))) % FOF formula (forall (X_23:x_a) (A_52:(x_a->Prop)) (B_23:(x_a->Prop)), (((ord_less_eq_a_o A_52) B_23)->(((member_a X_23) A_52)->((member_a X_23) B_23)))) of role axiom named fact_286_set__mp % A new axiom: (forall (X_23:x_a) (A_52:(x_a->Prop)) (B_23:(x_a->Prop)), (((ord_less_eq_a_o A_52) B_23)->(((member_a X_23) A_52)->((member_a X_23) B_23)))) % FOF formula (forall (X_22:nat) (A_51:(nat->Prop)), ((iff ((member_nat X_22) A_51)) (A_51 X_22))) of role axiom named fact_287_mem__def % A new axiom: (forall (X_22:nat) (A_51:(nat->Prop)), ((iff ((member_nat X_22) A_51)) (A_51 X_22))) % FOF formula (forall (X_22:pname) (A_51:(pname->Prop)), ((iff ((member_pname X_22) A_51)) (A_51 X_22))) of role axiom named fact_288_mem__def % A new axiom: (forall (X_22:pname) (A_51:(pname->Prop)), ((iff ((member_pname X_22) A_51)) (A_51 X_22))) % FOF formula (forall (X_22:x_a) (A_51:(x_a->Prop)), ((iff ((member_a X_22) A_51)) (A_51 X_22))) of role axiom named fact_289_mem__def % A new axiom: (forall (X_22:x_a) (A_51:(x_a->Prop)), ((iff ((member_a X_22) A_51)) (A_51 X_22))) % FOF formula (forall (P_7:(x_a->Prop)), (((eq (x_a->Prop)) (collect_a P_7)) P_7)) of role axiom named fact_290_Collect__def % A new axiom: (forall (P_7:(x_a->Prop)), (((eq (x_a->Prop)) (collect_a P_7)) P_7)) % FOF formula (forall (P_7:(pname->Prop)), (((eq (pname->Prop)) (collect_pname P_7)) P_7)) of role axiom named fact_291_Collect__def % A new axiom: (forall (P_7:(pname->Prop)), (((eq (pname->Prop)) (collect_pname P_7)) P_7)) % FOF formula (forall (P_7:((nat->Prop)->Prop)), (((eq ((nat->Prop)->Prop)) (collect_nat_o P_7)) P_7)) of role axiom named fact_292_Collect__def % A new axiom: (forall (P_7:((nat->Prop)->Prop)), (((eq ((nat->Prop)->Prop)) (collect_nat_o P_7)) P_7)) % FOF formula (forall (P_7:((pname->Prop)->Prop)), (((eq ((pname->Prop)->Prop)) (collect_pname_o P_7)) P_7)) of role axiom named fact_293_Collect__def % A new axiom: (forall (P_7:((pname->Prop)->Prop)), (((eq ((pname->Prop)->Prop)) (collect_pname_o P_7)) P_7)) % FOF formula (forall (P_7:((x_a->Prop)->Prop)), (((eq ((x_a->Prop)->Prop)) (collect_a_o P_7)) P_7)) of role axiom named fact_294_Collect__def % A new axiom: (forall (P_7:((x_a->Prop)->Prop)), (((eq ((x_a->Prop)->Prop)) (collect_a_o P_7)) P_7)) % FOF formula (forall (P_7:(nat->Prop)), (((eq (nat->Prop)) (collect_nat P_7)) P_7)) of role axiom named fact_295_Collect__def % A new axiom: (forall (P_7:(nat->Prop)), (((eq (nat->Prop)) (collect_nat P_7)) P_7)) % FOF formula (forall (C_10:(pname->Prop)) (A_50:(pname->Prop)) (B_22:(pname->Prop)), (((ord_less_eq_pname_o A_50) B_22)->(((ord_less_eq_pname_o B_22) C_10)->((ord_less_eq_pname_o A_50) C_10)))) of role axiom named fact_296_subset__trans % A new axiom: (forall (C_10:(pname->Prop)) (A_50:(pname->Prop)) (B_22:(pname->Prop)), (((ord_less_eq_pname_o A_50) B_22)->(((ord_less_eq_pname_o B_22) C_10)->((ord_less_eq_pname_o A_50) C_10)))) % FOF formula (forall (C_10:(nat->Prop)) (A_50:(nat->Prop)) (B_22:(nat->Prop)), (((ord_less_eq_nat_o A_50) B_22)->(((ord_less_eq_nat_o B_22) C_10)->((ord_less_eq_nat_o A_50) C_10)))) of role axiom named fact_297_subset__trans % A new axiom: (forall (C_10:(nat->Prop)) (A_50:(nat->Prop)) (B_22:(nat->Prop)), (((ord_less_eq_nat_o A_50) B_22)->(((ord_less_eq_nat_o B_22) C_10)->((ord_less_eq_nat_o A_50) C_10)))) % FOF formula (forall (C_10:(x_a->Prop)) (A_50:(x_a->Prop)) (B_22:(x_a->Prop)), (((ord_less_eq_a_o A_50) B_22)->(((ord_less_eq_a_o B_22) C_10)->((ord_less_eq_a_o A_50) C_10)))) of role axiom named fact_298_subset__trans % A new axiom: (forall (C_10:(x_a->Prop)) (A_50:(x_a->Prop)) (B_22:(x_a->Prop)), (((ord_less_eq_a_o A_50) B_22)->(((ord_less_eq_a_o B_22) C_10)->((ord_less_eq_a_o A_50) C_10)))) % FOF formula (forall (A_49:(pname->Prop)) (B_21:(pname->Prop)), ((((eq (pname->Prop)) A_49) B_21)->((((ord_less_eq_pname_o A_49) B_21)->(((ord_less_eq_pname_o B_21) A_49)->False))->False))) of role axiom named fact_299_equalityE % A new axiom: (forall (A_49:(pname->Prop)) (B_21:(pname->Prop)), ((((eq (pname->Prop)) A_49) B_21)->((((ord_less_eq_pname_o A_49) B_21)->(((ord_less_eq_pname_o B_21) A_49)->False))->False))) % FOF formula (forall (A_49:(nat->Prop)) (B_21:(nat->Prop)), ((((eq (nat->Prop)) A_49) B_21)->((((ord_less_eq_nat_o A_49) B_21)->(((ord_less_eq_nat_o B_21) A_49)->False))->False))) of role axiom named fact_300_equalityE % A new axiom: (forall (A_49:(nat->Prop)) (B_21:(nat->Prop)), ((((eq (nat->Prop)) A_49) B_21)->((((ord_less_eq_nat_o A_49) B_21)->(((ord_less_eq_nat_o B_21) A_49)->False))->False))) % FOF formula (forall (A_49:(x_a->Prop)) (B_21:(x_a->Prop)), ((((eq (x_a->Prop)) A_49) B_21)->((((ord_less_eq_a_o A_49) B_21)->(((ord_less_eq_a_o B_21) A_49)->False))->False))) of role axiom named fact_301_equalityE % A new axiom: (forall (A_49:(x_a->Prop)) (B_21:(x_a->Prop)), ((((eq (x_a->Prop)) A_49) B_21)->((((ord_less_eq_a_o A_49) B_21)->(((ord_less_eq_a_o B_21) A_49)->False))->False))) % FOF formula (forall (Z_3:x_a) (F_18:(pname->x_a)) (A_48:(pname->Prop)), ((iff ((member_a Z_3) ((image_pname_a F_18) A_48))) ((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_48)) (((eq x_a) Z_3) (F_18 X_1))))))) of role axiom named fact_302_image__iff % A new axiom: (forall (Z_3:x_a) (F_18:(pname->x_a)) (A_48:(pname->Prop)), ((iff ((member_a Z_3) ((image_pname_a F_18) A_48))) ((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_48)) (((eq x_a) Z_3) (F_18 X_1))))))) % FOF formula (forall (F_17:(pname->nat)) (X_21:pname) (A_47:(pname->Prop)), (((member_pname X_21) A_47)->((member_nat (F_17 X_21)) ((image_pname_nat F_17) A_47)))) of role axiom named fact_303_imageI % A new axiom: (forall (F_17:(pname->nat)) (X_21:pname) (A_47:(pname->Prop)), (((member_pname X_21) A_47)->((member_nat (F_17 X_21)) ((image_pname_nat F_17) A_47)))) % FOF formula (forall (F_17:(x_a->nat)) (X_21:x_a) (A_47:(x_a->Prop)), (((member_a X_21) A_47)->((member_nat (F_17 X_21)) ((image_a_nat F_17) A_47)))) of role axiom named fact_304_imageI % A new axiom: (forall (F_17:(x_a->nat)) (X_21:x_a) (A_47:(x_a->Prop)), (((member_a X_21) A_47)->((member_nat (F_17 X_21)) ((image_a_nat F_17) A_47)))) % FOF formula (forall (F_17:(nat->pname)) (X_21:nat) (A_47:(nat->Prop)), (((member_nat X_21) A_47)->((member_pname (F_17 X_21)) ((image_nat_pname F_17) A_47)))) of role axiom named fact_305_imageI % A new axiom: (forall (F_17:(nat->pname)) (X_21:nat) (A_47:(nat->Prop)), (((member_nat X_21) A_47)->((member_pname (F_17 X_21)) ((image_nat_pname F_17) A_47)))) % FOF formula (forall (F_17:(nat->x_a)) (X_21:nat) (A_47:(nat->Prop)), (((member_nat X_21) A_47)->((member_a (F_17 X_21)) ((image_nat_a F_17) A_47)))) of role axiom named fact_306_imageI % A new axiom: (forall (F_17:(nat->x_a)) (X_21:nat) (A_47:(nat->Prop)), (((member_nat X_21) A_47)->((member_a (F_17 X_21)) ((image_nat_a F_17) A_47)))) % FOF formula (forall (F_17:(pname->x_a)) (X_21:pname) (A_47:(pname->Prop)), (((member_pname X_21) A_47)->((member_a (F_17 X_21)) ((image_pname_a F_17) A_47)))) of role axiom named fact_307_imageI % A new axiom: (forall (F_17:(pname->x_a)) (X_21:pname) (A_47:(pname->Prop)), (((member_pname X_21) A_47)->((member_a (F_17 X_21)) ((image_pname_a F_17) A_47)))) % FOF formula (forall (B_20:nat) (F_16:(pname->nat)) (X_20:pname) (A_46:(pname->Prop)), (((member_pname X_20) A_46)->((((eq nat) B_20) (F_16 X_20))->((member_nat B_20) ((image_pname_nat F_16) A_46))))) of role axiom named fact_308_rev__image__eqI % A new axiom: (forall (B_20:nat) (F_16:(pname->nat)) (X_20:pname) (A_46:(pname->Prop)), (((member_pname X_20) A_46)->((((eq nat) B_20) (F_16 X_20))->((member_nat B_20) ((image_pname_nat F_16) A_46))))) % FOF formula (forall (B_20:nat) (F_16:(x_a->nat)) (X_20:x_a) (A_46:(x_a->Prop)), (((member_a X_20) A_46)->((((eq nat) B_20) (F_16 X_20))->((member_nat B_20) ((image_a_nat F_16) A_46))))) of role axiom named fact_309_rev__image__eqI % A new axiom: (forall (B_20:nat) (F_16:(x_a->nat)) (X_20:x_a) (A_46:(x_a->Prop)), (((member_a X_20) A_46)->((((eq nat) B_20) (F_16 X_20))->((member_nat B_20) ((image_a_nat F_16) A_46))))) % FOF formula (forall (B_20:pname) (F_16:(nat->pname)) (X_20:nat) (A_46:(nat->Prop)), (((member_nat X_20) A_46)->((((eq pname) B_20) (F_16 X_20))->((member_pname B_20) ((image_nat_pname F_16) A_46))))) of role axiom named fact_310_rev__image__eqI % A new axiom: (forall (B_20:pname) (F_16:(nat->pname)) (X_20:nat) (A_46:(nat->Prop)), (((member_nat X_20) A_46)->((((eq pname) B_20) (F_16 X_20))->((member_pname B_20) ((image_nat_pname F_16) A_46))))) % FOF formula (forall (B_20:x_a) (F_16:(nat->x_a)) (X_20:nat) (A_46:(nat->Prop)), (((member_nat X_20) A_46)->((((eq x_a) B_20) (F_16 X_20))->((member_a B_20) ((image_nat_a F_16) A_46))))) of role axiom named fact_311_rev__image__eqI % A new axiom: (forall (B_20:x_a) (F_16:(nat->x_a)) (X_20:nat) (A_46:(nat->Prop)), (((member_nat X_20) A_46)->((((eq x_a) B_20) (F_16 X_20))->((member_a B_20) ((image_nat_a F_16) A_46))))) % FOF formula (forall (B_20:x_a) (F_16:(pname->x_a)) (X_20:pname) (A_46:(pname->Prop)), (((member_pname X_20) A_46)->((((eq x_a) B_20) (F_16 X_20))->((member_a B_20) ((image_pname_a F_16) A_46))))) of role axiom named fact_312_rev__image__eqI % A new axiom: (forall (B_20:x_a) (F_16:(pname->x_a)) (X_20:pname) (A_46:(pname->Prop)), (((member_pname X_20) A_46)->((((eq x_a) B_20) (F_16 X_20))->((member_a B_20) ((image_pname_a F_16) A_46))))) % FOF formula (forall (X_1:(nat->Prop)) (Xa:((nat->Prop)->Prop)), (((eq ((nat->Prop)->Prop)) ((insert_nat_o X_1) Xa)) (collect_nat_o (fun (Y_10:(nat->Prop))=> ((or (((eq (nat->Prop)) Y_10) X_1)) ((member_nat_o Y_10) Xa)))))) of role axiom named fact_313_insert__compr__raw % A new axiom: (forall (X_1:(nat->Prop)) (Xa:((nat->Prop)->Prop)), (((eq ((nat->Prop)->Prop)) ((insert_nat_o X_1) Xa)) (collect_nat_o (fun (Y_10:(nat->Prop))=> ((or (((eq (nat->Prop)) Y_10) X_1)) ((member_nat_o Y_10) Xa)))))) % FOF formula (forall (X_1:(pname->Prop)) (Xa:((pname->Prop)->Prop)), (((eq ((pname->Prop)->Prop)) ((insert_pname_o X_1) Xa)) (collect_pname_o (fun (Y_10:(pname->Prop))=> ((or (((eq (pname->Prop)) Y_10) X_1)) ((member_pname_o Y_10) Xa)))))) of role axiom named fact_314_insert__compr__raw % A new axiom: (forall (X_1:(pname->Prop)) (Xa:((pname->Prop)->Prop)), (((eq ((pname->Prop)->Prop)) ((insert_pname_o X_1) Xa)) (collect_pname_o (fun (Y_10:(pname->Prop))=> ((or (((eq (pname->Prop)) Y_10) X_1)) ((member_pname_o Y_10) Xa)))))) % FOF formula (forall (X_1:(x_a->Prop)) (Xa:((x_a->Prop)->Prop)), (((eq ((x_a->Prop)->Prop)) ((insert_a_o X_1) Xa)) (collect_a_o (fun (Y_10:(x_a->Prop))=> ((or (((eq (x_a->Prop)) Y_10) X_1)) ((member_a_o Y_10) Xa)))))) of role axiom named fact_315_insert__compr__raw % A new axiom: (forall (X_1:(x_a->Prop)) (Xa:((x_a->Prop)->Prop)), (((eq ((x_a->Prop)->Prop)) ((insert_a_o X_1) Xa)) (collect_a_o (fun (Y_10:(x_a->Prop))=> ((or (((eq (x_a->Prop)) Y_10) X_1)) ((member_a_o Y_10) Xa)))))) % FOF formula (forall (X_1:nat) (Xa:(nat->Prop)), (((eq (nat->Prop)) ((insert_nat X_1) Xa)) (collect_nat (fun (Y_10:nat)=> ((or (((eq nat) Y_10) X_1)) ((member_nat Y_10) Xa)))))) of role axiom named fact_316_insert__compr__raw % A new axiom: (forall (X_1:nat) (Xa:(nat->Prop)), (((eq (nat->Prop)) ((insert_nat X_1) Xa)) (collect_nat (fun (Y_10:nat)=> ((or (((eq nat) Y_10) X_1)) ((member_nat Y_10) Xa)))))) % FOF formula (forall (X_1:pname) (Xa:(pname->Prop)), (((eq (pname->Prop)) ((insert_pname X_1) Xa)) (collect_pname (fun (Y_10:pname)=> ((or (((eq pname) Y_10) X_1)) ((member_pname Y_10) Xa)))))) of role axiom named fact_317_insert__compr__raw % A new axiom: (forall (X_1:pname) (Xa:(pname->Prop)), (((eq (pname->Prop)) ((insert_pname X_1) Xa)) (collect_pname (fun (Y_10:pname)=> ((or (((eq pname) Y_10) X_1)) ((member_pname Y_10) Xa)))))) % FOF formula (forall (X_1:x_a) (Xa:(x_a->Prop)), (((eq (x_a->Prop)) ((insert_a X_1) Xa)) (collect_a (fun (Y_10:x_a)=> ((or (((eq x_a) Y_10) X_1)) ((member_a Y_10) Xa)))))) of role axiom named fact_318_insert__compr__raw % A new axiom: (forall (X_1:x_a) (Xa:(x_a->Prop)), (((eq (x_a->Prop)) ((insert_a X_1) Xa)) (collect_a (fun (Y_10:x_a)=> ((or (((eq x_a) Y_10) X_1)) ((member_a Y_10) Xa)))))) % FOF formula (forall (B_19:(pname->Prop)) (A_45:pname), ((ord_less_eq_pname_o B_19) ((insert_pname A_45) B_19))) of role axiom named fact_319_subset__insertI % A new axiom: (forall (B_19:(pname->Prop)) (A_45:pname), ((ord_less_eq_pname_o B_19) ((insert_pname A_45) B_19))) % FOF formula (forall (B_19:(nat->Prop)) (A_45:nat), ((ord_less_eq_nat_o B_19) ((insert_nat A_45) B_19))) of role axiom named fact_320_subset__insertI % A new axiom: (forall (B_19:(nat->Prop)) (A_45:nat), ((ord_less_eq_nat_o B_19) ((insert_nat A_45) B_19))) % FOF formula (forall (B_19:(x_a->Prop)) (A_45:x_a), ((ord_less_eq_a_o B_19) ((insert_a A_45) B_19))) of role axiom named fact_321_subset__insertI % A new axiom: (forall (B_19:(x_a->Prop)) (A_45:x_a), ((ord_less_eq_a_o B_19) ((insert_a A_45) B_19))) % FOF formula (forall (X_19:nat) (A_44:(nat->Prop)) (B_18:(nat->Prop)), ((iff ((ord_less_eq_nat_o ((insert_nat X_19) A_44)) B_18)) ((and ((member_nat X_19) B_18)) ((ord_less_eq_nat_o A_44) B_18)))) of role axiom named fact_322_insert__subset % A new axiom: (forall (X_19:nat) (A_44:(nat->Prop)) (B_18:(nat->Prop)), ((iff ((ord_less_eq_nat_o ((insert_nat X_19) A_44)) B_18)) ((and ((member_nat X_19) B_18)) ((ord_less_eq_nat_o A_44) B_18)))) % FOF formula (forall (X_19:pname) (A_44:(pname->Prop)) (B_18:(pname->Prop)), ((iff ((ord_less_eq_pname_o ((insert_pname X_19) A_44)) B_18)) ((and ((member_pname X_19) B_18)) ((ord_less_eq_pname_o A_44) B_18)))) of role axiom named fact_323_insert__subset % A new axiom: (forall (X_19:pname) (A_44:(pname->Prop)) (B_18:(pname->Prop)), ((iff ((ord_less_eq_pname_o ((insert_pname X_19) A_44)) B_18)) ((and ((member_pname X_19) B_18)) ((ord_less_eq_pname_o A_44) B_18)))) % FOF formula (forall (X_19:x_a) (A_44:(x_a->Prop)) (B_18:(x_a->Prop)), ((iff ((ord_less_eq_a_o ((insert_a X_19) A_44)) B_18)) ((and ((member_a X_19) B_18)) ((ord_less_eq_a_o A_44) B_18)))) of role axiom named fact_324_insert__subset % A new axiom: (forall (X_19:x_a) (A_44:(x_a->Prop)) (B_18:(x_a->Prop)), ((iff ((ord_less_eq_a_o ((insert_a X_19) A_44)) B_18)) ((and ((member_a X_19) B_18)) ((ord_less_eq_a_o A_44) B_18)))) % FOF formula (forall (B_17:(nat->Prop)) (X_18:nat) (A_43:(nat->Prop)), ((((member_nat X_18) A_43)->False)->((iff ((ord_less_eq_nat_o A_43) ((insert_nat X_18) B_17))) ((ord_less_eq_nat_o A_43) B_17)))) of role axiom named fact_325_subset__insert % A new axiom: (forall (B_17:(nat->Prop)) (X_18:nat) (A_43:(nat->Prop)), ((((member_nat X_18) A_43)->False)->((iff ((ord_less_eq_nat_o A_43) ((insert_nat X_18) B_17))) ((ord_less_eq_nat_o A_43) B_17)))) % FOF formula (forall (B_17:(pname->Prop)) (X_18:pname) (A_43:(pname->Prop)), ((((member_pname X_18) A_43)->False)->((iff ((ord_less_eq_pname_o A_43) ((insert_pname X_18) B_17))) ((ord_less_eq_pname_o A_43) B_17)))) of role axiom named fact_326_subset__insert % A new axiom: (forall (B_17:(pname->Prop)) (X_18:pname) (A_43:(pname->Prop)), ((((member_pname X_18) A_43)->False)->((iff ((ord_less_eq_pname_o A_43) ((insert_pname X_18) B_17))) ((ord_less_eq_pname_o A_43) B_17)))) % FOF formula (forall (B_17:(x_a->Prop)) (X_18:x_a) (A_43:(x_a->Prop)), ((((member_a X_18) A_43)->False)->((iff ((ord_less_eq_a_o A_43) ((insert_a X_18) B_17))) ((ord_less_eq_a_o A_43) B_17)))) of role axiom named fact_327_subset__insert % A new axiom: (forall (B_17:(x_a->Prop)) (X_18:x_a) (A_43:(x_a->Prop)), ((((member_a X_18) A_43)->False)->((iff ((ord_less_eq_a_o A_43) ((insert_a X_18) B_17))) ((ord_less_eq_a_o A_43) B_17)))) % FOF formula (forall (B_16:pname) (A_42:(pname->Prop)) (B_15:(pname->Prop)), (((ord_less_eq_pname_o A_42) B_15)->((ord_less_eq_pname_o A_42) ((insert_pname B_16) B_15)))) of role axiom named fact_328_subset__insertI2 % A new axiom: (forall (B_16:pname) (A_42:(pname->Prop)) (B_15:(pname->Prop)), (((ord_less_eq_pname_o A_42) B_15)->((ord_less_eq_pname_o A_42) ((insert_pname B_16) B_15)))) % FOF formula (forall (B_16:nat) (A_42:(nat->Prop)) (B_15:(nat->Prop)), (((ord_less_eq_nat_o A_42) B_15)->((ord_less_eq_nat_o A_42) ((insert_nat B_16) B_15)))) of role axiom named fact_329_subset__insertI2 % A new axiom: (forall (B_16:nat) (A_42:(nat->Prop)) (B_15:(nat->Prop)), (((ord_less_eq_nat_o A_42) B_15)->((ord_less_eq_nat_o A_42) ((insert_nat B_16) B_15)))) % FOF formula (forall (B_16:x_a) (A_42:(x_a->Prop)) (B_15:(x_a->Prop)), (((ord_less_eq_a_o A_42) B_15)->((ord_less_eq_a_o A_42) ((insert_a B_16) B_15)))) of role axiom named fact_330_subset__insertI2 % A new axiom: (forall (B_16:x_a) (A_42:(x_a->Prop)) (B_15:(x_a->Prop)), (((ord_less_eq_a_o A_42) B_15)->((ord_less_eq_a_o A_42) ((insert_a B_16) B_15)))) % FOF formula (forall (A_41:pname) (C_9:(pname->Prop)) (D_1:(pname->Prop)), (((ord_less_eq_pname_o C_9) D_1)->((ord_less_eq_pname_o ((insert_pname A_41) C_9)) ((insert_pname A_41) D_1)))) of role axiom named fact_331_insert__mono % A new axiom: (forall (A_41:pname) (C_9:(pname->Prop)) (D_1:(pname->Prop)), (((ord_less_eq_pname_o C_9) D_1)->((ord_less_eq_pname_o ((insert_pname A_41) C_9)) ((insert_pname A_41) D_1)))) % FOF formula (forall (A_41:nat) (C_9:(nat->Prop)) (D_1:(nat->Prop)), (((ord_less_eq_nat_o C_9) D_1)->((ord_less_eq_nat_o ((insert_nat A_41) C_9)) ((insert_nat A_41) D_1)))) of role axiom named fact_332_insert__mono % A new axiom: (forall (A_41:nat) (C_9:(nat->Prop)) (D_1:(nat->Prop)), (((ord_less_eq_nat_o C_9) D_1)->((ord_less_eq_nat_o ((insert_nat A_41) C_9)) ((insert_nat A_41) D_1)))) % FOF formula (forall (A_41:x_a) (C_9:(x_a->Prop)) (D_1:(x_a->Prop)), (((ord_less_eq_a_o C_9) D_1)->((ord_less_eq_a_o ((insert_a A_41) C_9)) ((insert_a A_41) D_1)))) of role axiom named fact_333_insert__mono % A new axiom: (forall (A_41:x_a) (C_9:(x_a->Prop)) (D_1:(x_a->Prop)), (((ord_less_eq_a_o C_9) D_1)->((ord_less_eq_a_o ((insert_a A_41) C_9)) ((insert_a A_41) D_1)))) % FOF formula (forall (F_15:(x_a->pname)) (A_40:x_a) (B_14:(x_a->Prop)), (((eq (pname->Prop)) ((image_a_pname F_15) ((insert_a A_40) B_14))) ((insert_pname (F_15 A_40)) ((image_a_pname F_15) B_14)))) of role axiom named fact_334_image__insert % A new axiom: (forall (F_15:(x_a->pname)) (A_40:x_a) (B_14:(x_a->Prop)), (((eq (pname->Prop)) ((image_a_pname F_15) ((insert_a A_40) B_14))) ((insert_pname (F_15 A_40)) ((image_a_pname F_15) B_14)))) % FOF formula (forall (F_15:(x_a->nat)) (A_40:x_a) (B_14:(x_a->Prop)), (((eq (nat->Prop)) ((image_a_nat F_15) ((insert_a A_40) B_14))) ((insert_nat (F_15 A_40)) ((image_a_nat F_15) B_14)))) of role axiom named fact_335_image__insert % A new axiom: (forall (F_15:(x_a->nat)) (A_40:x_a) (B_14:(x_a->Prop)), (((eq (nat->Prop)) ((image_a_nat F_15) ((insert_a A_40) B_14))) ((insert_nat (F_15 A_40)) ((image_a_nat F_15) B_14)))) % FOF formula (forall (F_15:(nat->x_a)) (A_40:nat) (B_14:(nat->Prop)), (((eq (x_a->Prop)) ((image_nat_a F_15) ((insert_nat A_40) B_14))) ((insert_a (F_15 A_40)) ((image_nat_a F_15) B_14)))) of role axiom named fact_336_image__insert % A new axiom: (forall (F_15:(nat->x_a)) (A_40:nat) (B_14:(nat->Prop)), (((eq (x_a->Prop)) ((image_nat_a F_15) ((insert_nat A_40) B_14))) ((insert_a (F_15 A_40)) ((image_nat_a F_15) B_14)))) % FOF formula (forall (F_15:(pname->x_a)) (A_40:pname) (B_14:(pname->Prop)), (((eq (x_a->Prop)) ((image_pname_a F_15) ((insert_pname A_40) B_14))) ((insert_a (F_15 A_40)) ((image_pname_a F_15) B_14)))) of role axiom named fact_337_image__insert % A new axiom: (forall (F_15:(pname->x_a)) (A_40:pname) (B_14:(pname->Prop)), (((eq (x_a->Prop)) ((image_pname_a F_15) ((insert_pname A_40) B_14))) ((insert_a (F_15 A_40)) ((image_pname_a F_15) B_14)))) % FOF formula (forall (F_14:(pname->pname)) (X_17:pname) (A_39:(pname->Prop)), (((member_pname X_17) A_39)->(((eq (pname->Prop)) ((insert_pname (F_14 X_17)) ((image_pname_pname F_14) A_39))) ((image_pname_pname F_14) A_39)))) of role axiom named fact_338_insert__image % A new axiom: (forall (F_14:(pname->pname)) (X_17:pname) (A_39:(pname->Prop)), (((member_pname X_17) A_39)->(((eq (pname->Prop)) ((insert_pname (F_14 X_17)) ((image_pname_pname F_14) A_39))) ((image_pname_pname F_14) A_39)))) % FOF formula (forall (F_14:(pname->nat)) (X_17:pname) (A_39:(pname->Prop)), (((member_pname X_17) A_39)->(((eq (nat->Prop)) ((insert_nat (F_14 X_17)) ((image_pname_nat F_14) A_39))) ((image_pname_nat F_14) A_39)))) of role axiom named fact_339_insert__image % A new axiom: (forall (F_14:(pname->nat)) (X_17:pname) (A_39:(pname->Prop)), (((member_pname X_17) A_39)->(((eq (nat->Prop)) ((insert_nat (F_14 X_17)) ((image_pname_nat F_14) A_39))) ((image_pname_nat F_14) A_39)))) % FOF formula (forall (F_14:(x_a->pname)) (X_17:x_a) (A_39:(x_a->Prop)), (((member_a X_17) A_39)->(((eq (pname->Prop)) ((insert_pname (F_14 X_17)) ((image_a_pname F_14) A_39))) ((image_a_pname F_14) A_39)))) of role axiom named fact_340_insert__image % A new axiom: (forall (F_14:(x_a->pname)) (X_17:x_a) (A_39:(x_a->Prop)), (((member_a X_17) A_39)->(((eq (pname->Prop)) ((insert_pname (F_14 X_17)) ((image_a_pname F_14) A_39))) ((image_a_pname F_14) A_39)))) % FOF formula (forall (F_14:(x_a->nat)) (X_17:x_a) (A_39:(x_a->Prop)), (((member_a X_17) A_39)->(((eq (nat->Prop)) ((insert_nat (F_14 X_17)) ((image_a_nat F_14) A_39))) ((image_a_nat F_14) A_39)))) of role axiom named fact_341_insert__image % A new axiom: (forall (F_14:(x_a->nat)) (X_17:x_a) (A_39:(x_a->Prop)), (((member_a X_17) A_39)->(((eq (nat->Prop)) ((insert_nat (F_14 X_17)) ((image_a_nat F_14) A_39))) ((image_a_nat F_14) A_39)))) % FOF formula (forall (F_14:(nat->x_a)) (X_17:nat) (A_39:(nat->Prop)), (((member_nat X_17) A_39)->(((eq (x_a->Prop)) ((insert_a (F_14 X_17)) ((image_nat_a F_14) A_39))) ((image_nat_a F_14) A_39)))) of role axiom named fact_342_insert__image % A new axiom: (forall (F_14:(nat->x_a)) (X_17:nat) (A_39:(nat->Prop)), (((member_nat X_17) A_39)->(((eq (x_a->Prop)) ((insert_a (F_14 X_17)) ((image_nat_a F_14) A_39))) ((image_nat_a F_14) A_39)))) % FOF formula (forall (F_14:(pname->x_a)) (X_17:pname) (A_39:(pname->Prop)), (((member_pname X_17) A_39)->(((eq (x_a->Prop)) ((insert_a (F_14 X_17)) ((image_pname_a F_14) A_39))) ((image_pname_a F_14) A_39)))) of role axiom named fact_343_insert__image % A new axiom: (forall (F_14:(pname->x_a)) (X_17:pname) (A_39:(pname->Prop)), (((member_pname X_17) A_39)->(((eq (x_a->Prop)) ((insert_a (F_14 X_17)) ((image_pname_a F_14) A_39))) ((image_pname_a F_14) A_39)))) % FOF formula (forall (B_13:(x_a->Prop)) (F_13:(nat->x_a)) (A_38:(nat->Prop)), ((iff ((ord_less_eq_a_o B_13) ((image_nat_a F_13) A_38))) ((ex (nat->Prop)) (fun (AA:(nat->Prop))=> ((and ((ord_less_eq_nat_o AA) A_38)) (((eq (x_a->Prop)) B_13) ((image_nat_a F_13) AA))))))) of role axiom named fact_344_subset__image__iff % A new axiom: (forall (B_13:(x_a->Prop)) (F_13:(nat->x_a)) (A_38:(nat->Prop)), ((iff ((ord_less_eq_a_o B_13) ((image_nat_a F_13) A_38))) ((ex (nat->Prop)) (fun (AA:(nat->Prop))=> ((and ((ord_less_eq_nat_o AA) A_38)) (((eq (x_a->Prop)) B_13) ((image_nat_a F_13) AA))))))) % FOF formula (forall (B_13:(pname->Prop)) (F_13:(x_a->pname)) (A_38:(x_a->Prop)), ((iff ((ord_less_eq_pname_o B_13) ((image_a_pname F_13) A_38))) ((ex (x_a->Prop)) (fun (AA:(x_a->Prop))=> ((and ((ord_less_eq_a_o AA) A_38)) (((eq (pname->Prop)) B_13) ((image_a_pname F_13) AA))))))) of role axiom named fact_345_subset__image__iff % A new axiom: (forall (B_13:(pname->Prop)) (F_13:(x_a->pname)) (A_38:(x_a->Prop)), ((iff ((ord_less_eq_pname_o B_13) ((image_a_pname F_13) A_38))) ((ex (x_a->Prop)) (fun (AA:(x_a->Prop))=> ((and ((ord_less_eq_a_o AA) A_38)) (((eq (pname->Prop)) B_13) ((image_a_pname F_13) AA))))))) % FOF formula (forall (B_13:(nat->Prop)) (F_13:(x_a->nat)) (A_38:(x_a->Prop)), ((iff ((ord_less_eq_nat_o B_13) ((image_a_nat F_13) A_38))) ((ex (x_a->Prop)) (fun (AA:(x_a->Prop))=> ((and ((ord_less_eq_a_o AA) A_38)) (((eq (nat->Prop)) B_13) ((image_a_nat F_13) AA))))))) of role axiom named fact_346_subset__image__iff % A new axiom: (forall (B_13:(nat->Prop)) (F_13:(x_a->nat)) (A_38:(x_a->Prop)), ((iff ((ord_less_eq_nat_o B_13) ((image_a_nat F_13) A_38))) ((ex (x_a->Prop)) (fun (AA:(x_a->Prop))=> ((and ((ord_less_eq_a_o AA) A_38)) (((eq (nat->Prop)) B_13) ((image_a_nat F_13) AA))))))) % FOF formula (forall (B_13:(x_a->Prop)) (F_13:(pname->x_a)) (A_38:(pname->Prop)), ((iff ((ord_less_eq_a_o B_13) ((image_pname_a F_13) A_38))) ((ex (pname->Prop)) (fun (AA:(pname->Prop))=> ((and ((ord_less_eq_pname_o AA) A_38)) (((eq (x_a->Prop)) B_13) ((image_pname_a F_13) AA))))))) of role axiom named fact_347_subset__image__iff % A new axiom: (forall (B_13:(x_a->Prop)) (F_13:(pname->x_a)) (A_38:(pname->Prop)), ((iff ((ord_less_eq_a_o B_13) ((image_pname_a F_13) A_38))) ((ex (pname->Prop)) (fun (AA:(pname->Prop))=> ((and ((ord_less_eq_pname_o AA) A_38)) (((eq (x_a->Prop)) B_13) ((image_pname_a F_13) AA))))))) % FOF formula (forall (F_12:(x_a->pname)) (A_37:(x_a->Prop)) (B_12:(x_a->Prop)), (((ord_less_eq_a_o A_37) B_12)->((ord_less_eq_pname_o ((image_a_pname F_12) A_37)) ((image_a_pname F_12) B_12)))) of role axiom named fact_348_image__mono % A new axiom: (forall (F_12:(x_a->pname)) (A_37:(x_a->Prop)) (B_12:(x_a->Prop)), (((ord_less_eq_a_o A_37) B_12)->((ord_less_eq_pname_o ((image_a_pname F_12) A_37)) ((image_a_pname F_12) B_12)))) % FOF formula (forall (F_12:(x_a->nat)) (A_37:(x_a->Prop)) (B_12:(x_a->Prop)), (((ord_less_eq_a_o A_37) B_12)->((ord_less_eq_nat_o ((image_a_nat F_12) A_37)) ((image_a_nat F_12) B_12)))) of role axiom named fact_349_image__mono % A new axiom: (forall (F_12:(x_a->nat)) (A_37:(x_a->Prop)) (B_12:(x_a->Prop)), (((ord_less_eq_a_o A_37) B_12)->((ord_less_eq_nat_o ((image_a_nat F_12) A_37)) ((image_a_nat F_12) B_12)))) % FOF formula (forall (F_12:(nat->x_a)) (A_37:(nat->Prop)) (B_12:(nat->Prop)), (((ord_less_eq_nat_o A_37) B_12)->((ord_less_eq_a_o ((image_nat_a F_12) A_37)) ((image_nat_a F_12) B_12)))) of role axiom named fact_350_image__mono % A new axiom: (forall (F_12:(nat->x_a)) (A_37:(nat->Prop)) (B_12:(nat->Prop)), (((ord_less_eq_nat_o A_37) B_12)->((ord_less_eq_a_o ((image_nat_a F_12) A_37)) ((image_nat_a F_12) B_12)))) % FOF formula (forall (F_12:(pname->x_a)) (A_37:(pname->Prop)) (B_12:(pname->Prop)), (((ord_less_eq_pname_o A_37) B_12)->((ord_less_eq_a_o ((image_pname_a F_12) A_37)) ((image_pname_a F_12) B_12)))) of role axiom named fact_351_image__mono % A new axiom: (forall (F_12:(pname->x_a)) (A_37:(pname->Prop)) (B_12:(pname->Prop)), (((ord_less_eq_pname_o A_37) B_12)->((ord_less_eq_a_o ((image_pname_a F_12) A_37)) ((image_pname_a F_12) B_12)))) % FOF formula (forall (B_11:pname) (F_11:(nat->pname)) (A_36:(nat->Prop)), (((member_pname B_11) ((image_nat_pname F_11) A_36))->((forall (X_1:nat), ((((eq pname) B_11) (F_11 X_1))->(((member_nat X_1) A_36)->False)))->False))) of role axiom named fact_352_imageE % A new axiom: (forall (B_11:pname) (F_11:(nat->pname)) (A_36:(nat->Prop)), (((member_pname B_11) ((image_nat_pname F_11) A_36))->((forall (X_1:nat), ((((eq pname) B_11) (F_11 X_1))->(((member_nat X_1) A_36)->False)))->False))) % FOF formula (forall (B_11:x_a) (F_11:(nat->x_a)) (A_36:(nat->Prop)), (((member_a B_11) ((image_nat_a F_11) A_36))->((forall (X_1:nat), ((((eq x_a) B_11) (F_11 X_1))->(((member_nat X_1) A_36)->False)))->False))) of role axiom named fact_353_imageE % A new axiom: (forall (B_11:x_a) (F_11:(nat->x_a)) (A_36:(nat->Prop)), (((member_a B_11) ((image_nat_a F_11) A_36))->((forall (X_1:nat), ((((eq x_a) B_11) (F_11 X_1))->(((member_nat X_1) A_36)->False)))->False))) % FOF formula (forall (B_11:nat) (F_11:(pname->nat)) (A_36:(pname->Prop)), (((member_nat B_11) ((image_pname_nat F_11) A_36))->((forall (X_1:pname), ((((eq nat) B_11) (F_11 X_1))->(((member_pname X_1) A_36)->False)))->False))) of role axiom named fact_354_imageE % A new axiom: (forall (B_11:nat) (F_11:(pname->nat)) (A_36:(pname->Prop)), (((member_nat B_11) ((image_pname_nat F_11) A_36))->((forall (X_1:pname), ((((eq nat) B_11) (F_11 X_1))->(((member_pname X_1) A_36)->False)))->False))) % FOF formula (forall (B_11:nat) (F_11:(x_a->nat)) (A_36:(x_a->Prop)), (((member_nat B_11) ((image_a_nat F_11) A_36))->((forall (X_1:x_a), ((((eq nat) B_11) (F_11 X_1))->(((member_a X_1) A_36)->False)))->False))) of role axiom named fact_355_imageE % A new axiom: (forall (B_11:nat) (F_11:(x_a->nat)) (A_36:(x_a->Prop)), (((member_nat B_11) ((image_a_nat F_11) A_36))->((forall (X_1:x_a), ((((eq nat) B_11) (F_11 X_1))->(((member_a X_1) A_36)->False)))->False))) % FOF formula (forall (B_11:x_a) (F_11:(pname->x_a)) (A_36:(pname->Prop)), (((member_a B_11) ((image_pname_a F_11) A_36))->((forall (X_1:pname), ((((eq x_a) B_11) (F_11 X_1))->(((member_pname X_1) A_36)->False)))->False))) of role axiom named fact_356_imageE % A new axiom: (forall (B_11:x_a) (F_11:(pname->x_a)) (A_36:(pname->Prop)), (((member_a B_11) ((image_pname_a F_11) A_36))->((forall (X_1:pname), ((((eq x_a) B_11) (F_11 X_1))->(((member_pname X_1) A_36)->False)))->False))) % FOF formula (forall (B_10:(nat->Prop)) (A_35:(nat->Prop)), ((forall (X_1:nat), (((member_nat X_1) A_35)->((member_nat X_1) B_10)))->((ord_less_eq_nat_o A_35) B_10))) of role axiom named fact_357_subsetI % A new axiom: (forall (B_10:(nat->Prop)) (A_35:(nat->Prop)), ((forall (X_1:nat), (((member_nat X_1) A_35)->((member_nat X_1) B_10)))->((ord_less_eq_nat_o A_35) B_10))) % FOF formula (forall (B_10:(pname->Prop)) (A_35:(pname->Prop)), ((forall (X_1:pname), (((member_pname X_1) A_35)->((member_pname X_1) B_10)))->((ord_less_eq_pname_o A_35) B_10))) of role axiom named fact_358_subsetI % A new axiom: (forall (B_10:(pname->Prop)) (A_35:(pname->Prop)), ((forall (X_1:pname), (((member_pname X_1) A_35)->((member_pname X_1) B_10)))->((ord_less_eq_pname_o A_35) B_10))) % FOF formula (forall (B_10:(x_a->Prop)) (A_35:(x_a->Prop)), ((forall (X_1:x_a), (((member_a X_1) A_35)->((member_a X_1) B_10)))->((ord_less_eq_a_o A_35) B_10))) of role axiom named fact_359_subsetI % A new axiom: (forall (B_10:(x_a->Prop)) (A_35:(x_a->Prop)), ((forall (X_1:x_a), (((member_a X_1) A_35)->((member_a X_1) B_10)))->((ord_less_eq_a_o A_35) B_10))) % FOF formula (forall (F_10:(pname->nat)) (B_9:(nat->Prop)) (A_34:(pname->Prop)), ((forall (X_1:pname), (((member_pname X_1) A_34)->((member_nat (F_10 X_1)) B_9)))->((ord_less_eq_nat_o ((image_pname_nat F_10) A_34)) B_9))) of role axiom named fact_360_image__subsetI % A new axiom: (forall (F_10:(pname->nat)) (B_9:(nat->Prop)) (A_34:(pname->Prop)), ((forall (X_1:pname), (((member_pname X_1) A_34)->((member_nat (F_10 X_1)) B_9)))->((ord_less_eq_nat_o ((image_pname_nat F_10) A_34)) B_9))) % FOF formula (forall (F_10:(pname->pname)) (B_9:(pname->Prop)) (A_34:(pname->Prop)), ((forall (X_1:pname), (((member_pname X_1) A_34)->((member_pname (F_10 X_1)) B_9)))->((ord_less_eq_pname_o ((image_pname_pname F_10) A_34)) B_9))) of role axiom named fact_361_image__subsetI % A new axiom: (forall (F_10:(pname->pname)) (B_9:(pname->Prop)) (A_34:(pname->Prop)), ((forall (X_1:pname), (((member_pname X_1) A_34)->((member_pname (F_10 X_1)) B_9)))->((ord_less_eq_pname_o ((image_pname_pname F_10) A_34)) B_9))) % FOF formula (forall (F_10:(x_a->nat)) (B_9:(nat->Prop)) (A_34:(x_a->Prop)), ((forall (X_1:x_a), (((member_a X_1) A_34)->((member_nat (F_10 X_1)) B_9)))->((ord_less_eq_nat_o ((image_a_nat F_10) A_34)) B_9))) of role axiom named fact_362_image__subsetI % A new axiom: (forall (F_10:(x_a->nat)) (B_9:(nat->Prop)) (A_34:(x_a->Prop)), ((forall (X_1:x_a), (((member_a X_1) A_34)->((member_nat (F_10 X_1)) B_9)))->((ord_less_eq_nat_o ((image_a_nat F_10) A_34)) B_9))) % FOF formula (forall (F_10:(x_a->pname)) (B_9:(pname->Prop)) (A_34:(x_a->Prop)), ((forall (X_1:x_a), (((member_a X_1) A_34)->((member_pname (F_10 X_1)) B_9)))->((ord_less_eq_pname_o ((image_a_pname F_10) A_34)) B_9))) of role axiom named fact_363_image__subsetI % A new axiom: (forall (F_10:(x_a->pname)) (B_9:(pname->Prop)) (A_34:(x_a->Prop)), ((forall (X_1:x_a), (((member_a X_1) A_34)->((member_pname (F_10 X_1)) B_9)))->((ord_less_eq_pname_o ((image_a_pname F_10) A_34)) B_9))) % FOF formula (forall (F_10:(nat->pname)) (B_9:(pname->Prop)) (A_34:(nat->Prop)), ((forall (X_1:nat), (((member_nat X_1) A_34)->((member_pname (F_10 X_1)) B_9)))->((ord_less_eq_pname_o ((image_nat_pname F_10) A_34)) B_9))) of role axiom named fact_364_image__subsetI % A new axiom: (forall (F_10:(nat->pname)) (B_9:(pname->Prop)) (A_34:(nat->Prop)), ((forall (X_1:nat), (((member_nat X_1) A_34)->((member_pname (F_10 X_1)) B_9)))->((ord_less_eq_pname_o ((image_nat_pname F_10) A_34)) B_9))) % FOF formula (forall (F_10:(nat->x_a)) (B_9:(x_a->Prop)) (A_34:(nat->Prop)), ((forall (X_1:nat), (((member_nat X_1) A_34)->((member_a (F_10 X_1)) B_9)))->((ord_less_eq_a_o ((image_nat_a F_10) A_34)) B_9))) of role axiom named fact_365_image__subsetI % A new axiom: (forall (F_10:(nat->x_a)) (B_9:(x_a->Prop)) (A_34:(nat->Prop)), ((forall (X_1:nat), (((member_nat X_1) A_34)->((member_a (F_10 X_1)) B_9)))->((ord_less_eq_a_o ((image_nat_a F_10) A_34)) B_9))) % FOF formula (forall (F_10:(pname->x_a)) (B_9:(x_a->Prop)) (A_34:(pname->Prop)), ((forall (X_1:pname), (((member_pname X_1) A_34)->((member_a (F_10 X_1)) B_9)))->((ord_less_eq_a_o ((image_pname_a F_10) A_34)) B_9))) of role axiom named fact_366_image__subsetI % A new axiom: (forall (F_10:(pname->x_a)) (B_9:(x_a->Prop)) (A_34:(pname->Prop)), ((forall (X_1:pname), (((member_pname X_1) A_34)->((member_a (F_10 X_1)) B_9)))->((ord_less_eq_a_o ((image_pname_a F_10) A_34)) B_9))) % FOF formula (forall (X_16:Prop), ((ord_less_eq_o X_16) X_16)) of role axiom named fact_367_order__refl % A new axiom: (forall (X_16:Prop), ((ord_less_eq_o X_16) X_16)) % FOF formula (forall (X_16:(pname->Prop)), ((ord_less_eq_pname_o X_16) X_16)) of role axiom named fact_368_order__refl % A new axiom: (forall (X_16:(pname->Prop)), ((ord_less_eq_pname_o X_16) X_16)) % FOF formula (forall (X_16:(nat->Prop)), ((ord_less_eq_nat_o X_16) X_16)) of role axiom named fact_369_order__refl % A new axiom: (forall (X_16:(nat->Prop)), ((ord_less_eq_nat_o X_16) X_16)) % FOF formula (forall (X_16:(x_a->Prop)), ((ord_less_eq_a_o X_16) X_16)) of role axiom named fact_370_order__refl % A new axiom: (forall (X_16:(x_a->Prop)), ((ord_less_eq_a_o X_16) X_16)) % FOF formula (forall (X_16:nat), ((ord_less_eq_nat X_16) X_16)) of role axiom named fact_371_order__refl % A new axiom: (forall (X_16:nat), ((ord_less_eq_nat X_16) X_16)) % FOF formula (forall (N_1:(nat->Prop)), ((iff (finite_finite_nat N_1)) ((ex nat) (fun (M_1:nat)=> (forall (X_1:nat), (((member_nat X_1) N_1)->((ord_less_eq_nat X_1) M_1))))))) of role axiom named fact_372_finite__nat__set__iff__bounded__le % A new axiom: (forall (N_1:(nat->Prop)), ((iff (finite_finite_nat N_1)) ((ex nat) (fun (M_1:nat)=> (forall (X_1:nat), (((member_nat X_1) N_1)->((ord_less_eq_nat X_1) M_1))))))) % FOF formula (forall (G:(x_a->Prop)) (C:com), ((wt C)->((forall (X_1:pname), (((member_pname X_1) u)->((p G) ((insert_a (mgt_call X_1)) bot_bot_a_o))))->((p G) ((insert_a (mgt C)) bot_bot_a_o))))) of role axiom named fact_373_assms_I3_J % A new axiom: (forall (G:(x_a->Prop)) (C:com), ((wt C)->((forall (X_1:pname), (((member_pname X_1) u)->((p G) ((insert_a (mgt_call X_1)) bot_bot_a_o))))->((p G) ((insert_a (mgt C)) bot_bot_a_o))))) % FOF formula (forall (F_9:(pname->Prop)) (G_3:(pname->Prop)), ((iff ((ord_less_eq_pname_o F_9) G_3)) (forall (X_1:pname), ((ord_less_eq_o (F_9 X_1)) (G_3 X_1))))) of role axiom named fact_374_le__fun__def % A new axiom: (forall (F_9:(pname->Prop)) (G_3:(pname->Prop)), ((iff ((ord_less_eq_pname_o F_9) G_3)) (forall (X_1:pname), ((ord_less_eq_o (F_9 X_1)) (G_3 X_1))))) % FOF formula (forall (F_9:(nat->Prop)) (G_3:(nat->Prop)), ((iff ((ord_less_eq_nat_o F_9) G_3)) (forall (X_1:nat), ((ord_less_eq_o (F_9 X_1)) (G_3 X_1))))) of role axiom named fact_375_le__fun__def % A new axiom: (forall (F_9:(nat->Prop)) (G_3:(nat->Prop)), ((iff ((ord_less_eq_nat_o F_9) G_3)) (forall (X_1:nat), ((ord_less_eq_o (F_9 X_1)) (G_3 X_1))))) % FOF formula (forall (F_9:(x_a->Prop)) (G_3:(x_a->Prop)), ((iff ((ord_less_eq_a_o F_9) G_3)) (forall (X_1:x_a), ((ord_less_eq_o (F_9 X_1)) (G_3 X_1))))) of role axiom named fact_376_le__fun__def % A new axiom: (forall (F_9:(x_a->Prop)) (G_3:(x_a->Prop)), ((iff ((ord_less_eq_a_o F_9) G_3)) (forall (X_1:x_a), ((ord_less_eq_o (F_9 X_1)) (G_3 X_1))))) % FOF formula (forall (X_15:pname) (F_8:(pname->Prop)) (G_2:(pname->Prop)), (((ord_less_eq_pname_o F_8) G_2)->((ord_less_eq_o (F_8 X_15)) (G_2 X_15)))) of role axiom named fact_377_le__funD % A new axiom: (forall (X_15:pname) (F_8:(pname->Prop)) (G_2:(pname->Prop)), (((ord_less_eq_pname_o F_8) G_2)->((ord_less_eq_o (F_8 X_15)) (G_2 X_15)))) % FOF formula (forall (X_15:nat) (F_8:(nat->Prop)) (G_2:(nat->Prop)), (((ord_less_eq_nat_o F_8) G_2)->((ord_less_eq_o (F_8 X_15)) (G_2 X_15)))) of role axiom named fact_378_le__funD % A new axiom: (forall (X_15:nat) (F_8:(nat->Prop)) (G_2:(nat->Prop)), (((ord_less_eq_nat_o F_8) G_2)->((ord_less_eq_o (F_8 X_15)) (G_2 X_15)))) % FOF formula (forall (X_15:x_a) (F_8:(x_a->Prop)) (G_2:(x_a->Prop)), (((ord_less_eq_a_o F_8) G_2)->((ord_less_eq_o (F_8 X_15)) (G_2 X_15)))) of role axiom named fact_379_le__funD % A new axiom: (forall (X_15:x_a) (F_8:(x_a->Prop)) (G_2:(x_a->Prop)), (((ord_less_eq_a_o F_8) G_2)->((ord_less_eq_o (F_8 X_15)) (G_2 X_15)))) % FOF formula (forall (X_14:pname) (F_7:(pname->Prop)) (G_1:(pname->Prop)), (((ord_less_eq_pname_o F_7) G_1)->((ord_less_eq_o (F_7 X_14)) (G_1 X_14)))) of role axiom named fact_380_le__funE % A new axiom: (forall (X_14:pname) (F_7:(pname->Prop)) (G_1:(pname->Prop)), (((ord_less_eq_pname_o F_7) G_1)->((ord_less_eq_o (F_7 X_14)) (G_1 X_14)))) % FOF formula (forall (X_14:nat) (F_7:(nat->Prop)) (G_1:(nat->Prop)), (((ord_less_eq_nat_o F_7) G_1)->((ord_less_eq_o (F_7 X_14)) (G_1 X_14)))) of role axiom named fact_381_le__funE % A new axiom: (forall (X_14:nat) (F_7:(nat->Prop)) (G_1:(nat->Prop)), (((ord_less_eq_nat_o F_7) G_1)->((ord_less_eq_o (F_7 X_14)) (G_1 X_14)))) % FOF formula (forall (X_14:x_a) (F_7:(x_a->Prop)) (G_1:(x_a->Prop)), (((ord_less_eq_a_o F_7) G_1)->((ord_less_eq_o (F_7 X_14)) (G_1 X_14)))) of role axiom named fact_382_le__funE % A new axiom: (forall (X_14:x_a) (F_7:(x_a->Prop)) (G_1:(x_a->Prop)), (((ord_less_eq_a_o F_7) G_1)->((ord_less_eq_o (F_7 X_14)) (G_1 X_14)))) % FOF formula (forall (A_33:nat), (((member_nat A_33) bot_bot_nat_o)->False)) of role axiom named fact_383_emptyE % A new axiom: (forall (A_33:nat), (((member_nat A_33) bot_bot_nat_o)->False)) % FOF formula (forall (A_33:pname), (((member_pname A_33) bot_bot_pname_o)->False)) of role axiom named fact_384_emptyE % A new axiom: (forall (A_33:pname), (((member_pname A_33) bot_bot_pname_o)->False)) % FOF formula (forall (A_33:x_a), (((member_a A_33) bot_bot_a_o)->False)) of role axiom named fact_385_emptyE % A new axiom: (forall (A_33:x_a), (((member_a A_33) bot_bot_a_o)->False)) % FOF formula (finite_finite_nat_o bot_bot_nat_o_o) of role axiom named fact_386_finite_OemptyI % A new axiom: (finite_finite_nat_o bot_bot_nat_o_o) % FOF formula (finite297249702name_o bot_bot_pname_o_o) of role axiom named fact_387_finite_OemptyI % A new axiom: (finite297249702name_o bot_bot_pname_o_o) % FOF formula (finite_finite_a_o bot_bot_a_o_o) of role axiom named fact_388_finite_OemptyI % A new axiom: (finite_finite_a_o bot_bot_a_o_o) % FOF formula (finite_finite_a bot_bot_a_o) of role axiom named fact_389_finite_OemptyI % A new axiom: (finite_finite_a bot_bot_a_o) % FOF formula (finite_finite_pname bot_bot_pname_o) of role axiom named fact_390_finite_OemptyI % A new axiom: (finite_finite_pname bot_bot_pname_o) % FOF formula (finite_finite_nat bot_bot_nat_o) of role axiom named fact_391_finite_OemptyI % A new axiom: (finite_finite_nat bot_bot_nat_o) % FOF formula (forall (A_32:(pname->Prop)), ((ord_less_eq_pname_o bot_bot_pname_o) A_32)) of role axiom named fact_392_empty__subsetI % A new axiom: (forall (A_32:(pname->Prop)), ((ord_less_eq_pname_o bot_bot_pname_o) A_32)) % FOF formula (forall (A_32:(nat->Prop)), ((ord_less_eq_nat_o bot_bot_nat_o) A_32)) of role axiom named fact_393_empty__subsetI % A new axiom: (forall (A_32:(nat->Prop)), ((ord_less_eq_nat_o bot_bot_nat_o) A_32)) % FOF formula (forall (A_32:(x_a->Prop)), ((ord_less_eq_a_o bot_bot_a_o) A_32)) of role axiom named fact_394_empty__subsetI % A new axiom: (forall (A_32:(x_a->Prop)), ((ord_less_eq_a_o bot_bot_a_o) A_32)) % FOF formula (forall (A_31:nat) (A_30:(nat->Prop)), ((((eq (nat->Prop)) A_30) bot_bot_nat_o)->(((member_nat A_31) A_30)->False))) of role axiom named fact_395_equals0D % A new axiom: (forall (A_31:nat) (A_30:(nat->Prop)), ((((eq (nat->Prop)) A_30) bot_bot_nat_o)->(((member_nat A_31) A_30)->False))) % FOF formula (forall (A_31:pname) (A_30:(pname->Prop)), ((((eq (pname->Prop)) A_30) bot_bot_pname_o)->(((member_pname A_31) A_30)->False))) of role axiom named fact_396_equals0D % A new axiom: (forall (A_31:pname) (A_30:(pname->Prop)), ((((eq (pname->Prop)) A_30) bot_bot_pname_o)->(((member_pname A_31) A_30)->False))) % FOF formula (forall (A_31:x_a) (A_30:(x_a->Prop)), ((((eq (x_a->Prop)) A_30) bot_bot_a_o)->(((member_a A_31) A_30)->False))) of role axiom named fact_397_equals0D % A new axiom: (forall (A_31:x_a) (A_30:(x_a->Prop)), ((((eq (x_a->Prop)) A_30) bot_bot_a_o)->(((member_a A_31) A_30)->False))) % FOF formula (forall (P_6:(pname->Prop)), ((iff (((eq (pname->Prop)) (collect_pname P_6)) bot_bot_pname_o)) (forall (X_1:pname), ((P_6 X_1)->False)))) of role axiom named fact_398_Collect__empty__eq % A new axiom: (forall (P_6:(pname->Prop)), ((iff (((eq (pname->Prop)) (collect_pname P_6)) bot_bot_pname_o)) (forall (X_1:pname), ((P_6 X_1)->False)))) % FOF formula (forall (P_6:((nat->Prop)->Prop)), ((iff (((eq ((nat->Prop)->Prop)) (collect_nat_o P_6)) bot_bot_nat_o_o)) (forall (X_1:(nat->Prop)), ((P_6 X_1)->False)))) of role axiom named fact_399_Collect__empty__eq % A new axiom: (forall (P_6:((nat->Prop)->Prop)), ((iff (((eq ((nat->Prop)->Prop)) (collect_nat_o P_6)) bot_bot_nat_o_o)) (forall (X_1:(nat->Prop)), ((P_6 X_1)->False)))) % FOF formula (forall (P_6:((pname->Prop)->Prop)), ((iff (((eq ((pname->Prop)->Prop)) (collect_pname_o P_6)) bot_bot_pname_o_o)) (forall (X_1:(pname->Prop)), ((P_6 X_1)->False)))) of role axiom named fact_400_Collect__empty__eq % A new axiom: (forall (P_6:((pname->Prop)->Prop)), ((iff (((eq ((pname->Prop)->Prop)) (collect_pname_o P_6)) bot_bot_pname_o_o)) (forall (X_1:(pname->Prop)), ((P_6 X_1)->False)))) % FOF formula (forall (P_6:((x_a->Prop)->Prop)), ((iff (((eq ((x_a->Prop)->Prop)) (collect_a_o P_6)) bot_bot_a_o_o)) (forall (X_1:(x_a->Prop)), ((P_6 X_1)->False)))) of role axiom named fact_401_Collect__empty__eq % A new axiom: (forall (P_6:((x_a->Prop)->Prop)), ((iff (((eq ((x_a->Prop)->Prop)) (collect_a_o P_6)) bot_bot_a_o_o)) (forall (X_1:(x_a->Prop)), ((P_6 X_1)->False)))) % FOF formula (forall (P_6:(x_a->Prop)), ((iff (((eq (x_a->Prop)) (collect_a P_6)) bot_bot_a_o)) (forall (X_1:x_a), ((P_6 X_1)->False)))) of role axiom named fact_402_Collect__empty__eq % A new axiom: (forall (P_6:(x_a->Prop)), ((iff (((eq (x_a->Prop)) (collect_a P_6)) bot_bot_a_o)) (forall (X_1:x_a), ((P_6 X_1)->False)))) % FOF formula (forall (P_6:(nat->Prop)), ((iff (((eq (nat->Prop)) (collect_nat P_6)) bot_bot_nat_o)) (forall (X_1:nat), ((P_6 X_1)->False)))) of role axiom named fact_403_Collect__empty__eq % A new axiom: (forall (P_6:(nat->Prop)), ((iff (((eq (nat->Prop)) (collect_nat P_6)) bot_bot_nat_o)) (forall (X_1:nat), ((P_6 X_1)->False)))) % FOF formula (forall (C_8:nat), (((member_nat C_8) bot_bot_nat_o)->False)) of role axiom named fact_404_empty__iff % A new axiom: (forall (C_8:nat), (((member_nat C_8) bot_bot_nat_o)->False)) % FOF formula (forall (C_8:pname), (((member_pname C_8) bot_bot_pname_o)->False)) of role axiom named fact_405_empty__iff % A new axiom: (forall (C_8:pname), (((member_pname C_8) bot_bot_pname_o)->False)) % FOF formula (forall (C_8:x_a), (((member_a C_8) bot_bot_a_o)->False)) of role axiom named fact_406_empty__iff % A new axiom: (forall (C_8:x_a), (((member_a C_8) bot_bot_a_o)->False)) % FOF formula (forall (P_5:(pname->Prop)), ((iff (((eq (pname->Prop)) bot_bot_pname_o) (collect_pname P_5))) (forall (X_1:pname), ((P_5 X_1)->False)))) of role axiom named fact_407_empty__Collect__eq % A new axiom: (forall (P_5:(pname->Prop)), ((iff (((eq (pname->Prop)) bot_bot_pname_o) (collect_pname P_5))) (forall (X_1:pname), ((P_5 X_1)->False)))) % FOF formula (forall (P_5:((nat->Prop)->Prop)), ((iff (((eq ((nat->Prop)->Prop)) bot_bot_nat_o_o) (collect_nat_o P_5))) (forall (X_1:(nat->Prop)), ((P_5 X_1)->False)))) of role axiom named fact_408_empty__Collect__eq % A new axiom: (forall (P_5:((nat->Prop)->Prop)), ((iff (((eq ((nat->Prop)->Prop)) bot_bot_nat_o_o) (collect_nat_o P_5))) (forall (X_1:(nat->Prop)), ((P_5 X_1)->False)))) % FOF formula (forall (P_5:((pname->Prop)->Prop)), ((iff (((eq ((pname->Prop)->Prop)) bot_bot_pname_o_o) (collect_pname_o P_5))) (forall (X_1:(pname->Prop)), ((P_5 X_1)->False)))) of role axiom named fact_409_empty__Collect__eq % A new axiom: (forall (P_5:((pname->Prop)->Prop)), ((iff (((eq ((pname->Prop)->Prop)) bot_bot_pname_o_o) (collect_pname_o P_5))) (forall (X_1:(pname->Prop)), ((P_5 X_1)->False)))) % FOF formula (forall (P_5:((x_a->Prop)->Prop)), ((iff (((eq ((x_a->Prop)->Prop)) bot_bot_a_o_o) (collect_a_o P_5))) (forall (X_1:(x_a->Prop)), ((P_5 X_1)->False)))) of role axiom named fact_410_empty__Collect__eq % A new axiom: (forall (P_5:((x_a->Prop)->Prop)), ((iff (((eq ((x_a->Prop)->Prop)) bot_bot_a_o_o) (collect_a_o P_5))) (forall (X_1:(x_a->Prop)), ((P_5 X_1)->False)))) % FOF formula (forall (P_5:(x_a->Prop)), ((iff (((eq (x_a->Prop)) bot_bot_a_o) (collect_a P_5))) (forall (X_1:x_a), ((P_5 X_1)->False)))) of role axiom named fact_411_empty__Collect__eq % A new axiom: (forall (P_5:(x_a->Prop)), ((iff (((eq (x_a->Prop)) bot_bot_a_o) (collect_a P_5))) (forall (X_1:x_a), ((P_5 X_1)->False)))) % FOF formula (forall (P_5:(nat->Prop)), ((iff (((eq (nat->Prop)) bot_bot_nat_o) (collect_nat P_5))) (forall (X_1:nat), ((P_5 X_1)->False)))) of role axiom named fact_412_empty__Collect__eq % A new axiom: (forall (P_5:(nat->Prop)), ((iff (((eq (nat->Prop)) bot_bot_nat_o) (collect_nat P_5))) (forall (X_1:nat), ((P_5 X_1)->False)))) % FOF formula (forall (A_29:(nat->Prop)), ((iff ((ex nat) (fun (X_1:nat)=> ((member_nat X_1) A_29)))) (not (((eq (nat->Prop)) A_29) bot_bot_nat_o)))) of role axiom named fact_413_ex__in__conv % A new axiom: (forall (A_29:(nat->Prop)), ((iff ((ex nat) (fun (X_1:nat)=> ((member_nat X_1) A_29)))) (not (((eq (nat->Prop)) A_29) bot_bot_nat_o)))) % FOF formula (forall (A_29:(pname->Prop)), ((iff ((ex pname) (fun (X_1:pname)=> ((member_pname X_1) A_29)))) (not (((eq (pname->Prop)) A_29) bot_bot_pname_o)))) of role axiom named fact_414_ex__in__conv % A new axiom: (forall (A_29:(pname->Prop)), ((iff ((ex pname) (fun (X_1:pname)=> ((member_pname X_1) A_29)))) (not (((eq (pname->Prop)) A_29) bot_bot_pname_o)))) % FOF formula (forall (A_29:(x_a->Prop)), ((iff ((ex x_a) (fun (X_1:x_a)=> ((member_a X_1) A_29)))) (not (((eq (x_a->Prop)) A_29) bot_bot_a_o)))) of role axiom named fact_415_ex__in__conv % A new axiom: (forall (A_29:(x_a->Prop)), ((iff ((ex x_a) (fun (X_1:x_a)=> ((member_a X_1) A_29)))) (not (((eq (x_a->Prop)) A_29) bot_bot_a_o)))) % FOF formula (forall (A_28:(nat->Prop)), ((iff (forall (X_1:nat), (((member_nat X_1) A_28)->False))) (((eq (nat->Prop)) A_28) bot_bot_nat_o))) of role axiom named fact_416_all__not__in__conv % A new axiom: (forall (A_28:(nat->Prop)), ((iff (forall (X_1:nat), (((member_nat X_1) A_28)->False))) (((eq (nat->Prop)) A_28) bot_bot_nat_o))) % FOF formula (forall (A_28:(pname->Prop)), ((iff (forall (X_1:pname), (((member_pname X_1) A_28)->False))) (((eq (pname->Prop)) A_28) bot_bot_pname_o))) of role axiom named fact_417_all__not__in__conv % A new axiom: (forall (A_28:(pname->Prop)), ((iff (forall (X_1:pname), (((member_pname X_1) A_28)->False))) (((eq (pname->Prop)) A_28) bot_bot_pname_o))) % FOF formula (forall (A_28:(x_a->Prop)), ((iff (forall (X_1:x_a), (((member_a X_1) A_28)->False))) (((eq (x_a->Prop)) A_28) bot_bot_a_o))) of role axiom named fact_418_all__not__in__conv % A new axiom: (forall (A_28:(x_a->Prop)), ((iff (forall (X_1:x_a), (((member_a X_1) A_28)->False))) (((eq (x_a->Prop)) A_28) bot_bot_a_o))) % FOF formula (((eq (pname->Prop)) bot_bot_pname_o) (collect_pname (fun (X_1:pname)=> False))) of role axiom named fact_419_empty__def % A new axiom: (((eq (pname->Prop)) bot_bot_pname_o) (collect_pname (fun (X_1:pname)=> False))) % FOF formula (((eq ((nat->Prop)->Prop)) bot_bot_nat_o_o) (collect_nat_o (fun (X_1:(nat->Prop))=> False))) of role axiom named fact_420_empty__def % A new axiom: (((eq ((nat->Prop)->Prop)) bot_bot_nat_o_o) (collect_nat_o (fun (X_1:(nat->Prop))=> False))) % FOF formula (((eq ((pname->Prop)->Prop)) bot_bot_pname_o_o) (collect_pname_o (fun (X_1:(pname->Prop))=> False))) of role axiom named fact_421_empty__def % A new axiom: (((eq ((pname->Prop)->Prop)) bot_bot_pname_o_o) (collect_pname_o (fun (X_1:(pname->Prop))=> False))) % FOF formula (((eq ((x_a->Prop)->Prop)) bot_bot_a_o_o) (collect_a_o (fun (X_1:(x_a->Prop))=> False))) of role axiom named fact_422_empty__def % A new axiom: (((eq ((x_a->Prop)->Prop)) bot_bot_a_o_o) (collect_a_o (fun (X_1:(x_a->Prop))=> False))) % FOF formula (((eq (x_a->Prop)) bot_bot_a_o) (collect_a (fun (X_1:x_a)=> False))) of role axiom named fact_423_empty__def % A new axiom: (((eq (x_a->Prop)) bot_bot_a_o) (collect_a (fun (X_1:x_a)=> False))) % FOF formula (((eq (nat->Prop)) bot_bot_nat_o) (collect_nat (fun (X_1:nat)=> False))) of role axiom named fact_424_empty__def % A new axiom: (((eq (nat->Prop)) bot_bot_nat_o) (collect_nat (fun (X_1:nat)=> False))) % FOF formula (forall (X_1:pname), ((iff (bot_bot_pname_o X_1)) bot_bot_o)) of role axiom named fact_425_bot__fun__def % A new axiom: (forall (X_1:pname), ((iff (bot_bot_pname_o X_1)) bot_bot_o)) % FOF formula (forall (X_1:nat), ((iff (bot_bot_nat_o X_1)) bot_bot_o)) of role axiom named fact_426_bot__fun__def % A new axiom: (forall (X_1:nat), ((iff (bot_bot_nat_o X_1)) bot_bot_o)) % FOF formula (forall (X_1:x_a), ((iff (bot_bot_a_o X_1)) bot_bot_o)) of role axiom named fact_427_bot__fun__def % A new axiom: (forall (X_1:x_a), ((iff (bot_bot_a_o X_1)) bot_bot_o)) % FOF formula (forall (X_13:pname), ((iff (bot_bot_pname_o X_13)) bot_bot_o)) of role axiom named fact_428_bot__apply % A new axiom: (forall (X_13:pname), ((iff (bot_bot_pname_o X_13)) bot_bot_o)) % FOF formula (forall (X_13:nat), ((iff (bot_bot_nat_o X_13)) bot_bot_o)) of role axiom named fact_429_bot__apply % A new axiom: (forall (X_13:nat), ((iff (bot_bot_nat_o X_13)) bot_bot_o)) % FOF formula (forall (X_13:x_a), ((iff (bot_bot_a_o X_13)) bot_bot_o)) of role axiom named fact_430_bot__apply % A new axiom: (forall (X_13:x_a), ((iff (bot_bot_a_o X_13)) bot_bot_o)) % FOF formula (forall (A_27:(pname->Prop)), (((ord_less_eq_pname_o A_27) bot_bot_pname_o)->(((eq (pname->Prop)) A_27) bot_bot_pname_o))) of role axiom named fact_431_le__bot % A new axiom: (forall (A_27:(pname->Prop)), (((ord_less_eq_pname_o A_27) bot_bot_pname_o)->(((eq (pname->Prop)) A_27) bot_bot_pname_o))) % FOF formula (forall (A_27:Prop), (((ord_less_eq_o A_27) bot_bot_o)->((iff A_27) bot_bot_o))) of role axiom named fact_432_le__bot % A new axiom: (forall (A_27:Prop), (((ord_less_eq_o A_27) bot_bot_o)->((iff A_27) bot_bot_o))) % FOF formula (forall (A_27:(nat->Prop)), (((ord_less_eq_nat_o A_27) bot_bot_nat_o)->(((eq (nat->Prop)) A_27) bot_bot_nat_o))) of role axiom named fact_433_le__bot % A new axiom: (forall (A_27:(nat->Prop)), (((ord_less_eq_nat_o A_27) bot_bot_nat_o)->(((eq (nat->Prop)) A_27) bot_bot_nat_o))) % FOF formula (forall (A_27:(x_a->Prop)), (((ord_less_eq_a_o A_27) bot_bot_a_o)->(((eq (x_a->Prop)) A_27) bot_bot_a_o))) of role axiom named fact_434_le__bot % A new axiom: (forall (A_27:(x_a->Prop)), (((ord_less_eq_a_o A_27) bot_bot_a_o)->(((eq (x_a->Prop)) A_27) bot_bot_a_o))) % FOF formula (forall (A_27:nat), (((ord_less_eq_nat A_27) bot_bot_nat)->(((eq nat) A_27) bot_bot_nat))) of role axiom named fact_435_le__bot % A new axiom: (forall (A_27:nat), (((ord_less_eq_nat A_27) bot_bot_nat)->(((eq nat) A_27) bot_bot_nat))) % FOF formula (forall (A_26:(pname->Prop)), ((iff ((ord_less_eq_pname_o A_26) bot_bot_pname_o)) (((eq (pname->Prop)) A_26) bot_bot_pname_o))) of role axiom named fact_436_bot__unique % A new axiom: (forall (A_26:(pname->Prop)), ((iff ((ord_less_eq_pname_o A_26) bot_bot_pname_o)) (((eq (pname->Prop)) A_26) bot_bot_pname_o))) % FOF formula (forall (A_26:Prop), ((iff ((ord_less_eq_o A_26) bot_bot_o)) ((iff A_26) bot_bot_o))) of role axiom named fact_437_bot__unique % A new axiom: (forall (A_26:Prop), ((iff ((ord_less_eq_o A_26) bot_bot_o)) ((iff A_26) bot_bot_o))) % FOF formula (forall (A_26:(nat->Prop)), ((iff ((ord_less_eq_nat_o A_26) bot_bot_nat_o)) (((eq (nat->Prop)) A_26) bot_bot_nat_o))) of role axiom named fact_438_bot__unique % A new axiom: (forall (A_26:(nat->Prop)), ((iff ((ord_less_eq_nat_o A_26) bot_bot_nat_o)) (((eq (nat->Prop)) A_26) bot_bot_nat_o))) % FOF formula (forall (A_26:(x_a->Prop)), ((iff ((ord_less_eq_a_o A_26) bot_bot_a_o)) (((eq (x_a->Prop)) A_26) bot_bot_a_o))) of role axiom named fact_439_bot__unique % A new axiom: (forall (A_26:(x_a->Prop)), ((iff ((ord_less_eq_a_o A_26) bot_bot_a_o)) (((eq (x_a->Prop)) A_26) bot_bot_a_o))) % FOF formula (forall (A_26:nat), ((iff ((ord_less_eq_nat A_26) bot_bot_nat)) (((eq nat) A_26) bot_bot_nat))) of role axiom named fact_440_bot__unique % A new axiom: (forall (A_26:nat), ((iff ((ord_less_eq_nat A_26) bot_bot_nat)) (((eq nat) A_26) bot_bot_nat))) % FOF formula (forall (A_25:(pname->Prop)), ((ord_less_eq_pname_o bot_bot_pname_o) A_25)) of role axiom named fact_441_bot__least % A new axiom: (forall (A_25:(pname->Prop)), ((ord_less_eq_pname_o bot_bot_pname_o) A_25)) % FOF formula (forall (A_25:Prop), ((ord_less_eq_o bot_bot_o) A_25)) of role axiom named fact_442_bot__least % A new axiom: (forall (A_25:Prop), ((ord_less_eq_o bot_bot_o) A_25)) % FOF formula (forall (A_25:(nat->Prop)), ((ord_less_eq_nat_o bot_bot_nat_o) A_25)) of role axiom named fact_443_bot__least % A new axiom: (forall (A_25:(nat->Prop)), ((ord_less_eq_nat_o bot_bot_nat_o) A_25)) % FOF formula (forall (A_25:(x_a->Prop)), ((ord_less_eq_a_o bot_bot_a_o) A_25)) of role axiom named fact_444_bot__least % A new axiom: (forall (A_25:(x_a->Prop)), ((ord_less_eq_a_o bot_bot_a_o) A_25)) % FOF formula (forall (A_25:nat), ((ord_less_eq_nat bot_bot_nat) A_25)) of role axiom named fact_445_bot__least % A new axiom: (forall (A_25:nat), ((ord_less_eq_nat bot_bot_nat) A_25)) % FOF formula (forall (A_24:pname) (B_8:pname), ((((eq (pname->Prop)) ((insert_pname A_24) bot_bot_pname_o)) ((insert_pname B_8) bot_bot_pname_o))->(((eq pname) A_24) B_8))) of role axiom named fact_446_singleton__inject % A new axiom: (forall (A_24:pname) (B_8:pname), ((((eq (pname->Prop)) ((insert_pname A_24) bot_bot_pname_o)) ((insert_pname B_8) bot_bot_pname_o))->(((eq pname) A_24) B_8))) % FOF formula (forall (A_24:nat) (B_8:nat), ((((eq (nat->Prop)) ((insert_nat A_24) bot_bot_nat_o)) ((insert_nat B_8) bot_bot_nat_o))->(((eq nat) A_24) B_8))) of role axiom named fact_447_singleton__inject % A new axiom: (forall (A_24:nat) (B_8:nat), ((((eq (nat->Prop)) ((insert_nat A_24) bot_bot_nat_o)) ((insert_nat B_8) bot_bot_nat_o))->(((eq nat) A_24) B_8))) % FOF formula (forall (A_24:x_a) (B_8:x_a), ((((eq (x_a->Prop)) ((insert_a A_24) bot_bot_a_o)) ((insert_a B_8) bot_bot_a_o))->(((eq x_a) A_24) B_8))) of role axiom named fact_448_singleton__inject % A new axiom: (forall (A_24:x_a) (B_8:x_a), ((((eq (x_a->Prop)) ((insert_a A_24) bot_bot_a_o)) ((insert_a B_8) bot_bot_a_o))->(((eq x_a) A_24) B_8))) % FOF formula (forall (B_7:nat) (A_23:nat), (((member_nat B_7) ((insert_nat A_23) bot_bot_nat_o))->(((eq nat) B_7) A_23))) of role axiom named fact_449_singletonE % A new axiom: (forall (B_7:nat) (A_23:nat), (((member_nat B_7) ((insert_nat A_23) bot_bot_nat_o))->(((eq nat) B_7) A_23))) % FOF formula (forall (B_7:pname) (A_23:pname), (((member_pname B_7) ((insert_pname A_23) bot_bot_pname_o))->(((eq pname) B_7) A_23))) of role axiom named fact_450_singletonE % A new axiom: (forall (B_7:pname) (A_23:pname), (((member_pname B_7) ((insert_pname A_23) bot_bot_pname_o))->(((eq pname) B_7) A_23))) % FOF formula (forall (B_7:x_a) (A_23:x_a), (((member_a B_7) ((insert_a A_23) bot_bot_a_o))->(((eq x_a) B_7) A_23))) of role axiom named fact_451_singletonE % A new axiom: (forall (B_7:x_a) (A_23:x_a), (((member_a B_7) ((insert_a A_23) bot_bot_a_o))->(((eq x_a) B_7) A_23))) % FOF formula (forall (A_22:pname) (B_6:pname) (C_7:pname) (D:pname), ((iff (((eq (pname->Prop)) ((insert_pname A_22) ((insert_pname B_6) bot_bot_pname_o))) ((insert_pname C_7) ((insert_pname D) bot_bot_pname_o)))) ((or ((and (((eq pname) A_22) C_7)) (((eq pname) B_6) D))) ((and (((eq pname) A_22) D)) (((eq pname) B_6) C_7))))) of role axiom named fact_452_doubleton__eq__iff % A new axiom: (forall (A_22:pname) (B_6:pname) (C_7:pname) (D:pname), ((iff (((eq (pname->Prop)) ((insert_pname A_22) ((insert_pname B_6) bot_bot_pname_o))) ((insert_pname C_7) ((insert_pname D) bot_bot_pname_o)))) ((or ((and (((eq pname) A_22) C_7)) (((eq pname) B_6) D))) ((and (((eq pname) A_22) D)) (((eq pname) B_6) C_7))))) % FOF formula (forall (A_22:nat) (B_6:nat) (C_7:nat) (D:nat), ((iff (((eq (nat->Prop)) ((insert_nat A_22) ((insert_nat B_6) bot_bot_nat_o))) ((insert_nat C_7) ((insert_nat D) bot_bot_nat_o)))) ((or ((and (((eq nat) A_22) C_7)) (((eq nat) B_6) D))) ((and (((eq nat) A_22) D)) (((eq nat) B_6) C_7))))) of role axiom named fact_453_doubleton__eq__iff % A new axiom: (forall (A_22:nat) (B_6:nat) (C_7:nat) (D:nat), ((iff (((eq (nat->Prop)) ((insert_nat A_22) ((insert_nat B_6) bot_bot_nat_o))) ((insert_nat C_7) ((insert_nat D) bot_bot_nat_o)))) ((or ((and (((eq nat) A_22) C_7)) (((eq nat) B_6) D))) ((and (((eq nat) A_22) D)) (((eq nat) B_6) C_7))))) % FOF formula (forall (A_22:x_a) (B_6:x_a) (C_7:x_a) (D:x_a), ((iff (((eq (x_a->Prop)) ((insert_a A_22) ((insert_a B_6) bot_bot_a_o))) ((insert_a C_7) ((insert_a D) bot_bot_a_o)))) ((or ((and (((eq x_a) A_22) C_7)) (((eq x_a) B_6) D))) ((and (((eq x_a) A_22) D)) (((eq x_a) B_6) C_7))))) of role axiom named fact_454_doubleton__eq__iff % A new axiom: (forall (A_22:x_a) (B_6:x_a) (C_7:x_a) (D:x_a), ((iff (((eq (x_a->Prop)) ((insert_a A_22) ((insert_a B_6) bot_bot_a_o))) ((insert_a C_7) ((insert_a D) bot_bot_a_o)))) ((or ((and (((eq x_a) A_22) C_7)) (((eq x_a) B_6) D))) ((and (((eq x_a) A_22) D)) (((eq x_a) B_6) C_7))))) % FOF formula (forall (B_5:nat) (A_21:nat), ((iff ((member_nat B_5) ((insert_nat A_21) bot_bot_nat_o))) (((eq nat) B_5) A_21))) of role axiom named fact_455_singleton__iff % A new axiom: (forall (B_5:nat) (A_21:nat), ((iff ((member_nat B_5) ((insert_nat A_21) bot_bot_nat_o))) (((eq nat) B_5) A_21))) % FOF formula (forall (B_5:pname) (A_21:pname), ((iff ((member_pname B_5) ((insert_pname A_21) bot_bot_pname_o))) (((eq pname) B_5) A_21))) of role axiom named fact_456_singleton__iff % A new axiom: (forall (B_5:pname) (A_21:pname), ((iff ((member_pname B_5) ((insert_pname A_21) bot_bot_pname_o))) (((eq pname) B_5) A_21))) % FOF formula (forall (B_5:x_a) (A_21:x_a), ((iff ((member_a B_5) ((insert_a A_21) bot_bot_a_o))) (((eq x_a) B_5) A_21))) of role axiom named fact_457_singleton__iff % A new axiom: (forall (B_5:x_a) (A_21:x_a), ((iff ((member_a B_5) ((insert_a A_21) bot_bot_a_o))) (((eq x_a) B_5) A_21))) % FOF formula (forall (A_20:pname) (A_19:(pname->Prop)), (not (((eq (pname->Prop)) ((insert_pname A_20) A_19)) bot_bot_pname_o))) of role axiom named fact_458_insert__not__empty % A new axiom: (forall (A_20:pname) (A_19:(pname->Prop)), (not (((eq (pname->Prop)) ((insert_pname A_20) A_19)) bot_bot_pname_o))) % FOF formula (forall (A_20:nat) (A_19:(nat->Prop)), (not (((eq (nat->Prop)) ((insert_nat A_20) A_19)) bot_bot_nat_o))) of role axiom named fact_459_insert__not__empty % A new axiom: (forall (A_20:nat) (A_19:(nat->Prop)), (not (((eq (nat->Prop)) ((insert_nat A_20) A_19)) bot_bot_nat_o))) % FOF formula (forall (A_20:x_a) (A_19:(x_a->Prop)), (not (((eq (x_a->Prop)) ((insert_a A_20) A_19)) bot_bot_a_o))) of role axiom named fact_460_insert__not__empty % A new axiom: (forall (A_20:x_a) (A_19:(x_a->Prop)), (not (((eq (x_a->Prop)) ((insert_a A_20) A_19)) bot_bot_a_o))) % FOF formula (forall (A_18:pname) (A_17:(pname->Prop)), (not (((eq (pname->Prop)) bot_bot_pname_o) ((insert_pname A_18) A_17)))) of role axiom named fact_461_empty__not__insert % A new axiom: (forall (A_18:pname) (A_17:(pname->Prop)), (not (((eq (pname->Prop)) bot_bot_pname_o) ((insert_pname A_18) A_17)))) % FOF formula (forall (A_18:nat) (A_17:(nat->Prop)), (not (((eq (nat->Prop)) bot_bot_nat_o) ((insert_nat A_18) A_17)))) of role axiom named fact_462_empty__not__insert % A new axiom: (forall (A_18:nat) (A_17:(nat->Prop)), (not (((eq (nat->Prop)) bot_bot_nat_o) ((insert_nat A_18) A_17)))) % FOF formula (forall (A_18:x_a) (A_17:(x_a->Prop)), (not (((eq (x_a->Prop)) bot_bot_a_o) ((insert_a A_18) A_17)))) of role axiom named fact_463_empty__not__insert % A new axiom: (forall (A_18:x_a) (A_17:(x_a->Prop)), (not (((eq (x_a->Prop)) bot_bot_a_o) ((insert_a A_18) A_17)))) % FOF formula (forall (A_16:(nat->Prop)), ((iff ((ord_less_eq_nat_o A_16) bot_bot_nat_o)) (((eq (nat->Prop)) A_16) bot_bot_nat_o))) of role axiom named fact_464_subset__empty % A new axiom: (forall (A_16:(nat->Prop)), ((iff ((ord_less_eq_nat_o A_16) bot_bot_nat_o)) (((eq (nat->Prop)) A_16) bot_bot_nat_o))) % FOF formula (forall (A_16:(pname->Prop)), ((iff ((ord_less_eq_pname_o A_16) bot_bot_pname_o)) (((eq (pname->Prop)) A_16) bot_bot_pname_o))) of role axiom named fact_465_subset__empty % A new axiom: (forall (A_16:(pname->Prop)), ((iff ((ord_less_eq_pname_o A_16) bot_bot_pname_o)) (((eq (pname->Prop)) A_16) bot_bot_pname_o))) % FOF formula (forall (A_16:(x_a->Prop)), ((iff ((ord_less_eq_a_o A_16) bot_bot_a_o)) (((eq (x_a->Prop)) A_16) bot_bot_a_o))) of role axiom named fact_466_subset__empty % A new axiom: (forall (A_16:(x_a->Prop)), ((iff ((ord_less_eq_a_o A_16) bot_bot_a_o)) (((eq (x_a->Prop)) A_16) bot_bot_a_o))) % FOF formula (forall (F_6:(pname->x_a)) (A_15:(pname->Prop)), ((iff (((eq (x_a->Prop)) ((image_pname_a F_6) A_15)) bot_bot_a_o)) (((eq (pname->Prop)) A_15) bot_bot_pname_o))) of role axiom named fact_467_image__is__empty % A new axiom: (forall (F_6:(pname->x_a)) (A_15:(pname->Prop)), ((iff (((eq (x_a->Prop)) ((image_pname_a F_6) A_15)) bot_bot_a_o)) (((eq (pname->Prop)) A_15) bot_bot_pname_o))) % FOF formula (forall (F_5:(pname->x_a)), (((eq (x_a->Prop)) ((image_pname_a F_5) bot_bot_pname_o)) bot_bot_a_o)) of role axiom named fact_468_image__empty % A new axiom: (forall (F_5:(pname->x_a)), (((eq (x_a->Prop)) ((image_pname_a F_5) bot_bot_pname_o)) bot_bot_a_o)) % FOF formula (forall (F_4:(pname->x_a)) (A_14:(pname->Prop)), ((iff (((eq (x_a->Prop)) bot_bot_a_o) ((image_pname_a F_4) A_14))) (((eq (pname->Prop)) A_14) bot_bot_pname_o))) of role axiom named fact_469_empty__is__image % A new axiom: (forall (F_4:(pname->x_a)) (A_14:(pname->Prop)), ((iff (((eq (x_a->Prop)) bot_bot_a_o) ((image_pname_a F_4) A_14))) (((eq (pname->Prop)) A_14) bot_bot_pname_o))) % FOF formula (forall (P_4:(nat->Prop)) (A_13:nat), ((and ((P_4 A_13)->(((eq (nat->Prop)) (collect_nat (fun (X_1:nat)=> ((and (((eq nat) X_1) A_13)) (P_4 X_1))))) ((insert_nat A_13) bot_bot_nat_o)))) (((P_4 A_13)->False)->(((eq (nat->Prop)) (collect_nat (fun (X_1:nat)=> ((and (((eq nat) X_1) A_13)) (P_4 X_1))))) bot_bot_nat_o)))) of role axiom named fact_470_Collect__conv__if % A new axiom: (forall (P_4:(nat->Prop)) (A_13:nat), ((and ((P_4 A_13)->(((eq (nat->Prop)) (collect_nat (fun (X_1:nat)=> ((and (((eq nat) X_1) A_13)) (P_4 X_1))))) ((insert_nat A_13) bot_bot_nat_o)))) (((P_4 A_13)->False)->(((eq (nat->Prop)) (collect_nat (fun (X_1:nat)=> ((and (((eq nat) X_1) A_13)) (P_4 X_1))))) bot_bot_nat_o)))) % FOF formula (forall (P_4:(pname->Prop)) (A_13:pname), ((and ((P_4 A_13)->(((eq (pname->Prop)) (collect_pname (fun (X_1:pname)=> ((and (((eq pname) X_1) A_13)) (P_4 X_1))))) ((insert_pname A_13) bot_bot_pname_o)))) (((P_4 A_13)->False)->(((eq (pname->Prop)) (collect_pname (fun (X_1:pname)=> ((and (((eq pname) X_1) A_13)) (P_4 X_1))))) bot_bot_pname_o)))) of role axiom named fact_471_Collect__conv__if % A new axiom: (forall (P_4:(pname->Prop)) (A_13:pname), ((and ((P_4 A_13)->(((eq (pname->Prop)) (collect_pname (fun (X_1:pname)=> ((and (((eq pname) X_1) A_13)) (P_4 X_1))))) ((insert_pname A_13) bot_bot_pname_o)))) (((P_4 A_13)->False)->(((eq (pname->Prop)) (collect_pname (fun (X_1:pname)=> ((and (((eq pname) X_1) A_13)) (P_4 X_1))))) bot_bot_pname_o)))) % FOF formula (forall (P_4:(x_a->Prop)) (A_13:x_a), ((and ((P_4 A_13)->(((eq (x_a->Prop)) (collect_a (fun (X_1:x_a)=> ((and (((eq x_a) X_1) A_13)) (P_4 X_1))))) ((insert_a A_13) bot_bot_a_o)))) (((P_4 A_13)->False)->(((eq (x_a->Prop)) (collect_a (fun (X_1:x_a)=> ((and (((eq x_a) X_1) A_13)) (P_4 X_1))))) bot_bot_a_o)))) of role axiom named fact_472_Collect__conv__if % A new axiom: (forall (P_4:(x_a->Prop)) (A_13:x_a), ((and ((P_4 A_13)->(((eq (x_a->Prop)) (collect_a (fun (X_1:x_a)=> ((and (((eq x_a) X_1) A_13)) (P_4 X_1))))) ((insert_a A_13) bot_bot_a_o)))) (((P_4 A_13)->False)->(((eq (x_a->Prop)) (collect_a (fun (X_1:x_a)=> ((and (((eq x_a) X_1) A_13)) (P_4 X_1))))) bot_bot_a_o)))) % FOF formula (forall (P_4:((nat->Prop)->Prop)) (A_13:(nat->Prop)), ((and ((P_4 A_13)->(((eq ((nat->Prop)->Prop)) (collect_nat_o (fun (X_1:(nat->Prop))=> ((and (((eq (nat->Prop)) X_1) A_13)) (P_4 X_1))))) ((insert_nat_o A_13) bot_bot_nat_o_o)))) (((P_4 A_13)->False)->(((eq ((nat->Prop)->Prop)) (collect_nat_o (fun (X_1:(nat->Prop))=> ((and (((eq (nat->Prop)) X_1) A_13)) (P_4 X_1))))) bot_bot_nat_o_o)))) of role axiom named fact_473_Collect__conv__if % A new axiom: (forall (P_4:((nat->Prop)->Prop)) (A_13:(nat->Prop)), ((and ((P_4 A_13)->(((eq ((nat->Prop)->Prop)) (collect_nat_o (fun (X_1:(nat->Prop))=> ((and (((eq (nat->Prop)) X_1) A_13)) (P_4 X_1))))) ((insert_nat_o A_13) bot_bot_nat_o_o)))) (((P_4 A_13)->False)->(((eq ((nat->Prop)->Prop)) (collect_nat_o (fun (X_1:(nat->Prop))=> ((and (((eq (nat->Prop)) X_1) A_13)) (P_4 X_1))))) bot_bot_nat_o_o)))) % FOF formula (forall (P_4:((pname->Prop)->Prop)) (A_13:(pname->Prop)), ((and ((P_4 A_13)->(((eq ((pname->Prop)->Prop)) (collect_pname_o (fun (X_1:(pname->Prop))=> ((and (((eq (pname->Prop)) X_1) A_13)) (P_4 X_1))))) ((insert_pname_o A_13) bot_bot_pname_o_o)))) (((P_4 A_13)->False)->(((eq ((pname->Prop)->Prop)) (collect_pname_o (fun (X_1:(pname->Prop))=> ((and (((eq (pname->Prop)) X_1) A_13)) (P_4 X_1))))) bot_bot_pname_o_o)))) of role axiom named fact_474_Collect__conv__if % A new axiom: (forall (P_4:((pname->Prop)->Prop)) (A_13:(pname->Prop)), ((and ((P_4 A_13)->(((eq ((pname->Prop)->Prop)) (collect_pname_o (fun (X_1:(pname->Prop))=> ((and (((eq (pname->Prop)) X_1) A_13)) (P_4 X_1))))) ((insert_pname_o A_13) bot_bot_pname_o_o)))) (((P_4 A_13)->False)->(((eq ((pname->Prop)->Prop)) (collect_pname_o (fun (X_1:(pname->Prop))=> ((and (((eq (pname->Prop)) X_1) A_13)) (P_4 X_1))))) bot_bot_pname_o_o)))) % FOF formula (forall (P_4:((x_a->Prop)->Prop)) (A_13:(x_a->Prop)), ((and ((P_4 A_13)->(((eq ((x_a->Prop)->Prop)) (collect_a_o (fun (X_1:(x_a->Prop))=> ((and (((eq (x_a->Prop)) X_1) A_13)) (P_4 X_1))))) ((insert_a_o A_13) bot_bot_a_o_o)))) (((P_4 A_13)->False)->(((eq ((x_a->Prop)->Prop)) (collect_a_o (fun (X_1:(x_a->Prop))=> ((and (((eq (x_a->Prop)) X_1) A_13)) (P_4 X_1))))) bot_bot_a_o_o)))) of role axiom named fact_475_Collect__conv__if % A new axiom: (forall (P_4:((x_a->Prop)->Prop)) (A_13:(x_a->Prop)), ((and ((P_4 A_13)->(((eq ((x_a->Prop)->Prop)) (collect_a_o (fun (X_1:(x_a->Prop))=> ((and (((eq (x_a->Prop)) X_1) A_13)) (P_4 X_1))))) ((insert_a_o A_13) bot_bot_a_o_o)))) (((P_4 A_13)->False)->(((eq ((x_a->Prop)->Prop)) (collect_a_o (fun (X_1:(x_a->Prop))=> ((and (((eq (x_a->Prop)) X_1) A_13)) (P_4 X_1))))) bot_bot_a_o_o)))) % FOF formula (forall (P_3:(nat->Prop)) (A_12:nat), ((and ((P_3 A_12)->(((eq (nat->Prop)) (collect_nat (fun (X_1:nat)=> ((and (((eq nat) A_12) X_1)) (P_3 X_1))))) ((insert_nat A_12) bot_bot_nat_o)))) (((P_3 A_12)->False)->(((eq (nat->Prop)) (collect_nat (fun (X_1:nat)=> ((and (((eq nat) A_12) X_1)) (P_3 X_1))))) bot_bot_nat_o)))) of role axiom named fact_476_Collect__conv__if2 % A new axiom: (forall (P_3:(nat->Prop)) (A_12:nat), ((and ((P_3 A_12)->(((eq (nat->Prop)) (collect_nat (fun (X_1:nat)=> ((and (((eq nat) A_12) X_1)) (P_3 X_1))))) ((insert_nat A_12) bot_bot_nat_o)))) (((P_3 A_12)->False)->(((eq (nat->Prop)) (collect_nat (fun (X_1:nat)=> ((and (((eq nat) A_12) X_1)) (P_3 X_1))))) bot_bot_nat_o)))) % FOF formula (forall (P_3:(pname->Prop)) (A_12:pname), ((and ((P_3 A_12)->(((eq (pname->Prop)) (collect_pname (fun (X_1:pname)=> ((and (((eq pname) A_12) X_1)) (P_3 X_1))))) ((insert_pname A_12) bot_bot_pname_o)))) (((P_3 A_12)->False)->(((eq (pname->Prop)) (collect_pname (fun (X_1:pname)=> ((and (((eq pname) A_12) X_1)) (P_3 X_1))))) bot_bot_pname_o)))) of role axiom named fact_477_Collect__conv__if2 % A new axiom: (forall (P_3:(pname->Prop)) (A_12:pname), ((and ((P_3 A_12)->(((eq (pname->Prop)) (collect_pname (fun (X_1:pname)=> ((and (((eq pname) A_12) X_1)) (P_3 X_1))))) ((insert_pname A_12) bot_bot_pname_o)))) (((P_3 A_12)->False)->(((eq (pname->Prop)) (collect_pname (fun (X_1:pname)=> ((and (((eq pname) A_12) X_1)) (P_3 X_1))))) bot_bot_pname_o)))) % FOF formula (forall (P_3:(x_a->Prop)) (A_12:x_a), ((and ((P_3 A_12)->(((eq (x_a->Prop)) (collect_a (fun (X_1:x_a)=> ((and (((eq x_a) A_12) X_1)) (P_3 X_1))))) ((insert_a A_12) bot_bot_a_o)))) (((P_3 A_12)->False)->(((eq (x_a->Prop)) (collect_a (fun (X_1:x_a)=> ((and (((eq x_a) A_12) X_1)) (P_3 X_1))))) bot_bot_a_o)))) of role axiom named fact_478_Collect__conv__if2 % A new axiom: (forall (P_3:(x_a->Prop)) (A_12:x_a), ((and ((P_3 A_12)->(((eq (x_a->Prop)) (collect_a (fun (X_1:x_a)=> ((and (((eq x_a) A_12) X_1)) (P_3 X_1))))) ((insert_a A_12) bot_bot_a_o)))) (((P_3 A_12)->False)->(((eq (x_a->Prop)) (collect_a (fun (X_1:x_a)=> ((and (((eq x_a) A_12) X_1)) (P_3 X_1))))) bot_bot_a_o)))) % FOF formula (forall (P_3:((nat->Prop)->Prop)) (A_12:(nat->Prop)), ((and ((P_3 A_12)->(((eq ((nat->Prop)->Prop)) (collect_nat_o (fun (X_1:(nat->Prop))=> ((and (((eq (nat->Prop)) A_12) X_1)) (P_3 X_1))))) ((insert_nat_o A_12) bot_bot_nat_o_o)))) (((P_3 A_12)->False)->(((eq ((nat->Prop)->Prop)) (collect_nat_o (fun (X_1:(nat->Prop))=> ((and (((eq (nat->Prop)) A_12) X_1)) (P_3 X_1))))) bot_bot_nat_o_o)))) of role axiom named fact_479_Collect__conv__if2 % A new axiom: (forall (P_3:((nat->Prop)->Prop)) (A_12:(nat->Prop)), ((and ((P_3 A_12)->(((eq ((nat->Prop)->Prop)) (collect_nat_o (fun (X_1:(nat->Prop))=> ((and (((eq (nat->Prop)) A_12) X_1)) (P_3 X_1))))) ((insert_nat_o A_12) bot_bot_nat_o_o)))) (((P_3 A_12)->False)->(((eq ((nat->Prop)->Prop)) (collect_nat_o (fun (X_1:(nat->Prop))=> ((and (((eq (nat->Prop)) A_12) X_1)) (P_3 X_1))))) bot_bot_nat_o_o)))) % FOF formula (forall (P_3:((pname->Prop)->Prop)) (A_12:(pname->Prop)), ((and ((P_3 A_12)->(((eq ((pname->Prop)->Prop)) (collect_pname_o (fun (X_1:(pname->Prop))=> ((and (((eq (pname->Prop)) A_12) X_1)) (P_3 X_1))))) ((insert_pname_o A_12) bot_bot_pname_o_o)))) (((P_3 A_12)->False)->(((eq ((pname->Prop)->Prop)) (collect_pname_o (fun (X_1:(pname->Prop))=> ((and (((eq (pname->Prop)) A_12) X_1)) (P_3 X_1))))) bot_bot_pname_o_o)))) of role axiom named fact_480_Collect__conv__if2 % A new axiom: (forall (P_3:((pname->Prop)->Prop)) (A_12:(pname->Prop)), ((and ((P_3 A_12)->(((eq ((pname->Prop)->Prop)) (collect_pname_o (fun (X_1:(pname->Prop))=> ((and (((eq (pname->Prop)) A_12) X_1)) (P_3 X_1))))) ((insert_pname_o A_12) bot_bot_pname_o_o)))) (((P_3 A_12)->False)->(((eq ((pname->Prop)->Prop)) (collect_pname_o (fun (X_1:(pname->Prop))=> ((and (((eq (pname->Prop)) A_12) X_1)) (P_3 X_1))))) bot_bot_pname_o_o)))) % FOF formula (forall (P_3:((x_a->Prop)->Prop)) (A_12:(x_a->Prop)), ((and ((P_3 A_12)->(((eq ((x_a->Prop)->Prop)) (collect_a_o (fun (X_1:(x_a->Prop))=> ((and (((eq (x_a->Prop)) A_12) X_1)) (P_3 X_1))))) ((insert_a_o A_12) bot_bot_a_o_o)))) (((P_3 A_12)->False)->(((eq ((x_a->Prop)->Prop)) (collect_a_o (fun (X_1:(x_a->Prop))=> ((and (((eq (x_a->Prop)) A_12) X_1)) (P_3 X_1))))) bot_bot_a_o_o)))) of role axiom named fact_481_Collect__conv__if2 % A new axiom: (forall (P_3:((x_a->Prop)->Prop)) (A_12:(x_a->Prop)), ((and ((P_3 A_12)->(((eq ((x_a->Prop)->Prop)) (collect_a_o (fun (X_1:(x_a->Prop))=> ((and (((eq (x_a->Prop)) A_12) X_1)) (P_3 X_1))))) ((insert_a_o A_12) bot_bot_a_o_o)))) (((P_3 A_12)->False)->(((eq ((x_a->Prop)->Prop)) (collect_a_o (fun (X_1:(x_a->Prop))=> ((and (((eq (x_a->Prop)) A_12) X_1)) (P_3 X_1))))) bot_bot_a_o_o)))) % FOF formula (forall (A_11:nat), (((eq (nat->Prop)) (collect_nat (fun (X_1:nat)=> (((eq nat) X_1) A_11)))) ((insert_nat A_11) bot_bot_nat_o))) of role axiom named fact_482_singleton__conv % A new axiom: (forall (A_11:nat), (((eq (nat->Prop)) (collect_nat (fun (X_1:nat)=> (((eq nat) X_1) A_11)))) ((insert_nat A_11) bot_bot_nat_o))) % FOF formula (forall (A_11:pname), (((eq (pname->Prop)) (collect_pname (fun (X_1:pname)=> (((eq pname) X_1) A_11)))) ((insert_pname A_11) bot_bot_pname_o))) of role axiom named fact_483_singleton__conv % A new axiom: (forall (A_11:pname), (((eq (pname->Prop)) (collect_pname (fun (X_1:pname)=> (((eq pname) X_1) A_11)))) ((insert_pname A_11) bot_bot_pname_o))) % FOF formula (forall (A_11:x_a), (((eq (x_a->Prop)) (collect_a (fun (X_1:x_a)=> (((eq x_a) X_1) A_11)))) ((insert_a A_11) bot_bot_a_o))) of role axiom named fact_484_singleton__conv % A new axiom: (forall (A_11:x_a), (((eq (x_a->Prop)) (collect_a (fun (X_1:x_a)=> (((eq x_a) X_1) A_11)))) ((insert_a A_11) bot_bot_a_o))) % FOF formula (forall (A_11:(nat->Prop)), (((eq ((nat->Prop)->Prop)) (collect_nat_o (fun (X_1:(nat->Prop))=> (((eq (nat->Prop)) X_1) A_11)))) ((insert_nat_o A_11) bot_bot_nat_o_o))) of role axiom named fact_485_singleton__conv % A new axiom: (forall (A_11:(nat->Prop)), (((eq ((nat->Prop)->Prop)) (collect_nat_o (fun (X_1:(nat->Prop))=> (((eq (nat->Prop)) X_1) A_11)))) ((insert_nat_o A_11) bot_bot_nat_o_o))) % FOF formula (forall (A_11:(pname->Prop)), (((eq ((pname->Prop)->Prop)) (collect_pname_o (fun (X_1:(pname->Prop))=> (((eq (pname->Prop)) X_1) A_11)))) ((insert_pname_o A_11) bot_bot_pname_o_o))) of role axiom named fact_486_singleton__conv % A new axiom: (forall (A_11:(pname->Prop)), (((eq ((pname->Prop)->Prop)) (collect_pname_o (fun (X_1:(pname->Prop))=> (((eq (pname->Prop)) X_1) A_11)))) ((insert_pname_o A_11) bot_bot_pname_o_o))) % FOF formula (forall (A_11:(x_a->Prop)), (((eq ((x_a->Prop)->Prop)) (collect_a_o (fun (X_1:(x_a->Prop))=> (((eq (x_a->Prop)) X_1) A_11)))) ((insert_a_o A_11) bot_bot_a_o_o))) of role axiom named fact_487_singleton__conv % A new axiom: (forall (A_11:(x_a->Prop)), (((eq ((x_a->Prop)->Prop)) (collect_a_o (fun (X_1:(x_a->Prop))=> (((eq (x_a->Prop)) X_1) A_11)))) ((insert_a_o A_11) bot_bot_a_o_o))) % FOF formula (forall (A_10:nat), (((eq (nat->Prop)) (collect_nat (fequal_nat A_10))) ((insert_nat A_10) bot_bot_nat_o))) of role axiom named fact_488_singleton__conv2 % A new axiom: (forall (A_10:nat), (((eq (nat->Prop)) (collect_nat (fequal_nat A_10))) ((insert_nat A_10) bot_bot_nat_o))) % FOF formula (forall (A_10:pname), (((eq (pname->Prop)) (collect_pname (fequal_pname A_10))) ((insert_pname A_10) bot_bot_pname_o))) of role axiom named fact_489_singleton__conv2 % A new axiom: (forall (A_10:pname), (((eq (pname->Prop)) (collect_pname (fequal_pname A_10))) ((insert_pname A_10) bot_bot_pname_o))) % FOF formula (forall (A_10:x_a), (((eq (x_a->Prop)) (collect_a (fequal_a A_10))) ((insert_a A_10) bot_bot_a_o))) of role axiom named fact_490_singleton__conv2 % A new axiom: (forall (A_10:x_a), (((eq (x_a->Prop)) (collect_a (fequal_a A_10))) ((insert_a A_10) bot_bot_a_o))) % FOF formula (forall (A_10:(nat->Prop)), (((eq ((nat->Prop)->Prop)) (collect_nat_o (fequal_nat_o A_10))) ((insert_nat_o A_10) bot_bot_nat_o_o))) of role axiom named fact_491_singleton__conv2 % A new axiom: (forall (A_10:(nat->Prop)), (((eq ((nat->Prop)->Prop)) (collect_nat_o (fequal_nat_o A_10))) ((insert_nat_o A_10) bot_bot_nat_o_o))) % FOF formula (forall (A_10:(pname->Prop)), (((eq ((pname->Prop)->Prop)) (collect_pname_o (fequal_pname_o A_10))) ((insert_pname_o A_10) bot_bot_pname_o_o))) of role axiom named fact_492_singleton__conv2 % A new axiom: (forall (A_10:(pname->Prop)), (((eq ((pname->Prop)->Prop)) (collect_pname_o (fequal_pname_o A_10))) ((insert_pname_o A_10) bot_bot_pname_o_o))) % FOF formula (forall (A_10:(x_a->Prop)), (((eq ((x_a->Prop)->Prop)) (collect_a_o (fequal_a_o A_10))) ((insert_a_o A_10) bot_bot_a_o_o))) of role axiom named fact_493_singleton__conv2 % A new axiom: (forall (A_10:(x_a->Prop)), (((eq ((x_a->Prop)->Prop)) (collect_a_o (fequal_a_o A_10))) ((insert_a_o A_10) bot_bot_a_o_o))) % FOF formula (forall (A_9:(nat->Prop)) (X_12:nat), (((ord_less_eq_nat_o A_9) ((insert_nat X_12) bot_bot_nat_o))->((or (((eq (nat->Prop)) A_9) bot_bot_nat_o)) (((eq (nat->Prop)) A_9) ((insert_nat X_12) bot_bot_nat_o))))) of role axiom named fact_494_subset__singletonD % A new axiom: (forall (A_9:(nat->Prop)) (X_12:nat), (((ord_less_eq_nat_o A_9) ((insert_nat X_12) bot_bot_nat_o))->((or (((eq (nat->Prop)) A_9) bot_bot_nat_o)) (((eq (nat->Prop)) A_9) ((insert_nat X_12) bot_bot_nat_o))))) % FOF formula (forall (A_9:(pname->Prop)) (X_12:pname), (((ord_less_eq_pname_o A_9) ((insert_pname X_12) bot_bot_pname_o))->((or (((eq (pname->Prop)) A_9) bot_bot_pname_o)) (((eq (pname->Prop)) A_9) ((insert_pname X_12) bot_bot_pname_o))))) of role axiom named fact_495_subset__singletonD % A new axiom: (forall (A_9:(pname->Prop)) (X_12:pname), (((ord_less_eq_pname_o A_9) ((insert_pname X_12) bot_bot_pname_o))->((or (((eq (pname->Prop)) A_9) bot_bot_pname_o)) (((eq (pname->Prop)) A_9) ((insert_pname X_12) bot_bot_pname_o))))) % FOF formula (forall (A_9:(x_a->Prop)) (X_12:x_a), (((ord_less_eq_a_o A_9) ((insert_a X_12) bot_bot_a_o))->((or (((eq (x_a->Prop)) A_9) bot_bot_a_o)) (((eq (x_a->Prop)) A_9) ((insert_a X_12) bot_bot_a_o))))) of role axiom named fact_496_subset__singletonD % A new axiom: (forall (A_9:(x_a->Prop)) (X_12:x_a), (((ord_less_eq_a_o A_9) ((insert_a X_12) bot_bot_a_o))->((or (((eq (x_a->Prop)) A_9) bot_bot_a_o)) (((eq (x_a->Prop)) A_9) ((insert_a X_12) bot_bot_a_o))))) % FOF formula (forall (C_6:x_a) (X_11:pname) (A_8:(pname->Prop)), (((member_pname X_11) A_8)->(((eq (x_a->Prop)) ((image_pname_a (fun (X_1:pname)=> C_6)) A_8)) ((insert_a C_6) bot_bot_a_o)))) of role axiom named fact_497_image__constant % A new axiom: (forall (C_6:x_a) (X_11:pname) (A_8:(pname->Prop)), (((member_pname X_11) A_8)->(((eq (x_a->Prop)) ((image_pname_a (fun (X_1:pname)=> C_6)) A_8)) ((insert_a C_6) bot_bot_a_o)))) % FOF formula (forall (C_5:x_a) (A_7:(pname->Prop)), ((and ((((eq (pname->Prop)) A_7) bot_bot_pname_o)->(((eq (x_a->Prop)) ((image_pname_a (fun (X_1:pname)=> C_5)) A_7)) bot_bot_a_o))) ((not (((eq (pname->Prop)) A_7) bot_bot_pname_o))->(((eq (x_a->Prop)) ((image_pname_a (fun (X_1:pname)=> C_5)) A_7)) ((insert_a C_5) bot_bot_a_o))))) of role axiom named fact_498_image__constant__conv % A new axiom: (forall (C_5:x_a) (A_7:(pname->Prop)), ((and ((((eq (pname->Prop)) A_7) bot_bot_pname_o)->(((eq (x_a->Prop)) ((image_pname_a (fun (X_1:pname)=> C_5)) A_7)) bot_bot_a_o))) ((not (((eq (pname->Prop)) A_7) bot_bot_pname_o))->(((eq (x_a->Prop)) ((image_pname_a (fun (X_1:pname)=> C_5)) A_7)) ((insert_a C_5) bot_bot_a_o))))) % FOF formula (forall (X_10:nat) (Y_9:nat), ((((ord_less_eq_nat X_10) Y_9)->False)->((ord_less_eq_nat Y_9) X_10))) of role axiom named fact_499_linorder__le__cases % A new axiom: (forall (X_10:nat) (Y_9:nat), ((((ord_less_eq_nat X_10) Y_9)->False)->((ord_less_eq_nat Y_9) X_10))) % FOF formula (forall (Z_2:Prop) (Y_8:Prop) (X_9:Prop), (((ord_less_eq_o Y_8) X_9)->(((ord_less_eq_o Z_2) Y_8)->((ord_less_eq_o Z_2) X_9)))) of role axiom named fact_500_xt1_I6_J % A new axiom: (forall (Z_2:Prop) (Y_8:Prop) (X_9:Prop), (((ord_less_eq_o Y_8) X_9)->(((ord_less_eq_o Z_2) Y_8)->((ord_less_eq_o Z_2) X_9)))) % FOF formula (forall (Z_2:(nat->Prop)) (Y_8:(nat->Prop)) (X_9:(nat->Prop)), (((ord_less_eq_nat_o Y_8) X_9)->(((ord_less_eq_nat_o Z_2) Y_8)->((ord_less_eq_nat_o Z_2) X_9)))) of role axiom named fact_501_xt1_I6_J % A new axiom: (forall (Z_2:(nat->Prop)) (Y_8:(nat->Prop)) (X_9:(nat->Prop)), (((ord_less_eq_nat_o Y_8) X_9)->(((ord_less_eq_nat_o Z_2) Y_8)->((ord_less_eq_nat_o Z_2) X_9)))) % FOF formula (forall (Z_2:(pname->Prop)) (Y_8:(pname->Prop)) (X_9:(pname->Prop)), (((ord_less_eq_pname_o Y_8) X_9)->(((ord_less_eq_pname_o Z_2) Y_8)->((ord_less_eq_pname_o Z_2) X_9)))) of role axiom named fact_502_xt1_I6_J % A new axiom: (forall (Z_2:(pname->Prop)) (Y_8:(pname->Prop)) (X_9:(pname->Prop)), (((ord_less_eq_pname_o Y_8) X_9)->(((ord_less_eq_pname_o Z_2) Y_8)->((ord_less_eq_pname_o Z_2) X_9)))) % FOF formula (forall (Z_2:nat) (Y_8:nat) (X_9:nat), (((ord_less_eq_nat Y_8) X_9)->(((ord_less_eq_nat Z_2) Y_8)->((ord_less_eq_nat Z_2) X_9)))) of role axiom named fact_503_xt1_I6_J % A new axiom: (forall (Z_2:nat) (Y_8:nat) (X_9:nat), (((ord_less_eq_nat Y_8) X_9)->(((ord_less_eq_nat Z_2) Y_8)->((ord_less_eq_nat Z_2) X_9)))) % FOF formula (forall (Z_2:(x_a->Prop)) (Y_8:(x_a->Prop)) (X_9:(x_a->Prop)), (((ord_less_eq_a_o Y_8) X_9)->(((ord_less_eq_a_o Z_2) Y_8)->((ord_less_eq_a_o Z_2) X_9)))) of role axiom named fact_504_xt1_I6_J % A new axiom: (forall (Z_2:(x_a->Prop)) (Y_8:(x_a->Prop)) (X_9:(x_a->Prop)), (((ord_less_eq_a_o Y_8) X_9)->(((ord_less_eq_a_o Z_2) Y_8)->((ord_less_eq_a_o Z_2) X_9)))) % FOF formula (forall (Y_7:Prop) (X_8:Prop), (((ord_less_eq_o Y_7) X_8)->(((ord_less_eq_o X_8) Y_7)->((iff X_8) Y_7)))) of role axiom named fact_505_xt1_I5_J % A new axiom: (forall (Y_7:Prop) (X_8:Prop), (((ord_less_eq_o Y_7) X_8)->(((ord_less_eq_o X_8) Y_7)->((iff X_8) Y_7)))) % FOF formula (forall (Y_7:(nat->Prop)) (X_8:(nat->Prop)), (((ord_less_eq_nat_o Y_7) X_8)->(((ord_less_eq_nat_o X_8) Y_7)->(((eq (nat->Prop)) X_8) Y_7)))) of role axiom named fact_506_xt1_I5_J % A new axiom: (forall (Y_7:(nat->Prop)) (X_8:(nat->Prop)), (((ord_less_eq_nat_o Y_7) X_8)->(((ord_less_eq_nat_o X_8) Y_7)->(((eq (nat->Prop)) X_8) Y_7)))) % FOF formula (forall (Y_7:(pname->Prop)) (X_8:(pname->Prop)), (((ord_less_eq_pname_o Y_7) X_8)->(((ord_less_eq_pname_o X_8) Y_7)->(((eq (pname->Prop)) X_8) Y_7)))) of role axiom named fact_507_xt1_I5_J % A new axiom: (forall (Y_7:(pname->Prop)) (X_8:(pname->Prop)), (((ord_less_eq_pname_o Y_7) X_8)->(((ord_less_eq_pname_o X_8) Y_7)->(((eq (pname->Prop)) X_8) Y_7)))) % FOF formula (forall (Y_7:nat) (X_8:nat), (((ord_less_eq_nat Y_7) X_8)->(((ord_less_eq_nat X_8) Y_7)->(((eq nat) X_8) Y_7)))) of role axiom named fact_508_xt1_I5_J % A new axiom: (forall (Y_7:nat) (X_8:nat), (((ord_less_eq_nat Y_7) X_8)->(((ord_less_eq_nat X_8) Y_7)->(((eq nat) X_8) Y_7)))) % FOF formula (forall (Y_7:(x_a->Prop)) (X_8:(x_a->Prop)), (((ord_less_eq_a_o Y_7) X_8)->(((ord_less_eq_a_o X_8) Y_7)->(((eq (x_a->Prop)) X_8) Y_7)))) of role axiom named fact_509_xt1_I5_J % A new axiom: (forall (Y_7:(x_a->Prop)) (X_8:(x_a->Prop)), (((ord_less_eq_a_o Y_7) X_8)->(((ord_less_eq_a_o X_8) Y_7)->(((eq (x_a->Prop)) X_8) Y_7)))) % FOF formula (forall (Z_1:Prop) (X_7:Prop) (Y_6:Prop), (((ord_less_eq_o X_7) Y_6)->(((ord_less_eq_o Y_6) Z_1)->((ord_less_eq_o X_7) Z_1)))) of role axiom named fact_510_order__trans % A new axiom: (forall (Z_1:Prop) (X_7:Prop) (Y_6:Prop), (((ord_less_eq_o X_7) Y_6)->(((ord_less_eq_o Y_6) Z_1)->((ord_less_eq_o X_7) Z_1)))) % FOF formula (forall (Z_1:(nat->Prop)) (X_7:(nat->Prop)) (Y_6:(nat->Prop)), (((ord_less_eq_nat_o X_7) Y_6)->(((ord_less_eq_nat_o Y_6) Z_1)->((ord_less_eq_nat_o X_7) Z_1)))) of role axiom named fact_511_order__trans % A new axiom: (forall (Z_1:(nat->Prop)) (X_7:(nat->Prop)) (Y_6:(nat->Prop)), (((ord_less_eq_nat_o X_7) Y_6)->(((ord_less_eq_nat_o Y_6) Z_1)->((ord_less_eq_nat_o X_7) Z_1)))) % FOF formula (forall (Z_1:(pname->Prop)) (X_7:(pname->Prop)) (Y_6:(pname->Prop)), (((ord_less_eq_pname_o X_7) Y_6)->(((ord_less_eq_pname_o Y_6) Z_1)->((ord_less_eq_pname_o X_7) Z_1)))) of role axiom named fact_512_order__trans % A new axiom: (forall (Z_1:(pname->Prop)) (X_7:(pname->Prop)) (Y_6:(pname->Prop)), (((ord_less_eq_pname_o X_7) Y_6)->(((ord_less_eq_pname_o Y_6) Z_1)->((ord_less_eq_pname_o X_7) Z_1)))) % FOF formula (forall (Z_1:nat) (X_7:nat) (Y_6:nat), (((ord_less_eq_nat X_7) Y_6)->(((ord_less_eq_nat Y_6) Z_1)->((ord_less_eq_nat X_7) Z_1)))) of role axiom named fact_513_order__trans % A new axiom: (forall (Z_1:nat) (X_7:nat) (Y_6:nat), (((ord_less_eq_nat X_7) Y_6)->(((ord_less_eq_nat Y_6) Z_1)->((ord_less_eq_nat X_7) Z_1)))) % FOF formula (forall (Z_1:(x_a->Prop)) (X_7:(x_a->Prop)) (Y_6:(x_a->Prop)), (((ord_less_eq_a_o X_7) Y_6)->(((ord_less_eq_a_o Y_6) Z_1)->((ord_less_eq_a_o X_7) Z_1)))) of role axiom named fact_514_order__trans % A new axiom: (forall (Z_1:(x_a->Prop)) (X_7:(x_a->Prop)) (Y_6:(x_a->Prop)), (((ord_less_eq_a_o X_7) Y_6)->(((ord_less_eq_a_o Y_6) Z_1)->((ord_less_eq_a_o X_7) Z_1)))) % FOF formula (forall (X_6:Prop) (Y_5:Prop), (((ord_less_eq_o X_6) Y_5)->(((ord_less_eq_o Y_5) X_6)->((iff X_6) Y_5)))) of role axiom named fact_515_order__antisym % A new axiom: (forall (X_6:Prop) (Y_5:Prop), (((ord_less_eq_o X_6) Y_5)->(((ord_less_eq_o Y_5) X_6)->((iff X_6) Y_5)))) % FOF formula (forall (X_6:(nat->Prop)) (Y_5:(nat->Prop)), (((ord_less_eq_nat_o X_6) Y_5)->(((ord_less_eq_nat_o Y_5) X_6)->(((eq (nat->Prop)) X_6) Y_5)))) of role axiom named fact_516_order__antisym % A new axiom: (forall (X_6:(nat->Prop)) (Y_5:(nat->Prop)), (((ord_less_eq_nat_o X_6) Y_5)->(((ord_less_eq_nat_o Y_5) X_6)->(((eq (nat->Prop)) X_6) Y_5)))) % FOF formula (forall (X_6:(pname->Prop)) (Y_5:(pname->Prop)), (((ord_less_eq_pname_o X_6) Y_5)->(((ord_less_eq_pname_o Y_5) X_6)->(((eq (pname->Prop)) X_6) Y_5)))) of role axiom named fact_517_order__antisym % A new axiom: (forall (X_6:(pname->Prop)) (Y_5:(pname->Prop)), (((ord_less_eq_pname_o X_6) Y_5)->(((ord_less_eq_pname_o Y_5) X_6)->(((eq (pname->Prop)) X_6) Y_5)))) % FOF formula (forall (X_6:nat) (Y_5:nat), (((ord_less_eq_nat X_6) Y_5)->(((ord_less_eq_nat Y_5) X_6)->(((eq nat) X_6) Y_5)))) of role axiom named fact_518_order__antisym % A new axiom: (forall (X_6:nat) (Y_5:nat), (((ord_less_eq_nat X_6) Y_5)->(((ord_less_eq_nat Y_5) X_6)->(((eq nat) X_6) Y_5)))) % FOF formula (forall (X_6:(x_a->Prop)) (Y_5:(x_a->Prop)), (((ord_less_eq_a_o X_6) Y_5)->(((ord_less_eq_a_o Y_5) X_6)->(((eq (x_a->Prop)) X_6) Y_5)))) of role axiom named fact_519_order__antisym % A new axiom: (forall (X_6:(x_a->Prop)) (Y_5:(x_a->Prop)), (((ord_less_eq_a_o X_6) Y_5)->(((ord_less_eq_a_o Y_5) X_6)->(((eq (x_a->Prop)) X_6) Y_5)))) % FOF formula (forall (C_4:Prop) (B_4:Prop) (A_6:Prop), (((ord_less_eq_o B_4) A_6)->(((iff B_4) C_4)->((ord_less_eq_o C_4) A_6)))) of role axiom named fact_520_xt1_I4_J % A new axiom: (forall (C_4:Prop) (B_4:Prop) (A_6:Prop), (((ord_less_eq_o B_4) A_6)->(((iff B_4) C_4)->((ord_less_eq_o C_4) A_6)))) % FOF formula (forall (C_4:(nat->Prop)) (B_4:(nat->Prop)) (A_6:(nat->Prop)), (((ord_less_eq_nat_o B_4) A_6)->((((eq (nat->Prop)) B_4) C_4)->((ord_less_eq_nat_o C_4) A_6)))) of role axiom named fact_521_xt1_I4_J % A new axiom: (forall (C_4:(nat->Prop)) (B_4:(nat->Prop)) (A_6:(nat->Prop)), (((ord_less_eq_nat_o B_4) A_6)->((((eq (nat->Prop)) B_4) C_4)->((ord_less_eq_nat_o C_4) A_6)))) % FOF formula (forall (C_4:(pname->Prop)) (B_4:(pname->Prop)) (A_6:(pname->Prop)), (((ord_less_eq_pname_o B_4) A_6)->((((eq (pname->Prop)) B_4) C_4)->((ord_less_eq_pname_o C_4) A_6)))) of role axiom named fact_522_xt1_I4_J % A new axiom: (forall (C_4:(pname->Prop)) (B_4:(pname->Prop)) (A_6:(pname->Prop)), (((ord_less_eq_pname_o B_4) A_6)->((((eq (pname->Prop)) B_4) C_4)->((ord_less_eq_pname_o C_4) A_6)))) % FOF formula (forall (C_4:nat) (B_4:nat) (A_6:nat), (((ord_less_eq_nat B_4) A_6)->((((eq nat) B_4) C_4)->((ord_less_eq_nat C_4) A_6)))) of role axiom named fact_523_xt1_I4_J % A new axiom: (forall (C_4:nat) (B_4:nat) (A_6:nat), (((ord_less_eq_nat B_4) A_6)->((((eq nat) B_4) C_4)->((ord_less_eq_nat C_4) A_6)))) % FOF formula (forall (C_4:(x_a->Prop)) (B_4:(x_a->Prop)) (A_6:(x_a->Prop)), (((ord_less_eq_a_o B_4) A_6)->((((eq (x_a->Prop)) B_4) C_4)->((ord_less_eq_a_o C_4) A_6)))) of role axiom named fact_524_xt1_I4_J % A new axiom: (forall (C_4:(x_a->Prop)) (B_4:(x_a->Prop)) (A_6:(x_a->Prop)), (((ord_less_eq_a_o B_4) A_6)->((((eq (x_a->Prop)) B_4) C_4)->((ord_less_eq_a_o C_4) A_6)))) % FOF formula (forall (C_3:Prop) (A_5:Prop) (B_3:Prop), (((ord_less_eq_o A_5) B_3)->(((iff B_3) C_3)->((ord_less_eq_o A_5) C_3)))) of role axiom named fact_525_ord__le__eq__trans % A new axiom: (forall (C_3:Prop) (A_5:Prop) (B_3:Prop), (((ord_less_eq_o A_5) B_3)->(((iff B_3) C_3)->((ord_less_eq_o A_5) C_3)))) % FOF formula (forall (C_3:(nat->Prop)) (A_5:(nat->Prop)) (B_3:(nat->Prop)), (((ord_less_eq_nat_o A_5) B_3)->((((eq (nat->Prop)) B_3) C_3)->((ord_less_eq_nat_o A_5) C_3)))) of role axiom named fact_526_ord__le__eq__trans % A new axiom: (forall (C_3:(nat->Prop)) (A_5:(nat->Prop)) (B_3:(nat->Prop)), (((ord_less_eq_nat_o A_5) B_3)->((((eq (nat->Prop)) B_3) C_3)->((ord_less_eq_nat_o A_5) C_3)))) % FOF formula (forall (C_3:(pname->Prop)) (A_5:(pname->Prop)) (B_3:(pname->Prop)), (((ord_less_eq_pname_o A_5) B_3)->((((eq (pname->Prop)) B_3) C_3)->((ord_less_eq_pname_o A_5) C_3)))) of role axiom named fact_527_ord__le__eq__trans % A new axiom: (forall (C_3:(pname->Prop)) (A_5:(pname->Prop)) (B_3:(pname->Prop)), (((ord_less_eq_pname_o A_5) B_3)->((((eq (pname->Prop)) B_3) C_3)->((ord_less_eq_pname_o A_5) C_3)))) % FOF formula (forall (C_3:nat) (A_5:nat) (B_3:nat), (((ord_less_eq_nat A_5) B_3)->((((eq nat) B_3) C_3)->((ord_less_eq_nat A_5) C_3)))) of role axiom named fact_528_ord__le__eq__trans % A new axiom: (forall (C_3:nat) (A_5:nat) (B_3:nat), (((ord_less_eq_nat A_5) B_3)->((((eq nat) B_3) C_3)->((ord_less_eq_nat A_5) C_3)))) % FOF formula (forall (C_3:(x_a->Prop)) (A_5:(x_a->Prop)) (B_3:(x_a->Prop)), (((ord_less_eq_a_o A_5) B_3)->((((eq (x_a->Prop)) B_3) C_3)->((ord_less_eq_a_o A_5) C_3)))) of role axiom named fact_529_ord__le__eq__trans % A new axiom: (forall (C_3:(x_a->Prop)) (A_5:(x_a->Prop)) (B_3:(x_a->Prop)), (((ord_less_eq_a_o A_5) B_3)->((((eq (x_a->Prop)) B_3) C_3)->((ord_less_eq_a_o A_5) C_3)))) % FOF formula (forall (C_2:Prop) (B_2:Prop) (A_4:Prop), (((iff A_4) B_2)->(((ord_less_eq_o C_2) B_2)->((ord_less_eq_o C_2) A_4)))) of role axiom named fact_530_xt1_I3_J % A new axiom: (forall (C_2:Prop) (B_2:Prop) (A_4:Prop), (((iff A_4) B_2)->(((ord_less_eq_o C_2) B_2)->((ord_less_eq_o C_2) A_4)))) % FOF formula (forall (C_2:(nat->Prop)) (A_4:(nat->Prop)) (B_2:(nat->Prop)), ((((eq (nat->Prop)) A_4) B_2)->(((ord_less_eq_nat_o C_2) B_2)->((ord_less_eq_nat_o C_2) A_4)))) of role axiom named fact_531_xt1_I3_J % A new axiom: (forall (C_2:(nat->Prop)) (A_4:(nat->Prop)) (B_2:(nat->Prop)), ((((eq (nat->Prop)) A_4) B_2)->(((ord_less_eq_nat_o C_2) B_2)->((ord_less_eq_nat_o C_2) A_4)))) % FOF formula (forall (C_2:(pname->Prop)) (A_4:(pname->Prop)) (B_2:(pname->Prop)), ((((eq (pname->Prop)) A_4) B_2)->(((ord_less_eq_pname_o C_2) B_2)->((ord_less_eq_pname_o C_2) A_4)))) of role axiom named fact_532_xt1_I3_J % A new axiom: (forall (C_2:(pname->Prop)) (A_4:(pname->Prop)) (B_2:(pname->Prop)), ((((eq (pname->Prop)) A_4) B_2)->(((ord_less_eq_pname_o C_2) B_2)->((ord_less_eq_pname_o C_2) A_4)))) % FOF formula (forall (C_2:nat) (A_4:nat) (B_2:nat), ((((eq nat) A_4) B_2)->(((ord_less_eq_nat C_2) B_2)->((ord_less_eq_nat C_2) A_4)))) of role axiom named fact_533_xt1_I3_J % A new axiom: (forall (C_2:nat) (A_4:nat) (B_2:nat), ((((eq nat) A_4) B_2)->(((ord_less_eq_nat C_2) B_2)->((ord_less_eq_nat C_2) A_4)))) % FOF formula (forall (C_2:(x_a->Prop)) (A_4:(x_a->Prop)) (B_2:(x_a->Prop)), ((((eq (x_a->Prop)) A_4) B_2)->(((ord_less_eq_a_o C_2) B_2)->((ord_less_eq_a_o C_2) A_4)))) of role axiom named fact_534_xt1_I3_J % A new axiom: (forall (C_2:(x_a->Prop)) (A_4:(x_a->Prop)) (B_2:(x_a->Prop)), ((((eq (x_a->Prop)) A_4) B_2)->(((ord_less_eq_a_o C_2) B_2)->((ord_less_eq_a_o C_2) A_4)))) % FOF formula (forall (C_1:Prop) (B_1:Prop) (A_3:Prop), (((iff A_3) B_1)->(((ord_less_eq_o B_1) C_1)->((ord_less_eq_o A_3) C_1)))) of role axiom named fact_535_ord__eq__le__trans % A new axiom: (forall (C_1:Prop) (B_1:Prop) (A_3:Prop), (((iff A_3) B_1)->(((ord_less_eq_o B_1) C_1)->((ord_less_eq_o A_3) C_1)))) % FOF formula (forall (C_1:(nat->Prop)) (A_3:(nat->Prop)) (B_1:(nat->Prop)), ((((eq (nat->Prop)) A_3) B_1)->(((ord_less_eq_nat_o B_1) C_1)->((ord_less_eq_nat_o A_3) C_1)))) of role axiom named fact_536_ord__eq__le__trans % A new axiom: (forall (C_1:(nat->Prop)) (A_3:(nat->Prop)) (B_1:(nat->Prop)), ((((eq (nat->Prop)) A_3) B_1)->(((ord_less_eq_nat_o B_1) C_1)->((ord_less_eq_nat_o A_3) C_1)))) % FOF formula (forall (C_1:(pname->Prop)) (A_3:(pname->Prop)) (B_1:(pname->Prop)), ((((eq (pname->Prop)) A_3) B_1)->(((ord_less_eq_pname_o B_1) C_1)->((ord_less_eq_pname_o A_3) C_1)))) of role axiom named fact_537_ord__eq__le__trans % A new axiom: (forall (C_1:(pname->Prop)) (A_3:(pname->Prop)) (B_1:(pname->Prop)), ((((eq (pname->Prop)) A_3) B_1)->(((ord_less_eq_pname_o B_1) C_1)->((ord_less_eq_pname_o A_3) C_1)))) % FOF formula (forall (C_1:nat) (A_3:nat) (B_1:nat), ((((eq nat) A_3) B_1)->(((ord_less_eq_nat B_1) C_1)->((ord_less_eq_nat A_3) C_1)))) of role axiom named fact_538_ord__eq__le__trans % A new axiom: (forall (C_1:nat) (A_3:nat) (B_1:nat), ((((eq nat) A_3) B_1)->(((ord_less_eq_nat B_1) C_1)->((ord_less_eq_nat A_3) C_1)))) % FOF formula (forall (C_1:(x_a->Prop)) (A_3:(x_a->Prop)) (B_1:(x_a->Prop)), ((((eq (x_a->Prop)) A_3) B_1)->(((ord_less_eq_a_o B_1) C_1)->((ord_less_eq_a_o A_3) C_1)))) of role axiom named fact_539_ord__eq__le__trans % A new axiom: (forall (C_1:(x_a->Prop)) (A_3:(x_a->Prop)) (B_1:(x_a->Prop)), ((((eq (x_a->Prop)) A_3) B_1)->(((ord_less_eq_a_o B_1) C_1)->((ord_less_eq_a_o A_3) C_1)))) % FOF formula (forall (Y_4:Prop) (X_5:Prop), (((ord_less_eq_o Y_4) X_5)->((iff ((ord_less_eq_o X_5) Y_4)) ((iff X_5) Y_4)))) of role axiom named fact_540_order__antisym__conv % A new axiom: (forall (Y_4:Prop) (X_5:Prop), (((ord_less_eq_o Y_4) X_5)->((iff ((ord_less_eq_o X_5) Y_4)) ((iff X_5) Y_4)))) % FOF formula (forall (Y_4:(nat->Prop)) (X_5:(nat->Prop)), (((ord_less_eq_nat_o Y_4) X_5)->((iff ((ord_less_eq_nat_o X_5) Y_4)) (((eq (nat->Prop)) X_5) Y_4)))) of role axiom named fact_541_order__antisym__conv % A new axiom: (forall (Y_4:(nat->Prop)) (X_5:(nat->Prop)), (((ord_less_eq_nat_o Y_4) X_5)->((iff ((ord_less_eq_nat_o X_5) Y_4)) (((eq (nat->Prop)) X_5) Y_4)))) % FOF formula (forall (Y_4:(pname->Prop)) (X_5:(pname->Prop)), (((ord_less_eq_pname_o Y_4) X_5)->((iff ((ord_less_eq_pname_o X_5) Y_4)) (((eq (pname->Prop)) X_5) Y_4)))) of role axiom named fact_542_order__antisym__conv % A new axiom: (forall (Y_4:(pname->Prop)) (X_5:(pname->Prop)), (((ord_less_eq_pname_o Y_4) X_5)->((iff ((ord_less_eq_pname_o X_5) Y_4)) (((eq (pname->Prop)) X_5) Y_4)))) % FOF formula (forall (Y_4:nat) (X_5:nat), (((ord_less_eq_nat Y_4) X_5)->((iff ((ord_less_eq_nat X_5) Y_4)) (((eq nat) X_5) Y_4)))) of role axiom named fact_543_order__antisym__conv % A new axiom: (forall (Y_4:nat) (X_5:nat), (((ord_less_eq_nat Y_4) X_5)->((iff ((ord_less_eq_nat X_5) Y_4)) (((eq nat) X_5) Y_4)))) % FOF formula (forall (Y_4:(x_a->Prop)) (X_5:(x_a->Prop)), (((ord_less_eq_a_o Y_4) X_5)->((iff ((ord_less_eq_a_o X_5) Y_4)) (((eq (x_a->Prop)) X_5) Y_4)))) of role axiom named fact_544_order__antisym__conv % A new axiom: (forall (Y_4:(x_a->Prop)) (X_5:(x_a->Prop)), (((ord_less_eq_a_o Y_4) X_5)->((iff ((ord_less_eq_a_o X_5) Y_4)) (((eq (x_a->Prop)) X_5) Y_4)))) % FOF formula (forall (Y_3:Prop) (X_4:Prop), (((iff X_4) Y_3)->((ord_less_eq_o X_4) Y_3))) of role axiom named fact_545_order__eq__refl % A new axiom: (forall (Y_3:Prop) (X_4:Prop), (((iff X_4) Y_3)->((ord_less_eq_o X_4) Y_3))) % FOF formula (forall (X_4:(nat->Prop)) (Y_3:(nat->Prop)), ((((eq (nat->Prop)) X_4) Y_3)->((ord_less_eq_nat_o X_4) Y_3))) of role axiom named fact_546_order__eq__refl % A new axiom: (forall (X_4:(nat->Prop)) (Y_3:(nat->Prop)), ((((eq (nat->Prop)) X_4) Y_3)->((ord_less_eq_nat_o X_4) Y_3))) % FOF formula (forall (X_4:(pname->Prop)) (Y_3:(pname->Prop)), ((((eq (pname->Prop)) X_4) Y_3)->((ord_less_eq_pname_o X_4) Y_3))) of role axiom named fact_547_order__eq__refl % A new axiom: (forall (X_4:(pname->Prop)) (Y_3:(pname->Prop)), ((((eq (pname->Prop)) X_4) Y_3)->((ord_less_eq_pname_o X_4) Y_3))) % FOF formula (forall (X_4:nat) (Y_3:nat), ((((eq nat) X_4) Y_3)->((ord_less_eq_nat X_4) Y_3))) of role axiom named fact_548_order__eq__refl % A new axiom: (forall (X_4:nat) (Y_3:nat), ((((eq nat) X_4) Y_3)->((ord_less_eq_nat X_4) Y_3))) % FOF formula (forall (X_4:(x_a->Prop)) (Y_3:(x_a->Prop)), ((((eq (x_a->Prop)) X_4) Y_3)->((ord_less_eq_a_o X_4) Y_3))) of role axiom named fact_549_order__eq__refl % A new axiom: (forall (X_4:(x_a->Prop)) (Y_3:(x_a->Prop)), ((((eq (x_a->Prop)) X_4) Y_3)->((ord_less_eq_a_o X_4) Y_3))) % FOF formula (forall (Y_2:Prop) (X_3:Prop), ((iff ((iff X_3) Y_2)) ((and ((ord_less_eq_o X_3) Y_2)) ((ord_less_eq_o Y_2) X_3)))) of role axiom named fact_550_order__eq__iff % A new axiom: (forall (Y_2:Prop) (X_3:Prop), ((iff ((iff X_3) Y_2)) ((and ((ord_less_eq_o X_3) Y_2)) ((ord_less_eq_o Y_2) X_3)))) % FOF formula (forall (X_3:(nat->Prop)) (Y_2:(nat->Prop)), ((iff (((eq (nat->Prop)) X_3) Y_2)) ((and ((ord_less_eq_nat_o X_3) Y_2)) ((ord_less_eq_nat_o Y_2) X_3)))) of role axiom named fact_551_order__eq__iff % A new axiom: (forall (X_3:(nat->Prop)) (Y_2:(nat->Prop)), ((iff (((eq (nat->Prop)) X_3) Y_2)) ((and ((ord_less_eq_nat_o X_3) Y_2)) ((ord_less_eq_nat_o Y_2) X_3)))) % FOF formula (forall (X_3:(pname->Prop)) (Y_2:(pname->Prop)), ((iff (((eq (pname->Prop)) X_3) Y_2)) ((and ((ord_less_eq_pname_o X_3) Y_2)) ((ord_less_eq_pname_o Y_2) X_3)))) of role axiom named fact_552_order__eq__iff % A new axiom: (forall (X_3:(pname->Prop)) (Y_2:(pname->Prop)), ((iff (((eq (pname->Prop)) X_3) Y_2)) ((and ((ord_less_eq_pname_o X_3) Y_2)) ((ord_less_eq_pname_o Y_2) X_3)))) % FOF formula (forall (X_3:nat) (Y_2:nat), ((iff (((eq nat) X_3) Y_2)) ((and ((ord_less_eq_nat X_3) Y_2)) ((ord_less_eq_nat Y_2) X_3)))) of role axiom named fact_553_order__eq__iff % A new axiom: (forall (X_3:nat) (Y_2:nat), ((iff (((eq nat) X_3) Y_2)) ((and ((ord_less_eq_nat X_3) Y_2)) ((ord_less_eq_nat Y_2) X_3)))) % FOF formula (forall (X_3:(x_a->Prop)) (Y_2:(x_a->Prop)), ((iff (((eq (x_a->Prop)) X_3) Y_2)) ((and ((ord_less_eq_a_o X_3) Y_2)) ((ord_less_eq_a_o Y_2) X_3)))) of role axiom named fact_554_order__eq__iff % A new axiom: (forall (X_3:(x_a->Prop)) (Y_2:(x_a->Prop)), ((iff (((eq (x_a->Prop)) X_3) Y_2)) ((and ((ord_less_eq_a_o X_3) Y_2)) ((ord_less_eq_a_o Y_2) X_3)))) % FOF formula (forall (X_2:nat) (Y_1:nat), ((or ((ord_less_eq_nat X_2) Y_1)) ((ord_less_eq_nat Y_1) X_2))) of role axiom named fact_555_linorder__linear % A new axiom: (forall (X_2:nat) (Y_1:nat), ((or ((ord_less_eq_nat X_2) Y_1)) ((ord_less_eq_nat Y_1) X_2))) % FOF formula (forall (P_2:((nat->Prop)->Prop)) (A_1:(nat->Prop)) (F_3:(nat->Prop)), ((finite_finite_nat F_3)->(((ord_less_eq_nat_o F_3) A_1)->((P_2 bot_bot_nat_o)->((forall (A_2:nat) (F_2:(nat->Prop)), ((finite_finite_nat F_2)->(((member_nat A_2) A_1)->((((member_nat A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_nat A_2) F_2)))))))->(P_2 F_3)))))) of role axiom named fact_556_finite__subset__induct % A new axiom: (forall (P_2:((nat->Prop)->Prop)) (A_1:(nat->Prop)) (F_3:(nat->Prop)), ((finite_finite_nat F_3)->(((ord_less_eq_nat_o F_3) A_1)->((P_2 bot_bot_nat_o)->((forall (A_2:nat) (F_2:(nat->Prop)), ((finite_finite_nat F_2)->(((member_nat A_2) A_1)->((((member_nat A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_nat A_2) F_2)))))))->(P_2 F_3)))))) % FOF formula (forall (P_2:((pname->Prop)->Prop)) (A_1:(pname->Prop)) (F_3:(pname->Prop)), ((finite_finite_pname F_3)->(((ord_less_eq_pname_o F_3) A_1)->((P_2 bot_bot_pname_o)->((forall (A_2:pname) (F_2:(pname->Prop)), ((finite_finite_pname F_2)->(((member_pname A_2) A_1)->((((member_pname A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_pname A_2) F_2)))))))->(P_2 F_3)))))) of role axiom named fact_557_finite__subset__induct % A new axiom: (forall (P_2:((pname->Prop)->Prop)) (A_1:(pname->Prop)) (F_3:(pname->Prop)), ((finite_finite_pname F_3)->(((ord_less_eq_pname_o F_3) A_1)->((P_2 bot_bot_pname_o)->((forall (A_2:pname) (F_2:(pname->Prop)), ((finite_finite_pname F_2)->(((member_pname A_2) A_1)->((((member_pname A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_pname A_2) F_2)))))))->(P_2 F_3)))))) % FOF formula (forall (P_2:((x_a->Prop)->Prop)) (A_1:(x_a->Prop)) (F_3:(x_a->Prop)), ((finite_finite_a F_3)->(((ord_less_eq_a_o F_3) A_1)->((P_2 bot_bot_a_o)->((forall (A_2:x_a) (F_2:(x_a->Prop)), ((finite_finite_a F_2)->(((member_a A_2) A_1)->((((member_a A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_a A_2) F_2)))))))->(P_2 F_3)))))) of role axiom named fact_558_finite__subset__induct % A new axiom: (forall (P_2:((x_a->Prop)->Prop)) (A_1:(x_a->Prop)) (F_3:(x_a->Prop)), ((finite_finite_a F_3)->(((ord_less_eq_a_o F_3) A_1)->((P_2 bot_bot_a_o)->((forall (A_2:x_a) (F_2:(x_a->Prop)), ((finite_finite_a F_2)->(((member_a A_2) A_1)->((((member_a A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_a A_2) F_2)))))))->(P_2 F_3)))))) % FOF formula (forall (P_2:(((nat->Prop)->Prop)->Prop)) (A_1:((nat->Prop)->Prop)) (F_3:((nat->Prop)->Prop)), ((finite_finite_nat_o F_3)->(((ord_less_eq_nat_o_o F_3) A_1)->((P_2 bot_bot_nat_o_o)->((forall (A_2:(nat->Prop)) (F_2:((nat->Prop)->Prop)), ((finite_finite_nat_o F_2)->(((member_nat_o A_2) A_1)->((((member_nat_o A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_nat_o A_2) F_2)))))))->(P_2 F_3)))))) of role axiom named fact_559_finite__subset__induct % A new axiom: (forall (P_2:(((nat->Prop)->Prop)->Prop)) (A_1:((nat->Prop)->Prop)) (F_3:((nat->Prop)->Prop)), ((finite_finite_nat_o F_3)->(((ord_less_eq_nat_o_o F_3) A_1)->((P_2 bot_bot_nat_o_o)->((forall (A_2:(nat->Prop)) (F_2:((nat->Prop)->Prop)), ((finite_finite_nat_o F_2)->(((member_nat_o A_2) A_1)->((((member_nat_o A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_nat_o A_2) F_2)))))))->(P_2 F_3)))))) % FOF formula (forall (P_2:(((pname->Prop)->Prop)->Prop)) (A_1:((pname->Prop)->Prop)) (F_3:((pname->Prop)->Prop)), ((finite297249702name_o F_3)->(((ord_le1205211808me_o_o F_3) A_1)->((P_2 bot_bot_pname_o_o)->((forall (A_2:(pname->Prop)) (F_2:((pname->Prop)->Prop)), ((finite297249702name_o F_2)->(((member_pname_o A_2) A_1)->((((member_pname_o A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_pname_o A_2) F_2)))))))->(P_2 F_3)))))) of role axiom named fact_560_finite__subset__induct % A new axiom: (forall (P_2:(((pname->Prop)->Prop)->Prop)) (A_1:((pname->Prop)->Prop)) (F_3:((pname->Prop)->Prop)), ((finite297249702name_o F_3)->(((ord_le1205211808me_o_o F_3) A_1)->((P_2 bot_bot_pname_o_o)->((forall (A_2:(pname->Prop)) (F_2:((pname->Prop)->Prop)), ((finite297249702name_o F_2)->(((member_pname_o A_2) A_1)->((((member_pname_o A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_pname_o A_2) F_2)))))))->(P_2 F_3)))))) % FOF formula (forall (P_2:(((x_a->Prop)->Prop)->Prop)) (A_1:((x_a->Prop)->Prop)) (F_3:((x_a->Prop)->Prop)), ((finite_finite_a_o F_3)->(((ord_less_eq_a_o_o F_3) A_1)->((P_2 bot_bot_a_o_o)->((forall (A_2:(x_a->Prop)) (F_2:((x_a->Prop)->Prop)), ((finite_finite_a_o F_2)->(((member_a_o A_2) A_1)->((((member_a_o A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_a_o A_2) F_2)))))))->(P_2 F_3)))))) of role axiom named fact_561_finite__subset__induct % A new axiom: (forall (P_2:(((x_a->Prop)->Prop)->Prop)) (A_1:((x_a->Prop)->Prop)) (F_3:((x_a->Prop)->Prop)), ((finite_finite_a_o F_3)->(((ord_less_eq_a_o_o F_3) A_1)->((P_2 bot_bot_a_o_o)->((forall (A_2:(x_a->Prop)) (F_2:((x_a->Prop)->Prop)), ((finite_finite_a_o F_2)->(((member_a_o A_2) A_1)->((((member_a_o A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_a_o A_2) F_2)))))))->(P_2 F_3)))))) % FOF formula (forall (Pn:pname) (G:(x_a->Prop)), (((p ((insert_a (mgt_call Pn)) G)) ((insert_a (mgt (the_com (body Pn)))) bot_bot_a_o))->((p G) ((insert_a (mgt_call Pn)) bot_bot_a_o)))) of role axiom named fact_562_assms_I2_J % A new axiom: (forall (Pn:pname) (G:(x_a->Prop)), (((p ((insert_a (mgt_call Pn)) G)) ((insert_a (mgt (the_com (body Pn)))) bot_bot_a_o))->((p G) ((insert_a (mgt_call Pn)) bot_bot_a_o)))) % FOF formula (forall (P_1:(((nat->Prop)->Prop)->Prop)) (F_1:((nat->Prop)->Prop)), ((finite_finite_nat_o F_1)->((P_1 bot_bot_nat_o_o)->((forall (X_1:(nat->Prop)) (F_2:((nat->Prop)->Prop)), ((finite_finite_nat_o F_2)->((((member_nat_o X_1) F_2)->False)->((P_1 F_2)->(P_1 ((insert_nat_o X_1) F_2))))))->(P_1 F_1))))) of role axiom named fact_563_finite__induct % A new axiom: (forall (P_1:(((nat->Prop)->Prop)->Prop)) (F_1:((nat->Prop)->Prop)), ((finite_finite_nat_o F_1)->((P_1 bot_bot_nat_o_o)->((forall (X_1:(nat->Prop)) (F_2:((nat->Prop)->Prop)), ((finite_finite_nat_o F_2)->((((member_nat_o X_1) F_2)->False)->((P_1 F_2)->(P_1 ((insert_nat_o X_1) F_2))))))->(P_1 F_1))))) % FOF formula (forall (P_1:(((pname->Prop)->Prop)->Prop)) (F_1:((pname->Prop)->Prop)), ((finite297249702name_o F_1)->((P_1 bot_bot_pname_o_o)->((forall (X_1:(pname->Prop)) (F_2:((pname->Prop)->Prop)), ((finite297249702name_o F_2)->((((member_pname_o X_1) F_2)->False)->((P_1 F_2)->(P_1 ((insert_pname_o X_1) F_2))))))->(P_1 F_1))))) of role axiom named fact_564_finite__induct % A new axiom: (forall (P_1:(((pname->Prop)->Prop)->Prop)) (F_1:((pname->Prop)->Prop)), ((finite297249702name_o F_1)->((P_1 bot_bot_pname_o_o)->((forall (X_1:(pname->Prop)) (F_2:((pname->Prop)->Prop)), ((finite297249702name_o F_2)->((((member_pname_o X_1) F_2)->False)->((P_1 F_2)->(P_1 ((insert_pname_o X_1) F_2))))))->(P_1 F_1))))) % FOF formula (forall (P_1:(((x_a->Prop)->Prop)->Prop)) (F_1:((x_a->Prop)->Prop)), ((finite_finite_a_o F_1)->((P_1 bot_bot_a_o_o)->((forall (X_1:(x_a->Prop)) (F_2:((x_a->Prop)->Prop)), ((finite_finite_a_o F_2)->((((member_a_o X_1) F_2)->False)->((P_1 F_2)->(P_1 ((insert_a_o X_1) F_2))))))->(P_1 F_1))))) of role axiom named fact_565_finite__induct % A new axiom: (forall (P_1:(((x_a->Prop)->Prop)->Prop)) (F_1:((x_a->Prop)->Prop)), ((finite_finite_a_o F_1)->((P_1 bot_bot_a_o_o)->((forall (X_1:(x_a->Prop)) (F_2:((x_a->Prop)->Prop)), ((finite_finite_a_o F_2)->((((member_a_o X_1) F_2)->False)->((P_1 F_2)->(P_1 ((insert_a_o X_1) F_2))))))->(P_1 F_1))))) % FOF formula (forall (U:nat) (F:(nat->nat)), ((forall (N_2:nat), ((ord_less_eq_nat N_2) (F N_2)))->(finite_finite_nat (collect_nat (fun (N_2:nat)=> ((ord_less_eq_nat (F N_2)) U)))))) of role axiom named fact_566_finite__less__ub % A new axiom: (forall (U:nat) (F:(nat->nat)), ((forall (N_2:nat), ((ord_less_eq_nat N_2) (F N_2)))->(finite_finite_nat (collect_nat (fun (N_2:nat)=> ((ord_less_eq_nat (F N_2)) U)))))) % FOF formula (forall (Pn:pname), (((member_pname Pn) u)->(wt (the_com (body Pn))))) of role axiom named fact_567_assms_I4_J % A new axiom: (forall (Pn:pname), (((member_pname Pn) u)->(wt (the_com (body Pn))))) % FOF formula (forall (M:nat) (N:nat), (((eq nat) ((minus_minus_nat M) (suc N))) ((minus_minus_nat ((minus_minus_nat M) one_one_nat)) N))) of role axiom named fact_568_diff__Suc__eq__diff__pred % A new axiom: (forall (M:nat) (N:nat), (((eq nat) ((minus_minus_nat M) (suc N))) ((minus_minus_nat ((minus_minus_nat M) one_one_nat)) N))) % FOF formula (forall (N:nat), (((eq nat) ((minus_minus_nat (suc N)) one_one_nat)) N)) of role axiom named fact_569_diff__Suc__1 % A new axiom: (forall (N:nat), (((eq nat) ((minus_minus_nat (suc N)) one_one_nat)) N)) % FOF formula (forall (M:nat) (N:nat), ((iff ((ord_less_eq_nat (suc M)) N)) (((nat_case_o False) (ord_less_eq_nat M)) N))) of role axiom named fact_570_less__eq__nat_Osimps_I2_J % A new axiom: (forall (M:nat) (N:nat), ((iff ((ord_less_eq_nat (suc M)) N)) (((nat_case_o False) (ord_less_eq_nat M)) N))) % FOF formula (forall (M:nat) (N:nat), (((eq nat) ((plus_plus_nat M) (suc N))) (suc ((plus_plus_nat M) N)))) of role axiom named fact_571_add__Suc__right % A new axiom: (forall (M:nat) (N:nat), (((eq nat) ((plus_plus_nat M) (suc N))) (suc ((plus_plus_nat M) N)))) % FOF formula (forall (M:nat) (N:nat), (((eq nat) ((plus_plus_nat (suc M)) N)) (suc ((plus_plus_nat M) N)))) of role axiom named fact_572_add__Suc % A new axiom: (forall (M:nat) (N:nat), (((eq nat) ((plus_plus_nat (suc M)) N)) (suc ((plus_plus_nat M) N)))) % FOF formula (forall (M:nat) (N:nat), (((eq nat) ((plus_plus_nat (suc M)) N)) ((plus_plus_nat M) (suc N)))) of role axiom named fact_573_add__Suc__shift % A new axiom: (forall (M:nat) (N:nat), (((eq nat) ((plus_plus_nat (suc M)) N)) ((plus_plus_nat M) (suc N)))) % FOF formula (forall (M:nat) (K:nat) (N:nat), ((iff (((eq nat) ((plus_plus_nat M) K)) ((plus_plus_nat N) K))) (((eq nat) M) N))) of role axiom named fact_574_nat__add__right__cancel % A new axiom: (forall (M:nat) (K:nat) (N:nat), ((iff (((eq nat) ((plus_plus_nat M) K)) ((plus_plus_nat N) K))) (((eq nat) M) N))) % FOF formula (forall (K:nat) (M:nat) (N:nat), ((iff (((eq nat) ((plus_plus_nat K) M)) ((plus_plus_nat K) N))) (((eq nat) M) N))) of role axiom named fact_575_nat__add__left__cancel % A new axiom: (forall (K:nat) (M:nat) (N:nat), ((iff (((eq nat) ((plus_plus_nat K) M)) ((plus_plus_nat K) N))) (((eq nat) M) N))) % FOF formula (forall (M:nat) (N:nat) (K:nat), (((eq nat) ((plus_plus_nat ((plus_plus_nat M) N)) K)) ((plus_plus_nat M) ((plus_plus_nat N) K)))) of role axiom named fact_576_nat__add__assoc % A new axiom: (forall (M:nat) (N:nat) (K:nat), (((eq nat) ((plus_plus_nat ((plus_plus_nat M) N)) K)) ((plus_plus_nat M) ((plus_plus_nat N) K)))) % FOF formula (forall (X:nat) (Y:nat) (Z:nat), (((eq nat) ((plus_plus_nat X) ((plus_plus_nat Y) Z))) ((plus_plus_nat Y) ((plus_plus_nat X) Z)))) of role axiom named fact_577_nat__add__left__commute % A new axiom: (forall (X:nat) (Y:nat) (Z:nat), (((eq nat) ((plus_plus_nat X) ((plus_plus_nat Y) Z))) ((plus_plus_nat Y) ((plus_plus_nat X) Z)))) % FOF formula (forall (M:nat) (N:nat), (((eq nat) ((plus_plus_nat M) N)) ((plus_plus_nat N) M))) of role axiom named fact_578_nat__add__commute % A new axiom: (forall (M:nat) (N:nat), (((eq nat) ((plus_plus_nat M) N)) ((plus_plus_nat N) M))) % FOF formula (forall (M:nat) (N:nat), (((eq nat) ((minus_minus_nat ((plus_plus_nat M) N)) N)) M)) of role axiom named fact_579_diff__add__inverse2 % A new axiom: (forall (M:nat) (N:nat), (((eq nat) ((minus_minus_nat ((plus_plus_nat M) N)) N)) M)) % FOF formula (forall (N:nat) (M:nat), (((eq nat) ((minus_minus_nat ((plus_plus_nat N) M)) N)) M)) of role axiom named fact_580_diff__add__inverse % A new axiom: (forall (N:nat) (M:nat), (((eq nat) ((minus_minus_nat ((plus_plus_nat N) M)) N)) M)) % FOF formula (forall (_TPTP_I:nat) (J_1:nat) (K:nat), (((eq nat) ((minus_minus_nat ((minus_minus_nat _TPTP_I) J_1)) K)) ((minus_minus_nat _TPTP_I) ((plus_plus_nat J_1) K)))) of role axiom named fact_581_diff__diff__left % A new axiom: (forall (_TPTP_I:nat) (J_1:nat) (K:nat), (((eq nat) ((minus_minus_nat ((minus_minus_nat _TPTP_I) J_1)) K)) ((minus_minus_nat _TPTP_I) ((plus_plus_nat J_1) K)))) % FOF formula (forall (K:nat) (M:nat) (N:nat), (((eq nat) ((minus_minus_nat ((plus_plus_nat K) M)) ((plus_plus_nat K) N))) ((minus_minus_nat M) N))) of role axiom named fact_582_diff__cancel % A new axiom: (forall (K:nat) (M:nat) (N:nat), (((eq nat) ((minus_minus_nat ((plus_plus_nat K) M)) ((plus_plus_nat K) N))) ((minus_minus_nat M) N))) % FOF formula (forall (M:nat) (K:nat) (N:nat), (((eq nat) ((minus_minus_nat ((plus_plus_nat M) K)) ((plus_plus_nat N) K))) ((minus_minus_nat M) N))) of role axiom named fact_583_diff__cancel2 % A new axiom: (forall (M:nat) (K:nat) (N:nat), (((eq nat) ((minus_minus_nat ((plus_plus_nat M) K)) ((plus_plus_nat N) K))) ((minus_minus_nat M) N))) % FOF formula (forall (N:nat) (M:nat), ((ord_less_eq_nat N) ((plus_plus_nat M) N))) of role axiom named fact_584_le__add2 % A new axiom: (forall (N:nat) (M:nat), ((ord_less_eq_nat N) ((plus_plus_nat M) N))) % FOF formula (forall (N:nat) (M:nat), ((ord_less_eq_nat N) ((plus_plus_nat N) M))) of role axiom named fact_585_le__add1 % A new axiom: (forall (N:nat) (M:nat), ((ord_less_eq_nat N) ((plus_plus_nat N) M))) % FOF formula (forall (M:nat) (N:nat), ((iff ((ord_less_eq_nat M) N)) ((ex nat) (fun (K_1:nat)=> (((eq nat) N) ((plus_plus_nat M) K_1)))))) of role axiom named fact_586_le__iff__add % A new axiom: (forall (M:nat) (N:nat), ((iff ((ord_less_eq_nat M) N)) ((ex nat) (fun (K_1:nat)=> (((eq nat) N) ((plus_plus_nat M) K_1)))))) % FOF formula (forall (K:nat) (M:nat) (N:nat), ((iff ((ord_less_eq_nat ((plus_plus_nat K) M)) ((plus_plus_nat K) N))) ((ord_less_eq_nat M) N))) of role axiom named fact_587_nat__add__left__cancel__le % A new axiom: (forall (K:nat) (M:nat) (N:nat), ((iff ((ord_less_eq_nat ((plus_plus_nat K) M)) ((plus_plus_nat K) N))) ((ord_less_eq_nat M) N))) % FOF formula (forall (M:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->((ord_less_eq_nat _TPTP_I) ((plus_plus_nat J_1) M)))) of role axiom named fact_588_trans__le__add1 % A new axiom: (forall (M:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->((ord_less_eq_nat _TPTP_I) ((plus_plus_nat J_1) M)))) % FOF formula (forall (M:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->((ord_less_eq_nat _TPTP_I) ((plus_plus_nat M) J_1)))) of role axiom named fact_589_trans__le__add2 % A new axiom: (forall (M:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->((ord_less_eq_nat _TPTP_I) ((plus_plus_nat M) J_1)))) % FOF formula (forall (K:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->((ord_less_eq_nat ((plus_plus_nat _TPTP_I) K)) ((plus_plus_nat J_1) K)))) of role axiom named fact_590_add__le__mono1 % A new axiom: (forall (K:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->((ord_less_eq_nat ((plus_plus_nat _TPTP_I) K)) ((plus_plus_nat J_1) K)))) % FOF formula (forall (K:nat) (L:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->(((ord_less_eq_nat K) L)->((ord_less_eq_nat ((plus_plus_nat _TPTP_I) K)) ((plus_plus_nat J_1) L))))) of role axiom named fact_591_add__le__mono % A new axiom: (forall (K:nat) (L:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->(((ord_less_eq_nat K) L)->((ord_less_eq_nat ((plus_plus_nat _TPTP_I) K)) ((plus_plus_nat J_1) L))))) % FOF formula (forall (M:nat) (K:nat) (N:nat), (((ord_less_eq_nat ((plus_plus_nat M) K)) N)->((ord_less_eq_nat K) N))) of role axiom named fact_592_add__leD2 % A new axiom: (forall (M:nat) (K:nat) (N:nat), (((ord_less_eq_nat ((plus_plus_nat M) K)) N)->((ord_less_eq_nat K) N))) % FOF formula (forall (M:nat) (K:nat) (N:nat), (((ord_less_eq_nat ((plus_plus_nat M) K)) N)->((ord_less_eq_nat M) N))) of role axiom named fact_593_add__leD1 % A new axiom: (forall (M:nat) (K:nat) (N:nat), (((ord_less_eq_nat ((plus_plus_nat M) K)) N)->((ord_less_eq_nat M) N))) % FOF formula (forall (M:nat) (K:nat) (N:nat), (((ord_less_eq_nat ((plus_plus_nat M) K)) N)->((((ord_less_eq_nat M) N)->(((ord_less_eq_nat K) N)->False))->False))) of role axiom named fact_594_add__leE % A new axiom: (forall (M:nat) (K:nat) (N:nat), (((ord_less_eq_nat ((plus_plus_nat M) K)) N)->((((ord_less_eq_nat M) N)->(((ord_less_eq_nat K) N)->False))->False))) % FOF formula (forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((minus_minus_nat ((plus_plus_nat J_1) _TPTP_I)) K)) ((plus_plus_nat ((minus_minus_nat J_1) K)) _TPTP_I)))) of role axiom named fact_595_diff__add__assoc2 % A new axiom: (forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((minus_minus_nat ((plus_plus_nat J_1) _TPTP_I)) K)) ((plus_plus_nat ((minus_minus_nat J_1) K)) _TPTP_I)))) % FOF formula (forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((plus_plus_nat ((minus_minus_nat J_1) K)) _TPTP_I)) ((minus_minus_nat ((plus_plus_nat J_1) _TPTP_I)) K)))) of role axiom named fact_596_add__diff__assoc2 % A new axiom: (forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((plus_plus_nat ((minus_minus_nat J_1) K)) _TPTP_I)) ((minus_minus_nat ((plus_plus_nat J_1) _TPTP_I)) K)))) % FOF formula (forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((minus_minus_nat ((plus_plus_nat _TPTP_I) J_1)) K)) ((plus_plus_nat _TPTP_I) ((minus_minus_nat J_1) K))))) of role axiom named fact_597_diff__add__assoc % A new axiom: (forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((minus_minus_nat ((plus_plus_nat _TPTP_I) J_1)) K)) ((plus_plus_nat _TPTP_I) ((minus_minus_nat J_1) K))))) % FOF formula (forall (K:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->((iff (((eq nat) ((minus_minus_nat J_1) _TPTP_I)) K)) (((eq nat) J_1) ((plus_plus_nat K) _TPTP_I))))) of role axiom named fact_598_le__imp__diff__is__add % A new axiom: (forall (K:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->((iff (((eq nat) ((minus_minus_nat J_1) _TPTP_I)) K)) (((eq nat) J_1) ((plus_plus_nat K) _TPTP_I))))) % FOF formula (forall (N:nat) (M:nat), (((ord_less_eq_nat N) M)->(((eq nat) ((plus_plus_nat ((minus_minus_nat M) N)) N)) M))) of role axiom named fact_599_le__add__diff__inverse2 % A new axiom: (forall (N:nat) (M:nat), (((ord_less_eq_nat N) M)->(((eq nat) ((plus_plus_nat ((minus_minus_nat M) N)) N)) M))) % FOF formula (forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->((iff ((ord_less_eq_nat _TPTP_I) ((minus_minus_nat J_1) K))) ((ord_less_eq_nat ((plus_plus_nat _TPTP_I) K)) J_1)))) of role axiom named fact_600_le__diff__conv2 % A new axiom: (forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->((iff ((ord_less_eq_nat _TPTP_I) ((minus_minus_nat J_1) K))) ((ord_less_eq_nat ((plus_plus_nat _TPTP_I) K)) J_1)))) % FOF formula (forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((plus_plus_nat _TPTP_I) ((minus_minus_nat J_1) K))) ((minus_minus_nat ((plus_plus_nat _TPTP_I) J_1)) K)))) of role axiom named fact_601_add__diff__assoc % A new axiom: (forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((plus_plus_nat _TPTP_I) ((minus_minus_nat J_1) K))) ((minus_minus_nat ((plus_plus_nat _TPTP_I) J_1)) K)))) % FOF formula (forall (N:nat) (M:nat), (((ord_less_eq_nat N) M)->(((eq nat) ((plus_plus_nat N) ((minus_minus_nat M) N))) M))) of role axiom named fact_602_le__add__diff__inverse % A new axiom: (forall (N:nat) (M:nat), (((ord_less_eq_nat N) M)->(((eq nat) ((plus_plus_nat N) ((minus_minus_nat M) N))) M))) % FOF formula (forall (M:nat) (K:nat) (N:nat), (((ord_less_eq_nat K) N)->((ord_less_eq_nat M) ((minus_minus_nat ((plus_plus_nat N) M)) K)))) of role axiom named fact_603_le__add__diff % A new axiom: (forall (M:nat) (K:nat) (N:nat), (((ord_less_eq_nat K) N)->((ord_less_eq_nat M) ((minus_minus_nat ((plus_plus_nat N) M)) K)))) % FOF formula (forall (J_1:nat) (K:nat) (_TPTP_I:nat), ((iff ((ord_less_eq_nat ((minus_minus_nat J_1) K)) _TPTP_I)) ((ord_less_eq_nat J_1) ((plus_plus_nat _TPTP_I) K)))) of role axiom named fact_604_le__diff__conv % A new axiom: (forall (J_1:nat) (K:nat) (_TPTP_I:nat), ((iff ((ord_less_eq_nat ((minus_minus_nat J_1) K)) _TPTP_I)) ((ord_less_eq_nat J_1) ((plus_plus_nat _TPTP_I) K)))) % FOF formula (forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((minus_minus_nat _TPTP_I) ((minus_minus_nat J_1) K))) ((minus_minus_nat ((plus_plus_nat _TPTP_I) K)) J_1)))) of role axiom named fact_605_diff__diff__right % A new axiom: (forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((minus_minus_nat _TPTP_I) ((minus_minus_nat J_1) K))) ((minus_minus_nat ((plus_plus_nat _TPTP_I) K)) J_1)))) % FOF formula (forall (N:nat), (((eq nat) (suc N)) ((plus_plus_nat N) one_one_nat))) of role axiom named fact_606_Suc__eq__plus1 % A new axiom: (forall (N:nat), (((eq nat) (suc N)) ((plus_plus_nat N) one_one_nat))) % FOF formula (forall (N:nat), (((eq nat) (suc N)) ((plus_plus_nat one_one_nat) N))) of role axiom named fact_607_Suc__eq__plus1__left % A new axiom: (forall (N:nat), (((eq nat) (suc N)) ((plus_plus_nat one_one_nat) N))) % FOF formula (forall (M:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((minus_minus_nat (suc ((minus_minus_nat J_1) K))) M)) ((minus_minus_nat (suc J_1)) ((plus_plus_nat K) M))))) of role axiom named fact_608_diff__Suc__diff__eq2 % A new axiom: (forall (M:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((minus_minus_nat (suc ((minus_minus_nat J_1) K))) M)) ((minus_minus_nat (suc J_1)) ((plus_plus_nat K) M))))) % FOF formula (forall (M:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((minus_minus_nat M) (suc ((minus_minus_nat J_1) K)))) ((minus_minus_nat ((plus_plus_nat M) K)) (suc J_1))))) of role axiom named fact_609_diff__Suc__diff__eq1 % A new axiom: (forall (M:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((minus_minus_nat M) (suc ((minus_minus_nat J_1) K)))) ((minus_minus_nat ((plus_plus_nat M) K)) (suc J_1))))) % FOF formula (forall (Z:nat) (X:nat) (Y:nat), (((ord_less_eq_nat X) Y)->((ord_less_eq_nat X) ((plus_plus_nat Y) Z)))) of role axiom named fact_610_termination__basic__simps_I3_J % A new axiom: (forall (Z:nat) (X:nat) (Y:nat), (((ord_less_eq_nat X) Y)->((ord_less_eq_nat X) ((plus_plus_nat Y) Z)))) % FOF formula (forall (Y:nat) (X:nat) (Z:nat), (((ord_less_eq_nat X) Z)->((ord_less_eq_nat X) ((plus_plus_nat Y) Z)))) of role axiom named fact_611_termination__basic__simps_I4_J % A new axiom: (forall (Y:nat) (X:nat) (Z:nat), (((ord_less_eq_nat X) Z)->((ord_less_eq_nat X) ((plus_plus_nat Y) Z)))) % FOF formula (forall (N:nat), ((ord_less_nat N) (suc N))) of role axiom named fact_612_lessI % A new axiom: (forall (N:nat), ((ord_less_nat N) (suc N))) % FOF formula (forall (M:nat) (N:nat), (((ord_less_nat M) N)->((ord_less_nat (suc M)) (suc N)))) of role axiom named fact_613_Suc__mono % A new axiom: (forall (M:nat) (N:nat), (((ord_less_nat M) N)->((ord_less_nat (suc M)) (suc N)))) % FOF formula (forall (K:nat), (finite_finite_nat (collect_nat (fun (N_2:nat)=> ((ord_less_nat N_2) K))))) of role axiom named fact_614_finite__Collect__less__nat % A new axiom: (forall (K:nat), (finite_finite_nat (collect_nat (fun (N_2:nat)=> ((ord_less_nat N_2) K))))) % FOF formula (forall (Z:nat) (X:nat) (Y:nat), (((ord_less_nat X) Y)->((ord_less_nat X) ((plus_plus_nat Y) Z)))) of role axiom named fact_615_termination__basic__simps_I1_J % A new axiom: (forall (Z:nat) (X:nat) (Y:nat), (((ord_less_nat X) Y)->((ord_less_nat X) ((plus_plus_nat Y) Z)))) % FOF formula (forall (Y:nat) (X:nat) (Z:nat), (((ord_less_nat X) Z)->((ord_less_nat X) ((plus_plus_nat Y) Z)))) of role axiom named fact_616_termination__basic__simps_I2_J % A new axiom: (forall (Y:nat) (X:nat) (Z:nat), (((ord_less_nat X) Z)->((ord_less_nat X) ((plus_plus_nat Y) Z)))) % FOF formula (forall (_TPTP_I:nat) (J_1:nat) (K:nat), (((ord_less_nat ((plus_plus_nat _TPTP_I) J_1)) K)->((ord_less_nat _TPTP_I) K))) of role axiom named fact_617_add__lessD1 % A new axiom: (forall (_TPTP_I:nat) (J_1:nat) (K:nat), (((ord_less_nat ((plus_plus_nat _TPTP_I) J_1)) K)->((ord_less_nat _TPTP_I) K))) % FOF formula (forall (M:nat) (N:nat) (K:nat) (L:nat), (((ord_less_nat K) L)->((((eq nat) ((plus_plus_nat M) L)) ((plus_plus_nat K) N))->((ord_less_nat M) N)))) of role axiom named fact_618_less__add__eq__less % A new axiom: (forall (M:nat) (N:nat) (K:nat) (L:nat), (((ord_less_nat K) L)->((((eq nat) ((plus_plus_nat M) L)) ((plus_plus_nat K) N))->((ord_less_nat M) N)))) % FOF formula (forall (K:nat) (L:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_nat _TPTP_I) J_1)->(((ord_less_nat K) L)->((ord_less_nat ((plus_plus_nat _TPTP_I) K)) ((plus_plus_nat J_1) L))))) of role axiom named fact_619_add__less__mono % A new axiom: (forall (K:nat) (L:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_nat _TPTP_I) J_1)->(((ord_less_nat K) L)->((ord_less_nat ((plus_plus_nat _TPTP_I) K)) ((plus_plus_nat J_1) L))))) % FOF formula (forall (K:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_nat _TPTP_I) J_1)->((ord_less_nat ((plus_plus_nat _TPTP_I) K)) ((plus_plus_nat J_1) K)))) of role axiom named fact_620_add__less__mono1 % A new axiom: (forall (K:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_nat _TPTP_I) J_1)->((ord_less_nat ((plus_plus_nat _TPTP_I) K)) ((plus_plus_nat J_1) K)))) % FOF formula (forall (M:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_nat _TPTP_I) J_1)->((ord_less_nat _TPTP_I) ((plus_plus_nat M) J_1)))) of role axiom named fact_621_trans__less__add2 % A new axiom: (forall (M:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_nat _TPTP_I) J_1)->((ord_less_nat _TPTP_I) ((plus_plus_nat M) J_1)))) % FOF formula (forall (M:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_nat _TPTP_I) J_1)->((ord_less_nat _TPTP_I) ((plus_plus_nat J_1) M)))) of role axiom named fact_622_trans__less__add1 % A new axiom: (forall (M:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_nat _TPTP_I) J_1)->((ord_less_nat _TPTP_I) ((plus_plus_nat J_1) M)))) % FOF formula (forall (K:nat) (M:nat) (N:nat), ((iff ((ord_less_nat ((plus_plus_nat K) M)) ((plus_plus_nat K) N))) ((ord_less_nat M) N))) of role axiom named fact_623_nat__add__left__cancel__less % A new axiom: (forall (K:nat) (M:nat) (N:nat), ((iff ((ord_less_nat ((plus_plus_nat K) M)) ((plus_plus_nat K) N))) ((ord_less_nat M) N))) % FOF formula (forall (J_1:nat) (_TPTP_I:nat), (((ord_less_nat ((plus_plus_nat J_1) _TPTP_I)) _TPTP_I)->False)) of role axiom named fact_624_not__add__less2 % A new axiom: (forall (J_1:nat) (_TPTP_I:nat), (((ord_less_nat ((plus_plus_nat J_1) _TPTP_I)) _TPTP_I)->False)) % FOF formula (forall (_TPTP_I:nat) (J_1:nat), (((ord_less_nat ((plus_plus_nat _TPTP_I) J_1)) _TPTP_I)->False)) of role axiom named fact_625_not__add__less1 % A new axiom: (forall (_TPTP_I:nat) (J_1:nat), (((ord_less_nat ((plus_plus_nat _TPTP_I) J_1)) _TPTP_I)->False)) % FOF formula (forall (M:nat) (N:nat), (((ord_less_nat (suc M)) (suc N))->((ord_less_nat M) N))) of role axiom named fact_626_Suc__less__SucD % A new axiom: (forall (M:nat) (N:nat), (((ord_less_nat (suc M)) (suc N))->((ord_less_nat M) N))) % FOF formula (forall (M:nat) (N:nat), (((ord_less_nat (suc M)) N)->((ord_less_nat M) N))) of role axiom named fact_627_Suc__lessD % A new axiom: (forall (M:nat) (N:nat), (((ord_less_nat (suc M)) N)->((ord_less_nat M) N))) % FOF formula (forall (M:nat) (N:nat), (((ord_less_nat M) (suc N))->((((ord_less_nat M) N)->False)->(((eq nat) M) N)))) of role axiom named fact_628_less__SucE % A new axiom: (forall (M:nat) (N:nat), (((ord_less_nat M) (suc N))->((((ord_less_nat M) N)->False)->(((eq nat) M) N)))) % FOF formula (forall (K:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_nat _TPTP_I) J_1)->(((ord_less_nat J_1) K)->((ord_less_nat (suc _TPTP_I)) K)))) of role axiom named fact_629_less__trans__Suc % A new axiom: (forall (K:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_nat _TPTP_I) J_1)->(((ord_less_nat J_1) K)->((ord_less_nat (suc _TPTP_I)) K)))) % FOF formula (forall (M:nat) (N:nat), (((ord_less_nat M) N)->((not (((eq nat) (suc M)) N))->((ord_less_nat (suc M)) N)))) of role axiom named fact_630_Suc__lessI % A new axiom: (forall (M:nat) (N:nat), (((ord_less_nat M) N)->((not (((eq nat) (suc M)) N))->((ord_less_nat (suc M)) N)))) % FOF formula (forall (M:nat) (N:nat), (((ord_less_nat M) N)->((ord_less_nat M) (suc N)))) of role axiom named fact_631_less__SucI % A new axiom: (forall (M:nat) (N:nat), (((ord_less_nat M) N)->((ord_less_nat M) (suc N)))) % FOF formula (forall (N:nat) (M:nat), ((((ord_less_nat N) M)->False)->(((ord_less_nat N) (suc M))->(((eq nat) M) N)))) of role axiom named fact_632_less__antisym % A new axiom: (forall (N:nat) (M:nat), ((((ord_less_nat N) M)->False)->(((ord_less_nat N) (suc M))->(((eq nat) M) N)))) % FOF formula (forall (N:nat) (M:nat), ((((ord_less_nat N) M)->False)->((iff ((ord_less_nat N) (suc M))) (((eq nat) N) M)))) of role axiom named fact_633_not__less__less__Suc__eq % A new axiom: (forall (N:nat) (M:nat), ((((ord_less_nat N) M)->False)->((iff ((ord_less_nat N) (suc M))) (((eq nat) N) M)))) % FOF formula (forall (M:nat) (N:nat), ((iff ((ord_less_nat (suc M)) (suc N))) ((ord_less_nat M) N))) of role axiom named fact_634_Suc__less__eq % A new axiom: (forall (M:nat) (N:nat), ((iff ((ord_less_nat (suc M)) (suc N))) ((ord_less_nat M) N))) % FOF formula (forall (M:nat) (N:nat), ((iff ((ord_less_nat M) (suc N))) ((or ((ord_less_nat M) N)) (((eq nat) M) N)))) of role axiom named fact_635_less__Suc__eq % A new axiom: (forall (M:nat) (N:nat), ((iff ((ord_less_nat M) (suc N))) ((or ((ord_less_nat M) N)) (((eq nat) M) N)))) % FOF formula (forall (M:nat) (N:nat), ((iff (((ord_less_nat M) N)->False)) ((ord_less_nat N) (suc M)))) of role axiom named fact_636_not__less__eq % A new axiom: (forall (M:nat) (N:nat), ((iff (((ord_less_nat M) N)->False)) ((ord_less_nat N) (suc M)))) % FOF formula (forall (M:nat) (N:nat), (((or ((ord_less_nat M) N)) (((eq nat) M) N))->((ord_less_eq_nat M) N))) of role axiom named fact_637_less__or__eq__imp__le % A new axiom: (forall (M:nat) (N:nat), (((or ((ord_less_nat M) N)) (((eq nat) M) N))->((ord_less_eq_nat M) N))) % FOF formula (forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((not (((eq nat) M) N))->((ord_less_nat M) N)))) of role axiom named fact_638_le__neq__implies__less % A new axiom: (forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((not (((eq nat) M) N))->((ord_less_nat M) N)))) % FOF formula (forall (M:nat) (N:nat), (((ord_less_nat M) N)->((ord_less_eq_nat M) N))) of role axiom named fact_639_less__imp__le__nat % A new axiom: (forall (M:nat) (N:nat), (((ord_less_nat M) N)->((ord_less_eq_nat M) N))) % FOF formula (forall (M:nat) (N:nat), ((iff ((ord_less_eq_nat M) N)) ((or ((ord_less_nat M) N)) (((eq nat) M) N)))) of role axiom named fact_640_le__eq__less__or__eq % A new axiom: (forall (M:nat) (N:nat), ((iff ((ord_less_eq_nat M) N)) ((or ((ord_less_nat M) N)) (((eq nat) M) N)))) % FOF formula (forall (M:nat) (N:nat), ((iff ((ord_less_nat M) N)) ((and ((ord_less_eq_nat M) N)) (not (((eq nat) M) N))))) of role axiom named fact_641_nat__less__le % A new axiom: (forall (M:nat) (N:nat), ((iff ((ord_less_nat M) N)) ((and ((ord_less_eq_nat M) N)) (not (((eq nat) M) N))))) % FOF formula (forall (L:nat) (M:nat) (N:nat), (((ord_less_nat M) N)->(((ord_less_nat M) L)->((ord_less_nat ((minus_minus_nat L) N)) ((minus_minus_nat L) M))))) of role axiom named fact_642_diff__less__mono2 % A new axiom: (forall (L:nat) (M:nat) (N:nat), (((ord_less_nat M) N)->(((ord_less_nat M) L)->((ord_less_nat ((minus_minus_nat L) N)) ((minus_minus_nat L) M))))) % FOF formula (forall (N:nat) (J_1:nat) (K:nat), (((ord_less_nat J_1) K)->((ord_less_nat ((minus_minus_nat J_1) N)) K))) of role axiom named fact_643_less__imp__diff__less % A new axiom: (forall (N:nat) (J_1:nat) (K:nat), (((ord_less_nat J_1) K)->((ord_less_nat ((minus_minus_nat J_1) N)) K))) % FOF formula (forall (X:nat) (Y:nat), (((ord_less_nat X) Y)->((ord_less_eq_nat X) Y))) of role axiom named fact_644_termination__basic__simps_I5_J % A new axiom: (forall (X:nat) (Y:nat), (((ord_less_nat X) Y)->((ord_less_eq_nat X) Y))) % FOF formula (forall (N:nat), (((ord_less_nat N) N)->False)) of role axiom named fact_645_less__not__refl % A new axiom: (forall (N:nat), (((ord_less_nat N) N)->False)) % FOF formula (forall (M:nat) (N:nat), ((iff (not (((eq nat) M) N))) ((or ((ord_less_nat M) N)) ((ord_less_nat N) M)))) of role axiom named fact_646_nat__neq__iff % A new axiom: (forall (M:nat) (N:nat), ((iff (not (((eq nat) M) N))) ((or ((ord_less_nat M) N)) ((ord_less_nat N) M)))) % FOF formula (forall (X:nat) (Y:nat), ((not (((eq nat) X) Y))->((((ord_less_nat X) Y)->False)->((ord_less_nat Y) X)))) of role axiom named fact_647_linorder__neqE__nat % A new axiom: (forall (X:nat) (Y:nat), ((not (((eq nat) X) Y))->((((ord_less_nat X) Y)->False)->((ord_less_nat Y) X)))) % FOF formula (forall (N:nat), (((ord_less_nat N) N)->False)) of role axiom named fact_648_less__irrefl__nat % A new axiom: (forall (N:nat), (((ord_less_nat N) N)->False)) % FOF formula (forall (N:nat) (M:nat), (((ord_less_nat N) M)->(not (((eq nat) M) N)))) of role axiom named fact_649_less__not__refl2 % A new axiom: (forall (N:nat) (M:nat), (((ord_less_nat N) M)->(not (((eq nat) M) N)))) % FOF formula (forall (S:nat) (T:nat), (((ord_less_nat S) T)->(not (((eq nat) S) T)))) of role axiom named fact_650_less__not__refl3 % A new axiom: (forall (S:nat) (T:nat), (((ord_less_nat S) T)->(not (((eq nat) S) T)))) % FOF formula (forall (P:(nat->(nat->Prop))) (M:nat) (N:nat), ((((ord_less_nat M) N)->((P N) M))->(((((eq nat) M) N)->((P N) M))->((((ord_less_nat N) M)->((P N) M))->((P N) M))))) of role axiom named fact_651_nat__less__cases % A new axiom: (forall (P:(nat->(nat->Prop))) (M:nat) (N:nat), ((((ord_less_nat M) N)->((P N) M))->(((((eq nat) M) N)->((P N) M))->((((ord_less_nat N) M)->((P N) M))->((P N) M))))) % FOF formula (forall (N_1:(nat->Prop)), ((iff (finite_finite_nat N_1)) ((ex nat) (fun (M_1:nat)=> (forall (X_1:nat), (((member_nat X_1) N_1)->((ord_less_nat X_1) M_1))))))) of role axiom named fact_652_finite__nat__set__iff__bounded % A new axiom: (forall (N_1:(nat->Prop)), ((iff (finite_finite_nat N_1)) ((ex nat) (fun (M_1:nat)=> (forall (X_1:nat), (((member_nat X_1) N_1)->((ord_less_nat X_1) M_1))))))) % FOF formula (forall (N:nat), (((eq nat) (finite_card_nat (collect_nat (fun (I_1:nat)=> ((ord_less_nat I_1) N))))) N)) of role axiom named fact_653_card__Collect__less__nat % A new axiom: (forall (N:nat), (((eq nat) (finite_card_nat (collect_nat (fun (I_1:nat)=> ((ord_less_nat I_1) N))))) N)) % FOF formula (forall (P:(nat->Prop)) (_TPTP_I:nat), (finite_finite_nat (collect_nat (fun (K_1:nat)=> ((and (P K_1)) ((ord_less_nat K_1) _TPTP_I)))))) of role axiom named fact_654_finite__M__bounded__by__nat % A new axiom: (forall (P:(nat->Prop)) (_TPTP_I:nat), (finite_finite_nat (collect_nat (fun (K_1:nat)=> ((and (P K_1)) ((ord_less_nat K_1) _TPTP_I)))))) % FOF formula (forall (_TPTP_I:nat) (M:nat), ((ord_less_nat _TPTP_I) (suc ((plus_plus_nat _TPTP_I) M)))) of role axiom named fact_655_less__add__Suc1 % A new axiom: (forall (_TPTP_I:nat) (M:nat), ((ord_less_nat _TPTP_I) (suc ((plus_plus_nat _TPTP_I) M)))) % FOF formula (forall (_TPTP_I:nat) (M:nat), ((ord_less_nat _TPTP_I) (suc ((plus_plus_nat M) _TPTP_I)))) of role axiom named fact_656_less__add__Suc2 % A new axiom: (forall (_TPTP_I:nat) (M:nat), ((ord_less_nat _TPTP_I) (suc ((plus_plus_nat M) _TPTP_I)))) % FOF formula (forall (M:nat) (N:nat), ((iff ((ord_less_nat M) N)) ((ex nat) (fun (K_1:nat)=> (((eq nat) N) (suc ((plus_plus_nat M) K_1))))))) of role axiom named fact_657_less__iff__Suc__add % A new axiom: (forall (M:nat) (N:nat), ((iff ((ord_less_nat M) N)) ((ex nat) (fun (K_1:nat)=> (((eq nat) N) (suc ((plus_plus_nat M) K_1))))))) % FOF formula (forall (N:nat) (M:nat), ((iff ((ord_less_nat N) M)) ((ord_less_eq_nat (suc N)) M))) of role axiom named fact_658_less__eq__Suc__le % A new axiom: (forall (N:nat) (M:nat), ((iff ((ord_less_nat N) M)) ((ord_less_eq_nat (suc N)) M))) % FOF formula (forall (M:nat) (N:nat), ((iff ((ord_less_nat M) (suc N))) ((ord_less_eq_nat M) N))) of role axiom named fact_659_less__Suc__eq__le % A new axiom: (forall (M:nat) (N:nat), ((iff ((ord_less_nat M) (suc N))) ((ord_less_eq_nat M) N))) % FOF formula (forall (M:nat) (N:nat), ((iff ((ord_less_eq_nat (suc M)) N)) ((ord_less_nat M) N))) of role axiom named fact_660_Suc__le__eq % A new axiom: (forall (M:nat) (N:nat), ((iff ((ord_less_eq_nat (suc M)) N)) ((ord_less_nat M) N))) % FOF formula (forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((ord_less_nat M) (suc N)))) of role axiom named fact_661_le__imp__less__Suc % A new axiom: (forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((ord_less_nat M) (suc N)))) % FOF formula (forall (M:nat) (N:nat), (((ord_less_nat M) N)->((ord_less_eq_nat (suc M)) N))) of role axiom named fact_662_Suc__leI % A new axiom: (forall (M:nat) (N:nat), (((ord_less_nat M) N)->((ord_less_eq_nat (suc M)) N))) % FOF formula (forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((iff ((ord_less_nat N) (suc M))) (((eq nat) N) M)))) of role axiom named fact_663_le__less__Suc__eq % A new axiom: (forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((iff ((ord_less_nat N) (suc M))) (((eq nat) N) M)))) % FOF formula (forall (M:nat) (N:nat), (((ord_less_eq_nat (suc M)) N)->((ord_less_nat M) N))) of role axiom named fact_664_Suc__le__lessD % A new axiom: (forall (M:nat) (N:nat), (((ord_less_eq_nat (suc M)) N)->((ord_less_nat M) N))) % FOF formula (forall (M:nat) (N:nat), ((ord_less_nat ((minus_minus_nat M) N)) (suc M))) of role axiom named fact_665_diff__less__Suc % A new axiom: (forall (M:nat) (N:nat), ((ord_less_nat ((minus_minus_nat M) N)) (suc M))) % FOF formula (forall (M:nat) (N:nat), ((((ord_less_nat M) N)->False)->(((eq nat) ((plus_plus_nat N) ((minus_minus_nat M) N))) M))) of role axiom named fact_666_add__diff__inverse % A new axiom: (forall (M:nat) (N:nat), ((((ord_less_nat M) N)->False)->(((eq nat) ((plus_plus_nat N) ((minus_minus_nat M) N))) M))) % FOF formula (forall (_TPTP_I:nat) (J_1:nat) (K:nat), ((iff ((ord_less_nat _TPTP_I) ((minus_minus_nat J_1) K))) ((ord_less_nat ((plus_plus_nat _TPTP_I) K)) J_1))) of role axiom named fact_667_less__diff__conv % A new axiom: (forall (_TPTP_I:nat) (J_1:nat) (K:nat), ((iff ((ord_less_nat _TPTP_I) ((minus_minus_nat J_1) K))) ((ord_less_nat ((plus_plus_nat _TPTP_I) K)) J_1))) % FOF formula (forall (C:nat) (A:nat) (B:nat), (((ord_less_nat A) B)->(((ord_less_eq_nat C) A)->((ord_less_nat ((minus_minus_nat A) C)) ((minus_minus_nat B) C))))) of role axiom named fact_668_diff__less__mono % A new axiom: (forall (C:nat) (A:nat) (B:nat), (((ord_less_nat A) B)->(((ord_less_eq_nat C) A)->((ord_less_nat ((minus_minus_nat A) C)) ((minus_minus_nat B) C))))) % FOF formula (forall (N:nat) (K:nat) (M:nat), (((ord_less_eq_nat K) M)->(((ord_less_eq_nat K) N)->((iff ((ord_less_nat ((minus_minus_nat M) K)) ((minus_minus_nat N) K))) ((ord_less_nat M) N))))) of role axiom named fact_669_less__diff__iff % A new axiom: (forall (N:nat) (K:nat) (M:nat), (((ord_less_eq_nat K) M)->(((ord_less_eq_nat K) N)->((iff ((ord_less_nat ((minus_minus_nat M) K)) ((minus_minus_nat N) K))) ((ord_less_nat M) N))))) % FOF formula (forall (X_1:nat), (((eq (nat->Prop)) (ord_less_nat X_1)) (ord_less_eq_nat (suc X_1)))) of role axiom named fact_670_less__eq__Suc__le__raw % A new axiom: (forall (X_1:nat), (((eq (nat->Prop)) (ord_less_nat X_1)) (ord_less_eq_nat (suc X_1)))) % FOF formula (forall (M:nat) (K:nat) (F:(nat->nat)), ((forall (M_1:nat) (N_2:nat), (((ord_less_nat M_1) N_2)->((ord_less_nat (F M_1)) (F N_2))))->((ord_less_eq_nat ((plus_plus_nat (F M)) K)) (F ((plus_plus_nat M) K))))) of role axiom named fact_671_mono__nat__linear__lb % A new axiom: (forall (M:nat) (K:nat) (F:(nat->nat)), ((forall (M_1:nat) (N_2:nat), (((ord_less_nat M_1) N_2)->((ord_less_nat (F M_1)) (F N_2))))->((ord_less_eq_nat ((plus_plus_nat (F M)) K)) (F ((plus_plus_nat M) K))))) % FOF formula (forall (P:(nat->Prop)) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->((P J_1)->((forall (I_1:nat), (((ord_less_nat I_1) J_1)->((P (suc I_1))->(P I_1))))->(P _TPTP_I))))) of role axiom named fact_672_inc__induct % A new axiom: (forall (P:(nat->Prop)) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->((P J_1)->((forall (I_1:nat), (((ord_less_nat I_1) J_1)->((P (suc I_1))->(P I_1))))->(P _TPTP_I))))) % FOF formula (forall (M:nat) (N:nat), (((ord_less_nat M) N)->((ex nat) (fun (K_1:nat)=> (((eq nat) N) (suc ((plus_plus_nat M) K_1))))))) of role axiom named fact_673_less__imp__Suc__add % A new axiom: (forall (M:nat) (N:nat), (((ord_less_nat M) N)->((ex nat) (fun (K_1:nat)=> (((eq nat) N) (suc ((plus_plus_nat M) K_1))))))) % FOF formula (forall (N:nat) (N_1:(nat->Prop)), ((forall (X_1:nat), (((member_nat X_1) N_1)->((ord_less_nat X_1) N)))->(finite_finite_nat N_1))) of role axiom named fact_674_bounded__nat__set__is__finite % A new axiom: (forall (N:nat) (N_1:(nat->Prop)), ((forall (X_1:nat), (((member_nat X_1) N_1)->((ord_less_nat X_1) N)))->(finite_finite_nat N_1))) % FOF formula (forall (_TPTP_I:nat) (J_1:nat) (F:(nat->nat)), ((forall (I_1:nat) (J:nat), (((ord_less_nat I_1) J)->((ord_less_nat (F I_1)) (F J))))->(((ord_less_eq_nat _TPTP_I) J_1)->((ord_less_eq_nat (F _TPTP_I)) (F J_1))))) of role axiom named fact_675_less__mono__imp__le__mono % A new axiom: (forall (_TPTP_I:nat) (J_1:nat) (F:(nat->nat)), ((forall (I_1:nat) (J:nat), (((ord_less_nat I_1) J)->((ord_less_nat (F I_1)) (F J))))->(((ord_less_eq_nat _TPTP_I) J_1)->((ord_less_eq_nat (F _TPTP_I)) (F J_1))))) % FOF formula (forall (_TPTP_I:nat) (K:nat), (((ord_less_nat (suc _TPTP_I)) K)->((forall (J:nat), (((ord_less_nat _TPTP_I) J)->(not (((eq nat) K) (suc J)))))->False))) of role axiom named fact_676_Suc__lessE % A new axiom: (forall (_TPTP_I:nat) (K:nat), (((ord_less_nat (suc _TPTP_I)) K)->((forall (J:nat), (((ord_less_nat _TPTP_I) J)->(not (((eq nat) K) (suc J)))))->False))) % FOF formula (forall (_TPTP_I:nat) (K:nat), (((ord_less_nat _TPTP_I) K)->((not (((eq nat) K) (suc _TPTP_I)))->((forall (J:nat), (((ord_less_nat _TPTP_I) J)->(not (((eq nat) K) (suc J)))))->False)))) of role axiom named fact_677_lessE % A new axiom: (forall (_TPTP_I:nat) (K:nat), (((ord_less_nat _TPTP_I) K)->((not (((eq nat) K) (suc _TPTP_I)))->((forall (J:nat), (((ord_less_nat _TPTP_I) J)->(not (((eq nat) K) (suc J)))))->False)))) % FOF formula (forall (N:nat), (((ord_less_nat N) zero_zero_nat)->False)) of role axiom named fact_678_less__zeroE % A new axiom: (forall (N:nat), (((ord_less_nat N) zero_zero_nat)->False)) % FOF formula (forall (N:nat), ((ord_less_eq_nat zero_zero_nat) N)) of role axiom named fact_679_le0 % A new axiom: (forall (N:nat), ((ord_less_eq_nat zero_zero_nat) N)) % FOF formula (forall (N:nat), ((ord_less_nat zero_zero_nat) (suc N))) of role axiom named fact_680_zero__less__Suc % A new axiom: (forall (N:nat), ((ord_less_nat zero_zero_nat) (suc N))) % FOF formula (forall (M:nat) (N:nat), ((iff (((eq nat) ((plus_plus_nat M) N)) (suc zero_zero_nat))) ((or ((and (((eq nat) M) (suc zero_zero_nat))) (((eq nat) N) zero_zero_nat))) ((and (((eq nat) M) zero_zero_nat)) (((eq nat) N) (suc zero_zero_nat)))))) of role axiom named fact_681_add__is__1 % A new axiom: (forall (M:nat) (N:nat), ((iff (((eq nat) ((plus_plus_nat M) N)) (suc zero_zero_nat))) ((or ((and (((eq nat) M) (suc zero_zero_nat))) (((eq nat) N) zero_zero_nat))) ((and (((eq nat) M) zero_zero_nat)) (((eq nat) N) (suc zero_zero_nat)))))) % FOF formula (forall (M:nat) (N:nat), ((iff (((eq nat) (suc zero_zero_nat)) ((plus_plus_nat M) N))) ((or ((and (((eq nat) M) (suc zero_zero_nat))) (((eq nat) N) zero_zero_nat))) ((and (((eq nat) M) zero_zero_nat)) (((eq nat) N) (suc zero_zero_nat)))))) of role axiom named fact_682_one__is__add % A new axiom: (forall (M:nat) (N:nat), ((iff (((eq nat) (suc zero_zero_nat)) ((plus_plus_nat M) N))) ((or ((and (((eq nat) M) (suc zero_zero_nat))) (((eq nat) N) zero_zero_nat))) ((and (((eq nat) M) zero_zero_nat)) (((eq nat) N) (suc zero_zero_nat)))))) % FOF formula (forall (N:nat) (M:nat), (((eq nat) ((minus_minus_nat N) ((plus_plus_nat N) M))) zero_zero_nat)) of role axiom named fact_683_diff__add__0 % A new axiom: (forall (N:nat) (M:nat), (((eq nat) ((minus_minus_nat N) ((plus_plus_nat N) M))) zero_zero_nat)) % FOF formula (forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->(((eq nat) ((minus_minus_nat M) N)) zero_zero_nat))) of role axiom named fact_684_diff__is__0__eq_H % A new axiom: (forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->(((eq nat) ((minus_minus_nat M) N)) zero_zero_nat))) % FOF formula (forall (M:nat) (N:nat), ((iff (((eq nat) ((minus_minus_nat M) N)) zero_zero_nat)) ((ord_less_eq_nat M) N))) of role axiom named fact_685_diff__is__0__eq % A new axiom: (forall (M:nat) (N:nat), ((iff (((eq nat) ((minus_minus_nat M) N)) zero_zero_nat)) ((ord_less_eq_nat M) N))) % FOF formula (((eq nat) one_one_nat) (suc zero_zero_nat)) of role axiom named fact_686_One__nat__def % A new axiom: (((eq nat) one_one_nat) (suc zero_zero_nat)) % FOF formula (forall (M:nat) (N:nat), ((((eq nat) ((minus_minus_nat M) N)) zero_zero_nat)->((((eq nat) ((minus_minus_nat N) M)) zero_zero_nat)->(((eq nat) M) N)))) of role axiom named fact_687_diffs0__imp__equal % A new axiom: (forall (M:nat) (N:nat), ((((eq nat) ((minus_minus_nat M) N)) zero_zero_nat)->((((eq nat) ((minus_minus_nat N) M)) zero_zero_nat)->(((eq nat) M) N)))) % FOF formula (forall (M:nat), (((eq nat) ((minus_minus_nat M) M)) zero_zero_nat)) of role axiom named fact_688_diff__self__eq__0 % A new axiom: (forall (M:nat), (((eq nat) ((minus_minus_nat M) M)) zero_zero_nat)) % FOF formula (forall (M:nat), (((eq nat) ((minus_minus_nat M) zero_zero_nat)) M)) of role axiom named fact_689_minus__nat_Odiff__0 % A new axiom: (forall (M:nat), (((eq nat) ((minus_minus_nat M) zero_zero_nat)) M)) % FOF formula (forall (N:nat), (((eq nat) ((minus_minus_nat zero_zero_nat) N)) zero_zero_nat)) of role axiom named fact_690_diff__0__eq__0 % A new axiom: (forall (N:nat), (((eq nat) ((minus_minus_nat zero_zero_nat) N)) zero_zero_nat)) % FOF formula (forall (M:nat), (not (((eq nat) (suc M)) zero_zero_nat))) of role axiom named fact_691_Suc__neq__Zero % A new axiom: (forall (M:nat), (not (((eq nat) (suc M)) zero_zero_nat))) % FOF formula (forall (M:nat), (not (((eq nat) zero_zero_nat) (suc M)))) of role axiom named fact_692_Zero__neq__Suc % A new axiom: (forall (M:nat), (not (((eq nat) zero_zero_nat) (suc M)))) % FOF formula (forall (Nat_1:nat), (not (((eq nat) (suc Nat_1)) zero_zero_nat))) of role axiom named fact_693_nat_Osimps_I3_J % A new axiom: (forall (Nat_1:nat), (not (((eq nat) (suc Nat_1)) zero_zero_nat))) % FOF formula (forall (M:nat), (not (((eq nat) (suc M)) zero_zero_nat))) of role axiom named fact_694_Suc__not__Zero % A new axiom: (forall (M:nat), (not (((eq nat) (suc M)) zero_zero_nat))) % FOF formula (forall (Nat:nat), (not (((eq nat) zero_zero_nat) (suc Nat)))) of role axiom named fact_695_nat_Osimps_I2_J % A new axiom: (forall (Nat:nat), (not (((eq nat) zero_zero_nat) (suc Nat)))) % FOF formula (forall (M:nat), (not (((eq nat) zero_zero_nat) (suc M)))) of role axiom named fact_696_Zero__not__Suc % A new axiom: (forall (M:nat), (not (((eq nat) zero_zero_nat) (suc M)))) % FOF formula (((eq nat) bot_bot_nat) zero_zero_nat) of role axiom named fact_697_bot__nat__def % A new axiom: (((eq nat) bot_bot_nat) zero_zero_nat) % FOF formula (forall (N:nat), ((iff ((ord_less_eq_nat N) zero_zero_nat)) (((eq nat) N) zero_zero_nat))) of role axiom named fact_698_le__0__eq % A new axiom: (forall (N:nat), ((iff ((ord_less_eq_nat N) zero_zero_nat)) (((eq nat) N) zero_zero_nat))) % FOF formula (forall (N:nat), ((ord_less_eq_nat zero_zero_nat) N)) of role axiom named fact_699_less__eq__nat_Osimps_I1_J % A new axiom: (forall (N:nat), ((ord_less_eq_nat zero_zero_nat) N)) % FOF formula (forall (X:x_a) (Y:x_a), ((or (((fequal_a X) Y)->False)) (((eq x_a) X) Y))) of role axiom named help_fequal_1_1_fequal_000t__a_T % A new axiom: (forall (X:x_a) (Y:x_a), ((or (((fequal_a X) Y)->False)) (((eq x_a) X) Y))) % FOF formula (forall (X:x_a) (Y:x_a), ((or (not (((eq x_a) X) Y))) ((fequal_a X) Y))) of role axiom named help_fequal_2_1_fequal_000t__a_T % A new axiom: (forall (X:x_a) (Y:x_a), ((or (not (((eq x_a) X) Y))) ((fequal_a X) Y))) % FOF formula (forall (X:nat) (Y:nat), ((or (((fequal_nat X) Y)->False)) (((eq nat) X) Y))) of role axiom named help_fequal_1_1_fequal_000tc__Nat__Onat_T % A new axiom: (forall (X:nat) (Y:nat), ((or (((fequal_nat X) Y)->False)) (((eq nat) X) Y))) % FOF formula (forall (X:nat) (Y:nat), ((or (not (((eq nat) X) Y))) ((fequal_nat X) Y))) of role axiom named help_fequal_2_1_fequal_000tc__Nat__Onat_T % A new axiom: (forall (X:nat) (Y:nat), ((or (not (((eq nat) X) Y))) ((fequal_nat X) Y))) % FOF formula (forall (X:pname) (Y:pname), ((or (((fequal_pname X) Y)->False)) (((eq pname) X) Y))) of role axiom named help_fequal_1_1_fequal_000tc__Com__Opname_T % A new axiom: (forall (X:pname) (Y:pname), ((or (((fequal_pname X) Y)->False)) (((eq pname) X) Y))) % FOF formula (forall (X:pname) (Y:pname), ((or (not (((eq pname) X) Y))) ((fequal_pname X) Y))) of role axiom named help_fequal_2_1_fequal_000tc__Com__Opname_T % A new axiom: (forall (X:pname) (Y:pname), ((or (not (((eq pname) X) Y))) ((fequal_pname X) Y))) % FOF formula (forall (X:(x_a->Prop)) (Y:(x_a->Prop)), ((or (((fequal_a_o X) Y)->False)) (((eq (x_a->Prop)) X) Y))) of role axiom named help_fequal_1_1_fequal_000_062_It__a_M_Eo_J_T % A new axiom: (forall (X:(x_a->Prop)) (Y:(x_a->Prop)), ((or (((fequal_a_o X) Y)->False)) (((eq (x_a->Prop)) X) Y))) % FOF formula (forall (X:(x_a->Prop)) (Y:(x_a->Prop)), ((or (not (((eq (x_a->Prop)) X) Y))) ((fequal_a_o X) Y))) of role axiom named help_fequal_2_1_fequal_000_062_It__a_M_Eo_J_T % A new axiom: (forall (X:(x_a->Prop)) (Y:(x_a->Prop)), ((or (not (((eq (x_a->Prop)) X) Y))) ((fequal_a_o X) Y))) % FOF formula (forall (X:(nat->Prop)) (Y:(nat->Prop)), ((or (((fequal_nat_o X) Y)->False)) (((eq (nat->Prop)) X) Y))) of role axiom named help_fequal_1_1_fequal_000_062_Itc__Nat__Onat_M_Eo_J_T % A new axiom: (forall (X:(nat->Prop)) (Y:(nat->Prop)), ((or (((fequal_nat_o X) Y)->False)) (((eq (nat->Prop)) X) Y))) % FOF formula (forall (X:(nat->Prop)) (Y:(nat->Prop)), ((or (not (((eq (nat->Prop)) X) Y))) ((fequal_nat_o X) Y))) of role axiom named help_fequal_2_1_fequal_000_062_Itc__Nat__Onat_M_Eo_J_T % A new axiom: (forall (X:(nat->Prop)) (Y:(nat->Prop)), ((or (not (((eq (nat->Prop)) X) Y))) ((fequal_nat_o X) Y))) % FOF formula (forall (X:(pname->Prop)) (Y:(pname->Prop)), ((or (((fequal_pname_o X) Y)->False)) (((eq (pname->Prop)) X) Y))) of role axiom named help_fequal_1_1_fequal_000_062_Itc__Com__Opname_M_Eo_J_T % A new axiom: (forall (X:(pname->Prop)) (Y:(pname->Prop)), ((or (((fequal_pname_o X) Y)->False)) (((eq (pname->Prop)) X) Y))) % FOF formula (forall (X:(pname->Prop)) (Y:(pname->Prop)), ((or (not (((eq (pname->Prop)) X) Y))) ((fequal_pname_o X) Y))) of role axiom named help_fequal_2_1_fequal_000_062_Itc__Com__Opname_M_Eo_J_T % A new axiom: (forall (X:(pname->Prop)) (Y:(pname->Prop)), ((or (not (((eq (pname->Prop)) X) Y))) ((fequal_pname_o X) Y))) % FOF formula (finite_finite_pname u) of role hypothesis named conj_0 % A new axiom: (finite_finite_pname u) % FOF formula ((ord_less_eq_a_o g) ((image_pname_a mgt_call) u)) of role hypothesis named conj_1 % A new axiom: ((ord_less_eq_a_o g) ((image_pname_a mgt_call) u)) % FOF formula ((ord_less_eq_nat (suc na)) (finite_card_a ((image_pname_a mgt_call) u))) of role hypothesis named conj_2 % A new axiom: ((ord_less_eq_nat (suc na)) (finite_card_a ((image_pname_a mgt_call) u))) % FOF formula (((eq nat) (finite_card_a g)) ((minus_minus_nat (finite_card_a ((image_pname_a mgt_call) u))) (suc na))) of role hypothesis named conj_3 % A new axiom: (((eq nat) (finite_card_a g)) ((minus_minus_nat (finite_card_a ((image_pname_a mgt_call) u))) (suc na))) % FOF formula ((member_pname pn) u) of role hypothesis named conj_4 % A new axiom: ((member_pname pn) u) % FOF formula (((member_a (mgt_call pn)) g)->False) of role hypothesis named conj_5 % A new axiom: (((member_a (mgt_call pn)) g)->False) % FOF formula ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((image_pname_a mgt_call) u)) of role conjecture named conj_6 % Conjecture to prove = ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((image_pname_a mgt_call) u)):Prop % Parameter x_a_DUMMY:x_a. % Parameter com_DUMMY:com. % Parameter option_com_DUMMY:option_com. % We need to prove ['((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((image_pname_a mgt_call) u))'] % Parameter x_a:Type. % Parameter com:Type. % Parameter pname:Type. % Parameter nat:Type. % Parameter option_com:Type. % Parameter body:(pname->option_com). % Parameter finite_card_a_o:(((x_a->Prop)->Prop)->nat). % Parameter finite_card_pname_o:(((pname->Prop)->Prop)->nat). % Parameter finite_card_nat_o:(((nat->Prop)->Prop)->nat). % Parameter finite_card_a:((x_a->Prop)->nat). % Parameter finite_card_pname:((pname->Prop)->nat). % Parameter finite_card_nat:((nat->Prop)->nat). % Parameter finite_finite_a_o_o:((((x_a->Prop)->Prop)->Prop)->Prop). % Parameter finite1066544169me_o_o:((((pname->Prop)->Prop)->Prop)->Prop). % Parameter finite1676163439at_o_o:((((nat->Prop)->Prop)->Prop)->Prop). % Parameter finite_finite_a_o:(((x_a->Prop)->Prop)->Prop). % Parameter finite297249702name_o:(((pname->Prop)->Prop)->Prop). % Parameter finite_finite_nat_o:(((nat->Prop)->Prop)->Prop). % Parameter finite_finite_a:((x_a->Prop)->Prop). % Parameter finite_finite_pname:((pname->Prop)->Prop). % Parameter finite_finite_nat:((nat->Prop)->Prop). % Parameter minus_minus_nat:(nat->(nat->nat)). % Parameter one_one_nat:nat. % Parameter plus_plus_nat:(nat->(nat->nat)). % Parameter zero_zero_nat:nat. % Parameter suc:(nat->nat). % Parameter nat_case_o:(Prop->((nat->Prop)->(nat->Prop))). % Parameter the_com:(option_com->com). % Parameter bot_bot_a_o_o:((x_a->Prop)->Prop). % Parameter bot_bot_pname_o_o:((pname->Prop)->Prop). % Parameter bot_bot_nat_o_o:((nat->Prop)->Prop). % Parameter bot_bot_a_o:(x_a->Prop). % Parameter bot_bot_pname_o:(pname->Prop). % Parameter bot_bot_nat_o:(nat->Prop). % Parameter bot_bot_o:Prop. % Parameter bot_bot_nat:nat. % Parameter ord_less_nat:(nat->(nat->Prop)). % Parameter ord_less_eq_a_o_o:(((x_a->Prop)->Prop)->(((x_a->Prop)->Prop)->Prop)). % Parameter ord_le1205211808me_o_o:(((pname->Prop)->Prop)->(((pname->Prop)->Prop)->Prop)). % Parameter ord_less_eq_nat_o_o:(((nat->Prop)->Prop)->(((nat->Prop)->Prop)->Prop)). % Parameter ord_less_eq_a_o:((x_a->Prop)->((x_a->Prop)->Prop)). % Parameter ord_less_eq_pname_o:((pname->Prop)->((pname->Prop)->Prop)). % Parameter ord_less_eq_nat_o:((nat->Prop)->((nat->Prop)->Prop)). % Parameter ord_less_eq_o:(Prop->(Prop->Prop)). % Parameter ord_less_eq_nat:(nat->(nat->Prop)). % Parameter collect_a_o_o:((((x_a->Prop)->Prop)->Prop)->(((x_a->Prop)->Prop)->Prop)). % Parameter collect_pname_o_o:((((pname->Prop)->Prop)->Prop)->(((pname->Prop)->Prop)->Prop)). % Parameter collect_nat_o_o:((((nat->Prop)->Prop)->Prop)->(((nat->Prop)->Prop)->Prop)). % Parameter collect_a_o:(((x_a->Prop)->Prop)->((x_a->Prop)->Prop)). % Parameter collect_pname_o:(((pname->Prop)->Prop)->((pname->Prop)->Prop)). % Parameter collect_nat_o:(((nat->Prop)->Prop)->((nat->Prop)->Prop)). % Parameter collect_a:((x_a->Prop)->(x_a->Prop)). % Parameter collect_pname:((pname->Prop)->(pname->Prop)). % Parameter collect_nat:((nat->Prop)->(nat->Prop)). % Parameter image_a_o_a:(((x_a->Prop)->x_a)->(((x_a->Prop)->Prop)->(x_a->Prop))). % Parameter image_a_o_pname:(((x_a->Prop)->pname)->(((x_a->Prop)->Prop)->(pname->Prop))). % Parameter image_a_o_nat:(((x_a->Prop)->nat)->(((x_a->Prop)->Prop)->(nat->Prop))). % Parameter image_pname_o_a:(((pname->Prop)->x_a)->(((pname->Prop)->Prop)->(x_a->Prop))). % Parameter image_pname_o_pname:(((pname->Prop)->pname)->(((pname->Prop)->Prop)->(pname->Prop))). % Parameter image_pname_o_nat:(((pname->Prop)->nat)->(((pname->Prop)->Prop)->(nat->Prop))). % Parameter image_nat_o_a:(((nat->Prop)->x_a)->(((nat->Prop)->Prop)->(x_a->Prop))). % Parameter image_nat_o_pname:(((nat->Prop)->pname)->(((nat->Prop)->Prop)->(pname->Prop))). % Parameter image_nat_o_nat:(((nat->Prop)->nat)->(((nat->Prop)->Prop)->(nat->Prop))). % Parameter image_a_a_o:((x_a->(x_a->Prop))->((x_a->Prop)->((x_a->Prop)->Prop))). % Parameter image_a_pname_o:((x_a->(pname->Prop))->((x_a->Prop)->((pname->Prop)->Prop))). % Parameter image_a_nat_o:((x_a->(nat->Prop))->((x_a->Prop)->((nat->Prop)->Prop))). % Parameter image_a_a:((x_a->x_a)->((x_a->Prop)->(x_a->Prop))). % Parameter image_a_pname:((x_a->pname)->((x_a->Prop)->(pname->Prop))). % Parameter image_a_nat:((x_a->nat)->((x_a->Prop)->(nat->Prop))). % Parameter image_pname_a_o:((pname->(x_a->Prop))->((pname->Prop)->((x_a->Prop)->Prop))). % Parameter image_pname_pname_o:((pname->(pname->Prop))->((pname->Prop)->((pname->Prop)->Prop))). % Parameter image_pname_nat_o:((pname->(nat->Prop))->((pname->Prop)->((nat->Prop)->Prop))). % Parameter image_pname_a:((pname->x_a)->((pname->Prop)->(x_a->Prop))). % Parameter image_pname_pname:((pname->pname)->((pname->Prop)->(pname->Prop))). % Parameter image_pname_nat:((pname->nat)->((pname->Prop)->(nat->Prop))). % Parameter image_nat_a_o:((nat->(x_a->Prop))->((nat->Prop)->((x_a->Prop)->Prop))). % Parameter image_nat_pname_o:((nat->(pname->Prop))->((nat->Prop)->((pname->Prop)->Prop))). % Parameter image_nat_nat_o:((nat->(nat->Prop))->((nat->Prop)->((nat->Prop)->Prop))). % Parameter image_nat_a:((nat->x_a)->((nat->Prop)->(x_a->Prop))). % Parameter image_nat_pname:((nat->pname)->((nat->Prop)->(pname->Prop))). % Parameter image_nat_nat:((nat->nat)->((nat->Prop)->(nat->Prop))). % Parameter insert_a_o:((x_a->Prop)->(((x_a->Prop)->Prop)->((x_a->Prop)->Prop))). % Parameter insert_pname_o:((pname->Prop)->(((pname->Prop)->Prop)->((pname->Prop)->Prop))). % Parameter insert_nat_o:((nat->Prop)->(((nat->Prop)->Prop)->((nat->Prop)->Prop))). % Parameter insert_a:(x_a->((x_a->Prop)->(x_a->Prop))). % Parameter insert_pname:(pname->((pname->Prop)->(pname->Prop))). % Parameter insert_nat:(nat->((nat->Prop)->(nat->Prop))). % Parameter fequal_a_o:((x_a->Prop)->((x_a->Prop)->Prop)). % Parameter fequal_pname_o:((pname->Prop)->((pname->Prop)->Prop)). % Parameter fequal_nat_o:((nat->Prop)->((nat->Prop)->Prop)). % Parameter fequal_a:(x_a->(x_a->Prop)). % Parameter fequal_pname:(pname->(pname->Prop)). % Parameter fequal_nat:(nat->(nat->Prop)). % Parameter member_a_o:((x_a->Prop)->(((x_a->Prop)->Prop)->Prop)). % Parameter member_pname_o:((pname->Prop)->(((pname->Prop)->Prop)->Prop)). % Parameter member_nat_o:((nat->Prop)->(((nat->Prop)->Prop)->Prop)). % Parameter member_a:(x_a->((x_a->Prop)->Prop)). % Parameter member_pname:(pname->((pname->Prop)->Prop)). % Parameter member_nat:(nat->((nat->Prop)->Prop)). % Parameter g:(x_a->Prop). % Parameter p:((x_a->Prop)->((x_a->Prop)->Prop)). % Parameter u:(pname->Prop). % Parameter mgt:(com->x_a). % Parameter mgt_call:(pname->x_a). % Parameter na:nat. % Parameter pn:pname. % Parameter wt:(com->Prop). % Axiom fact_0_assms_I1_J:(forall (Ts:(x_a->Prop)) (G:(x_a->Prop)), (((ord_less_eq_a_o Ts) G)->((p G) Ts))). % Axiom fact_1_finite__Collect__subsets:(forall (A_92:((nat->Prop)->Prop)), ((finite_finite_nat_o A_92)->(finite1676163439at_o_o (collect_nat_o_o (fun (B_47:((nat->Prop)->Prop))=> ((ord_less_eq_nat_o_o B_47) A_92)))))). % Axiom fact_2_finite__Collect__subsets:(forall (A_92:((pname->Prop)->Prop)), ((finite297249702name_o A_92)->(finite1066544169me_o_o (collect_pname_o_o (fun (B_47:((pname->Prop)->Prop))=> ((ord_le1205211808me_o_o B_47) A_92)))))). % Axiom fact_3_finite__Collect__subsets:(forall (A_92:((x_a->Prop)->Prop)), ((finite_finite_a_o A_92)->(finite_finite_a_o_o (collect_a_o_o (fun (B_47:((x_a->Prop)->Prop))=> ((ord_less_eq_a_o_o B_47) A_92)))))). % Axiom fact_4_finite__Collect__subsets:(forall (A_92:(x_a->Prop)), ((finite_finite_a A_92)->(finite_finite_a_o (collect_a_o (fun (B_47:(x_a->Prop))=> ((ord_less_eq_a_o B_47) A_92)))))). % Axiom fact_5_finite__Collect__subsets:(forall (A_92:(pname->Prop)), ((finite_finite_pname A_92)->(finite297249702name_o (collect_pname_o (fun (B_47:(pname->Prop))=> ((ord_less_eq_pname_o B_47) A_92)))))). % Axiom fact_6_finite__Collect__subsets:(forall (A_92:(nat->Prop)), ((finite_finite_nat A_92)->(finite_finite_nat_o (collect_nat_o (fun (B_47:(nat->Prop))=> ((ord_less_eq_nat_o B_47) A_92)))))). % Axiom fact_7_finite__imageI:(forall (H:(pname->(nat->Prop))) (F_25:(pname->Prop)), ((finite_finite_pname F_25)->(finite_finite_nat_o ((image_pname_nat_o H) F_25)))). % Axiom fact_8_finite__imageI:(forall (H:(pname->(pname->Prop))) (F_25:(pname->Prop)), ((finite_finite_pname F_25)->(finite297249702name_o ((image_pname_pname_o H) F_25)))). % Axiom fact_9_finite__imageI:(forall (H:(pname->(x_a->Prop))) (F_25:(pname->Prop)), ((finite_finite_pname F_25)->(finite_finite_a_o ((image_pname_a_o H) F_25)))). % Axiom fact_10_finite__imageI:(forall (H:(nat->x_a)) (F_25:(nat->Prop)), ((finite_finite_nat F_25)->(finite_finite_a ((image_nat_a H) F_25)))). % Axiom fact_11_finite__imageI:(forall (H:(nat->(nat->Prop))) (F_25:(nat->Prop)), ((finite_finite_nat F_25)->(finite_finite_nat_o ((image_nat_nat_o H) F_25)))). % Axiom fact_12_finite__imageI:(forall (H:(nat->(pname->Prop))) (F_25:(nat->Prop)), ((finite_finite_nat F_25)->(finite297249702name_o ((image_nat_pname_o H) F_25)))). % Axiom fact_13_finite__imageI:(forall (H:(nat->(x_a->Prop))) (F_25:(nat->Prop)), ((finite_finite_nat F_25)->(finite_finite_a_o ((image_nat_a_o H) F_25)))). % Axiom fact_14_finite__imageI:(forall (H:(x_a->pname)) (F_25:(x_a->Prop)), ((finite_finite_a F_25)->(finite_finite_pname ((image_a_pname H) F_25)))). % Axiom fact_15_finite__imageI:(forall (H:((nat->Prop)->pname)) (F_25:((nat->Prop)->Prop)), ((finite_finite_nat_o F_25)->(finite_finite_pname ((image_nat_o_pname H) F_25)))). % Axiom fact_16_finite__imageI:(forall (H:((pname->Prop)->pname)) (F_25:((pname->Prop)->Prop)), ((finite297249702name_o F_25)->(finite_finite_pname ((image_pname_o_pname H) F_25)))). % Axiom fact_17_finite__imageI:(forall (H:((x_a->Prop)->pname)) (F_25:((x_a->Prop)->Prop)), ((finite_finite_a_o F_25)->(finite_finite_pname ((image_a_o_pname H) F_25)))). % Axiom fact_18_finite__imageI:(forall (H:(x_a->nat)) (F_25:(x_a->Prop)), ((finite_finite_a F_25)->(finite_finite_nat ((image_a_nat H) F_25)))). % Axiom fact_19_finite__imageI:(forall (H:((nat->Prop)->nat)) (F_25:((nat->Prop)->Prop)), ((finite_finite_nat_o F_25)->(finite_finite_nat ((image_nat_o_nat H) F_25)))). % Axiom fact_20_finite__imageI:(forall (H:((pname->Prop)->nat)) (F_25:((pname->Prop)->Prop)), ((finite297249702name_o F_25)->(finite_finite_nat ((image_pname_o_nat H) F_25)))). % Axiom fact_21_finite__imageI:(forall (H:((x_a->Prop)->nat)) (F_25:((x_a->Prop)->Prop)), ((finite_finite_a_o F_25)->(finite_finite_nat ((image_a_o_nat H) F_25)))). % Axiom fact_22_finite__imageI:(forall (H:(pname->x_a)) (F_25:(pname->Prop)), ((finite_finite_pname F_25)->(finite_finite_a ((image_pname_a H) F_25)))). % Axiom fact_23_finite_OinsertI:(forall (A_91:(nat->Prop)) (A_90:((nat->Prop)->Prop)), ((finite_finite_nat_o A_90)->(finite_finite_nat_o ((insert_nat_o A_91) A_90)))). % Axiom fact_24_finite_OinsertI:(forall (A_91:(pname->Prop)) (A_90:((pname->Prop)->Prop)), ((finite297249702name_o A_90)->(finite297249702name_o ((insert_pname_o A_91) A_90)))). % Axiom fact_25_finite_OinsertI:(forall (A_91:(x_a->Prop)) (A_90:((x_a->Prop)->Prop)), ((finite_finite_a_o A_90)->(finite_finite_a_o ((insert_a_o A_91) A_90)))). % Axiom fact_26_finite_OinsertI:(forall (A_91:pname) (A_90:(pname->Prop)), ((finite_finite_pname A_90)->(finite_finite_pname ((insert_pname A_91) A_90)))). % Axiom fact_27_finite_OinsertI:(forall (A_91:nat) (A_90:(nat->Prop)), ((finite_finite_nat A_90)->(finite_finite_nat ((insert_nat A_91) A_90)))). % Axiom fact_28_finite_OinsertI:(forall (A_91:x_a) (A_90:(x_a->Prop)), ((finite_finite_a A_90)->(finite_finite_a ((insert_a A_91) A_90)))). % Axiom fact_29_card__image__le:(forall (F_24:(pname->pname)) (A_89:(pname->Prop)), ((finite_finite_pname A_89)->((ord_less_eq_nat (finite_card_pname ((image_pname_pname F_24) A_89))) (finite_card_pname A_89)))). % Axiom fact_30_card__image__le:(forall (F_24:(x_a->x_a)) (A_89:(x_a->Prop)), ((finite_finite_a A_89)->((ord_less_eq_nat (finite_card_a ((image_a_a F_24) A_89))) (finite_card_a A_89)))). % Axiom fact_31_card__image__le:(forall (F_24:((nat->Prop)->x_a)) (A_89:((nat->Prop)->Prop)), ((finite_finite_nat_o A_89)->((ord_less_eq_nat (finite_card_a ((image_nat_o_a F_24) A_89))) (finite_card_nat_o A_89)))). % Axiom fact_32_card__image__le:(forall (F_24:((pname->Prop)->x_a)) (A_89:((pname->Prop)->Prop)), ((finite297249702name_o A_89)->((ord_less_eq_nat (finite_card_a ((image_pname_o_a F_24) A_89))) (finite_card_pname_o A_89)))). % Axiom fact_33_card__image__le:(forall (F_24:((x_a->Prop)->x_a)) (A_89:((x_a->Prop)->Prop)), ((finite_finite_a_o A_89)->((ord_less_eq_nat (finite_card_a ((image_a_o_a F_24) A_89))) (finite_card_a_o A_89)))). % Axiom fact_34_card__image__le:(forall (F_24:(pname->nat)) (A_89:(pname->Prop)), ((finite_finite_pname A_89)->((ord_less_eq_nat (finite_card_nat ((image_pname_nat F_24) A_89))) (finite_card_pname A_89)))). % Axiom fact_35_card__image__le:(forall (F_24:(x_a->nat)) (A_89:(x_a->Prop)), ((finite_finite_a A_89)->((ord_less_eq_nat (finite_card_nat ((image_a_nat F_24) A_89))) (finite_card_a A_89)))). % Axiom fact_36_card__image__le:(forall (F_24:((nat->Prop)->nat)) (A_89:((nat->Prop)->Prop)), ((finite_finite_nat_o A_89)->((ord_less_eq_nat (finite_card_nat ((image_nat_o_nat F_24) A_89))) (finite_card_nat_o A_89)))). % Axiom fact_37_card__image__le:(forall (F_24:((pname->Prop)->nat)) (A_89:((pname->Prop)->Prop)), ((finite297249702name_o A_89)->((ord_less_eq_nat (finite_card_nat ((image_pname_o_nat F_24) A_89))) (finite_card_pname_o A_89)))). % Axiom fact_38_card__image__le:(forall (F_24:((x_a->Prop)->nat)) (A_89:((x_a->Prop)->Prop)), ((finite_finite_a_o A_89)->((ord_less_eq_nat (finite_card_nat ((image_a_o_nat F_24) A_89))) (finite_card_a_o A_89)))). % Axiom fact_39_card__image__le:(forall (F_24:(x_a->pname)) (A_89:(x_a->Prop)), ((finite_finite_a A_89)->((ord_less_eq_nat (finite_card_pname ((image_a_pname F_24) A_89))) (finite_card_a A_89)))). % Axiom fact_40_card__image__le:(forall (F_24:(nat->pname)) (A_89:(nat->Prop)), ((finite_finite_nat A_89)->((ord_less_eq_nat (finite_card_pname ((image_nat_pname F_24) A_89))) (finite_card_nat A_89)))). % Axiom fact_41_card__image__le:(forall (F_24:(pname->x_a)) (A_89:(pname->Prop)), ((finite_finite_pname A_89)->((ord_less_eq_nat (finite_card_a ((image_pname_a F_24) A_89))) (finite_card_pname A_89)))). % Axiom fact_42_card__mono:(forall (A_88:((nat->Prop)->Prop)) (B_46:((nat->Prop)->Prop)), ((finite_finite_nat_o B_46)->(((ord_less_eq_nat_o_o A_88) B_46)->((ord_less_eq_nat (finite_card_nat_o A_88)) (finite_card_nat_o B_46))))). % Axiom fact_43_card__mono:(forall (A_88:((pname->Prop)->Prop)) (B_46:((pname->Prop)->Prop)), ((finite297249702name_o B_46)->(((ord_le1205211808me_o_o A_88) B_46)->((ord_less_eq_nat (finite_card_pname_o A_88)) (finite_card_pname_o B_46))))). % Axiom fact_44_card__mono:(forall (A_88:((x_a->Prop)->Prop)) (B_46:((x_a->Prop)->Prop)), ((finite_finite_a_o B_46)->(((ord_less_eq_a_o_o A_88) B_46)->((ord_less_eq_nat (finite_card_a_o A_88)) (finite_card_a_o B_46))))). % Axiom fact_45_card__mono:(forall (A_88:(pname->Prop)) (B_46:(pname->Prop)), ((finite_finite_pname B_46)->(((ord_less_eq_pname_o A_88) B_46)->((ord_less_eq_nat (finite_card_pname A_88)) (finite_card_pname B_46))))). % Axiom fact_46_card__mono:(forall (A_88:(x_a->Prop)) (B_46:(x_a->Prop)), ((finite_finite_a B_46)->(((ord_less_eq_a_o A_88) B_46)->((ord_less_eq_nat (finite_card_a A_88)) (finite_card_a B_46))))). % Axiom fact_47_card__mono:(forall (A_88:(nat->Prop)) (B_46:(nat->Prop)), ((finite_finite_nat B_46)->(((ord_less_eq_nat_o A_88) B_46)->((ord_less_eq_nat (finite_card_nat A_88)) (finite_card_nat B_46))))). % Axiom fact_48_card__seteq:(forall (A_87:((nat->Prop)->Prop)) (B_45:((nat->Prop)->Prop)), ((finite_finite_nat_o B_45)->(((ord_less_eq_nat_o_o A_87) B_45)->(((ord_less_eq_nat (finite_card_nat_o B_45)) (finite_card_nat_o A_87))->(((eq ((nat->Prop)->Prop)) A_87) B_45))))). % Axiom fact_49_card__seteq:(forall (A_87:((pname->Prop)->Prop)) (B_45:((pname->Prop)->Prop)), ((finite297249702name_o B_45)->(((ord_le1205211808me_o_o A_87) B_45)->(((ord_less_eq_nat (finite_card_pname_o B_45)) (finite_card_pname_o A_87))->(((eq ((pname->Prop)->Prop)) A_87) B_45))))). % Axiom fact_50_card__seteq:(forall (A_87:((x_a->Prop)->Prop)) (B_45:((x_a->Prop)->Prop)), ((finite_finite_a_o B_45)->(((ord_less_eq_a_o_o A_87) B_45)->(((ord_less_eq_nat (finite_card_a_o B_45)) (finite_card_a_o A_87))->(((eq ((x_a->Prop)->Prop)) A_87) B_45))))). % Axiom fact_51_card__seteq:(forall (A_87:(pname->Prop)) (B_45:(pname->Prop)), ((finite_finite_pname B_45)->(((ord_less_eq_pname_o A_87) B_45)->(((ord_less_eq_nat (finite_card_pname B_45)) (finite_card_pname A_87))->(((eq (pname->Prop)) A_87) B_45))))). % Axiom fact_52_card__seteq:(forall (A_87:(x_a->Prop)) (B_45:(x_a->Prop)), ((finite_finite_a B_45)->(((ord_less_eq_a_o A_87) B_45)->(((ord_less_eq_nat (finite_card_a B_45)) (finite_card_a A_87))->(((eq (x_a->Prop)) A_87) B_45))))). % Axiom fact_53_card__seteq:(forall (A_87:(nat->Prop)) (B_45:(nat->Prop)), ((finite_finite_nat B_45)->(((ord_less_eq_nat_o A_87) B_45)->(((ord_less_eq_nat (finite_card_nat B_45)) (finite_card_nat A_87))->(((eq (nat->Prop)) A_87) B_45))))). % Axiom fact_54_card__insert__le:(forall (X_33:(nat->Prop)) (A_86:((nat->Prop)->Prop)), ((finite_finite_nat_o A_86)->((ord_less_eq_nat (finite_card_nat_o A_86)) (finite_card_nat_o ((insert_nat_o X_33) A_86))))). % Axiom fact_55_card__insert__le:(forall (X_33:(pname->Prop)) (A_86:((pname->Prop)->Prop)), ((finite297249702name_o A_86)->((ord_less_eq_nat (finite_card_pname_o A_86)) (finite_card_pname_o ((insert_pname_o X_33) A_86))))). % Axiom fact_56_card__insert__le:(forall (X_33:(x_a->Prop)) (A_86:((x_a->Prop)->Prop)), ((finite_finite_a_o A_86)->((ord_less_eq_nat (finite_card_a_o A_86)) (finite_card_a_o ((insert_a_o X_33) A_86))))). % Axiom fact_57_card__insert__le:(forall (X_33:pname) (A_86:(pname->Prop)), ((finite_finite_pname A_86)->((ord_less_eq_nat (finite_card_pname A_86)) (finite_card_pname ((insert_pname X_33) A_86))))). % Axiom fact_58_card__insert__le:(forall (X_33:nat) (A_86:(nat->Prop)), ((finite_finite_nat A_86)->((ord_less_eq_nat (finite_card_nat A_86)) (finite_card_nat ((insert_nat X_33) A_86))))). % Axiom fact_59_card__insert__le:(forall (X_33:x_a) (A_86:(x_a->Prop)), ((finite_finite_a A_86)->((ord_less_eq_nat (finite_card_a A_86)) (finite_card_a ((insert_a X_33) A_86))))). % Axiom fact_60_card__insert__if:(forall (X_32:(nat->Prop)) (A_85:((nat->Prop)->Prop)), ((finite_finite_nat_o A_85)->((and (((member_nat_o X_32) A_85)->(((eq nat) (finite_card_nat_o ((insert_nat_o X_32) A_85))) (finite_card_nat_o A_85)))) ((((member_nat_o X_32) A_85)->False)->(((eq nat) (finite_card_nat_o ((insert_nat_o X_32) A_85))) (suc (finite_card_nat_o A_85))))))). % Axiom fact_61_card__insert__if:(forall (X_32:(pname->Prop)) (A_85:((pname->Prop)->Prop)), ((finite297249702name_o A_85)->((and (((member_pname_o X_32) A_85)->(((eq nat) (finite_card_pname_o ((insert_pname_o X_32) A_85))) (finite_card_pname_o A_85)))) ((((member_pname_o X_32) A_85)->False)->(((eq nat) (finite_card_pname_o ((insert_pname_o X_32) A_85))) (suc (finite_card_pname_o A_85))))))). % Axiom fact_62_card__insert__if:(forall (X_32:(x_a->Prop)) (A_85:((x_a->Prop)->Prop)), ((finite_finite_a_o A_85)->((and (((member_a_o X_32) A_85)->(((eq nat) (finite_card_a_o ((insert_a_o X_32) A_85))) (finite_card_a_o A_85)))) ((((member_a_o X_32) A_85)->False)->(((eq nat) (finite_card_a_o ((insert_a_o X_32) A_85))) (suc (finite_card_a_o A_85))))))). % Axiom fact_63_card__insert__if:(forall (X_32:nat) (A_85:(nat->Prop)), ((finite_finite_nat A_85)->((and (((member_nat X_32) A_85)->(((eq nat) (finite_card_nat ((insert_nat X_32) A_85))) (finite_card_nat A_85)))) ((((member_nat X_32) A_85)->False)->(((eq nat) (finite_card_nat ((insert_nat X_32) A_85))) (suc (finite_card_nat A_85))))))). % Axiom fact_64_card__insert__if:(forall (X_32:pname) (A_85:(pname->Prop)), ((finite_finite_pname A_85)->((and (((member_pname X_32) A_85)->(((eq nat) (finite_card_pname ((insert_pname X_32) A_85))) (finite_card_pname A_85)))) ((((member_pname X_32) A_85)->False)->(((eq nat) (finite_card_pname ((insert_pname X_32) A_85))) (suc (finite_card_pname A_85))))))). % Axiom fact_65_card__insert__if:(forall (X_32:x_a) (A_85:(x_a->Prop)), ((finite_finite_a A_85)->((and (((member_a X_32) A_85)->(((eq nat) (finite_card_a ((insert_a X_32) A_85))) (finite_card_a A_85)))) ((((member_a X_32) A_85)->False)->(((eq nat) (finite_card_a ((insert_a X_32) A_85))) (suc (finite_card_a A_85))))))). % Axiom fact_66_card__insert__disjoint:(forall (X_31:(nat->Prop)) (A_84:((nat->Prop)->Prop)), ((finite_finite_nat_o A_84)->((((member_nat_o X_31) A_84)->False)->(((eq nat) (finite_card_nat_o ((insert_nat_o X_31) A_84))) (suc (finite_card_nat_o A_84)))))). % Axiom fact_67_card__insert__disjoint:(forall (X_31:(pname->Prop)) (A_84:((pname->Prop)->Prop)), ((finite297249702name_o A_84)->((((member_pname_o X_31) A_84)->False)->(((eq nat) (finite_card_pname_o ((insert_pname_o X_31) A_84))) (suc (finite_card_pname_o A_84)))))). % Axiom fact_68_card__insert__disjoint:(forall (X_31:(x_a->Prop)) (A_84:((x_a->Prop)->Prop)), ((finite_finite_a_o A_84)->((((member_a_o X_31) A_84)->False)->(((eq nat) (finite_card_a_o ((insert_a_o X_31) A_84))) (suc (finite_card_a_o A_84)))))). % Axiom fact_69_card__insert__disjoint:(forall (X_31:nat) (A_84:(nat->Prop)), ((finite_finite_nat A_84)->((((member_nat X_31) A_84)->False)->(((eq nat) (finite_card_nat ((insert_nat X_31) A_84))) (suc (finite_card_nat A_84)))))). % Axiom fact_70_card__insert__disjoint:(forall (X_31:pname) (A_84:(pname->Prop)), ((finite_finite_pname A_84)->((((member_pname X_31) A_84)->False)->(((eq nat) (finite_card_pname ((insert_pname X_31) A_84))) (suc (finite_card_pname A_84)))))). % Axiom fact_71_card__insert__disjoint:(forall (X_31:x_a) (A_84:(x_a->Prop)), ((finite_finite_a A_84)->((((member_a X_31) A_84)->False)->(((eq nat) (finite_card_a ((insert_a X_31) A_84))) (suc (finite_card_a A_84)))))). % Axiom fact_72_finite__Collect__conjI:(forall (Q_1:(x_a->Prop)) (P_10:(x_a->Prop)), (((or (finite_finite_a (collect_a P_10))) (finite_finite_a (collect_a Q_1)))->(finite_finite_a (collect_a (fun (X_1:x_a)=> ((and (P_10 X_1)) (Q_1 X_1))))))). % Axiom fact_73_finite__Collect__conjI:(forall (Q_1:((nat->Prop)->Prop)) (P_10:((nat->Prop)->Prop)), (((or (finite_finite_nat_o (collect_nat_o P_10))) (finite_finite_nat_o (collect_nat_o Q_1)))->(finite_finite_nat_o (collect_nat_o (fun (X_1:(nat->Prop))=> ((and (P_10 X_1)) (Q_1 X_1))))))). % Axiom fact_74_finite__Collect__conjI:(forall (Q_1:((pname->Prop)->Prop)) (P_10:((pname->Prop)->Prop)), (((or (finite297249702name_o (collect_pname_o P_10))) (finite297249702name_o (collect_pname_o Q_1)))->(finite297249702name_o (collect_pname_o (fun (X_1:(pname->Prop))=> ((and (P_10 X_1)) (Q_1 X_1))))))). % Axiom fact_75_finite__Collect__conjI:(forall (Q_1:((x_a->Prop)->Prop)) (P_10:((x_a->Prop)->Prop)), (((or (finite_finite_a_o (collect_a_o P_10))) (finite_finite_a_o (collect_a_o Q_1)))->(finite_finite_a_o (collect_a_o (fun (X_1:(x_a->Prop))=> ((and (P_10 X_1)) (Q_1 X_1))))))). % Axiom fact_76_finite__Collect__conjI:(forall (Q_1:(pname->Prop)) (P_10:(pname->Prop)), (((or (finite_finite_pname (collect_pname P_10))) (finite_finite_pname (collect_pname Q_1)))->(finite_finite_pname (collect_pname (fun (X_1:pname)=> ((and (P_10 X_1)) (Q_1 X_1))))))). % Axiom fact_77_finite__Collect__conjI:(forall (Q_1:(nat->Prop)) (P_10:(nat->Prop)), (((or (finite_finite_nat (collect_nat P_10))) (finite_finite_nat (collect_nat Q_1)))->(finite_finite_nat (collect_nat (fun (X_1:nat)=> ((and (P_10 X_1)) (Q_1 X_1))))))). % Axiom fact_78_Suc__diff__le:(forall (N:nat) (M:nat), (((ord_less_eq_nat N) M)->(((eq nat) ((minus_minus_nat (suc M)) N)) (suc ((minus_minus_nat M) N))))). % Axiom fact_79_finite__Collect__le__nat:(forall (K:nat), (finite_finite_nat (collect_nat (fun (N_2:nat)=> ((ord_less_eq_nat N_2) K))))). % Axiom fact_80_card__Collect__le__nat:(forall (N:nat), (((eq nat) (finite_card_nat (collect_nat (fun (I_1:nat)=> ((ord_less_eq_nat I_1) N))))) (suc N))). % Axiom fact_81_Suc__inject:(forall (X:nat) (Y:nat), ((((eq nat) (suc X)) (suc Y))->(((eq nat) X) Y))). % Axiom fact_82_nat_Oinject:(forall (Nat_2:nat) (Nat:nat), ((iff (((eq nat) (suc Nat_2)) (suc Nat))) (((eq nat) Nat_2) Nat))). % Axiom fact_83_Suc__n__not__n:(forall (N:nat), (not (((eq nat) (suc N)) N))). % Axiom fact_84_n__not__Suc__n:(forall (N:nat), (not (((eq nat) N) (suc N)))). % Axiom fact_85_le__antisym:(forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->(((ord_less_eq_nat N) M)->(((eq nat) M) N)))). % Axiom fact_86_le__trans:(forall (K:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->(((ord_less_eq_nat J_1) K)->((ord_less_eq_nat _TPTP_I) K)))). % Axiom fact_87_eq__imp__le:(forall (M:nat) (N:nat), ((((eq nat) M) N)->((ord_less_eq_nat M) N))). % Axiom fact_88_nat__le__linear:(forall (M:nat) (N:nat), ((or ((ord_less_eq_nat M) N)) ((ord_less_eq_nat N) M))). % Axiom fact_89_le__refl:(forall (N:nat), ((ord_less_eq_nat N) N)). % Axiom fact_90_diff__commute:(forall (_TPTP_I:nat) (J_1:nat) (K:nat), (((eq nat) ((minus_minus_nat ((minus_minus_nat _TPTP_I) J_1)) K)) ((minus_minus_nat ((minus_minus_nat _TPTP_I) K)) J_1))). % Axiom fact_91_finite__Collect__disjI:(forall (P_9:(x_a->Prop)) (Q:(x_a->Prop)), ((iff (finite_finite_a (collect_a (fun (X_1:x_a)=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite_finite_a (collect_a P_9))) (finite_finite_a (collect_a Q))))). % Axiom fact_92_finite__Collect__disjI:(forall (P_9:((nat->Prop)->Prop)) (Q:((nat->Prop)->Prop)), ((iff (finite_finite_nat_o (collect_nat_o (fun (X_1:(nat->Prop))=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite_finite_nat_o (collect_nat_o P_9))) (finite_finite_nat_o (collect_nat_o Q))))). % Axiom fact_93_finite__Collect__disjI:(forall (P_9:((pname->Prop)->Prop)) (Q:((pname->Prop)->Prop)), ((iff (finite297249702name_o (collect_pname_o (fun (X_1:(pname->Prop))=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite297249702name_o (collect_pname_o P_9))) (finite297249702name_o (collect_pname_o Q))))). % Axiom fact_94_finite__Collect__disjI:(forall (P_9:((x_a->Prop)->Prop)) (Q:((x_a->Prop)->Prop)), ((iff (finite_finite_a_o (collect_a_o (fun (X_1:(x_a->Prop))=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite_finite_a_o (collect_a_o P_9))) (finite_finite_a_o (collect_a_o Q))))). % Axiom fact_95_finite__Collect__disjI:(forall (P_9:(pname->Prop)) (Q:(pname->Prop)), ((iff (finite_finite_pname (collect_pname (fun (X_1:pname)=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite_finite_pname (collect_pname P_9))) (finite_finite_pname (collect_pname Q))))). % Axiom fact_96_finite__Collect__disjI:(forall (P_9:(nat->Prop)) (Q:(nat->Prop)), ((iff (finite_finite_nat (collect_nat (fun (X_1:nat)=> ((or (P_9 X_1)) (Q X_1)))))) ((and (finite_finite_nat (collect_nat P_9))) (finite_finite_nat (collect_nat Q))))). % Axiom fact_97_finite__insert:(forall (A_83:(nat->Prop)) (A_82:((nat->Prop)->Prop)), ((iff (finite_finite_nat_o ((insert_nat_o A_83) A_82))) (finite_finite_nat_o A_82))). % Axiom fact_98_finite__insert:(forall (A_83:(pname->Prop)) (A_82:((pname->Prop)->Prop)), ((iff (finite297249702name_o ((insert_pname_o A_83) A_82))) (finite297249702name_o A_82))). % Axiom fact_99_finite__insert:(forall (A_83:(x_a->Prop)) (A_82:((x_a->Prop)->Prop)), ((iff (finite_finite_a_o ((insert_a_o A_83) A_82))) (finite_finite_a_o A_82))). % Axiom fact_100_finite__insert:(forall (A_83:pname) (A_82:(pname->Prop)), ((iff (finite_finite_pname ((insert_pname A_83) A_82))) (finite_finite_pname A_82))). % Axiom fact_101_finite__insert:(forall (A_83:nat) (A_82:(nat->Prop)), ((iff (finite_finite_nat ((insert_nat A_83) A_82))) (finite_finite_nat A_82))). % Axiom fact_102_finite__insert:(forall (A_83:x_a) (A_82:(x_a->Prop)), ((iff (finite_finite_a ((insert_a A_83) A_82))) (finite_finite_a A_82))). % Axiom fact_103_finite__subset:(forall (A_81:((nat->Prop)->Prop)) (B_44:((nat->Prop)->Prop)), (((ord_less_eq_nat_o_o A_81) B_44)->((finite_finite_nat_o B_44)->(finite_finite_nat_o A_81)))). % Axiom fact_104_finite__subset:(forall (A_81:((pname->Prop)->Prop)) (B_44:((pname->Prop)->Prop)), (((ord_le1205211808me_o_o A_81) B_44)->((finite297249702name_o B_44)->(finite297249702name_o A_81)))). % Axiom fact_105_finite__subset:(forall (A_81:((x_a->Prop)->Prop)) (B_44:((x_a->Prop)->Prop)), (((ord_less_eq_a_o_o A_81) B_44)->((finite_finite_a_o B_44)->(finite_finite_a_o A_81)))). % Axiom fact_106_finite__subset:(forall (A_81:(x_a->Prop)) (B_44:(x_a->Prop)), (((ord_less_eq_a_o A_81) B_44)->((finite_finite_a B_44)->(finite_finite_a A_81)))). % Axiom fact_107_finite__subset:(forall (A_81:(pname->Prop)) (B_44:(pname->Prop)), (((ord_less_eq_pname_o A_81) B_44)->((finite_finite_pname B_44)->(finite_finite_pname A_81)))). % Axiom fact_108_finite__subset:(forall (A_81:(nat->Prop)) (B_44:(nat->Prop)), (((ord_less_eq_nat_o A_81) B_44)->((finite_finite_nat B_44)->(finite_finite_nat A_81)))). % Axiom fact_109_rev__finite__subset:(forall (A_80:((nat->Prop)->Prop)) (B_43:((nat->Prop)->Prop)), ((finite_finite_nat_o B_43)->(((ord_less_eq_nat_o_o A_80) B_43)->(finite_finite_nat_o A_80)))). % Axiom fact_110_rev__finite__subset:(forall (A_80:((pname->Prop)->Prop)) (B_43:((pname->Prop)->Prop)), ((finite297249702name_o B_43)->(((ord_le1205211808me_o_o A_80) B_43)->(finite297249702name_o A_80)))). % Axiom fact_111_rev__finite__subset:(forall (A_80:((x_a->Prop)->Prop)) (B_43:((x_a->Prop)->Prop)), ((finite_finite_a_o B_43)->(((ord_less_eq_a_o_o A_80) B_43)->(finite_finite_a_o A_80)))). % Axiom fact_112_rev__finite__subset:(forall (A_80:(x_a->Prop)) (B_43:(x_a->Prop)), ((finite_finite_a B_43)->(((ord_less_eq_a_o A_80) B_43)->(finite_finite_a A_80)))). % Axiom fact_113_rev__finite__subset:(forall (A_80:(pname->Prop)) (B_43:(pname->Prop)), ((finite_finite_pname B_43)->(((ord_less_eq_pname_o A_80) B_43)->(finite_finite_pname A_80)))). % Axiom fact_114_rev__finite__subset:(forall (A_80:(nat->Prop)) (B_43:(nat->Prop)), ((finite_finite_nat B_43)->(((ord_less_eq_nat_o A_80) B_43)->(finite_finite_nat A_80)))). % Axiom fact_115_Suc__leD:(forall (M:nat) (N:nat), (((ord_less_eq_nat (suc M)) N)->((ord_less_eq_nat M) N))). % Axiom fact_116_le__SucE:(forall (M:nat) (N:nat), (((ord_less_eq_nat M) (suc N))->((((ord_less_eq_nat M) N)->False)->(((eq nat) M) (suc N))))). % Axiom fact_117_le__SucI:(forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((ord_less_eq_nat M) (suc N)))). % Axiom fact_118_Suc__le__mono:(forall (N:nat) (M:nat), ((iff ((ord_less_eq_nat (suc N)) (suc M))) ((ord_less_eq_nat N) M))). % Axiom fact_119_le__Suc__eq:(forall (M:nat) (N:nat), ((iff ((ord_less_eq_nat M) (suc N))) ((or ((ord_less_eq_nat M) N)) (((eq nat) M) (suc N))))). % Axiom fact_120_not__less__eq__eq:(forall (M:nat) (N:nat), ((iff (((ord_less_eq_nat M) N)->False)) ((ord_less_eq_nat (suc N)) M))). % Axiom fact_121_Suc__n__not__le__n:(forall (N:nat), (((ord_less_eq_nat (suc N)) N)->False)). % Axiom fact_122_Suc__diff__diff:(forall (M:nat) (N:nat) (K:nat), (((eq nat) ((minus_minus_nat ((minus_minus_nat (suc M)) N)) (suc K))) ((minus_minus_nat ((minus_minus_nat M) N)) K))). % Axiom fact_123_diff__Suc__Suc:(forall (M:nat) (N:nat), (((eq nat) ((minus_minus_nat (suc M)) (suc N))) ((minus_minus_nat M) N))). % Axiom fact_124_le__diff__iff:(forall (N:nat) (K:nat) (M:nat), (((ord_less_eq_nat K) M)->(((ord_less_eq_nat K) N)->((iff ((ord_less_eq_nat ((minus_minus_nat M) K)) ((minus_minus_nat N) K))) ((ord_less_eq_nat M) N))))). % Axiom fact_125_Nat_Odiff__diff__eq:(forall (N:nat) (K:nat) (M:nat), (((ord_less_eq_nat K) M)->(((ord_less_eq_nat K) N)->(((eq nat) ((minus_minus_nat ((minus_minus_nat M) K)) ((minus_minus_nat N) K))) ((minus_minus_nat M) N))))). % Axiom fact_126_eq__diff__iff:(forall (N:nat) (K:nat) (M:nat), (((ord_less_eq_nat K) M)->(((ord_less_eq_nat K) N)->((iff (((eq nat) ((minus_minus_nat M) K)) ((minus_minus_nat N) K))) (((eq nat) M) N))))). % Axiom fact_127_diff__diff__cancel:(forall (_TPTP_I:nat) (N:nat), (((ord_less_eq_nat _TPTP_I) N)->(((eq nat) ((minus_minus_nat N) ((minus_minus_nat N) _TPTP_I))) _TPTP_I))). % Axiom fact_128_diff__le__mono:(forall (L:nat) (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((ord_less_eq_nat ((minus_minus_nat M) L)) ((minus_minus_nat N) L)))). % Axiom fact_129_diff__le__mono2:(forall (L:nat) (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((ord_less_eq_nat ((minus_minus_nat L) N)) ((minus_minus_nat L) M)))). % Axiom fact_130_diff__le__self:(forall (M:nat) (N:nat), ((ord_less_eq_nat ((minus_minus_nat M) N)) M)). % Axiom fact_131_finite__surj:(forall (B_42:(x_a->Prop)) (F_23:(x_a->x_a)) (A_79:(x_a->Prop)), ((finite_finite_a A_79)->(((ord_less_eq_a_o B_42) ((image_a_a F_23) A_79))->(finite_finite_a B_42)))). % Axiom fact_132_finite__surj:(forall (B_42:(x_a->Prop)) (F_23:((nat->Prop)->x_a)) (A_79:((nat->Prop)->Prop)), ((finite_finite_nat_o A_79)->(((ord_less_eq_a_o B_42) ((image_nat_o_a F_23) A_79))->(finite_finite_a B_42)))). % Axiom fact_133_finite__surj:(forall (B_42:(x_a->Prop)) (F_23:((pname->Prop)->x_a)) (A_79:((pname->Prop)->Prop)), ((finite297249702name_o A_79)->(((ord_less_eq_a_o B_42) ((image_pname_o_a F_23) A_79))->(finite_finite_a B_42)))). % Axiom fact_134_finite__surj:(forall (B_42:(x_a->Prop)) (F_23:((x_a->Prop)->x_a)) (A_79:((x_a->Prop)->Prop)), ((finite_finite_a_o A_79)->(((ord_less_eq_a_o B_42) ((image_a_o_a F_23) A_79))->(finite_finite_a B_42)))). % Axiom fact_135_finite__surj:(forall (B_42:((nat->Prop)->Prop)) (F_23:(pname->(nat->Prop))) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_less_eq_nat_o_o B_42) ((image_pname_nat_o F_23) A_79))->(finite_finite_nat_o B_42)))). % Axiom fact_136_finite__surj:(forall (B_42:((pname->Prop)->Prop)) (F_23:(pname->(pname->Prop))) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_le1205211808me_o_o B_42) ((image_pname_pname_o F_23) A_79))->(finite297249702name_o B_42)))). % Axiom fact_137_finite__surj:(forall (B_42:((x_a->Prop)->Prop)) (F_23:(pname->(x_a->Prop))) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_less_eq_a_o_o B_42) ((image_pname_a_o F_23) A_79))->(finite_finite_a_o B_42)))). % Axiom fact_138_finite__surj:(forall (B_42:(pname->Prop)) (F_23:(pname->pname)) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_less_eq_pname_o B_42) ((image_pname_pname F_23) A_79))->(finite_finite_pname B_42)))). % Axiom fact_139_finite__surj:(forall (B_42:(nat->Prop)) (F_23:(pname->nat)) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_less_eq_nat_o B_42) ((image_pname_nat F_23) A_79))->(finite_finite_nat B_42)))). % Axiom fact_140_finite__surj:(forall (B_42:(x_a->Prop)) (F_23:(nat->x_a)) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_less_eq_a_o B_42) ((image_nat_a F_23) A_79))->(finite_finite_a B_42)))). % Axiom fact_141_finite__surj:(forall (B_42:((nat->Prop)->Prop)) (F_23:(nat->(nat->Prop))) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_less_eq_nat_o_o B_42) ((image_nat_nat_o F_23) A_79))->(finite_finite_nat_o B_42)))). % Axiom fact_142_finite__surj:(forall (B_42:((pname->Prop)->Prop)) (F_23:(nat->(pname->Prop))) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_le1205211808me_o_o B_42) ((image_nat_pname_o F_23) A_79))->(finite297249702name_o B_42)))). % Axiom fact_143_finite__surj:(forall (B_42:((x_a->Prop)->Prop)) (F_23:(nat->(x_a->Prop))) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_less_eq_a_o_o B_42) ((image_nat_a_o F_23) A_79))->(finite_finite_a_o B_42)))). % Axiom fact_144_finite__surj:(forall (B_42:(pname->Prop)) (F_23:(nat->pname)) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_less_eq_pname_o B_42) ((image_nat_pname F_23) A_79))->(finite_finite_pname B_42)))). % Axiom fact_145_finite__surj:(forall (B_42:(nat->Prop)) (F_23:(nat->nat)) (A_79:(nat->Prop)), ((finite_finite_nat A_79)->(((ord_less_eq_nat_o B_42) ((image_nat_nat F_23) A_79))->(finite_finite_nat B_42)))). % Axiom fact_146_finite__surj:(forall (B_42:(pname->Prop)) (F_23:(x_a->pname)) (A_79:(x_a->Prop)), ((finite_finite_a A_79)->(((ord_less_eq_pname_o B_42) ((image_a_pname F_23) A_79))->(finite_finite_pname B_42)))). % Axiom fact_147_finite__surj:(forall (B_42:(pname->Prop)) (F_23:((nat->Prop)->pname)) (A_79:((nat->Prop)->Prop)), ((finite_finite_nat_o A_79)->(((ord_less_eq_pname_o B_42) ((image_nat_o_pname F_23) A_79))->(finite_finite_pname B_42)))). % Axiom fact_148_finite__surj:(forall (B_42:(pname->Prop)) (F_23:((pname->Prop)->pname)) (A_79:((pname->Prop)->Prop)), ((finite297249702name_o A_79)->(((ord_less_eq_pname_o B_42) ((image_pname_o_pname F_23) A_79))->(finite_finite_pname B_42)))). % Axiom fact_149_finite__surj:(forall (B_42:(pname->Prop)) (F_23:((x_a->Prop)->pname)) (A_79:((x_a->Prop)->Prop)), ((finite_finite_a_o A_79)->(((ord_less_eq_pname_o B_42) ((image_a_o_pname F_23) A_79))->(finite_finite_pname B_42)))). % Axiom fact_150_finite__surj:(forall (B_42:(nat->Prop)) (F_23:(x_a->nat)) (A_79:(x_a->Prop)), ((finite_finite_a A_79)->(((ord_less_eq_nat_o B_42) ((image_a_nat F_23) A_79))->(finite_finite_nat B_42)))). % Axiom fact_151_finite__surj:(forall (B_42:(nat->Prop)) (F_23:((nat->Prop)->nat)) (A_79:((nat->Prop)->Prop)), ((finite_finite_nat_o A_79)->(((ord_less_eq_nat_o B_42) ((image_nat_o_nat F_23) A_79))->(finite_finite_nat B_42)))). % Axiom fact_152_finite__surj:(forall (B_42:(nat->Prop)) (F_23:((pname->Prop)->nat)) (A_79:((pname->Prop)->Prop)), ((finite297249702name_o A_79)->(((ord_less_eq_nat_o B_42) ((image_pname_o_nat F_23) A_79))->(finite_finite_nat B_42)))). % Axiom fact_153_finite__surj:(forall (B_42:(nat->Prop)) (F_23:((x_a->Prop)->nat)) (A_79:((x_a->Prop)->Prop)), ((finite_finite_a_o A_79)->(((ord_less_eq_nat_o B_42) ((image_a_o_nat F_23) A_79))->(finite_finite_nat B_42)))). % Axiom fact_154_finite__surj:(forall (B_42:(x_a->Prop)) (F_23:(pname->x_a)) (A_79:(pname->Prop)), ((finite_finite_pname A_79)->(((ord_less_eq_a_o B_42) ((image_pname_a F_23) A_79))->(finite_finite_a B_42)))). % Axiom fact_155_finite__subset__image:(forall (F_22:((nat->Prop)->x_a)) (A_78:((nat->Prop)->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_nat_o_a F_22) A_78))->((ex ((nat->Prop)->Prop)) (fun (C_12:((nat->Prop)->Prop))=> ((and ((and ((ord_less_eq_nat_o_o C_12) A_78)) (finite_finite_nat_o C_12))) (((eq (x_a->Prop)) B_41) ((image_nat_o_a F_22) C_12)))))))). % Axiom fact_156_finite__subset__image:(forall (F_22:((pname->Prop)->x_a)) (A_78:((pname->Prop)->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_pname_o_a F_22) A_78))->((ex ((pname->Prop)->Prop)) (fun (C_12:((pname->Prop)->Prop))=> ((and ((and ((ord_le1205211808me_o_o C_12) A_78)) (finite297249702name_o C_12))) (((eq (x_a->Prop)) B_41) ((image_pname_o_a F_22) C_12)))))))). % Axiom fact_157_finite__subset__image:(forall (F_22:((x_a->Prop)->x_a)) (A_78:((x_a->Prop)->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_a_o_a F_22) A_78))->((ex ((x_a->Prop)->Prop)) (fun (C_12:((x_a->Prop)->Prop))=> ((and ((and ((ord_less_eq_a_o_o C_12) A_78)) (finite_finite_a_o C_12))) (((eq (x_a->Prop)) B_41) ((image_a_o_a F_22) C_12)))))))). % Axiom fact_158_finite__subset__image:(forall (F_22:(x_a->x_a)) (A_78:(x_a->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_a_a F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq (x_a->Prop)) B_41) ((image_a_a F_22) C_12)))))))). % Axiom fact_159_finite__subset__image:(forall (F_22:(x_a->(nat->Prop))) (A_78:(x_a->Prop)) (B_41:((nat->Prop)->Prop)), ((finite_finite_nat_o B_41)->(((ord_less_eq_nat_o_o B_41) ((image_a_nat_o F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq ((nat->Prop)->Prop)) B_41) ((image_a_nat_o F_22) C_12)))))))). % Axiom fact_160_finite__subset__image:(forall (F_22:(x_a->(pname->Prop))) (A_78:(x_a->Prop)) (B_41:((pname->Prop)->Prop)), ((finite297249702name_o B_41)->(((ord_le1205211808me_o_o B_41) ((image_a_pname_o F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq ((pname->Prop)->Prop)) B_41) ((image_a_pname_o F_22) C_12)))))))). % Axiom fact_161_finite__subset__image:(forall (F_22:(x_a->(x_a->Prop))) (A_78:(x_a->Prop)) (B_41:((x_a->Prop)->Prop)), ((finite_finite_a_o B_41)->(((ord_less_eq_a_o_o B_41) ((image_a_a_o F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq ((x_a->Prop)->Prop)) B_41) ((image_a_a_o F_22) C_12)))))))). % Axiom fact_162_finite__subset__image:(forall (F_22:(x_a->pname)) (A_78:(x_a->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_a_pname F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq (pname->Prop)) B_41) ((image_a_pname F_22) C_12)))))))). % Axiom fact_163_finite__subset__image:(forall (F_22:((nat->Prop)->pname)) (A_78:((nat->Prop)->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_nat_o_pname F_22) A_78))->((ex ((nat->Prop)->Prop)) (fun (C_12:((nat->Prop)->Prop))=> ((and ((and ((ord_less_eq_nat_o_o C_12) A_78)) (finite_finite_nat_o C_12))) (((eq (pname->Prop)) B_41) ((image_nat_o_pname F_22) C_12)))))))). % Axiom fact_164_finite__subset__image:(forall (F_22:((pname->Prop)->pname)) (A_78:((pname->Prop)->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_pname_o_pname F_22) A_78))->((ex ((pname->Prop)->Prop)) (fun (C_12:((pname->Prop)->Prop))=> ((and ((and ((ord_le1205211808me_o_o C_12) A_78)) (finite297249702name_o C_12))) (((eq (pname->Prop)) B_41) ((image_pname_o_pname F_22) C_12)))))))). % Axiom fact_165_finite__subset__image:(forall (F_22:((x_a->Prop)->pname)) (A_78:((x_a->Prop)->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_a_o_pname F_22) A_78))->((ex ((x_a->Prop)->Prop)) (fun (C_12:((x_a->Prop)->Prop))=> ((and ((and ((ord_less_eq_a_o_o C_12) A_78)) (finite_finite_a_o C_12))) (((eq (pname->Prop)) B_41) ((image_a_o_pname F_22) C_12)))))))). % Axiom fact_166_finite__subset__image:(forall (F_22:(x_a->nat)) (A_78:(x_a->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_a_nat F_22) A_78))->((ex (x_a->Prop)) (fun (C_12:(x_a->Prop))=> ((and ((and ((ord_less_eq_a_o C_12) A_78)) (finite_finite_a C_12))) (((eq (nat->Prop)) B_41) ((image_a_nat F_22) C_12)))))))). % Axiom fact_167_finite__subset__image:(forall (F_22:((nat->Prop)->nat)) (A_78:((nat->Prop)->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_nat_o_nat F_22) A_78))->((ex ((nat->Prop)->Prop)) (fun (C_12:((nat->Prop)->Prop))=> ((and ((and ((ord_less_eq_nat_o_o C_12) A_78)) (finite_finite_nat_o C_12))) (((eq (nat->Prop)) B_41) ((image_nat_o_nat F_22) C_12)))))))). % Axiom fact_168_finite__subset__image:(forall (F_22:((pname->Prop)->nat)) (A_78:((pname->Prop)->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_pname_o_nat F_22) A_78))->((ex ((pname->Prop)->Prop)) (fun (C_12:((pname->Prop)->Prop))=> ((and ((and ((ord_le1205211808me_o_o C_12) A_78)) (finite297249702name_o C_12))) (((eq (nat->Prop)) B_41) ((image_pname_o_nat F_22) C_12)))))))). % Axiom fact_169_finite__subset__image:(forall (F_22:((x_a->Prop)->nat)) (A_78:((x_a->Prop)->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_a_o_nat F_22) A_78))->((ex ((x_a->Prop)->Prop)) (fun (C_12:((x_a->Prop)->Prop))=> ((and ((and ((ord_less_eq_a_o_o C_12) A_78)) (finite_finite_a_o C_12))) (((eq (nat->Prop)) B_41) ((image_a_o_nat F_22) C_12)))))))). % Axiom fact_170_finite__subset__image:(forall (F_22:(pname->(nat->Prop))) (A_78:(pname->Prop)) (B_41:((nat->Prop)->Prop)), ((finite_finite_nat_o B_41)->(((ord_less_eq_nat_o_o B_41) ((image_pname_nat_o F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq ((nat->Prop)->Prop)) B_41) ((image_pname_nat_o F_22) C_12)))))))). % Axiom fact_171_finite__subset__image:(forall (F_22:(pname->(pname->Prop))) (A_78:(pname->Prop)) (B_41:((pname->Prop)->Prop)), ((finite297249702name_o B_41)->(((ord_le1205211808me_o_o B_41) ((image_pname_pname_o F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq ((pname->Prop)->Prop)) B_41) ((image_pname_pname_o F_22) C_12)))))))). % Axiom fact_172_finite__subset__image:(forall (F_22:(pname->(x_a->Prop))) (A_78:(pname->Prop)) (B_41:((x_a->Prop)->Prop)), ((finite_finite_a_o B_41)->(((ord_less_eq_a_o_o B_41) ((image_pname_a_o F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq ((x_a->Prop)->Prop)) B_41) ((image_pname_a_o F_22) C_12)))))))). % Axiom fact_173_finite__subset__image:(forall (F_22:(pname->pname)) (A_78:(pname->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_pname_pname F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq (pname->Prop)) B_41) ((image_pname_pname F_22) C_12)))))))). % Axiom fact_174_finite__subset__image:(forall (F_22:(pname->nat)) (A_78:(pname->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_pname_nat F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq (nat->Prop)) B_41) ((image_pname_nat F_22) C_12)))))))). % Axiom fact_175_finite__subset__image:(forall (F_22:(nat->x_a)) (A_78:(nat->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_nat_a F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq (x_a->Prop)) B_41) ((image_nat_a F_22) C_12)))))))). % Axiom fact_176_finite__subset__image:(forall (F_22:(nat->(nat->Prop))) (A_78:(nat->Prop)) (B_41:((nat->Prop)->Prop)), ((finite_finite_nat_o B_41)->(((ord_less_eq_nat_o_o B_41) ((image_nat_nat_o F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq ((nat->Prop)->Prop)) B_41) ((image_nat_nat_o F_22) C_12)))))))). % Axiom fact_177_finite__subset__image:(forall (F_22:(nat->(pname->Prop))) (A_78:(nat->Prop)) (B_41:((pname->Prop)->Prop)), ((finite297249702name_o B_41)->(((ord_le1205211808me_o_o B_41) ((image_nat_pname_o F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq ((pname->Prop)->Prop)) B_41) ((image_nat_pname_o F_22) C_12)))))))). % Axiom fact_178_finite__subset__image:(forall (F_22:(nat->(x_a->Prop))) (A_78:(nat->Prop)) (B_41:((x_a->Prop)->Prop)), ((finite_finite_a_o B_41)->(((ord_less_eq_a_o_o B_41) ((image_nat_a_o F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq ((x_a->Prop)->Prop)) B_41) ((image_nat_a_o F_22) C_12)))))))). % Axiom fact_179_finite__subset__image:(forall (F_22:(nat->pname)) (A_78:(nat->Prop)) (B_41:(pname->Prop)), ((finite_finite_pname B_41)->(((ord_less_eq_pname_o B_41) ((image_nat_pname F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq (pname->Prop)) B_41) ((image_nat_pname F_22) C_12)))))))). % Axiom fact_180_finite__subset__image:(forall (F_22:(nat->nat)) (A_78:(nat->Prop)) (B_41:(nat->Prop)), ((finite_finite_nat B_41)->(((ord_less_eq_nat_o B_41) ((image_nat_nat F_22) A_78))->((ex (nat->Prop)) (fun (C_12:(nat->Prop))=> ((and ((and ((ord_less_eq_nat_o C_12) A_78)) (finite_finite_nat C_12))) (((eq (nat->Prop)) B_41) ((image_nat_nat F_22) C_12)))))))). % Axiom fact_181_finite__subset__image:(forall (F_22:(pname->x_a)) (A_78:(pname->Prop)) (B_41:(x_a->Prop)), ((finite_finite_a B_41)->(((ord_less_eq_a_o B_41) ((image_pname_a F_22) A_78))->((ex (pname->Prop)) (fun (C_12:(pname->Prop))=> ((and ((and ((ord_less_eq_pname_o C_12) A_78)) (finite_finite_pname C_12))) (((eq (x_a->Prop)) B_41) ((image_pname_a F_22) C_12)))))))). % Axiom fact_182_lift__Suc__mono__le:(forall (N_4:nat) (N_3:nat) (F_21:(nat->Prop)), ((forall (N_2:nat), ((ord_less_eq_o (F_21 N_2)) (F_21 (suc N_2))))->(((ord_less_eq_nat N_4) N_3)->((ord_less_eq_o (F_21 N_4)) (F_21 N_3))))). % Axiom fact_183_lift__Suc__mono__le:(forall (N_4:nat) (N_3:nat) (F_21:(nat->(pname->Prop))), ((forall (N_2:nat), ((ord_less_eq_pname_o (F_21 N_2)) (F_21 (suc N_2))))->(((ord_less_eq_nat N_4) N_3)->((ord_less_eq_pname_o (F_21 N_4)) (F_21 N_3))))). % Axiom fact_184_lift__Suc__mono__le:(forall (N_4:nat) (N_3:nat) (F_21:(nat->(nat->Prop))), ((forall (N_2:nat), ((ord_less_eq_nat_o (F_21 N_2)) (F_21 (suc N_2))))->(((ord_less_eq_nat N_4) N_3)->((ord_less_eq_nat_o (F_21 N_4)) (F_21 N_3))))). % Axiom fact_185_lift__Suc__mono__le:(forall (N_4:nat) (N_3:nat) (F_21:(nat->(x_a->Prop))), ((forall (N_2:nat), ((ord_less_eq_a_o (F_21 N_2)) (F_21 (suc N_2))))->(((ord_less_eq_nat N_4) N_3)->((ord_less_eq_a_o (F_21 N_4)) (F_21 N_3))))). % Axiom fact_186_lift__Suc__mono__le:(forall (N_4:nat) (N_3:nat) (F_21:(nat->nat)), ((forall (N_2:nat), ((ord_less_eq_nat (F_21 N_2)) (F_21 (suc N_2))))->(((ord_less_eq_nat N_4) N_3)->((ord_less_eq_nat (F_21 N_4)) (F_21 N_3))))). % Axiom fact_187_pigeonhole__infinite:(forall (F_20:(nat->pname)) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite_finite_pname ((image_nat_pname F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_188_pigeonhole__infinite:(forall (F_20:(x_a->pname)) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite_finite_pname ((image_a_pname F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_189_pigeonhole__infinite:(forall (F_20:(pname->pname)) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite_finite_pname ((image_pname_pname F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_190_pigeonhole__infinite:(forall (F_20:((nat->Prop)->pname)) (A_77:((nat->Prop)->Prop)), (((finite_finite_nat_o A_77)->False)->((finite_finite_pname ((image_nat_o_pname F_20) A_77))->((ex (nat->Prop)) (fun (X_1:(nat->Prop))=> ((and ((member_nat_o X_1) A_77)) ((finite_finite_nat_o (collect_nat_o (fun (A_2:(nat->Prop))=> ((and ((member_nat_o A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_191_pigeonhole__infinite:(forall (F_20:((pname->Prop)->pname)) (A_77:((pname->Prop)->Prop)), (((finite297249702name_o A_77)->False)->((finite_finite_pname ((image_pname_o_pname F_20) A_77))->((ex (pname->Prop)) (fun (X_1:(pname->Prop))=> ((and ((member_pname_o X_1) A_77)) ((finite297249702name_o (collect_pname_o (fun (A_2:(pname->Prop))=> ((and ((member_pname_o A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_192_pigeonhole__infinite:(forall (F_20:((x_a->Prop)->pname)) (A_77:((x_a->Prop)->Prop)), (((finite_finite_a_o A_77)->False)->((finite_finite_pname ((image_a_o_pname F_20) A_77))->((ex (x_a->Prop)) (fun (X_1:(x_a->Prop))=> ((and ((member_a_o X_1) A_77)) ((finite_finite_a_o (collect_a_o (fun (A_2:(x_a->Prop))=> ((and ((member_a_o A_2) A_77)) (((eq pname) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_193_pigeonhole__infinite:(forall (F_20:(nat->nat)) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite_finite_nat ((image_nat_nat F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_194_pigeonhole__infinite:(forall (F_20:(x_a->nat)) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite_finite_nat ((image_a_nat F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_195_pigeonhole__infinite:(forall (F_20:(pname->nat)) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite_finite_nat ((image_pname_nat F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_196_pigeonhole__infinite:(forall (F_20:((nat->Prop)->nat)) (A_77:((nat->Prop)->Prop)), (((finite_finite_nat_o A_77)->False)->((finite_finite_nat ((image_nat_o_nat F_20) A_77))->((ex (nat->Prop)) (fun (X_1:(nat->Prop))=> ((and ((member_nat_o X_1) A_77)) ((finite_finite_nat_o (collect_nat_o (fun (A_2:(nat->Prop))=> ((and ((member_nat_o A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_197_pigeonhole__infinite:(forall (F_20:((pname->Prop)->nat)) (A_77:((pname->Prop)->Prop)), (((finite297249702name_o A_77)->False)->((finite_finite_nat ((image_pname_o_nat F_20) A_77))->((ex (pname->Prop)) (fun (X_1:(pname->Prop))=> ((and ((member_pname_o X_1) A_77)) ((finite297249702name_o (collect_pname_o (fun (A_2:(pname->Prop))=> ((and ((member_pname_o A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_198_pigeonhole__infinite:(forall (F_20:((x_a->Prop)->nat)) (A_77:((x_a->Prop)->Prop)), (((finite_finite_a_o A_77)->False)->((finite_finite_nat ((image_a_o_nat F_20) A_77))->((ex (x_a->Prop)) (fun (X_1:(x_a->Prop))=> ((and ((member_a_o X_1) A_77)) ((finite_finite_a_o (collect_a_o (fun (A_2:(x_a->Prop))=> ((and ((member_a_o A_2) A_77)) (((eq nat) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_199_pigeonhole__infinite:(forall (F_20:(nat->x_a)) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite_finite_a ((image_nat_a F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq x_a) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_200_pigeonhole__infinite:(forall (F_20:(nat->(nat->Prop))) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite_finite_nat_o ((image_nat_nat_o F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq (nat->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_201_pigeonhole__infinite:(forall (F_20:(nat->(pname->Prop))) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite297249702name_o ((image_nat_pname_o F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq (pname->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_202_pigeonhole__infinite:(forall (F_20:(nat->(x_a->Prop))) (A_77:(nat->Prop)), (((finite_finite_nat A_77)->False)->((finite_finite_a_o ((image_nat_a_o F_20) A_77))->((ex nat) (fun (X_1:nat)=> ((and ((member_nat X_1) A_77)) ((finite_finite_nat (collect_nat (fun (A_2:nat)=> ((and ((member_nat A_2) A_77)) (((eq (x_a->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_203_pigeonhole__infinite:(forall (F_20:(pname->(nat->Prop))) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite_finite_nat_o ((image_pname_nat_o F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq (nat->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_204_pigeonhole__infinite:(forall (F_20:(pname->(pname->Prop))) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite297249702name_o ((image_pname_pname_o F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq (pname->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_205_pigeonhole__infinite:(forall (F_20:(pname->(x_a->Prop))) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite_finite_a_o ((image_pname_a_o F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq (x_a->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_206_pigeonhole__infinite:(forall (F_20:(x_a->x_a)) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite_finite_a ((image_a_a F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq x_a) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_207_pigeonhole__infinite:(forall (F_20:(x_a->(nat->Prop))) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite_finite_nat_o ((image_a_nat_o F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq (nat->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_208_pigeonhole__infinite:(forall (F_20:(x_a->(pname->Prop))) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite297249702name_o ((image_a_pname_o F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq (pname->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_209_pigeonhole__infinite:(forall (F_20:(x_a->(x_a->Prop))) (A_77:(x_a->Prop)), (((finite_finite_a A_77)->False)->((finite_finite_a_o ((image_a_a_o F_20) A_77))->((ex x_a) (fun (X_1:x_a)=> ((and ((member_a X_1) A_77)) ((finite_finite_a (collect_a (fun (A_2:x_a)=> ((and ((member_a A_2) A_77)) (((eq (x_a->Prop)) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_210_pigeonhole__infinite:(forall (F_20:(pname->x_a)) (A_77:(pname->Prop)), (((finite_finite_pname A_77)->False)->((finite_finite_a ((image_pname_a F_20) A_77))->((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_77)) ((finite_finite_pname (collect_pname (fun (A_2:pname)=> ((and ((member_pname A_2) A_77)) (((eq x_a) (F_20 A_2)) (F_20 X_1))))))->False))))))). % Axiom fact_211_image__eqI:(forall (A_76:(pname->Prop)) (B_40:nat) (F_19:(pname->nat)) (X_30:pname), ((((eq nat) B_40) (F_19 X_30))->(((member_pname X_30) A_76)->((member_nat B_40) ((image_pname_nat F_19) A_76))))). % Axiom fact_212_image__eqI:(forall (A_76:(x_a->Prop)) (B_40:nat) (F_19:(x_a->nat)) (X_30:x_a), ((((eq nat) B_40) (F_19 X_30))->(((member_a X_30) A_76)->((member_nat B_40) ((image_a_nat F_19) A_76))))). % Axiom fact_213_image__eqI:(forall (A_76:(nat->Prop)) (B_40:pname) (F_19:(nat->pname)) (X_30:nat), ((((eq pname) B_40) (F_19 X_30))->(((member_nat X_30) A_76)->((member_pname B_40) ((image_nat_pname F_19) A_76))))). % Axiom fact_214_image__eqI:(forall (A_76:(nat->Prop)) (B_40:x_a) (F_19:(nat->x_a)) (X_30:nat), ((((eq x_a) B_40) (F_19 X_30))->(((member_nat X_30) A_76)->((member_a B_40) ((image_nat_a F_19) A_76))))). % Axiom fact_215_image__eqI:(forall (A_76:(pname->Prop)) (B_40:x_a) (F_19:(pname->x_a)) (X_30:pname), ((((eq x_a) B_40) (F_19 X_30))->(((member_pname X_30) A_76)->((member_a B_40) ((image_pname_a F_19) A_76))))). % Axiom fact_216_equalityI:(forall (A_75:(pname->Prop)) (B_39:(pname->Prop)), (((ord_less_eq_pname_o A_75) B_39)->(((ord_less_eq_pname_o B_39) A_75)->(((eq (pname->Prop)) A_75) B_39)))). % Axiom fact_217_equalityI:(forall (A_75:(nat->Prop)) (B_39:(nat->Prop)), (((ord_less_eq_nat_o A_75) B_39)->(((ord_less_eq_nat_o B_39) A_75)->(((eq (nat->Prop)) A_75) B_39)))). % Axiom fact_218_equalityI:(forall (A_75:(x_a->Prop)) (B_39:(x_a->Prop)), (((ord_less_eq_a_o A_75) B_39)->(((ord_less_eq_a_o B_39) A_75)->(((eq (x_a->Prop)) A_75) B_39)))). % Axiom fact_219_subsetD:(forall (C_11:nat) (A_74:(nat->Prop)) (B_38:(nat->Prop)), (((ord_less_eq_nat_o A_74) B_38)->(((member_nat C_11) A_74)->((member_nat C_11) B_38)))). % Axiom fact_220_subsetD:(forall (C_11:pname) (A_74:(pname->Prop)) (B_38:(pname->Prop)), (((ord_less_eq_pname_o A_74) B_38)->(((member_pname C_11) A_74)->((member_pname C_11) B_38)))). % Axiom fact_221_subsetD:(forall (C_11:x_a) (A_74:(x_a->Prop)) (B_38:(x_a->Prop)), (((ord_less_eq_a_o A_74) B_38)->(((member_a C_11) A_74)->((member_a C_11) B_38)))). % Axiom fact_222_insertCI:(forall (B_37:nat) (A_73:nat) (B_36:(nat->Prop)), (((((member_nat A_73) B_36)->False)->(((eq nat) A_73) B_37))->((member_nat A_73) ((insert_nat B_37) B_36)))). % Axiom fact_223_insertCI:(forall (B_37:pname) (A_73:pname) (B_36:(pname->Prop)), (((((member_pname A_73) B_36)->False)->(((eq pname) A_73) B_37))->((member_pname A_73) ((insert_pname B_37) B_36)))). % Axiom fact_224_insertCI:(forall (B_37:x_a) (A_73:x_a) (B_36:(x_a->Prop)), (((((member_a A_73) B_36)->False)->(((eq x_a) A_73) B_37))->((member_a A_73) ((insert_a B_37) B_36)))). % Axiom fact_225_insertE:(forall (A_72:nat) (B_35:nat) (A_71:(nat->Prop)), (((member_nat A_72) ((insert_nat B_35) A_71))->((not (((eq nat) A_72) B_35))->((member_nat A_72) A_71)))). % Axiom fact_226_insertE:(forall (A_72:pname) (B_35:pname) (A_71:(pname->Prop)), (((member_pname A_72) ((insert_pname B_35) A_71))->((not (((eq pname) A_72) B_35))->((member_pname A_72) A_71)))). % Axiom fact_227_insertE:(forall (A_72:x_a) (B_35:x_a) (A_71:(x_a->Prop)), (((member_a A_72) ((insert_a B_35) A_71))->((not (((eq x_a) A_72) B_35))->((member_a A_72) A_71)))). % Axiom fact_228_zero__induct__lemma:(forall (_TPTP_I:nat) (P:(nat->Prop)) (K:nat), ((P K)->((forall (N_2:nat), ((P (suc N_2))->(P N_2)))->(P ((minus_minus_nat K) _TPTP_I))))). % Axiom fact_229_Suc__le__D:(forall (N:nat) (M_2:nat), (((ord_less_eq_nat (suc N)) M_2)->((ex nat) (fun (M_1:nat)=> (((eq nat) M_2) (suc M_1)))))). % Axiom fact_230_insertI1:(forall (A_70:nat) (B_34:(nat->Prop)), ((member_nat A_70) ((insert_nat A_70) B_34))). % Axiom fact_231_insertI1:(forall (A_70:pname) (B_34:(pname->Prop)), ((member_pname A_70) ((insert_pname A_70) B_34))). % Axiom fact_232_insertI1:(forall (A_70:x_a) (B_34:(x_a->Prop)), ((member_a A_70) ((insert_a A_70) B_34))). % Axiom fact_233_insert__compr:(forall (A_69:(nat->Prop)) (B_33:((nat->Prop)->Prop)), (((eq ((nat->Prop)->Prop)) ((insert_nat_o A_69) B_33)) (collect_nat_o (fun (X_1:(nat->Prop))=> ((or (((eq (nat->Prop)) X_1) A_69)) ((member_nat_o X_1) B_33)))))). % Axiom fact_234_insert__compr:(forall (A_69:(pname->Prop)) (B_33:((pname->Prop)->Prop)), (((eq ((pname->Prop)->Prop)) ((insert_pname_o A_69) B_33)) (collect_pname_o (fun (X_1:(pname->Prop))=> ((or (((eq (pname->Prop)) X_1) A_69)) ((member_pname_o X_1) B_33)))))). % Axiom fact_235_insert__compr:(forall (A_69:(x_a->Prop)) (B_33:((x_a->Prop)->Prop)), (((eq ((x_a->Prop)->Prop)) ((insert_a_o A_69) B_33)) (collect_a_o (fun (X_1:(x_a->Prop))=> ((or (((eq (x_a->Prop)) X_1) A_69)) ((member_a_o X_1) B_33)))))). % Axiom fact_236_insert__compr:(forall (A_69:nat) (B_33:(nat->Prop)), (((eq (nat->Prop)) ((insert_nat A_69) B_33)) (collect_nat (fun (X_1:nat)=> ((or (((eq nat) X_1) A_69)) ((member_nat X_1) B_33)))))). % Axiom fact_237_insert__compr:(forall (A_69:pname) (B_33:(pname->Prop)), (((eq (pname->Prop)) ((insert_pname A_69) B_33)) (collect_pname (fun (X_1:pname)=> ((or (((eq pname) X_1) A_69)) ((member_pname X_1) B_33)))))). % Axiom fact_238_insert__compr:(forall (A_69:x_a) (B_33:(x_a->Prop)), (((eq (x_a->Prop)) ((insert_a A_69) B_33)) (collect_a (fun (X_1:x_a)=> ((or (((eq x_a) X_1) A_69)) ((member_a X_1) B_33)))))). % Axiom fact_239_insert__Collect:(forall (A_68:pname) (P_8:(pname->Prop)), (((eq (pname->Prop)) ((insert_pname A_68) (collect_pname P_8))) (collect_pname (fun (U_1:pname)=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq pname) U_1) A_68))) (P_8 U_1)))))). % Axiom fact_240_insert__Collect:(forall (A_68:(nat->Prop)) (P_8:((nat->Prop)->Prop)), (((eq ((nat->Prop)->Prop)) ((insert_nat_o A_68) (collect_nat_o P_8))) (collect_nat_o (fun (U_1:(nat->Prop))=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq (nat->Prop)) U_1) A_68))) (P_8 U_1)))))). % Axiom fact_241_insert__Collect:(forall (A_68:(pname->Prop)) (P_8:((pname->Prop)->Prop)), (((eq ((pname->Prop)->Prop)) ((insert_pname_o A_68) (collect_pname_o P_8))) (collect_pname_o (fun (U_1:(pname->Prop))=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq (pname->Prop)) U_1) A_68))) (P_8 U_1)))))). % Axiom fact_242_insert__Collect:(forall (A_68:(x_a->Prop)) (P_8:((x_a->Prop)->Prop)), (((eq ((x_a->Prop)->Prop)) ((insert_a_o A_68) (collect_a_o P_8))) (collect_a_o (fun (U_1:(x_a->Prop))=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq (x_a->Prop)) U_1) A_68))) (P_8 U_1)))))). % Axiom fact_243_insert__Collect:(forall (A_68:nat) (P_8:(nat->Prop)), (((eq (nat->Prop)) ((insert_nat A_68) (collect_nat P_8))) (collect_nat (fun (U_1:nat)=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq nat) U_1) A_68))) (P_8 U_1)))))). % Axiom fact_244_insert__Collect:(forall (A_68:x_a) (P_8:(x_a->Prop)), (((eq (x_a->Prop)) ((insert_a A_68) (collect_a P_8))) (collect_a (fun (U_1:x_a)=> (((fun (X:Prop) (Y:Prop)=> (X->Y)) (not (((eq x_a) U_1) A_68))) (P_8 U_1)))))). % Axiom fact_245_insert__absorb2:(forall (X_29:pname) (A_67:(pname->Prop)), (((eq (pname->Prop)) ((insert_pname X_29) ((insert_pname X_29) A_67))) ((insert_pname X_29) A_67))). % Axiom fact_246_insert__absorb2:(forall (X_29:nat) (A_67:(nat->Prop)), (((eq (nat->Prop)) ((insert_nat X_29) ((insert_nat X_29) A_67))) ((insert_nat X_29) A_67))). % Axiom fact_247_insert__absorb2:(forall (X_29:x_a) (A_67:(x_a->Prop)), (((eq (x_a->Prop)) ((insert_a X_29) ((insert_a X_29) A_67))) ((insert_a X_29) A_67))). % Axiom fact_248_insert__commute:(forall (X_28:pname) (Y_12:pname) (A_66:(pname->Prop)), (((eq (pname->Prop)) ((insert_pname X_28) ((insert_pname Y_12) A_66))) ((insert_pname Y_12) ((insert_pname X_28) A_66)))). % Axiom fact_249_insert__commute:(forall (X_28:nat) (Y_12:nat) (A_66:(nat->Prop)), (((eq (nat->Prop)) ((insert_nat X_28) ((insert_nat Y_12) A_66))) ((insert_nat Y_12) ((insert_nat X_28) A_66)))). % Axiom fact_250_insert__commute:(forall (X_28:x_a) (Y_12:x_a) (A_66:(x_a->Prop)), (((eq (x_a->Prop)) ((insert_a X_28) ((insert_a Y_12) A_66))) ((insert_a Y_12) ((insert_a X_28) A_66)))). % Axiom fact_251_insert__iff:(forall (A_65:nat) (B_32:nat) (A_64:(nat->Prop)), ((iff ((member_nat A_65) ((insert_nat B_32) A_64))) ((or (((eq nat) A_65) B_32)) ((member_nat A_65) A_64)))). % Axiom fact_252_insert__iff:(forall (A_65:pname) (B_32:pname) (A_64:(pname->Prop)), ((iff ((member_pname A_65) ((insert_pname B_32) A_64))) ((or (((eq pname) A_65) B_32)) ((member_pname A_65) A_64)))). % Axiom fact_253_insert__iff:(forall (A_65:x_a) (B_32:x_a) (A_64:(x_a->Prop)), ((iff ((member_a A_65) ((insert_a B_32) A_64))) ((or (((eq x_a) A_65) B_32)) ((member_a A_65) A_64)))). % Axiom fact_254_insert__code:(forall (Y_11:pname) (A_63:(pname->Prop)) (X_27:pname), ((iff (((insert_pname Y_11) A_63) X_27)) ((or (((eq pname) Y_11) X_27)) (A_63 X_27)))). % Axiom fact_255_insert__code:(forall (Y_11:nat) (A_63:(nat->Prop)) (X_27:nat), ((iff (((insert_nat Y_11) A_63) X_27)) ((or (((eq nat) Y_11) X_27)) (A_63 X_27)))). % Axiom fact_256_insert__code:(forall (Y_11:x_a) (A_63:(x_a->Prop)) (X_27:x_a), ((iff (((insert_a Y_11) A_63) X_27)) ((or (((eq x_a) Y_11) X_27)) (A_63 X_27)))). % Axiom fact_257_insert__ident:(forall (B_31:(nat->Prop)) (X_26:nat) (A_62:(nat->Prop)), ((((member_nat X_26) A_62)->False)->((((member_nat X_26) B_31)->False)->((iff (((eq (nat->Prop)) ((insert_nat X_26) A_62)) ((insert_nat X_26) B_31))) (((eq (nat->Prop)) A_62) B_31))))). % Axiom fact_258_insert__ident:(forall (B_31:(pname->Prop)) (X_26:pname) (A_62:(pname->Prop)), ((((member_pname X_26) A_62)->False)->((((member_pname X_26) B_31)->False)->((iff (((eq (pname->Prop)) ((insert_pname X_26) A_62)) ((insert_pname X_26) B_31))) (((eq (pname->Prop)) A_62) B_31))))). % Axiom fact_259_insert__ident:(forall (B_31:(x_a->Prop)) (X_26:x_a) (A_62:(x_a->Prop)), ((((member_a X_26) A_62)->False)->((((member_a X_26) B_31)->False)->((iff (((eq (x_a->Prop)) ((insert_a X_26) A_62)) ((insert_a X_26) B_31))) (((eq (x_a->Prop)) A_62) B_31))))). % Axiom fact_260_insertI2:(forall (B_30:nat) (A_61:nat) (B_29:(nat->Prop)), (((member_nat A_61) B_29)->((member_nat A_61) ((insert_nat B_30) B_29)))). % Axiom fact_261_insertI2:(forall (B_30:pname) (A_61:pname) (B_29:(pname->Prop)), (((member_pname A_61) B_29)->((member_pname A_61) ((insert_pname B_30) B_29)))). % Axiom fact_262_insertI2:(forall (B_30:x_a) (A_61:x_a) (B_29:(x_a->Prop)), (((member_a A_61) B_29)->((member_a A_61) ((insert_a B_30) B_29)))). % Axiom fact_263_insert__absorb:(forall (A_60:nat) (A_59:(nat->Prop)), (((member_nat A_60) A_59)->(((eq (nat->Prop)) ((insert_nat A_60) A_59)) A_59))). % Axiom fact_264_insert__absorb:(forall (A_60:pname) (A_59:(pname->Prop)), (((member_pname A_60) A_59)->(((eq (pname->Prop)) ((insert_pname A_60) A_59)) A_59))). % Axiom fact_265_insert__absorb:(forall (A_60:x_a) (A_59:(x_a->Prop)), (((member_a A_60) A_59)->(((eq (x_a->Prop)) ((insert_a A_60) A_59)) A_59))). % Axiom fact_266_subset__refl:(forall (A_58:(pname->Prop)), ((ord_less_eq_pname_o A_58) A_58)). % Axiom fact_267_subset__refl:(forall (A_58:(nat->Prop)), ((ord_less_eq_nat_o A_58) A_58)). % Axiom fact_268_subset__refl:(forall (A_58:(x_a->Prop)), ((ord_less_eq_a_o A_58) A_58)). % Axiom fact_269_set__eq__subset:(forall (A_57:(pname->Prop)) (B_28:(pname->Prop)), ((iff (((eq (pname->Prop)) A_57) B_28)) ((and ((ord_less_eq_pname_o A_57) B_28)) ((ord_less_eq_pname_o B_28) A_57)))). % Axiom fact_270_set__eq__subset:(forall (A_57:(nat->Prop)) (B_28:(nat->Prop)), ((iff (((eq (nat->Prop)) A_57) B_28)) ((and ((ord_less_eq_nat_o A_57) B_28)) ((ord_less_eq_nat_o B_28) A_57)))). % Axiom fact_271_set__eq__subset:(forall (A_57:(x_a->Prop)) (B_28:(x_a->Prop)), ((iff (((eq (x_a->Prop)) A_57) B_28)) ((and ((ord_less_eq_a_o A_57) B_28)) ((ord_less_eq_a_o B_28) A_57)))). % Axiom fact_272_equalityD1:(forall (A_56:(pname->Prop)) (B_27:(pname->Prop)), ((((eq (pname->Prop)) A_56) B_27)->((ord_less_eq_pname_o A_56) B_27))). % Axiom fact_273_equalityD1:(forall (A_56:(nat->Prop)) (B_27:(nat->Prop)), ((((eq (nat->Prop)) A_56) B_27)->((ord_less_eq_nat_o A_56) B_27))). % Axiom fact_274_equalityD1:(forall (A_56:(x_a->Prop)) (B_27:(x_a->Prop)), ((((eq (x_a->Prop)) A_56) B_27)->((ord_less_eq_a_o A_56) B_27))). % Axiom fact_275_equalityD2:(forall (A_55:(pname->Prop)) (B_26:(pname->Prop)), ((((eq (pname->Prop)) A_55) B_26)->((ord_less_eq_pname_o B_26) A_55))). % Axiom fact_276_equalityD2:(forall (A_55:(nat->Prop)) (B_26:(nat->Prop)), ((((eq (nat->Prop)) A_55) B_26)->((ord_less_eq_nat_o B_26) A_55))). % Axiom fact_277_equalityD2:(forall (A_55:(x_a->Prop)) (B_26:(x_a->Prop)), ((((eq (x_a->Prop)) A_55) B_26)->((ord_less_eq_a_o B_26) A_55))). % Axiom fact_278_in__mono:(forall (X_25:nat) (A_54:(nat->Prop)) (B_25:(nat->Prop)), (((ord_less_eq_nat_o A_54) B_25)->(((member_nat X_25) A_54)->((member_nat X_25) B_25)))). % Axiom fact_279_in__mono:(forall (X_25:pname) (A_54:(pname->Prop)) (B_25:(pname->Prop)), (((ord_less_eq_pname_o A_54) B_25)->(((member_pname X_25) A_54)->((member_pname X_25) B_25)))). % Axiom fact_280_in__mono:(forall (X_25:x_a) (A_54:(x_a->Prop)) (B_25:(x_a->Prop)), (((ord_less_eq_a_o A_54) B_25)->(((member_a X_25) A_54)->((member_a X_25) B_25)))). % Axiom fact_281_set__rev__mp:(forall (B_24:(nat->Prop)) (X_24:nat) (A_53:(nat->Prop)), (((member_nat X_24) A_53)->(((ord_less_eq_nat_o A_53) B_24)->((member_nat X_24) B_24)))). % Axiom fact_282_set__rev__mp:(forall (B_24:(pname->Prop)) (X_24:pname) (A_53:(pname->Prop)), (((member_pname X_24) A_53)->(((ord_less_eq_pname_o A_53) B_24)->((member_pname X_24) B_24)))). % Axiom fact_283_set__rev__mp:(forall (B_24:(x_a->Prop)) (X_24:x_a) (A_53:(x_a->Prop)), (((member_a X_24) A_53)->(((ord_less_eq_a_o A_53) B_24)->((member_a X_24) B_24)))). % Axiom fact_284_set__mp:(forall (X_23:nat) (A_52:(nat->Prop)) (B_23:(nat->Prop)), (((ord_less_eq_nat_o A_52) B_23)->(((member_nat X_23) A_52)->((member_nat X_23) B_23)))). % Axiom fact_285_set__mp:(forall (X_23:pname) (A_52:(pname->Prop)) (B_23:(pname->Prop)), (((ord_less_eq_pname_o A_52) B_23)->(((member_pname X_23) A_52)->((member_pname X_23) B_23)))). % Axiom fact_286_set__mp:(forall (X_23:x_a) (A_52:(x_a->Prop)) (B_23:(x_a->Prop)), (((ord_less_eq_a_o A_52) B_23)->(((member_a X_23) A_52)->((member_a X_23) B_23)))). % Axiom fact_287_mem__def:(forall (X_22:nat) (A_51:(nat->Prop)), ((iff ((member_nat X_22) A_51)) (A_51 X_22))). % Axiom fact_288_mem__def:(forall (X_22:pname) (A_51:(pname->Prop)), ((iff ((member_pname X_22) A_51)) (A_51 X_22))). % Axiom fact_289_mem__def:(forall (X_22:x_a) (A_51:(x_a->Prop)), ((iff ((member_a X_22) A_51)) (A_51 X_22))). % Axiom fact_290_Collect__def:(forall (P_7:(x_a->Prop)), (((eq (x_a->Prop)) (collect_a P_7)) P_7)). % Axiom fact_291_Collect__def:(forall (P_7:(pname->Prop)), (((eq (pname->Prop)) (collect_pname P_7)) P_7)). % Axiom fact_292_Collect__def:(forall (P_7:((nat->Prop)->Prop)), (((eq ((nat->Prop)->Prop)) (collect_nat_o P_7)) P_7)). % Axiom fact_293_Collect__def:(forall (P_7:((pname->Prop)->Prop)), (((eq ((pname->Prop)->Prop)) (collect_pname_o P_7)) P_7)). % Axiom fact_294_Collect__def:(forall (P_7:((x_a->Prop)->Prop)), (((eq ((x_a->Prop)->Prop)) (collect_a_o P_7)) P_7)). % Axiom fact_295_Collect__def:(forall (P_7:(nat->Prop)), (((eq (nat->Prop)) (collect_nat P_7)) P_7)). % Axiom fact_296_subset__trans:(forall (C_10:(pname->Prop)) (A_50:(pname->Prop)) (B_22:(pname->Prop)), (((ord_less_eq_pname_o A_50) B_22)->(((ord_less_eq_pname_o B_22) C_10)->((ord_less_eq_pname_o A_50) C_10)))). % Axiom fact_297_subset__trans:(forall (C_10:(nat->Prop)) (A_50:(nat->Prop)) (B_22:(nat->Prop)), (((ord_less_eq_nat_o A_50) B_22)->(((ord_less_eq_nat_o B_22) C_10)->((ord_less_eq_nat_o A_50) C_10)))). % Axiom fact_298_subset__trans:(forall (C_10:(x_a->Prop)) (A_50:(x_a->Prop)) (B_22:(x_a->Prop)), (((ord_less_eq_a_o A_50) B_22)->(((ord_less_eq_a_o B_22) C_10)->((ord_less_eq_a_o A_50) C_10)))). % Axiom fact_299_equalityE:(forall (A_49:(pname->Prop)) (B_21:(pname->Prop)), ((((eq (pname->Prop)) A_49) B_21)->((((ord_less_eq_pname_o A_49) B_21)->(((ord_less_eq_pname_o B_21) A_49)->False))->False))). % Axiom fact_300_equalityE:(forall (A_49:(nat->Prop)) (B_21:(nat->Prop)), ((((eq (nat->Prop)) A_49) B_21)->((((ord_less_eq_nat_o A_49) B_21)->(((ord_less_eq_nat_o B_21) A_49)->False))->False))). % Axiom fact_301_equalityE:(forall (A_49:(x_a->Prop)) (B_21:(x_a->Prop)), ((((eq (x_a->Prop)) A_49) B_21)->((((ord_less_eq_a_o A_49) B_21)->(((ord_less_eq_a_o B_21) A_49)->False))->False))). % Axiom fact_302_image__iff:(forall (Z_3:x_a) (F_18:(pname->x_a)) (A_48:(pname->Prop)), ((iff ((member_a Z_3) ((image_pname_a F_18) A_48))) ((ex pname) (fun (X_1:pname)=> ((and ((member_pname X_1) A_48)) (((eq x_a) Z_3) (F_18 X_1))))))). % Axiom fact_303_imageI:(forall (F_17:(pname->nat)) (X_21:pname) (A_47:(pname->Prop)), (((member_pname X_21) A_47)->((member_nat (F_17 X_21)) ((image_pname_nat F_17) A_47)))). % Axiom fact_304_imageI:(forall (F_17:(x_a->nat)) (X_21:x_a) (A_47:(x_a->Prop)), (((member_a X_21) A_47)->((member_nat (F_17 X_21)) ((image_a_nat F_17) A_47)))). % Axiom fact_305_imageI:(forall (F_17:(nat->pname)) (X_21:nat) (A_47:(nat->Prop)), (((member_nat X_21) A_47)->((member_pname (F_17 X_21)) ((image_nat_pname F_17) A_47)))). % Axiom fact_306_imageI:(forall (F_17:(nat->x_a)) (X_21:nat) (A_47:(nat->Prop)), (((member_nat X_21) A_47)->((member_a (F_17 X_21)) ((image_nat_a F_17) A_47)))). % Axiom fact_307_imageI:(forall (F_17:(pname->x_a)) (X_21:pname) (A_47:(pname->Prop)), (((member_pname X_21) A_47)->((member_a (F_17 X_21)) ((image_pname_a F_17) A_47)))). % Axiom fact_308_rev__image__eqI:(forall (B_20:nat) (F_16:(pname->nat)) (X_20:pname) (A_46:(pname->Prop)), (((member_pname X_20) A_46)->((((eq nat) B_20) (F_16 X_20))->((member_nat B_20) ((image_pname_nat F_16) A_46))))). % Axiom fact_309_rev__image__eqI:(forall (B_20:nat) (F_16:(x_a->nat)) (X_20:x_a) (A_46:(x_a->Prop)), (((member_a X_20) A_46)->((((eq nat) B_20) (F_16 X_20))->((member_nat B_20) ((image_a_nat F_16) A_46))))). % Axiom fact_310_rev__image__eqI:(forall (B_20:pname) (F_16:(nat->pname)) (X_20:nat) (A_46:(nat->Prop)), (((member_nat X_20) A_46)->((((eq pname) B_20) (F_16 X_20))->((member_pname B_20) ((image_nat_pname F_16) A_46))))). % Axiom fact_311_rev__image__eqI:(forall (B_20:x_a) (F_16:(nat->x_a)) (X_20:nat) (A_46:(nat->Prop)), (((member_nat X_20) A_46)->((((eq x_a) B_20) (F_16 X_20))->((member_a B_20) ((image_nat_a F_16) A_46))))). % Axiom fact_312_rev__image__eqI:(forall (B_20:x_a) (F_16:(pname->x_a)) (X_20:pname) (A_46:(pname->Prop)), (((member_pname X_20) A_46)->((((eq x_a) B_20) (F_16 X_20))->((member_a B_20) ((image_pname_a F_16) A_46))))). % Axiom fact_313_insert__compr__raw:(forall (X_1:(nat->Prop)) (Xa:((nat->Prop)->Prop)), (((eq ((nat->Prop)->Prop)) ((insert_nat_o X_1) Xa)) (collect_nat_o (fun (Y_10:(nat->Prop))=> ((or (((eq (nat->Prop)) Y_10) X_1)) ((member_nat_o Y_10) Xa)))))). % Axiom fact_314_insert__compr__raw:(forall (X_1:(pname->Prop)) (Xa:((pname->Prop)->Prop)), (((eq ((pname->Prop)->Prop)) ((insert_pname_o X_1) Xa)) (collect_pname_o (fun (Y_10:(pname->Prop))=> ((or (((eq (pname->Prop)) Y_10) X_1)) ((member_pname_o Y_10) Xa)))))). % Axiom fact_315_insert__compr__raw:(forall (X_1:(x_a->Prop)) (Xa:((x_a->Prop)->Prop)), (((eq ((x_a->Prop)->Prop)) ((insert_a_o X_1) Xa)) (collect_a_o (fun (Y_10:(x_a->Prop))=> ((or (((eq (x_a->Prop)) Y_10) X_1)) ((member_a_o Y_10) Xa)))))). % Axiom fact_316_insert__compr__raw:(forall (X_1:nat) (Xa:(nat->Prop)), (((eq (nat->Prop)) ((insert_nat X_1) Xa)) (collect_nat (fun (Y_10:nat)=> ((or (((eq nat) Y_10) X_1)) ((member_nat Y_10) Xa)))))). % Axiom fact_317_insert__compr__raw:(forall (X_1:pname) (Xa:(pname->Prop)), (((eq (pname->Prop)) ((insert_pname X_1) Xa)) (collect_pname (fun (Y_10:pname)=> ((or (((eq pname) Y_10) X_1)) ((member_pname Y_10) Xa)))))). % Axiom fact_318_insert__compr__raw:(forall (X_1:x_a) (Xa:(x_a->Prop)), (((eq (x_a->Prop)) ((insert_a X_1) Xa)) (collect_a (fun (Y_10:x_a)=> ((or (((eq x_a) Y_10) X_1)) ((member_a Y_10) Xa)))))). % Axiom fact_319_subset__insertI:(forall (B_19:(pname->Prop)) (A_45:pname), ((ord_less_eq_pname_o B_19) ((insert_pname A_45) B_19))). % Axiom fact_320_subset__insertI:(forall (B_19:(nat->Prop)) (A_45:nat), ((ord_less_eq_nat_o B_19) ((insert_nat A_45) B_19))). % Axiom fact_321_subset__insertI:(forall (B_19:(x_a->Prop)) (A_45:x_a), ((ord_less_eq_a_o B_19) ((insert_a A_45) B_19))). % Axiom fact_322_insert__subset:(forall (X_19:nat) (A_44:(nat->Prop)) (B_18:(nat->Prop)), ((iff ((ord_less_eq_nat_o ((insert_nat X_19) A_44)) B_18)) ((and ((member_nat X_19) B_18)) ((ord_less_eq_nat_o A_44) B_18)))). % Axiom fact_323_insert__subset:(forall (X_19:pname) (A_44:(pname->Prop)) (B_18:(pname->Prop)), ((iff ((ord_less_eq_pname_o ((insert_pname X_19) A_44)) B_18)) ((and ((member_pname X_19) B_18)) ((ord_less_eq_pname_o A_44) B_18)))). % Axiom fact_324_insert__subset:(forall (X_19:x_a) (A_44:(x_a->Prop)) (B_18:(x_a->Prop)), ((iff ((ord_less_eq_a_o ((insert_a X_19) A_44)) B_18)) ((and ((member_a X_19) B_18)) ((ord_less_eq_a_o A_44) B_18)))). % Axiom fact_325_subset__insert:(forall (B_17:(nat->Prop)) (X_18:nat) (A_43:(nat->Prop)), ((((member_nat X_18) A_43)->False)->((iff ((ord_less_eq_nat_o A_43) ((insert_nat X_18) B_17))) ((ord_less_eq_nat_o A_43) B_17)))). % Axiom fact_326_subset__insert:(forall (B_17:(pname->Prop)) (X_18:pname) (A_43:(pname->Prop)), ((((member_pname X_18) A_43)->False)->((iff ((ord_less_eq_pname_o A_43) ((insert_pname X_18) B_17))) ((ord_less_eq_pname_o A_43) B_17)))). % Axiom fact_327_subset__insert:(forall (B_17:(x_a->Prop)) (X_18:x_a) (A_43:(x_a->Prop)), ((((member_a X_18) A_43)->False)->((iff ((ord_less_eq_a_o A_43) ((insert_a X_18) B_17))) ((ord_less_eq_a_o A_43) B_17)))). % Axiom fact_328_subset__insertI2:(forall (B_16:pname) (A_42:(pname->Prop)) (B_15:(pname->Prop)), (((ord_less_eq_pname_o A_42) B_15)->((ord_less_eq_pname_o A_42) ((insert_pname B_16) B_15)))). % Axiom fact_329_subset__insertI2:(forall (B_16:nat) (A_42:(nat->Prop)) (B_15:(nat->Prop)), (((ord_less_eq_nat_o A_42) B_15)->((ord_less_eq_nat_o A_42) ((insert_nat B_16) B_15)))). % Axiom fact_330_subset__insertI2:(forall (B_16:x_a) (A_42:(x_a->Prop)) (B_15:(x_a->Prop)), (((ord_less_eq_a_o A_42) B_15)->((ord_less_eq_a_o A_42) ((insert_a B_16) B_15)))). % Axiom fact_331_insert__mono:(forall (A_41:pname) (C_9:(pname->Prop)) (D_1:(pname->Prop)), (((ord_less_eq_pname_o C_9) D_1)->((ord_less_eq_pname_o ((insert_pname A_41) C_9)) ((insert_pname A_41) D_1)))). % Axiom fact_332_insert__mono:(forall (A_41:nat) (C_9:(nat->Prop)) (D_1:(nat->Prop)), (((ord_less_eq_nat_o C_9) D_1)->((ord_less_eq_nat_o ((insert_nat A_41) C_9)) ((insert_nat A_41) D_1)))). % Axiom fact_333_insert__mono:(forall (A_41:x_a) (C_9:(x_a->Prop)) (D_1:(x_a->Prop)), (((ord_less_eq_a_o C_9) D_1)->((ord_less_eq_a_o ((insert_a A_41) C_9)) ((insert_a A_41) D_1)))). % Axiom fact_334_image__insert:(forall (F_15:(x_a->pname)) (A_40:x_a) (B_14:(x_a->Prop)), (((eq (pname->Prop)) ((image_a_pname F_15) ((insert_a A_40) B_14))) ((insert_pname (F_15 A_40)) ((image_a_pname F_15) B_14)))). % Axiom fact_335_image__insert:(forall (F_15:(x_a->nat)) (A_40:x_a) (B_14:(x_a->Prop)), (((eq (nat->Prop)) ((image_a_nat F_15) ((insert_a A_40) B_14))) ((insert_nat (F_15 A_40)) ((image_a_nat F_15) B_14)))). % Axiom fact_336_image__insert:(forall (F_15:(nat->x_a)) (A_40:nat) (B_14:(nat->Prop)), (((eq (x_a->Prop)) ((image_nat_a F_15) ((insert_nat A_40) B_14))) ((insert_a (F_15 A_40)) ((image_nat_a F_15) B_14)))). % Axiom fact_337_image__insert:(forall (F_15:(pname->x_a)) (A_40:pname) (B_14:(pname->Prop)), (((eq (x_a->Prop)) ((image_pname_a F_15) ((insert_pname A_40) B_14))) ((insert_a (F_15 A_40)) ((image_pname_a F_15) B_14)))). % Axiom fact_338_insert__image:(forall (F_14:(pname->pname)) (X_17:pname) (A_39:(pname->Prop)), (((member_pname X_17) A_39)->(((eq (pname->Prop)) ((insert_pname (F_14 X_17)) ((image_pname_pname F_14) A_39))) ((image_pname_pname F_14) A_39)))). % Axiom fact_339_insert__image:(forall (F_14:(pname->nat)) (X_17:pname) (A_39:(pname->Prop)), (((member_pname X_17) A_39)->(((eq (nat->Prop)) ((insert_nat (F_14 X_17)) ((image_pname_nat F_14) A_39))) ((image_pname_nat F_14) A_39)))). % Axiom fact_340_insert__image:(forall (F_14:(x_a->pname)) (X_17:x_a) (A_39:(x_a->Prop)), (((member_a X_17) A_39)->(((eq (pname->Prop)) ((insert_pname (F_14 X_17)) ((image_a_pname F_14) A_39))) ((image_a_pname F_14) A_39)))). % Axiom fact_341_insert__image:(forall (F_14:(x_a->nat)) (X_17:x_a) (A_39:(x_a->Prop)), (((member_a X_17) A_39)->(((eq (nat->Prop)) ((insert_nat (F_14 X_17)) ((image_a_nat F_14) A_39))) ((image_a_nat F_14) A_39)))). % Axiom fact_342_insert__image:(forall (F_14:(nat->x_a)) (X_17:nat) (A_39:(nat->Prop)), (((member_nat X_17) A_39)->(((eq (x_a->Prop)) ((insert_a (F_14 X_17)) ((image_nat_a F_14) A_39))) ((image_nat_a F_14) A_39)))). % Axiom fact_343_insert__image:(forall (F_14:(pname->x_a)) (X_17:pname) (A_39:(pname->Prop)), (((member_pname X_17) A_39)->(((eq (x_a->Prop)) ((insert_a (F_14 X_17)) ((image_pname_a F_14) A_39))) ((image_pname_a F_14) A_39)))). % Axiom fact_344_subset__image__iff:(forall (B_13:(x_a->Prop)) (F_13:(nat->x_a)) (A_38:(nat->Prop)), ((iff ((ord_less_eq_a_o B_13) ((image_nat_a F_13) A_38))) ((ex (nat->Prop)) (fun (AA:(nat->Prop))=> ((and ((ord_less_eq_nat_o AA) A_38)) (((eq (x_a->Prop)) B_13) ((image_nat_a F_13) AA))))))). % Axiom fact_345_subset__image__iff:(forall (B_13:(pname->Prop)) (F_13:(x_a->pname)) (A_38:(x_a->Prop)), ((iff ((ord_less_eq_pname_o B_13) ((image_a_pname F_13) A_38))) ((ex (x_a->Prop)) (fun (AA:(x_a->Prop))=> ((and ((ord_less_eq_a_o AA) A_38)) (((eq (pname->Prop)) B_13) ((image_a_pname F_13) AA))))))). % Axiom fact_346_subset__image__iff:(forall (B_13:(nat->Prop)) (F_13:(x_a->nat)) (A_38:(x_a->Prop)), ((iff ((ord_less_eq_nat_o B_13) ((image_a_nat F_13) A_38))) ((ex (x_a->Prop)) (fun (AA:(x_a->Prop))=> ((and ((ord_less_eq_a_o AA) A_38)) (((eq (nat->Prop)) B_13) ((image_a_nat F_13) AA))))))). % Axiom fact_347_subset__image__iff:(forall (B_13:(x_a->Prop)) (F_13:(pname->x_a)) (A_38:(pname->Prop)), ((iff ((ord_less_eq_a_o B_13) ((image_pname_a F_13) A_38))) ((ex (pname->Prop)) (fun (AA:(pname->Prop))=> ((and ((ord_less_eq_pname_o AA) A_38)) (((eq (x_a->Prop)) B_13) ((image_pname_a F_13) AA))))))). % Axiom fact_348_image__mono:(forall (F_12:(x_a->pname)) (A_37:(x_a->Prop)) (B_12:(x_a->Prop)), (((ord_less_eq_a_o A_37) B_12)->((ord_less_eq_pname_o ((image_a_pname F_12) A_37)) ((image_a_pname F_12) B_12)))). % Axiom fact_349_image__mono:(forall (F_12:(x_a->nat)) (A_37:(x_a->Prop)) (B_12:(x_a->Prop)), (((ord_less_eq_a_o A_37) B_12)->((ord_less_eq_nat_o ((image_a_nat F_12) A_37)) ((image_a_nat F_12) B_12)))). % Axiom fact_350_image__mono:(forall (F_12:(nat->x_a)) (A_37:(nat->Prop)) (B_12:(nat->Prop)), (((ord_less_eq_nat_o A_37) B_12)->((ord_less_eq_a_o ((image_nat_a F_12) A_37)) ((image_nat_a F_12) B_12)))). % Axiom fact_351_image__mono:(forall (F_12:(pname->x_a)) (A_37:(pname->Prop)) (B_12:(pname->Prop)), (((ord_less_eq_pname_o A_37) B_12)->((ord_less_eq_a_o ((image_pname_a F_12) A_37)) ((image_pname_a F_12) B_12)))). % Axiom fact_352_imageE:(forall (B_11:pname) (F_11:(nat->pname)) (A_36:(nat->Prop)), (((member_pname B_11) ((image_nat_pname F_11) A_36))->((forall (X_1:nat), ((((eq pname) B_11) (F_11 X_1))->(((member_nat X_1) A_36)->False)))->False))). % Axiom fact_353_imageE:(forall (B_11:x_a) (F_11:(nat->x_a)) (A_36:(nat->Prop)), (((member_a B_11) ((image_nat_a F_11) A_36))->((forall (X_1:nat), ((((eq x_a) B_11) (F_11 X_1))->(((member_nat X_1) A_36)->False)))->False))). % Axiom fact_354_imageE:(forall (B_11:nat) (F_11:(pname->nat)) (A_36:(pname->Prop)), (((member_nat B_11) ((image_pname_nat F_11) A_36))->((forall (X_1:pname), ((((eq nat) B_11) (F_11 X_1))->(((member_pname X_1) A_36)->False)))->False))). % Axiom fact_355_imageE:(forall (B_11:nat) (F_11:(x_a->nat)) (A_36:(x_a->Prop)), (((member_nat B_11) ((image_a_nat F_11) A_36))->((forall (X_1:x_a), ((((eq nat) B_11) (F_11 X_1))->(((member_a X_1) A_36)->False)))->False))). % Axiom fact_356_imageE:(forall (B_11:x_a) (F_11:(pname->x_a)) (A_36:(pname->Prop)), (((member_a B_11) ((image_pname_a F_11) A_36))->((forall (X_1:pname), ((((eq x_a) B_11) (F_11 X_1))->(((member_pname X_1) A_36)->False)))->False))). % Axiom fact_357_subsetI:(forall (B_10:(nat->Prop)) (A_35:(nat->Prop)), ((forall (X_1:nat), (((member_nat X_1) A_35)->((member_nat X_1) B_10)))->((ord_less_eq_nat_o A_35) B_10))). % Axiom fact_358_subsetI:(forall (B_10:(pname->Prop)) (A_35:(pname->Prop)), ((forall (X_1:pname), (((member_pname X_1) A_35)->((member_pname X_1) B_10)))->((ord_less_eq_pname_o A_35) B_10))). % Axiom fact_359_subsetI:(forall (B_10:(x_a->Prop)) (A_35:(x_a->Prop)), ((forall (X_1:x_a), (((member_a X_1) A_35)->((member_a X_1) B_10)))->((ord_less_eq_a_o A_35) B_10))). % Axiom fact_360_image__subsetI:(forall (F_10:(pname->nat)) (B_9:(nat->Prop)) (A_34:(pname->Prop)), ((forall (X_1:pname), (((member_pname X_1) A_34)->((member_nat (F_10 X_1)) B_9)))->((ord_less_eq_nat_o ((image_pname_nat F_10) A_34)) B_9))). % Axiom fact_361_image__subsetI:(forall (F_10:(pname->pname)) (B_9:(pname->Prop)) (A_34:(pname->Prop)), ((forall (X_1:pname), (((member_pname X_1) A_34)->((member_pname (F_10 X_1)) B_9)))->((ord_less_eq_pname_o ((image_pname_pname F_10) A_34)) B_9))). % Axiom fact_362_image__subsetI:(forall (F_10:(x_a->nat)) (B_9:(nat->Prop)) (A_34:(x_a->Prop)), ((forall (X_1:x_a), (((member_a X_1) A_34)->((member_nat (F_10 X_1)) B_9)))->((ord_less_eq_nat_o ((image_a_nat F_10) A_34)) B_9))). % Axiom fact_363_image__subsetI:(forall (F_10:(x_a->pname)) (B_9:(pname->Prop)) (A_34:(x_a->Prop)), ((forall (X_1:x_a), (((member_a X_1) A_34)->((member_pname (F_10 X_1)) B_9)))->((ord_less_eq_pname_o ((image_a_pname F_10) A_34)) B_9))). % Axiom fact_364_image__subsetI:(forall (F_10:(nat->pname)) (B_9:(pname->Prop)) (A_34:(nat->Prop)), ((forall (X_1:nat), (((member_nat X_1) A_34)->((member_pname (F_10 X_1)) B_9)))->((ord_less_eq_pname_o ((image_nat_pname F_10) A_34)) B_9))). % Axiom fact_365_image__subsetI:(forall (F_10:(nat->x_a)) (B_9:(x_a->Prop)) (A_34:(nat->Prop)), ((forall (X_1:nat), (((member_nat X_1) A_34)->((member_a (F_10 X_1)) B_9)))->((ord_less_eq_a_o ((image_nat_a F_10) A_34)) B_9))). % Axiom fact_366_image__subsetI:(forall (F_10:(pname->x_a)) (B_9:(x_a->Prop)) (A_34:(pname->Prop)), ((forall (X_1:pname), (((member_pname X_1) A_34)->((member_a (F_10 X_1)) B_9)))->((ord_less_eq_a_o ((image_pname_a F_10) A_34)) B_9))). % Axiom fact_367_order__refl:(forall (X_16:Prop), ((ord_less_eq_o X_16) X_16)). % Axiom fact_368_order__refl:(forall (X_16:(pname->Prop)), ((ord_less_eq_pname_o X_16) X_16)). % Axiom fact_369_order__refl:(forall (X_16:(nat->Prop)), ((ord_less_eq_nat_o X_16) X_16)). % Axiom fact_370_order__refl:(forall (X_16:(x_a->Prop)), ((ord_less_eq_a_o X_16) X_16)). % Axiom fact_371_order__refl:(forall (X_16:nat), ((ord_less_eq_nat X_16) X_16)). % Axiom fact_372_finite__nat__set__iff__bounded__le:(forall (N_1:(nat->Prop)), ((iff (finite_finite_nat N_1)) ((ex nat) (fun (M_1:nat)=> (forall (X_1:nat), (((member_nat X_1) N_1)->((ord_less_eq_nat X_1) M_1))))))). % Axiom fact_373_assms_I3_J:(forall (G:(x_a->Prop)) (C:com), ((wt C)->((forall (X_1:pname), (((member_pname X_1) u)->((p G) ((insert_a (mgt_call X_1)) bot_bot_a_o))))->((p G) ((insert_a (mgt C)) bot_bot_a_o))))). % Axiom fact_374_le__fun__def:(forall (F_9:(pname->Prop)) (G_3:(pname->Prop)), ((iff ((ord_less_eq_pname_o F_9) G_3)) (forall (X_1:pname), ((ord_less_eq_o (F_9 X_1)) (G_3 X_1))))). % Axiom fact_375_le__fun__def:(forall (F_9:(nat->Prop)) (G_3:(nat->Prop)), ((iff ((ord_less_eq_nat_o F_9) G_3)) (forall (X_1:nat), ((ord_less_eq_o (F_9 X_1)) (G_3 X_1))))). % Axiom fact_376_le__fun__def:(forall (F_9:(x_a->Prop)) (G_3:(x_a->Prop)), ((iff ((ord_less_eq_a_o F_9) G_3)) (forall (X_1:x_a), ((ord_less_eq_o (F_9 X_1)) (G_3 X_1))))). % Axiom fact_377_le__funD:(forall (X_15:pname) (F_8:(pname->Prop)) (G_2:(pname->Prop)), (((ord_less_eq_pname_o F_8) G_2)->((ord_less_eq_o (F_8 X_15)) (G_2 X_15)))). % Axiom fact_378_le__funD:(forall (X_15:nat) (F_8:(nat->Prop)) (G_2:(nat->Prop)), (((ord_less_eq_nat_o F_8) G_2)->((ord_less_eq_o (F_8 X_15)) (G_2 X_15)))). % Axiom fact_379_le__funD:(forall (X_15:x_a) (F_8:(x_a->Prop)) (G_2:(x_a->Prop)), (((ord_less_eq_a_o F_8) G_2)->((ord_less_eq_o (F_8 X_15)) (G_2 X_15)))). % Axiom fact_380_le__funE:(forall (X_14:pname) (F_7:(pname->Prop)) (G_1:(pname->Prop)), (((ord_less_eq_pname_o F_7) G_1)->((ord_less_eq_o (F_7 X_14)) (G_1 X_14)))). % Axiom fact_381_le__funE:(forall (X_14:nat) (F_7:(nat->Prop)) (G_1:(nat->Prop)), (((ord_less_eq_nat_o F_7) G_1)->((ord_less_eq_o (F_7 X_14)) (G_1 X_14)))). % Axiom fact_382_le__funE:(forall (X_14:x_a) (F_7:(x_a->Prop)) (G_1:(x_a->Prop)), (((ord_less_eq_a_o F_7) G_1)->((ord_less_eq_o (F_7 X_14)) (G_1 X_14)))). % Axiom fact_383_emptyE:(forall (A_33:nat), (((member_nat A_33) bot_bot_nat_o)->False)). % Axiom fact_384_emptyE:(forall (A_33:pname), (((member_pname A_33) bot_bot_pname_o)->False)). % Axiom fact_385_emptyE:(forall (A_33:x_a), (((member_a A_33) bot_bot_a_o)->False)). % Axiom fact_386_finite_OemptyI:(finite_finite_nat_o bot_bot_nat_o_o). % Axiom fact_387_finite_OemptyI:(finite297249702name_o bot_bot_pname_o_o). % Axiom fact_388_finite_OemptyI:(finite_finite_a_o bot_bot_a_o_o). % Axiom fact_389_finite_OemptyI:(finite_finite_a bot_bot_a_o). % Axiom fact_390_finite_OemptyI:(finite_finite_pname bot_bot_pname_o). % Axiom fact_391_finite_OemptyI:(finite_finite_nat bot_bot_nat_o). % Axiom fact_392_empty__subsetI:(forall (A_32:(pname->Prop)), ((ord_less_eq_pname_o bot_bot_pname_o) A_32)). % Axiom fact_393_empty__subsetI:(forall (A_32:(nat->Prop)), ((ord_less_eq_nat_o bot_bot_nat_o) A_32)). % Axiom fact_394_empty__subsetI:(forall (A_32:(x_a->Prop)), ((ord_less_eq_a_o bot_bot_a_o) A_32)). % Axiom fact_395_equals0D:(forall (A_31:nat) (A_30:(nat->Prop)), ((((eq (nat->Prop)) A_30) bot_bot_nat_o)->(((member_nat A_31) A_30)->False))). % Axiom fact_396_equals0D:(forall (A_31:pname) (A_30:(pname->Prop)), ((((eq (pname->Prop)) A_30) bot_bot_pname_o)->(((member_pname A_31) A_30)->False))). % Axiom fact_397_equals0D:(forall (A_31:x_a) (A_30:(x_a->Prop)), ((((eq (x_a->Prop)) A_30) bot_bot_a_o)->(((member_a A_31) A_30)->False))). % Axiom fact_398_Collect__empty__eq:(forall (P_6:(pname->Prop)), ((iff (((eq (pname->Prop)) (collect_pname P_6)) bot_bot_pname_o)) (forall (X_1:pname), ((P_6 X_1)->False)))). % Axiom fact_399_Collect__empty__eq:(forall (P_6:((nat->Prop)->Prop)), ((iff (((eq ((nat->Prop)->Prop)) (collect_nat_o P_6)) bot_bot_nat_o_o)) (forall (X_1:(nat->Prop)), ((P_6 X_1)->False)))). % Axiom fact_400_Collect__empty__eq:(forall (P_6:((pname->Prop)->Prop)), ((iff (((eq ((pname->Prop)->Prop)) (collect_pname_o P_6)) bot_bot_pname_o_o)) (forall (X_1:(pname->Prop)), ((P_6 X_1)->False)))). % Axiom fact_401_Collect__empty__eq:(forall (P_6:((x_a->Prop)->Prop)), ((iff (((eq ((x_a->Prop)->Prop)) (collect_a_o P_6)) bot_bot_a_o_o)) (forall (X_1:(x_a->Prop)), ((P_6 X_1)->False)))). % Axiom fact_402_Collect__empty__eq:(forall (P_6:(x_a->Prop)), ((iff (((eq (x_a->Prop)) (collect_a P_6)) bot_bot_a_o)) (forall (X_1:x_a), ((P_6 X_1)->False)))). % Axiom fact_403_Collect__empty__eq:(forall (P_6:(nat->Prop)), ((iff (((eq (nat->Prop)) (collect_nat P_6)) bot_bot_nat_o)) (forall (X_1:nat), ((P_6 X_1)->False)))). % Axiom fact_404_empty__iff:(forall (C_8:nat), (((member_nat C_8) bot_bot_nat_o)->False)). % Axiom fact_405_empty__iff:(forall (C_8:pname), (((member_pname C_8) bot_bot_pname_o)->False)). % Axiom fact_406_empty__iff:(forall (C_8:x_a), (((member_a C_8) bot_bot_a_o)->False)). % Axiom fact_407_empty__Collect__eq:(forall (P_5:(pname->Prop)), ((iff (((eq (pname->Prop)) bot_bot_pname_o) (collect_pname P_5))) (forall (X_1:pname), ((P_5 X_1)->False)))). % Axiom fact_408_empty__Collect__eq:(forall (P_5:((nat->Prop)->Prop)), ((iff (((eq ((nat->Prop)->Prop)) bot_bot_nat_o_o) (collect_nat_o P_5))) (forall (X_1:(nat->Prop)), ((P_5 X_1)->False)))). % Axiom fact_409_empty__Collect__eq:(forall (P_5:((pname->Prop)->Prop)), ((iff (((eq ((pname->Prop)->Prop)) bot_bot_pname_o_o) (collect_pname_o P_5))) (forall (X_1:(pname->Prop)), ((P_5 X_1)->False)))). % Axiom fact_410_empty__Collect__eq:(forall (P_5:((x_a->Prop)->Prop)), ((iff (((eq ((x_a->Prop)->Prop)) bot_bot_a_o_o) (collect_a_o P_5))) (forall (X_1:(x_a->Prop)), ((P_5 X_1)->False)))). % Axiom fact_411_empty__Collect__eq:(forall (P_5:(x_a->Prop)), ((iff (((eq (x_a->Prop)) bot_bot_a_o) (collect_a P_5))) (forall (X_1:x_a), ((P_5 X_1)->False)))). % Axiom fact_412_empty__Collect__eq:(forall (P_5:(nat->Prop)), ((iff (((eq (nat->Prop)) bot_bot_nat_o) (collect_nat P_5))) (forall (X_1:nat), ((P_5 X_1)->False)))). % Axiom fact_413_ex__in__conv:(forall (A_29:(nat->Prop)), ((iff ((ex nat) (fun (X_1:nat)=> ((member_nat X_1) A_29)))) (not (((eq (nat->Prop)) A_29) bot_bot_nat_o)))). % Axiom fact_414_ex__in__conv:(forall (A_29:(pname->Prop)), ((iff ((ex pname) (fun (X_1:pname)=> ((member_pname X_1) A_29)))) (not (((eq (pname->Prop)) A_29) bot_bot_pname_o)))). % Axiom fact_415_ex__in__conv:(forall (A_29:(x_a->Prop)), ((iff ((ex x_a) (fun (X_1:x_a)=> ((member_a X_1) A_29)))) (not (((eq (x_a->Prop)) A_29) bot_bot_a_o)))). % Axiom fact_416_all__not__in__conv:(forall (A_28:(nat->Prop)), ((iff (forall (X_1:nat), (((member_nat X_1) A_28)->False))) (((eq (nat->Prop)) A_28) bot_bot_nat_o))). % Axiom fact_417_all__not__in__conv:(forall (A_28:(pname->Prop)), ((iff (forall (X_1:pname), (((member_pname X_1) A_28)->False))) (((eq (pname->Prop)) A_28) bot_bot_pname_o))). % Axiom fact_418_all__not__in__conv:(forall (A_28:(x_a->Prop)), ((iff (forall (X_1:x_a), (((member_a X_1) A_28)->False))) (((eq (x_a->Prop)) A_28) bot_bot_a_o))). % Axiom fact_419_empty__def:(((eq (pname->Prop)) bot_bot_pname_o) (collect_pname (fun (X_1:pname)=> False))). % Axiom fact_420_empty__def:(((eq ((nat->Prop)->Prop)) bot_bot_nat_o_o) (collect_nat_o (fun (X_1:(nat->Prop))=> False))). % Axiom fact_421_empty__def:(((eq ((pname->Prop)->Prop)) bot_bot_pname_o_o) (collect_pname_o (fun (X_1:(pname->Prop))=> False))). % Axiom fact_422_empty__def:(((eq ((x_a->Prop)->Prop)) bot_bot_a_o_o) (collect_a_o (fun (X_1:(x_a->Prop))=> False))). % Axiom fact_423_empty__def:(((eq (x_a->Prop)) bot_bot_a_o) (collect_a (fun (X_1:x_a)=> False))). % Axiom fact_424_empty__def:(((eq (nat->Prop)) bot_bot_nat_o) (collect_nat (fun (X_1:nat)=> False))). % Axiom fact_425_bot__fun__def:(forall (X_1:pname), ((iff (bot_bot_pname_o X_1)) bot_bot_o)). % Axiom fact_426_bot__fun__def:(forall (X_1:nat), ((iff (bot_bot_nat_o X_1)) bot_bot_o)). % Axiom fact_427_bot__fun__def:(forall (X_1:x_a), ((iff (bot_bot_a_o X_1)) bot_bot_o)). % Axiom fact_428_bot__apply:(forall (X_13:pname), ((iff (bot_bot_pname_o X_13)) bot_bot_o)). % Axiom fact_429_bot__apply:(forall (X_13:nat), ((iff (bot_bot_nat_o X_13)) bot_bot_o)). % Axiom fact_430_bot__apply:(forall (X_13:x_a), ((iff (bot_bot_a_o X_13)) bot_bot_o)). % Axiom fact_431_le__bot:(forall (A_27:(pname->Prop)), (((ord_less_eq_pname_o A_27) bot_bot_pname_o)->(((eq (pname->Prop)) A_27) bot_bot_pname_o))). % Axiom fact_432_le__bot:(forall (A_27:Prop), (((ord_less_eq_o A_27) bot_bot_o)->((iff A_27) bot_bot_o))). % Axiom fact_433_le__bot:(forall (A_27:(nat->Prop)), (((ord_less_eq_nat_o A_27) bot_bot_nat_o)->(((eq (nat->Prop)) A_27) bot_bot_nat_o))). % Axiom fact_434_le__bot:(forall (A_27:(x_a->Prop)), (((ord_less_eq_a_o A_27) bot_bot_a_o)->(((eq (x_a->Prop)) A_27) bot_bot_a_o))). % Axiom fact_435_le__bot:(forall (A_27:nat), (((ord_less_eq_nat A_27) bot_bot_nat)->(((eq nat) A_27) bot_bot_nat))). % Axiom fact_436_bot__unique:(forall (A_26:(pname->Prop)), ((iff ((ord_less_eq_pname_o A_26) bot_bot_pname_o)) (((eq (pname->Prop)) A_26) bot_bot_pname_o))). % Axiom fact_437_bot__unique:(forall (A_26:Prop), ((iff ((ord_less_eq_o A_26) bot_bot_o)) ((iff A_26) bot_bot_o))). % Axiom fact_438_bot__unique:(forall (A_26:(nat->Prop)), ((iff ((ord_less_eq_nat_o A_26) bot_bot_nat_o)) (((eq (nat->Prop)) A_26) bot_bot_nat_o))). % Axiom fact_439_bot__unique:(forall (A_26:(x_a->Prop)), ((iff ((ord_less_eq_a_o A_26) bot_bot_a_o)) (((eq (x_a->Prop)) A_26) bot_bot_a_o))). % Axiom fact_440_bot__unique:(forall (A_26:nat), ((iff ((ord_less_eq_nat A_26) bot_bot_nat)) (((eq nat) A_26) bot_bot_nat))). % Axiom fact_441_bot__least:(forall (A_25:(pname->Prop)), ((ord_less_eq_pname_o bot_bot_pname_o) A_25)). % Axiom fact_442_bot__least:(forall (A_25:Prop), ((ord_less_eq_o bot_bot_o) A_25)). % Axiom fact_443_bot__least:(forall (A_25:(nat->Prop)), ((ord_less_eq_nat_o bot_bot_nat_o) A_25)). % Axiom fact_444_bot__least:(forall (A_25:(x_a->Prop)), ((ord_less_eq_a_o bot_bot_a_o) A_25)). % Axiom fact_445_bot__least:(forall (A_25:nat), ((ord_less_eq_nat bot_bot_nat) A_25)). % Axiom fact_446_singleton__inject:(forall (A_24:pname) (B_8:pname), ((((eq (pname->Prop)) ((insert_pname A_24) bot_bot_pname_o)) ((insert_pname B_8) bot_bot_pname_o))->(((eq pname) A_24) B_8))). % Axiom fact_447_singleton__inject:(forall (A_24:nat) (B_8:nat), ((((eq (nat->Prop)) ((insert_nat A_24) bot_bot_nat_o)) ((insert_nat B_8) bot_bot_nat_o))->(((eq nat) A_24) B_8))). % Axiom fact_448_singleton__inject:(forall (A_24:x_a) (B_8:x_a), ((((eq (x_a->Prop)) ((insert_a A_24) bot_bot_a_o)) ((insert_a B_8) bot_bot_a_o))->(((eq x_a) A_24) B_8))). % Axiom fact_449_singletonE:(forall (B_7:nat) (A_23:nat), (((member_nat B_7) ((insert_nat A_23) bot_bot_nat_o))->(((eq nat) B_7) A_23))). % Axiom fact_450_singletonE:(forall (B_7:pname) (A_23:pname), (((member_pname B_7) ((insert_pname A_23) bot_bot_pname_o))->(((eq pname) B_7) A_23))). % Axiom fact_451_singletonE:(forall (B_7:x_a) (A_23:x_a), (((member_a B_7) ((insert_a A_23) bot_bot_a_o))->(((eq x_a) B_7) A_23))). % Axiom fact_452_doubleton__eq__iff:(forall (A_22:pname) (B_6:pname) (C_7:pname) (D:pname), ((iff (((eq (pname->Prop)) ((insert_pname A_22) ((insert_pname B_6) bot_bot_pname_o))) ((insert_pname C_7) ((insert_pname D) bot_bot_pname_o)))) ((or ((and (((eq pname) A_22) C_7)) (((eq pname) B_6) D))) ((and (((eq pname) A_22) D)) (((eq pname) B_6) C_7))))). % Axiom fact_453_doubleton__eq__iff:(forall (A_22:nat) (B_6:nat) (C_7:nat) (D:nat), ((iff (((eq (nat->Prop)) ((insert_nat A_22) ((insert_nat B_6) bot_bot_nat_o))) ((insert_nat C_7) ((insert_nat D) bot_bot_nat_o)))) ((or ((and (((eq nat) A_22) C_7)) (((eq nat) B_6) D))) ((and (((eq nat) A_22) D)) (((eq nat) B_6) C_7))))). % Axiom fact_454_doubleton__eq__iff:(forall (A_22:x_a) (B_6:x_a) (C_7:x_a) (D:x_a), ((iff (((eq (x_a->Prop)) ((insert_a A_22) ((insert_a B_6) bot_bot_a_o))) ((insert_a C_7) ((insert_a D) bot_bot_a_o)))) ((or ((and (((eq x_a) A_22) C_7)) (((eq x_a) B_6) D))) ((and (((eq x_a) A_22) D)) (((eq x_a) B_6) C_7))))). % Axiom fact_455_singleton__iff:(forall (B_5:nat) (A_21:nat), ((iff ((member_nat B_5) ((insert_nat A_21) bot_bot_nat_o))) (((eq nat) B_5) A_21))). % Axiom fact_456_singleton__iff:(forall (B_5:pname) (A_21:pname), ((iff ((member_pname B_5) ((insert_pname A_21) bot_bot_pname_o))) (((eq pname) B_5) A_21))). % Axiom fact_457_singleton__iff:(forall (B_5:x_a) (A_21:x_a), ((iff ((member_a B_5) ((insert_a A_21) bot_bot_a_o))) (((eq x_a) B_5) A_21))). % Axiom fact_458_insert__not__empty:(forall (A_20:pname) (A_19:(pname->Prop)), (not (((eq (pname->Prop)) ((insert_pname A_20) A_19)) bot_bot_pname_o))). % Axiom fact_459_insert__not__empty:(forall (A_20:nat) (A_19:(nat->Prop)), (not (((eq (nat->Prop)) ((insert_nat A_20) A_19)) bot_bot_nat_o))). % Axiom fact_460_insert__not__empty:(forall (A_20:x_a) (A_19:(x_a->Prop)), (not (((eq (x_a->Prop)) ((insert_a A_20) A_19)) bot_bot_a_o))). % Axiom fact_461_empty__not__insert:(forall (A_18:pname) (A_17:(pname->Prop)), (not (((eq (pname->Prop)) bot_bot_pname_o) ((insert_pname A_18) A_17)))). % Axiom fact_462_empty__not__insert:(forall (A_18:nat) (A_17:(nat->Prop)), (not (((eq (nat->Prop)) bot_bot_nat_o) ((insert_nat A_18) A_17)))). % Axiom fact_463_empty__not__insert:(forall (A_18:x_a) (A_17:(x_a->Prop)), (not (((eq (x_a->Prop)) bot_bot_a_o) ((insert_a A_18) A_17)))). % Axiom fact_464_subset__empty:(forall (A_16:(nat->Prop)), ((iff ((ord_less_eq_nat_o A_16) bot_bot_nat_o)) (((eq (nat->Prop)) A_16) bot_bot_nat_o))). % Axiom fact_465_subset__empty:(forall (A_16:(pname->Prop)), ((iff ((ord_less_eq_pname_o A_16) bot_bot_pname_o)) (((eq (pname->Prop)) A_16) bot_bot_pname_o))). % Axiom fact_466_subset__empty:(forall (A_16:(x_a->Prop)), ((iff ((ord_less_eq_a_o A_16) bot_bot_a_o)) (((eq (x_a->Prop)) A_16) bot_bot_a_o))). % Axiom fact_467_image__is__empty:(forall (F_6:(pname->x_a)) (A_15:(pname->Prop)), ((iff (((eq (x_a->Prop)) ((image_pname_a F_6) A_15)) bot_bot_a_o)) (((eq (pname->Prop)) A_15) bot_bot_pname_o))). % Axiom fact_468_image__empty:(forall (F_5:(pname->x_a)), (((eq (x_a->Prop)) ((image_pname_a F_5) bot_bot_pname_o)) bot_bot_a_o)). % Axiom fact_469_empty__is__image:(forall (F_4:(pname->x_a)) (A_14:(pname->Prop)), ((iff (((eq (x_a->Prop)) bot_bot_a_o) ((image_pname_a F_4) A_14))) (((eq (pname->Prop)) A_14) bot_bot_pname_o))). % Axiom fact_470_Collect__conv__if:(forall (P_4:(nat->Prop)) (A_13:nat), ((and ((P_4 A_13)->(((eq (nat->Prop)) (collect_nat (fun (X_1:nat)=> ((and (((eq nat) X_1) A_13)) (P_4 X_1))))) ((insert_nat A_13) bot_bot_nat_o)))) (((P_4 A_13)->False)->(((eq (nat->Prop)) (collect_nat (fun (X_1:nat)=> ((and (((eq nat) X_1) A_13)) (P_4 X_1))))) bot_bot_nat_o)))). % Axiom fact_471_Collect__conv__if:(forall (P_4:(pname->Prop)) (A_13:pname), ((and ((P_4 A_13)->(((eq (pname->Prop)) (collect_pname (fun (X_1:pname)=> ((and (((eq pname) X_1) A_13)) (P_4 X_1))))) ((insert_pname A_13) bot_bot_pname_o)))) (((P_4 A_13)->False)->(((eq (pname->Prop)) (collect_pname (fun (X_1:pname)=> ((and (((eq pname) X_1) A_13)) (P_4 X_1))))) bot_bot_pname_o)))). % Axiom fact_472_Collect__conv__if:(forall (P_4:(x_a->Prop)) (A_13:x_a), ((and ((P_4 A_13)->(((eq (x_a->Prop)) (collect_a (fun (X_1:x_a)=> ((and (((eq x_a) X_1) A_13)) (P_4 X_1))))) ((insert_a A_13) bot_bot_a_o)))) (((P_4 A_13)->False)->(((eq (x_a->Prop)) (collect_a (fun (X_1:x_a)=> ((and (((eq x_a) X_1) A_13)) (P_4 X_1))))) bot_bot_a_o)))). % Axiom fact_473_Collect__conv__if:(forall (P_4:((nat->Prop)->Prop)) (A_13:(nat->Prop)), ((and ((P_4 A_13)->(((eq ((nat->Prop)->Prop)) (collect_nat_o (fun (X_1:(nat->Prop))=> ((and (((eq (nat->Prop)) X_1) A_13)) (P_4 X_1))))) ((insert_nat_o A_13) bot_bot_nat_o_o)))) (((P_4 A_13)->False)->(((eq ((nat->Prop)->Prop)) (collect_nat_o (fun (X_1:(nat->Prop))=> ((and (((eq (nat->Prop)) X_1) A_13)) (P_4 X_1))))) bot_bot_nat_o_o)))). % Axiom fact_474_Collect__conv__if:(forall (P_4:((pname->Prop)->Prop)) (A_13:(pname->Prop)), ((and ((P_4 A_13)->(((eq ((pname->Prop)->Prop)) (collect_pname_o (fun (X_1:(pname->Prop))=> ((and (((eq (pname->Prop)) X_1) A_13)) (P_4 X_1))))) ((insert_pname_o A_13) bot_bot_pname_o_o)))) (((P_4 A_13)->False)->(((eq ((pname->Prop)->Prop)) (collect_pname_o (fun (X_1:(pname->Prop))=> ((and (((eq (pname->Prop)) X_1) A_13)) (P_4 X_1))))) bot_bot_pname_o_o)))). % Axiom fact_475_Collect__conv__if:(forall (P_4:((x_a->Prop)->Prop)) (A_13:(x_a->Prop)), ((and ((P_4 A_13)->(((eq ((x_a->Prop)->Prop)) (collect_a_o (fun (X_1:(x_a->Prop))=> ((and (((eq (x_a->Prop)) X_1) A_13)) (P_4 X_1))))) ((insert_a_o A_13) bot_bot_a_o_o)))) (((P_4 A_13)->False)->(((eq ((x_a->Prop)->Prop)) (collect_a_o (fun (X_1:(x_a->Prop))=> ((and (((eq (x_a->Prop)) X_1) A_13)) (P_4 X_1))))) bot_bot_a_o_o)))). % Axiom fact_476_Collect__conv__if2:(forall (P_3:(nat->Prop)) (A_12:nat), ((and ((P_3 A_12)->(((eq (nat->Prop)) (collect_nat (fun (X_1:nat)=> ((and (((eq nat) A_12) X_1)) (P_3 X_1))))) ((insert_nat A_12) bot_bot_nat_o)))) (((P_3 A_12)->False)->(((eq (nat->Prop)) (collect_nat (fun (X_1:nat)=> ((and (((eq nat) A_12) X_1)) (P_3 X_1))))) bot_bot_nat_o)))). % Axiom fact_477_Collect__conv__if2:(forall (P_3:(pname->Prop)) (A_12:pname), ((and ((P_3 A_12)->(((eq (pname->Prop)) (collect_pname (fun (X_1:pname)=> ((and (((eq pname) A_12) X_1)) (P_3 X_1))))) ((insert_pname A_12) bot_bot_pname_o)))) (((P_3 A_12)->False)->(((eq (pname->Prop)) (collect_pname (fun (X_1:pname)=> ((and (((eq pname) A_12) X_1)) (P_3 X_1))))) bot_bot_pname_o)))). % Axiom fact_478_Collect__conv__if2:(forall (P_3:(x_a->Prop)) (A_12:x_a), ((and ((P_3 A_12)->(((eq (x_a->Prop)) (collect_a (fun (X_1:x_a)=> ((and (((eq x_a) A_12) X_1)) (P_3 X_1))))) ((insert_a A_12) bot_bot_a_o)))) (((P_3 A_12)->False)->(((eq (x_a->Prop)) (collect_a (fun (X_1:x_a)=> ((and (((eq x_a) A_12) X_1)) (P_3 X_1))))) bot_bot_a_o)))). % Axiom fact_479_Collect__conv__if2:(forall (P_3:((nat->Prop)->Prop)) (A_12:(nat->Prop)), ((and ((P_3 A_12)->(((eq ((nat->Prop)->Prop)) (collect_nat_o (fun (X_1:(nat->Prop))=> ((and (((eq (nat->Prop)) A_12) X_1)) (P_3 X_1))))) ((insert_nat_o A_12) bot_bot_nat_o_o)))) (((P_3 A_12)->False)->(((eq ((nat->Prop)->Prop)) (collect_nat_o (fun (X_1:(nat->Prop))=> ((and (((eq (nat->Prop)) A_12) X_1)) (P_3 X_1))))) bot_bot_nat_o_o)))). % Axiom fact_480_Collect__conv__if2:(forall (P_3:((pname->Prop)->Prop)) (A_12:(pname->Prop)), ((and ((P_3 A_12)->(((eq ((pname->Prop)->Prop)) (collect_pname_o (fun (X_1:(pname->Prop))=> ((and (((eq (pname->Prop)) A_12) X_1)) (P_3 X_1))))) ((insert_pname_o A_12) bot_bot_pname_o_o)))) (((P_3 A_12)->False)->(((eq ((pname->Prop)->Prop)) (collect_pname_o (fun (X_1:(pname->Prop))=> ((and (((eq (pname->Prop)) A_12) X_1)) (P_3 X_1))))) bot_bot_pname_o_o)))). % Axiom fact_481_Collect__conv__if2:(forall (P_3:((x_a->Prop)->Prop)) (A_12:(x_a->Prop)), ((and ((P_3 A_12)->(((eq ((x_a->Prop)->Prop)) (collect_a_o (fun (X_1:(x_a->Prop))=> ((and (((eq (x_a->Prop)) A_12) X_1)) (P_3 X_1))))) ((insert_a_o A_12) bot_bot_a_o_o)))) (((P_3 A_12)->False)->(((eq ((x_a->Prop)->Prop)) (collect_a_o (fun (X_1:(x_a->Prop))=> ((and (((eq (x_a->Prop)) A_12) X_1)) (P_3 X_1))))) bot_bot_a_o_o)))). % Axiom fact_482_singleton__conv:(forall (A_11:nat), (((eq (nat->Prop)) (collect_nat (fun (X_1:nat)=> (((eq nat) X_1) A_11)))) ((insert_nat A_11) bot_bot_nat_o))). % Axiom fact_483_singleton__conv:(forall (A_11:pname), (((eq (pname->Prop)) (collect_pname (fun (X_1:pname)=> (((eq pname) X_1) A_11)))) ((insert_pname A_11) bot_bot_pname_o))). % Axiom fact_484_singleton__conv:(forall (A_11:x_a), (((eq (x_a->Prop)) (collect_a (fun (X_1:x_a)=> (((eq x_a) X_1) A_11)))) ((insert_a A_11) bot_bot_a_o))). % Axiom fact_485_singleton__conv:(forall (A_11:(nat->Prop)), (((eq ((nat->Prop)->Prop)) (collect_nat_o (fun (X_1:(nat->Prop))=> (((eq (nat->Prop)) X_1) A_11)))) ((insert_nat_o A_11) bot_bot_nat_o_o))). % Axiom fact_486_singleton__conv:(forall (A_11:(pname->Prop)), (((eq ((pname->Prop)->Prop)) (collect_pname_o (fun (X_1:(pname->Prop))=> (((eq (pname->Prop)) X_1) A_11)))) ((insert_pname_o A_11) bot_bot_pname_o_o))). % Axiom fact_487_singleton__conv:(forall (A_11:(x_a->Prop)), (((eq ((x_a->Prop)->Prop)) (collect_a_o (fun (X_1:(x_a->Prop))=> (((eq (x_a->Prop)) X_1) A_11)))) ((insert_a_o A_11) bot_bot_a_o_o))). % Axiom fact_488_singleton__conv2:(forall (A_10:nat), (((eq (nat->Prop)) (collect_nat (fequal_nat A_10))) ((insert_nat A_10) bot_bot_nat_o))). % Axiom fact_489_singleton__conv2:(forall (A_10:pname), (((eq (pname->Prop)) (collect_pname (fequal_pname A_10))) ((insert_pname A_10) bot_bot_pname_o))). % Axiom fact_490_singleton__conv2:(forall (A_10:x_a), (((eq (x_a->Prop)) (collect_a (fequal_a A_10))) ((insert_a A_10) bot_bot_a_o))). % Axiom fact_491_singleton__conv2:(forall (A_10:(nat->Prop)), (((eq ((nat->Prop)->Prop)) (collect_nat_o (fequal_nat_o A_10))) ((insert_nat_o A_10) bot_bot_nat_o_o))). % Axiom fact_492_singleton__conv2:(forall (A_10:(pname->Prop)), (((eq ((pname->Prop)->Prop)) (collect_pname_o (fequal_pname_o A_10))) ((insert_pname_o A_10) bot_bot_pname_o_o))). % Axiom fact_493_singleton__conv2:(forall (A_10:(x_a->Prop)), (((eq ((x_a->Prop)->Prop)) (collect_a_o (fequal_a_o A_10))) ((insert_a_o A_10) bot_bot_a_o_o))). % Axiom fact_494_subset__singletonD:(forall (A_9:(nat->Prop)) (X_12:nat), (((ord_less_eq_nat_o A_9) ((insert_nat X_12) bot_bot_nat_o))->((or (((eq (nat->Prop)) A_9) bot_bot_nat_o)) (((eq (nat->Prop)) A_9) ((insert_nat X_12) bot_bot_nat_o))))). % Axiom fact_495_subset__singletonD:(forall (A_9:(pname->Prop)) (X_12:pname), (((ord_less_eq_pname_o A_9) ((insert_pname X_12) bot_bot_pname_o))->((or (((eq (pname->Prop)) A_9) bot_bot_pname_o)) (((eq (pname->Prop)) A_9) ((insert_pname X_12) bot_bot_pname_o))))). % Axiom fact_496_subset__singletonD:(forall (A_9:(x_a->Prop)) (X_12:x_a), (((ord_less_eq_a_o A_9) ((insert_a X_12) bot_bot_a_o))->((or (((eq (x_a->Prop)) A_9) bot_bot_a_o)) (((eq (x_a->Prop)) A_9) ((insert_a X_12) bot_bot_a_o))))). % Axiom fact_497_image__constant:(forall (C_6:x_a) (X_11:pname) (A_8:(pname->Prop)), (((member_pname X_11) A_8)->(((eq (x_a->Prop)) ((image_pname_a (fun (X_1:pname)=> C_6)) A_8)) ((insert_a C_6) bot_bot_a_o)))). % Axiom fact_498_image__constant__conv:(forall (C_5:x_a) (A_7:(pname->Prop)), ((and ((((eq (pname->Prop)) A_7) bot_bot_pname_o)->(((eq (x_a->Prop)) ((image_pname_a (fun (X_1:pname)=> C_5)) A_7)) bot_bot_a_o))) ((not (((eq (pname->Prop)) A_7) bot_bot_pname_o))->(((eq (x_a->Prop)) ((image_pname_a (fun (X_1:pname)=> C_5)) A_7)) ((insert_a C_5) bot_bot_a_o))))). % Axiom fact_499_linorder__le__cases:(forall (X_10:nat) (Y_9:nat), ((((ord_less_eq_nat X_10) Y_9)->False)->((ord_less_eq_nat Y_9) X_10))). % Axiom fact_500_xt1_I6_J:(forall (Z_2:Prop) (Y_8:Prop) (X_9:Prop), (((ord_less_eq_o Y_8) X_9)->(((ord_less_eq_o Z_2) Y_8)->((ord_less_eq_o Z_2) X_9)))). % Axiom fact_501_xt1_I6_J:(forall (Z_2:(nat->Prop)) (Y_8:(nat->Prop)) (X_9:(nat->Prop)), (((ord_less_eq_nat_o Y_8) X_9)->(((ord_less_eq_nat_o Z_2) Y_8)->((ord_less_eq_nat_o Z_2) X_9)))). % Axiom fact_502_xt1_I6_J:(forall (Z_2:(pname->Prop)) (Y_8:(pname->Prop)) (X_9:(pname->Prop)), (((ord_less_eq_pname_o Y_8) X_9)->(((ord_less_eq_pname_o Z_2) Y_8)->((ord_less_eq_pname_o Z_2) X_9)))). % Axiom fact_503_xt1_I6_J:(forall (Z_2:nat) (Y_8:nat) (X_9:nat), (((ord_less_eq_nat Y_8) X_9)->(((ord_less_eq_nat Z_2) Y_8)->((ord_less_eq_nat Z_2) X_9)))). % Axiom fact_504_xt1_I6_J:(forall (Z_2:(x_a->Prop)) (Y_8:(x_a->Prop)) (X_9:(x_a->Prop)), (((ord_less_eq_a_o Y_8) X_9)->(((ord_less_eq_a_o Z_2) Y_8)->((ord_less_eq_a_o Z_2) X_9)))). % Axiom fact_505_xt1_I5_J:(forall (Y_7:Prop) (X_8:Prop), (((ord_less_eq_o Y_7) X_8)->(((ord_less_eq_o X_8) Y_7)->((iff X_8) Y_7)))). % Axiom fact_506_xt1_I5_J:(forall (Y_7:(nat->Prop)) (X_8:(nat->Prop)), (((ord_less_eq_nat_o Y_7) X_8)->(((ord_less_eq_nat_o X_8) Y_7)->(((eq (nat->Prop)) X_8) Y_7)))). % Axiom fact_507_xt1_I5_J:(forall (Y_7:(pname->Prop)) (X_8:(pname->Prop)), (((ord_less_eq_pname_o Y_7) X_8)->(((ord_less_eq_pname_o X_8) Y_7)->(((eq (pname->Prop)) X_8) Y_7)))). % Axiom fact_508_xt1_I5_J:(forall (Y_7:nat) (X_8:nat), (((ord_less_eq_nat Y_7) X_8)->(((ord_less_eq_nat X_8) Y_7)->(((eq nat) X_8) Y_7)))). % Axiom fact_509_xt1_I5_J:(forall (Y_7:(x_a->Prop)) (X_8:(x_a->Prop)), (((ord_less_eq_a_o Y_7) X_8)->(((ord_less_eq_a_o X_8) Y_7)->(((eq (x_a->Prop)) X_8) Y_7)))). % Axiom fact_510_order__trans:(forall (Z_1:Prop) (X_7:Prop) (Y_6:Prop), (((ord_less_eq_o X_7) Y_6)->(((ord_less_eq_o Y_6) Z_1)->((ord_less_eq_o X_7) Z_1)))). % Axiom fact_511_order__trans:(forall (Z_1:(nat->Prop)) (X_7:(nat->Prop)) (Y_6:(nat->Prop)), (((ord_less_eq_nat_o X_7) Y_6)->(((ord_less_eq_nat_o Y_6) Z_1)->((ord_less_eq_nat_o X_7) Z_1)))). % Axiom fact_512_order__trans:(forall (Z_1:(pname->Prop)) (X_7:(pname->Prop)) (Y_6:(pname->Prop)), (((ord_less_eq_pname_o X_7) Y_6)->(((ord_less_eq_pname_o Y_6) Z_1)->((ord_less_eq_pname_o X_7) Z_1)))). % Axiom fact_513_order__trans:(forall (Z_1:nat) (X_7:nat) (Y_6:nat), (((ord_less_eq_nat X_7) Y_6)->(((ord_less_eq_nat Y_6) Z_1)->((ord_less_eq_nat X_7) Z_1)))). % Axiom fact_514_order__trans:(forall (Z_1:(x_a->Prop)) (X_7:(x_a->Prop)) (Y_6:(x_a->Prop)), (((ord_less_eq_a_o X_7) Y_6)->(((ord_less_eq_a_o Y_6) Z_1)->((ord_less_eq_a_o X_7) Z_1)))). % Axiom fact_515_order__antisym:(forall (X_6:Prop) (Y_5:Prop), (((ord_less_eq_o X_6) Y_5)->(((ord_less_eq_o Y_5) X_6)->((iff X_6) Y_5)))). % Axiom fact_516_order__antisym:(forall (X_6:(nat->Prop)) (Y_5:(nat->Prop)), (((ord_less_eq_nat_o X_6) Y_5)->(((ord_less_eq_nat_o Y_5) X_6)->(((eq (nat->Prop)) X_6) Y_5)))). % Axiom fact_517_order__antisym:(forall (X_6:(pname->Prop)) (Y_5:(pname->Prop)), (((ord_less_eq_pname_o X_6) Y_5)->(((ord_less_eq_pname_o Y_5) X_6)->(((eq (pname->Prop)) X_6) Y_5)))). % Axiom fact_518_order__antisym:(forall (X_6:nat) (Y_5:nat), (((ord_less_eq_nat X_6) Y_5)->(((ord_less_eq_nat Y_5) X_6)->(((eq nat) X_6) Y_5)))). % Axiom fact_519_order__antisym:(forall (X_6:(x_a->Prop)) (Y_5:(x_a->Prop)), (((ord_less_eq_a_o X_6) Y_5)->(((ord_less_eq_a_o Y_5) X_6)->(((eq (x_a->Prop)) X_6) Y_5)))). % Axiom fact_520_xt1_I4_J:(forall (C_4:Prop) (B_4:Prop) (A_6:Prop), (((ord_less_eq_o B_4) A_6)->(((iff B_4) C_4)->((ord_less_eq_o C_4) A_6)))). % Axiom fact_521_xt1_I4_J:(forall (C_4:(nat->Prop)) (B_4:(nat->Prop)) (A_6:(nat->Prop)), (((ord_less_eq_nat_o B_4) A_6)->((((eq (nat->Prop)) B_4) C_4)->((ord_less_eq_nat_o C_4) A_6)))). % Axiom fact_522_xt1_I4_J:(forall (C_4:(pname->Prop)) (B_4:(pname->Prop)) (A_6:(pname->Prop)), (((ord_less_eq_pname_o B_4) A_6)->((((eq (pname->Prop)) B_4) C_4)->((ord_less_eq_pname_o C_4) A_6)))). % Axiom fact_523_xt1_I4_J:(forall (C_4:nat) (B_4:nat) (A_6:nat), (((ord_less_eq_nat B_4) A_6)->((((eq nat) B_4) C_4)->((ord_less_eq_nat C_4) A_6)))). % Axiom fact_524_xt1_I4_J:(forall (C_4:(x_a->Prop)) (B_4:(x_a->Prop)) (A_6:(x_a->Prop)), (((ord_less_eq_a_o B_4) A_6)->((((eq (x_a->Prop)) B_4) C_4)->((ord_less_eq_a_o C_4) A_6)))). % Axiom fact_525_ord__le__eq__trans:(forall (C_3:Prop) (A_5:Prop) (B_3:Prop), (((ord_less_eq_o A_5) B_3)->(((iff B_3) C_3)->((ord_less_eq_o A_5) C_3)))). % Axiom fact_526_ord__le__eq__trans:(forall (C_3:(nat->Prop)) (A_5:(nat->Prop)) (B_3:(nat->Prop)), (((ord_less_eq_nat_o A_5) B_3)->((((eq (nat->Prop)) B_3) C_3)->((ord_less_eq_nat_o A_5) C_3)))). % Axiom fact_527_ord__le__eq__trans:(forall (C_3:(pname->Prop)) (A_5:(pname->Prop)) (B_3:(pname->Prop)), (((ord_less_eq_pname_o A_5) B_3)->((((eq (pname->Prop)) B_3) C_3)->((ord_less_eq_pname_o A_5) C_3)))). % Axiom fact_528_ord__le__eq__trans:(forall (C_3:nat) (A_5:nat) (B_3:nat), (((ord_less_eq_nat A_5) B_3)->((((eq nat) B_3) C_3)->((ord_less_eq_nat A_5) C_3)))). % Axiom fact_529_ord__le__eq__trans:(forall (C_3:(x_a->Prop)) (A_5:(x_a->Prop)) (B_3:(x_a->Prop)), (((ord_less_eq_a_o A_5) B_3)->((((eq (x_a->Prop)) B_3) C_3)->((ord_less_eq_a_o A_5) C_3)))). % Axiom fact_530_xt1_I3_J:(forall (C_2:Prop) (B_2:Prop) (A_4:Prop), (((iff A_4) B_2)->(((ord_less_eq_o C_2) B_2)->((ord_less_eq_o C_2) A_4)))). % Axiom fact_531_xt1_I3_J:(forall (C_2:(nat->Prop)) (A_4:(nat->Prop)) (B_2:(nat->Prop)), ((((eq (nat->Prop)) A_4) B_2)->(((ord_less_eq_nat_o C_2) B_2)->((ord_less_eq_nat_o C_2) A_4)))). % Axiom fact_532_xt1_I3_J:(forall (C_2:(pname->Prop)) (A_4:(pname->Prop)) (B_2:(pname->Prop)), ((((eq (pname->Prop)) A_4) B_2)->(((ord_less_eq_pname_o C_2) B_2)->((ord_less_eq_pname_o C_2) A_4)))). % Axiom fact_533_xt1_I3_J:(forall (C_2:nat) (A_4:nat) (B_2:nat), ((((eq nat) A_4) B_2)->(((ord_less_eq_nat C_2) B_2)->((ord_less_eq_nat C_2) A_4)))). % Axiom fact_534_xt1_I3_J:(forall (C_2:(x_a->Prop)) (A_4:(x_a->Prop)) (B_2:(x_a->Prop)), ((((eq (x_a->Prop)) A_4) B_2)->(((ord_less_eq_a_o C_2) B_2)->((ord_less_eq_a_o C_2) A_4)))). % Axiom fact_535_ord__eq__le__trans:(forall (C_1:Prop) (B_1:Prop) (A_3:Prop), (((iff A_3) B_1)->(((ord_less_eq_o B_1) C_1)->((ord_less_eq_o A_3) C_1)))). % Axiom fact_536_ord__eq__le__trans:(forall (C_1:(nat->Prop)) (A_3:(nat->Prop)) (B_1:(nat->Prop)), ((((eq (nat->Prop)) A_3) B_1)->(((ord_less_eq_nat_o B_1) C_1)->((ord_less_eq_nat_o A_3) C_1)))). % Axiom fact_537_ord__eq__le__trans:(forall (C_1:(pname->Prop)) (A_3:(pname->Prop)) (B_1:(pname->Prop)), ((((eq (pname->Prop)) A_3) B_1)->(((ord_less_eq_pname_o B_1) C_1)->((ord_less_eq_pname_o A_3) C_1)))). % Axiom fact_538_ord__eq__le__trans:(forall (C_1:nat) (A_3:nat) (B_1:nat), ((((eq nat) A_3) B_1)->(((ord_less_eq_nat B_1) C_1)->((ord_less_eq_nat A_3) C_1)))). % Axiom fact_539_ord__eq__le__trans:(forall (C_1:(x_a->Prop)) (A_3:(x_a->Prop)) (B_1:(x_a->Prop)), ((((eq (x_a->Prop)) A_3) B_1)->(((ord_less_eq_a_o B_1) C_1)->((ord_less_eq_a_o A_3) C_1)))). % Axiom fact_540_order__antisym__conv:(forall (Y_4:Prop) (X_5:Prop), (((ord_less_eq_o Y_4) X_5)->((iff ((ord_less_eq_o X_5) Y_4)) ((iff X_5) Y_4)))). % Axiom fact_541_order__antisym__conv:(forall (Y_4:(nat->Prop)) (X_5:(nat->Prop)), (((ord_less_eq_nat_o Y_4) X_5)->((iff ((ord_less_eq_nat_o X_5) Y_4)) (((eq (nat->Prop)) X_5) Y_4)))). % Axiom fact_542_order__antisym__conv:(forall (Y_4:(pname->Prop)) (X_5:(pname->Prop)), (((ord_less_eq_pname_o Y_4) X_5)->((iff ((ord_less_eq_pname_o X_5) Y_4)) (((eq (pname->Prop)) X_5) Y_4)))). % Axiom fact_543_order__antisym__conv:(forall (Y_4:nat) (X_5:nat), (((ord_less_eq_nat Y_4) X_5)->((iff ((ord_less_eq_nat X_5) Y_4)) (((eq nat) X_5) Y_4)))). % Axiom fact_544_order__antisym__conv:(forall (Y_4:(x_a->Prop)) (X_5:(x_a->Prop)), (((ord_less_eq_a_o Y_4) X_5)->((iff ((ord_less_eq_a_o X_5) Y_4)) (((eq (x_a->Prop)) X_5) Y_4)))). % Axiom fact_545_order__eq__refl:(forall (Y_3:Prop) (X_4:Prop), (((iff X_4) Y_3)->((ord_less_eq_o X_4) Y_3))). % Axiom fact_546_order__eq__refl:(forall (X_4:(nat->Prop)) (Y_3:(nat->Prop)), ((((eq (nat->Prop)) X_4) Y_3)->((ord_less_eq_nat_o X_4) Y_3))). % Axiom fact_547_order__eq__refl:(forall (X_4:(pname->Prop)) (Y_3:(pname->Prop)), ((((eq (pname->Prop)) X_4) Y_3)->((ord_less_eq_pname_o X_4) Y_3))). % Axiom fact_548_order__eq__refl:(forall (X_4:nat) (Y_3:nat), ((((eq nat) X_4) Y_3)->((ord_less_eq_nat X_4) Y_3))). % Axiom fact_549_order__eq__refl:(forall (X_4:(x_a->Prop)) (Y_3:(x_a->Prop)), ((((eq (x_a->Prop)) X_4) Y_3)->((ord_less_eq_a_o X_4) Y_3))). % Axiom fact_550_order__eq__iff:(forall (Y_2:Prop) (X_3:Prop), ((iff ((iff X_3) Y_2)) ((and ((ord_less_eq_o X_3) Y_2)) ((ord_less_eq_o Y_2) X_3)))). % Axiom fact_551_order__eq__iff:(forall (X_3:(nat->Prop)) (Y_2:(nat->Prop)), ((iff (((eq (nat->Prop)) X_3) Y_2)) ((and ((ord_less_eq_nat_o X_3) Y_2)) ((ord_less_eq_nat_o Y_2) X_3)))). % Axiom fact_552_order__eq__iff:(forall (X_3:(pname->Prop)) (Y_2:(pname->Prop)), ((iff (((eq (pname->Prop)) X_3) Y_2)) ((and ((ord_less_eq_pname_o X_3) Y_2)) ((ord_less_eq_pname_o Y_2) X_3)))). % Axiom fact_553_order__eq__iff:(forall (X_3:nat) (Y_2:nat), ((iff (((eq nat) X_3) Y_2)) ((and ((ord_less_eq_nat X_3) Y_2)) ((ord_less_eq_nat Y_2) X_3)))). % Axiom fact_554_order__eq__iff:(forall (X_3:(x_a->Prop)) (Y_2:(x_a->Prop)), ((iff (((eq (x_a->Prop)) X_3) Y_2)) ((and ((ord_less_eq_a_o X_3) Y_2)) ((ord_less_eq_a_o Y_2) X_3)))). % Axiom fact_555_linorder__linear:(forall (X_2:nat) (Y_1:nat), ((or ((ord_less_eq_nat X_2) Y_1)) ((ord_less_eq_nat Y_1) X_2))). % Axiom fact_556_finite__subset__induct:(forall (P_2:((nat->Prop)->Prop)) (A_1:(nat->Prop)) (F_3:(nat->Prop)), ((finite_finite_nat F_3)->(((ord_less_eq_nat_o F_3) A_1)->((P_2 bot_bot_nat_o)->((forall (A_2:nat) (F_2:(nat->Prop)), ((finite_finite_nat F_2)->(((member_nat A_2) A_1)->((((member_nat A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_nat A_2) F_2)))))))->(P_2 F_3)))))). % Axiom fact_557_finite__subset__induct:(forall (P_2:((pname->Prop)->Prop)) (A_1:(pname->Prop)) (F_3:(pname->Prop)), ((finite_finite_pname F_3)->(((ord_less_eq_pname_o F_3) A_1)->((P_2 bot_bot_pname_o)->((forall (A_2:pname) (F_2:(pname->Prop)), ((finite_finite_pname F_2)->(((member_pname A_2) A_1)->((((member_pname A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_pname A_2) F_2)))))))->(P_2 F_3)))))). % Axiom fact_558_finite__subset__induct:(forall (P_2:((x_a->Prop)->Prop)) (A_1:(x_a->Prop)) (F_3:(x_a->Prop)), ((finite_finite_a F_3)->(((ord_less_eq_a_o F_3) A_1)->((P_2 bot_bot_a_o)->((forall (A_2:x_a) (F_2:(x_a->Prop)), ((finite_finite_a F_2)->(((member_a A_2) A_1)->((((member_a A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_a A_2) F_2)))))))->(P_2 F_3)))))). % Axiom fact_559_finite__subset__induct:(forall (P_2:(((nat->Prop)->Prop)->Prop)) (A_1:((nat->Prop)->Prop)) (F_3:((nat->Prop)->Prop)), ((finite_finite_nat_o F_3)->(((ord_less_eq_nat_o_o F_3) A_1)->((P_2 bot_bot_nat_o_o)->((forall (A_2:(nat->Prop)) (F_2:((nat->Prop)->Prop)), ((finite_finite_nat_o F_2)->(((member_nat_o A_2) A_1)->((((member_nat_o A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_nat_o A_2) F_2)))))))->(P_2 F_3)))))). % Axiom fact_560_finite__subset__induct:(forall (P_2:(((pname->Prop)->Prop)->Prop)) (A_1:((pname->Prop)->Prop)) (F_3:((pname->Prop)->Prop)), ((finite297249702name_o F_3)->(((ord_le1205211808me_o_o F_3) A_1)->((P_2 bot_bot_pname_o_o)->((forall (A_2:(pname->Prop)) (F_2:((pname->Prop)->Prop)), ((finite297249702name_o F_2)->(((member_pname_o A_2) A_1)->((((member_pname_o A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_pname_o A_2) F_2)))))))->(P_2 F_3)))))). % Axiom fact_561_finite__subset__induct:(forall (P_2:(((x_a->Prop)->Prop)->Prop)) (A_1:((x_a->Prop)->Prop)) (F_3:((x_a->Prop)->Prop)), ((finite_finite_a_o F_3)->(((ord_less_eq_a_o_o F_3) A_1)->((P_2 bot_bot_a_o_o)->((forall (A_2:(x_a->Prop)) (F_2:((x_a->Prop)->Prop)), ((finite_finite_a_o F_2)->(((member_a_o A_2) A_1)->((((member_a_o A_2) F_2)->False)->((P_2 F_2)->(P_2 ((insert_a_o A_2) F_2)))))))->(P_2 F_3)))))). % Axiom fact_562_assms_I2_J:(forall (Pn:pname) (G:(x_a->Prop)), (((p ((insert_a (mgt_call Pn)) G)) ((insert_a (mgt (the_com (body Pn)))) bot_bot_a_o))->((p G) ((insert_a (mgt_call Pn)) bot_bot_a_o)))). % Axiom fact_563_finite__induct:(forall (P_1:(((nat->Prop)->Prop)->Prop)) (F_1:((nat->Prop)->Prop)), ((finite_finite_nat_o F_1)->((P_1 bot_bot_nat_o_o)->((forall (X_1:(nat->Prop)) (F_2:((nat->Prop)->Prop)), ((finite_finite_nat_o F_2)->((((member_nat_o X_1) F_2)->False)->((P_1 F_2)->(P_1 ((insert_nat_o X_1) F_2))))))->(P_1 F_1))))). % Axiom fact_564_finite__induct:(forall (P_1:(((pname->Prop)->Prop)->Prop)) (F_1:((pname->Prop)->Prop)), ((finite297249702name_o F_1)->((P_1 bot_bot_pname_o_o)->((forall (X_1:(pname->Prop)) (F_2:((pname->Prop)->Prop)), ((finite297249702name_o F_2)->((((member_pname_o X_1) F_2)->False)->((P_1 F_2)->(P_1 ((insert_pname_o X_1) F_2))))))->(P_1 F_1))))). % Axiom fact_565_finite__induct:(forall (P_1:(((x_a->Prop)->Prop)->Prop)) (F_1:((x_a->Prop)->Prop)), ((finite_finite_a_o F_1)->((P_1 bot_bot_a_o_o)->((forall (X_1:(x_a->Prop)) (F_2:((x_a->Prop)->Prop)), ((finite_finite_a_o F_2)->((((member_a_o X_1) F_2)->False)->((P_1 F_2)->(P_1 ((insert_a_o X_1) F_2))))))->(P_1 F_1))))). % Axiom fact_566_finite__less__ub:(forall (U:nat) (F:(nat->nat)), ((forall (N_2:nat), ((ord_less_eq_nat N_2) (F N_2)))->(finite_finite_nat (collect_nat (fun (N_2:nat)=> ((ord_less_eq_nat (F N_2)) U)))))). % Axiom fact_567_assms_I4_J:(forall (Pn:pname), (((member_pname Pn) u)->(wt (the_com (body Pn))))). % Axiom fact_568_diff__Suc__eq__diff__pred:(forall (M:nat) (N:nat), (((eq nat) ((minus_minus_nat M) (suc N))) ((minus_minus_nat ((minus_minus_nat M) one_one_nat)) N))). % Axiom fact_569_diff__Suc__1:(forall (N:nat), (((eq nat) ((minus_minus_nat (suc N)) one_one_nat)) N)). % Axiom fact_570_less__eq__nat_Osimps_I2_J:(forall (M:nat) (N:nat), ((iff ((ord_less_eq_nat (suc M)) N)) (((nat_case_o False) (ord_less_eq_nat M)) N))). % Axiom fact_571_add__Suc__right:(forall (M:nat) (N:nat), (((eq nat) ((plus_plus_nat M) (suc N))) (suc ((plus_plus_nat M) N)))). % Axiom fact_572_add__Suc:(forall (M:nat) (N:nat), (((eq nat) ((plus_plus_nat (suc M)) N)) (suc ((plus_plus_nat M) N)))). % Axiom fact_573_add__Suc__shift:(forall (M:nat) (N:nat), (((eq nat) ((plus_plus_nat (suc M)) N)) ((plus_plus_nat M) (suc N)))). % Axiom fact_574_nat__add__right__cancel:(forall (M:nat) (K:nat) (N:nat), ((iff (((eq nat) ((plus_plus_nat M) K)) ((plus_plus_nat N) K))) (((eq nat) M) N))). % Axiom fact_575_nat__add__left__cancel:(forall (K:nat) (M:nat) (N:nat), ((iff (((eq nat) ((plus_plus_nat K) M)) ((plus_plus_nat K) N))) (((eq nat) M) N))). % Axiom fact_576_nat__add__assoc:(forall (M:nat) (N:nat) (K:nat), (((eq nat) ((plus_plus_nat ((plus_plus_nat M) N)) K)) ((plus_plus_nat M) ((plus_plus_nat N) K)))). % Axiom fact_577_nat__add__left__commute:(forall (X:nat) (Y:nat) (Z:nat), (((eq nat) ((plus_plus_nat X) ((plus_plus_nat Y) Z))) ((plus_plus_nat Y) ((plus_plus_nat X) Z)))). % Axiom fact_578_nat__add__commute:(forall (M:nat) (N:nat), (((eq nat) ((plus_plus_nat M) N)) ((plus_plus_nat N) M))). % Axiom fact_579_diff__add__inverse2:(forall (M:nat) (N:nat), (((eq nat) ((minus_minus_nat ((plus_plus_nat M) N)) N)) M)). % Axiom fact_580_diff__add__inverse:(forall (N:nat) (M:nat), (((eq nat) ((minus_minus_nat ((plus_plus_nat N) M)) N)) M)). % Axiom fact_581_diff__diff__left:(forall (_TPTP_I:nat) (J_1:nat) (K:nat), (((eq nat) ((minus_minus_nat ((minus_minus_nat _TPTP_I) J_1)) K)) ((minus_minus_nat _TPTP_I) ((plus_plus_nat J_1) K)))). % Axiom fact_582_diff__cancel:(forall (K:nat) (M:nat) (N:nat), (((eq nat) ((minus_minus_nat ((plus_plus_nat K) M)) ((plus_plus_nat K) N))) ((minus_minus_nat M) N))). % Axiom fact_583_diff__cancel2:(forall (M:nat) (K:nat) (N:nat), (((eq nat) ((minus_minus_nat ((plus_plus_nat M) K)) ((plus_plus_nat N) K))) ((minus_minus_nat M) N))). % Axiom fact_584_le__add2:(forall (N:nat) (M:nat), ((ord_less_eq_nat N) ((plus_plus_nat M) N))). % Axiom fact_585_le__add1:(forall (N:nat) (M:nat), ((ord_less_eq_nat N) ((plus_plus_nat N) M))). % Axiom fact_586_le__iff__add:(forall (M:nat) (N:nat), ((iff ((ord_less_eq_nat M) N)) ((ex nat) (fun (K_1:nat)=> (((eq nat) N) ((plus_plus_nat M) K_1)))))). % Axiom fact_587_nat__add__left__cancel__le:(forall (K:nat) (M:nat) (N:nat), ((iff ((ord_less_eq_nat ((plus_plus_nat K) M)) ((plus_plus_nat K) N))) ((ord_less_eq_nat M) N))). % Axiom fact_588_trans__le__add1:(forall (M:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->((ord_less_eq_nat _TPTP_I) ((plus_plus_nat J_1) M)))). % Axiom fact_589_trans__le__add2:(forall (M:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->((ord_less_eq_nat _TPTP_I) ((plus_plus_nat M) J_1)))). % Axiom fact_590_add__le__mono1:(forall (K:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->((ord_less_eq_nat ((plus_plus_nat _TPTP_I) K)) ((plus_plus_nat J_1) K)))). % Axiom fact_591_add__le__mono:(forall (K:nat) (L:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->(((ord_less_eq_nat K) L)->((ord_less_eq_nat ((plus_plus_nat _TPTP_I) K)) ((plus_plus_nat J_1) L))))). % Axiom fact_592_add__leD2:(forall (M:nat) (K:nat) (N:nat), (((ord_less_eq_nat ((plus_plus_nat M) K)) N)->((ord_less_eq_nat K) N))). % Axiom fact_593_add__leD1:(forall (M:nat) (K:nat) (N:nat), (((ord_less_eq_nat ((plus_plus_nat M) K)) N)->((ord_less_eq_nat M) N))). % Axiom fact_594_add__leE:(forall (M:nat) (K:nat) (N:nat), (((ord_less_eq_nat ((plus_plus_nat M) K)) N)->((((ord_less_eq_nat M) N)->(((ord_less_eq_nat K) N)->False))->False))). % Axiom fact_595_diff__add__assoc2:(forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((minus_minus_nat ((plus_plus_nat J_1) _TPTP_I)) K)) ((plus_plus_nat ((minus_minus_nat J_1) K)) _TPTP_I)))). % Axiom fact_596_add__diff__assoc2:(forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((plus_plus_nat ((minus_minus_nat J_1) K)) _TPTP_I)) ((minus_minus_nat ((plus_plus_nat J_1) _TPTP_I)) K)))). % Axiom fact_597_diff__add__assoc:(forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((minus_minus_nat ((plus_plus_nat _TPTP_I) J_1)) K)) ((plus_plus_nat _TPTP_I) ((minus_minus_nat J_1) K))))). % Axiom fact_598_le__imp__diff__is__add:(forall (K:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->((iff (((eq nat) ((minus_minus_nat J_1) _TPTP_I)) K)) (((eq nat) J_1) ((plus_plus_nat K) _TPTP_I))))). % Axiom fact_599_le__add__diff__inverse2:(forall (N:nat) (M:nat), (((ord_less_eq_nat N) M)->(((eq nat) ((plus_plus_nat ((minus_minus_nat M) N)) N)) M))). % Axiom fact_600_le__diff__conv2:(forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->((iff ((ord_less_eq_nat _TPTP_I) ((minus_minus_nat J_1) K))) ((ord_less_eq_nat ((plus_plus_nat _TPTP_I) K)) J_1)))). % Axiom fact_601_add__diff__assoc:(forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((plus_plus_nat _TPTP_I) ((minus_minus_nat J_1) K))) ((minus_minus_nat ((plus_plus_nat _TPTP_I) J_1)) K)))). % Axiom fact_602_le__add__diff__inverse:(forall (N:nat) (M:nat), (((ord_less_eq_nat N) M)->(((eq nat) ((plus_plus_nat N) ((minus_minus_nat M) N))) M))). % Axiom fact_603_le__add__diff:(forall (M:nat) (K:nat) (N:nat), (((ord_less_eq_nat K) N)->((ord_less_eq_nat M) ((minus_minus_nat ((plus_plus_nat N) M)) K)))). % Axiom fact_604_le__diff__conv:(forall (J_1:nat) (K:nat) (_TPTP_I:nat), ((iff ((ord_less_eq_nat ((minus_minus_nat J_1) K)) _TPTP_I)) ((ord_less_eq_nat J_1) ((plus_plus_nat _TPTP_I) K)))). % Axiom fact_605_diff__diff__right:(forall (_TPTP_I:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((minus_minus_nat _TPTP_I) ((minus_minus_nat J_1) K))) ((minus_minus_nat ((plus_plus_nat _TPTP_I) K)) J_1)))). % Axiom fact_606_Suc__eq__plus1:(forall (N:nat), (((eq nat) (suc N)) ((plus_plus_nat N) one_one_nat))). % Axiom fact_607_Suc__eq__plus1__left:(forall (N:nat), (((eq nat) (suc N)) ((plus_plus_nat one_one_nat) N))). % Axiom fact_608_diff__Suc__diff__eq2:(forall (M:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((minus_minus_nat (suc ((minus_minus_nat J_1) K))) M)) ((minus_minus_nat (suc J_1)) ((plus_plus_nat K) M))))). % Axiom fact_609_diff__Suc__diff__eq1:(forall (M:nat) (K:nat) (J_1:nat), (((ord_less_eq_nat K) J_1)->(((eq nat) ((minus_minus_nat M) (suc ((minus_minus_nat J_1) K)))) ((minus_minus_nat ((plus_plus_nat M) K)) (suc J_1))))). % Axiom fact_610_termination__basic__simps_I3_J:(forall (Z:nat) (X:nat) (Y:nat), (((ord_less_eq_nat X) Y)->((ord_less_eq_nat X) ((plus_plus_nat Y) Z)))). % Axiom fact_611_termination__basic__simps_I4_J:(forall (Y:nat) (X:nat) (Z:nat), (((ord_less_eq_nat X) Z)->((ord_less_eq_nat X) ((plus_plus_nat Y) Z)))). % Axiom fact_612_lessI:(forall (N:nat), ((ord_less_nat N) (suc N))). % Axiom fact_613_Suc__mono:(forall (M:nat) (N:nat), (((ord_less_nat M) N)->((ord_less_nat (suc M)) (suc N)))). % Axiom fact_614_finite__Collect__less__nat:(forall (K:nat), (finite_finite_nat (collect_nat (fun (N_2:nat)=> ((ord_less_nat N_2) K))))). % Axiom fact_615_termination__basic__simps_I1_J:(forall (Z:nat) (X:nat) (Y:nat), (((ord_less_nat X) Y)->((ord_less_nat X) ((plus_plus_nat Y) Z)))). % Axiom fact_616_termination__basic__simps_I2_J:(forall (Y:nat) (X:nat) (Z:nat), (((ord_less_nat X) Z)->((ord_less_nat X) ((plus_plus_nat Y) Z)))). % Axiom fact_617_add__lessD1:(forall (_TPTP_I:nat) (J_1:nat) (K:nat), (((ord_less_nat ((plus_plus_nat _TPTP_I) J_1)) K)->((ord_less_nat _TPTP_I) K))). % Axiom fact_618_less__add__eq__less:(forall (M:nat) (N:nat) (K:nat) (L:nat), (((ord_less_nat K) L)->((((eq nat) ((plus_plus_nat M) L)) ((plus_plus_nat K) N))->((ord_less_nat M) N)))). % Axiom fact_619_add__less__mono:(forall (K:nat) (L:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_nat _TPTP_I) J_1)->(((ord_less_nat K) L)->((ord_less_nat ((plus_plus_nat _TPTP_I) K)) ((plus_plus_nat J_1) L))))). % Axiom fact_620_add__less__mono1:(forall (K:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_nat _TPTP_I) J_1)->((ord_less_nat ((plus_plus_nat _TPTP_I) K)) ((plus_plus_nat J_1) K)))). % Axiom fact_621_trans__less__add2:(forall (M:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_nat _TPTP_I) J_1)->((ord_less_nat _TPTP_I) ((plus_plus_nat M) J_1)))). % Axiom fact_622_trans__less__add1:(forall (M:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_nat _TPTP_I) J_1)->((ord_less_nat _TPTP_I) ((plus_plus_nat J_1) M)))). % Axiom fact_623_nat__add__left__cancel__less:(forall (K:nat) (M:nat) (N:nat), ((iff ((ord_less_nat ((plus_plus_nat K) M)) ((plus_plus_nat K) N))) ((ord_less_nat M) N))). % Axiom fact_624_not__add__less2:(forall (J_1:nat) (_TPTP_I:nat), (((ord_less_nat ((plus_plus_nat J_1) _TPTP_I)) _TPTP_I)->False)). % Axiom fact_625_not__add__less1:(forall (_TPTP_I:nat) (J_1:nat), (((ord_less_nat ((plus_plus_nat _TPTP_I) J_1)) _TPTP_I)->False)). % Axiom fact_626_Suc__less__SucD:(forall (M:nat) (N:nat), (((ord_less_nat (suc M)) (suc N))->((ord_less_nat M) N))). % Axiom fact_627_Suc__lessD:(forall (M:nat) (N:nat), (((ord_less_nat (suc M)) N)->((ord_less_nat M) N))). % Axiom fact_628_less__SucE:(forall (M:nat) (N:nat), (((ord_less_nat M) (suc N))->((((ord_less_nat M) N)->False)->(((eq nat) M) N)))). % Axiom fact_629_less__trans__Suc:(forall (K:nat) (_TPTP_I:nat) (J_1:nat), (((ord_less_nat _TPTP_I) J_1)->(((ord_less_nat J_1) K)->((ord_less_nat (suc _TPTP_I)) K)))). % Axiom fact_630_Suc__lessI:(forall (M:nat) (N:nat), (((ord_less_nat M) N)->((not (((eq nat) (suc M)) N))->((ord_less_nat (suc M)) N)))). % Axiom fact_631_less__SucI:(forall (M:nat) (N:nat), (((ord_less_nat M) N)->((ord_less_nat M) (suc N)))). % Axiom fact_632_less__antisym:(forall (N:nat) (M:nat), ((((ord_less_nat N) M)->False)->(((ord_less_nat N) (suc M))->(((eq nat) M) N)))). % Axiom fact_633_not__less__less__Suc__eq:(forall (N:nat) (M:nat), ((((ord_less_nat N) M)->False)->((iff ((ord_less_nat N) (suc M))) (((eq nat) N) M)))). % Axiom fact_634_Suc__less__eq:(forall (M:nat) (N:nat), ((iff ((ord_less_nat (suc M)) (suc N))) ((ord_less_nat M) N))). % Axiom fact_635_less__Suc__eq:(forall (M:nat) (N:nat), ((iff ((ord_less_nat M) (suc N))) ((or ((ord_less_nat M) N)) (((eq nat) M) N)))). % Axiom fact_636_not__less__eq:(forall (M:nat) (N:nat), ((iff (((ord_less_nat M) N)->False)) ((ord_less_nat N) (suc M)))). % Axiom fact_637_less__or__eq__imp__le:(forall (M:nat) (N:nat), (((or ((ord_less_nat M) N)) (((eq nat) M) N))->((ord_less_eq_nat M) N))). % Axiom fact_638_le__neq__implies__less:(forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((not (((eq nat) M) N))->((ord_less_nat M) N)))). % Axiom fact_639_less__imp__le__nat:(forall (M:nat) (N:nat), (((ord_less_nat M) N)->((ord_less_eq_nat M) N))). % Axiom fact_640_le__eq__less__or__eq:(forall (M:nat) (N:nat), ((iff ((ord_less_eq_nat M) N)) ((or ((ord_less_nat M) N)) (((eq nat) M) N)))). % Axiom fact_641_nat__less__le:(forall (M:nat) (N:nat), ((iff ((ord_less_nat M) N)) ((and ((ord_less_eq_nat M) N)) (not (((eq nat) M) N))))). % Axiom fact_642_diff__less__mono2:(forall (L:nat) (M:nat) (N:nat), (((ord_less_nat M) N)->(((ord_less_nat M) L)->((ord_less_nat ((minus_minus_nat L) N)) ((minus_minus_nat L) M))))). % Axiom fact_643_less__imp__diff__less:(forall (N:nat) (J_1:nat) (K:nat), (((ord_less_nat J_1) K)->((ord_less_nat ((minus_minus_nat J_1) N)) K))). % Axiom fact_644_termination__basic__simps_I5_J:(forall (X:nat) (Y:nat), (((ord_less_nat X) Y)->((ord_less_eq_nat X) Y))). % Axiom fact_645_less__not__refl:(forall (N:nat), (((ord_less_nat N) N)->False)). % Axiom fact_646_nat__neq__iff:(forall (M:nat) (N:nat), ((iff (not (((eq nat) M) N))) ((or ((ord_less_nat M) N)) ((ord_less_nat N) M)))). % Axiom fact_647_linorder__neqE__nat:(forall (X:nat) (Y:nat), ((not (((eq nat) X) Y))->((((ord_less_nat X) Y)->False)->((ord_less_nat Y) X)))). % Axiom fact_648_less__irrefl__nat:(forall (N:nat), (((ord_less_nat N) N)->False)). % Axiom fact_649_less__not__refl2:(forall (N:nat) (M:nat), (((ord_less_nat N) M)->(not (((eq nat) M) N)))). % Axiom fact_650_less__not__refl3:(forall (S:nat) (T:nat), (((ord_less_nat S) T)->(not (((eq nat) S) T)))). % Axiom fact_651_nat__less__cases:(forall (P:(nat->(nat->Prop))) (M:nat) (N:nat), ((((ord_less_nat M) N)->((P N) M))->(((((eq nat) M) N)->((P N) M))->((((ord_less_nat N) M)->((P N) M))->((P N) M))))). % Axiom fact_652_finite__nat__set__iff__bounded:(forall (N_1:(nat->Prop)), ((iff (finite_finite_nat N_1)) ((ex nat) (fun (M_1:nat)=> (forall (X_1:nat), (((member_nat X_1) N_1)->((ord_less_nat X_1) M_1))))))). % Axiom fact_653_card__Collect__less__nat:(forall (N:nat), (((eq nat) (finite_card_nat (collect_nat (fun (I_1:nat)=> ((ord_less_nat I_1) N))))) N)). % Axiom fact_654_finite__M__bounded__by__nat:(forall (P:(nat->Prop)) (_TPTP_I:nat), (finite_finite_nat (collect_nat (fun (K_1:nat)=> ((and (P K_1)) ((ord_less_nat K_1) _TPTP_I)))))). % Axiom fact_655_less__add__Suc1:(forall (_TPTP_I:nat) (M:nat), ((ord_less_nat _TPTP_I) (suc ((plus_plus_nat _TPTP_I) M)))). % Axiom fact_656_less__add__Suc2:(forall (_TPTP_I:nat) (M:nat), ((ord_less_nat _TPTP_I) (suc ((plus_plus_nat M) _TPTP_I)))). % Axiom fact_657_less__iff__Suc__add:(forall (M:nat) (N:nat), ((iff ((ord_less_nat M) N)) ((ex nat) (fun (K_1:nat)=> (((eq nat) N) (suc ((plus_plus_nat M) K_1))))))). % Axiom fact_658_less__eq__Suc__le:(forall (N:nat) (M:nat), ((iff ((ord_less_nat N) M)) ((ord_less_eq_nat (suc N)) M))). % Axiom fact_659_less__Suc__eq__le:(forall (M:nat) (N:nat), ((iff ((ord_less_nat M) (suc N))) ((ord_less_eq_nat M) N))). % Axiom fact_660_Suc__le__eq:(forall (M:nat) (N:nat), ((iff ((ord_less_eq_nat (suc M)) N)) ((ord_less_nat M) N))). % Axiom fact_661_le__imp__less__Suc:(forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((ord_less_nat M) (suc N)))). % Axiom fact_662_Suc__leI:(forall (M:nat) (N:nat), (((ord_less_nat M) N)->((ord_less_eq_nat (suc M)) N))). % Axiom fact_663_le__less__Suc__eq:(forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->((iff ((ord_less_nat N) (suc M))) (((eq nat) N) M)))). % Axiom fact_664_Suc__le__lessD:(forall (M:nat) (N:nat), (((ord_less_eq_nat (suc M)) N)->((ord_less_nat M) N))). % Axiom fact_665_diff__less__Suc:(forall (M:nat) (N:nat), ((ord_less_nat ((minus_minus_nat M) N)) (suc M))). % Axiom fact_666_add__diff__inverse:(forall (M:nat) (N:nat), ((((ord_less_nat M) N)->False)->(((eq nat) ((plus_plus_nat N) ((minus_minus_nat M) N))) M))). % Axiom fact_667_less__diff__conv:(forall (_TPTP_I:nat) (J_1:nat) (K:nat), ((iff ((ord_less_nat _TPTP_I) ((minus_minus_nat J_1) K))) ((ord_less_nat ((plus_plus_nat _TPTP_I) K)) J_1))). % Axiom fact_668_diff__less__mono:(forall (C:nat) (A:nat) (B:nat), (((ord_less_nat A) B)->(((ord_less_eq_nat C) A)->((ord_less_nat ((minus_minus_nat A) C)) ((minus_minus_nat B) C))))). % Axiom fact_669_less__diff__iff:(forall (N:nat) (K:nat) (M:nat), (((ord_less_eq_nat K) M)->(((ord_less_eq_nat K) N)->((iff ((ord_less_nat ((minus_minus_nat M) K)) ((minus_minus_nat N) K))) ((ord_less_nat M) N))))). % Axiom fact_670_less__eq__Suc__le__raw:(forall (X_1:nat), (((eq (nat->Prop)) (ord_less_nat X_1)) (ord_less_eq_nat (suc X_1)))). % Axiom fact_671_mono__nat__linear__lb:(forall (M:nat) (K:nat) (F:(nat->nat)), ((forall (M_1:nat) (N_2:nat), (((ord_less_nat M_1) N_2)->((ord_less_nat (F M_1)) (F N_2))))->((ord_less_eq_nat ((plus_plus_nat (F M)) K)) (F ((plus_plus_nat M) K))))). % Axiom fact_672_inc__induct:(forall (P:(nat->Prop)) (_TPTP_I:nat) (J_1:nat), (((ord_less_eq_nat _TPTP_I) J_1)->((P J_1)->((forall (I_1:nat), (((ord_less_nat I_1) J_1)->((P (suc I_1))->(P I_1))))->(P _TPTP_I))))). % Axiom fact_673_less__imp__Suc__add:(forall (M:nat) (N:nat), (((ord_less_nat M) N)->((ex nat) (fun (K_1:nat)=> (((eq nat) N) (suc ((plus_plus_nat M) K_1))))))). % Axiom fact_674_bounded__nat__set__is__finite:(forall (N:nat) (N_1:(nat->Prop)), ((forall (X_1:nat), (((member_nat X_1) N_1)->((ord_less_nat X_1) N)))->(finite_finite_nat N_1))). % Axiom fact_675_less__mono__imp__le__mono:(forall (_TPTP_I:nat) (J_1:nat) (F:(nat->nat)), ((forall (I_1:nat) (J:nat), (((ord_less_nat I_1) J)->((ord_less_nat (F I_1)) (F J))))->(((ord_less_eq_nat _TPTP_I) J_1)->((ord_less_eq_nat (F _TPTP_I)) (F J_1))))). % Axiom fact_676_Suc__lessE:(forall (_TPTP_I:nat) (K:nat), (((ord_less_nat (suc _TPTP_I)) K)->((forall (J:nat), (((ord_less_nat _TPTP_I) J)->(not (((eq nat) K) (suc J)))))->False))). % Axiom fact_677_lessE:(forall (_TPTP_I:nat) (K:nat), (((ord_less_nat _TPTP_I) K)->((not (((eq nat) K) (suc _TPTP_I)))->((forall (J:nat), (((ord_less_nat _TPTP_I) J)->(not (((eq nat) K) (suc J)))))->False)))). % Axiom fact_678_less__zeroE:(forall (N:nat), (((ord_less_nat N) zero_zero_nat)->False)). % Axiom fact_679_le0:(forall (N:nat), ((ord_less_eq_nat zero_zero_nat) N)). % Axiom fact_680_zero__less__Suc:(forall (N:nat), ((ord_less_nat zero_zero_nat) (suc N))). % Axiom fact_681_add__is__1:(forall (M:nat) (N:nat), ((iff (((eq nat) ((plus_plus_nat M) N)) (suc zero_zero_nat))) ((or ((and (((eq nat) M) (suc zero_zero_nat))) (((eq nat) N) zero_zero_nat))) ((and (((eq nat) M) zero_zero_nat)) (((eq nat) N) (suc zero_zero_nat)))))). % Axiom fact_682_one__is__add:(forall (M:nat) (N:nat), ((iff (((eq nat) (suc zero_zero_nat)) ((plus_plus_nat M) N))) ((or ((and (((eq nat) M) (suc zero_zero_nat))) (((eq nat) N) zero_zero_nat))) ((and (((eq nat) M) zero_zero_nat)) (((eq nat) N) (suc zero_zero_nat)))))). % Axiom fact_683_diff__add__0:(forall (N:nat) (M:nat), (((eq nat) ((minus_minus_nat N) ((plus_plus_nat N) M))) zero_zero_nat)). % Axiom fact_684_diff__is__0__eq_H:(forall (M:nat) (N:nat), (((ord_less_eq_nat M) N)->(((eq nat) ((minus_minus_nat M) N)) zero_zero_nat))). % Axiom fact_685_diff__is__0__eq:(forall (M:nat) (N:nat), ((iff (((eq nat) ((minus_minus_nat M) N)) zero_zero_nat)) ((ord_less_eq_nat M) N))). % Axiom fact_686_One__nat__def:(((eq nat) one_one_nat) (suc zero_zero_nat)). % Axiom fact_687_diffs0__imp__equal:(forall (M:nat) (N:nat), ((((eq nat) ((minus_minus_nat M) N)) zero_zero_nat)->((((eq nat) ((minus_minus_nat N) M)) zero_zero_nat)->(((eq nat) M) N)))). % Axiom fact_688_diff__self__eq__0:(forall (M:nat), (((eq nat) ((minus_minus_nat M) M)) zero_zero_nat)). % Axiom fact_689_minus__nat_Odiff__0:(forall (M:nat), (((eq nat) ((minus_minus_nat M) zero_zero_nat)) M)). % Axiom fact_690_diff__0__eq__0:(forall (N:nat), (((eq nat) ((minus_minus_nat zero_zero_nat) N)) zero_zero_nat)). % Axiom fact_691_Suc__neq__Zero:(forall (M:nat), (not (((eq nat) (suc M)) zero_zero_nat))). % Axiom fact_692_Zero__neq__Suc:(forall (M:nat), (not (((eq nat) zero_zero_nat) (suc M)))). % Axiom fact_693_nat_Osimps_I3_J:(forall (Nat_1:nat), (not (((eq nat) (suc Nat_1)) zero_zero_nat))). % Axiom fact_694_Suc__not__Zero:(forall (M:nat), (not (((eq nat) (suc M)) zero_zero_nat))). % Axiom fact_695_nat_Osimps_I2_J:(forall (Nat:nat), (not (((eq nat) zero_zero_nat) (suc Nat)))). % Axiom fact_696_Zero__not__Suc:(forall (M:nat), (not (((eq nat) zero_zero_nat) (suc M)))). % Axiom fact_697_bot__nat__def:(((eq nat) bot_bot_nat) zero_zero_nat). % Axiom fact_698_le__0__eq:(forall (N:nat), ((iff ((ord_less_eq_nat N) zero_zero_nat)) (((eq nat) N) zero_zero_nat))). % Axiom fact_699_less__eq__nat_Osimps_I1_J:(forall (N:nat), ((ord_less_eq_nat zero_zero_nat) N)). % Axiom help_fequal_1_1_fequal_000t__a_T:(forall (X:x_a) (Y:x_a), ((or (((fequal_a X) Y)->False)) (((eq x_a) X) Y))). % Axiom help_fequal_2_1_fequal_000t__a_T:(forall (X:x_a) (Y:x_a), ((or (not (((eq x_a) X) Y))) ((fequal_a X) Y))). % Axiom help_fequal_1_1_fequal_000tc__Nat__Onat_T:(forall (X:nat) (Y:nat), ((or (((fequal_nat X) Y)->False)) (((eq nat) X) Y))). % Axiom help_fequal_2_1_fequal_000tc__Nat__Onat_T:(forall (X:nat) (Y:nat), ((or (not (((eq nat) X) Y))) ((fequal_nat X) Y))). % Axiom help_fequal_1_1_fequal_000tc__Com__Opname_T:(forall (X:pname) (Y:pname), ((or (((fequal_pname X) Y)->False)) (((eq pname) X) Y))). % Axiom help_fequal_2_1_fequal_000tc__Com__Opname_T:(forall (X:pname) (Y:pname), ((or (not (((eq pname) X) Y))) ((fequal_pname X) Y))). % Axiom help_fequal_1_1_fequal_000_062_It__a_M_Eo_J_T:(forall (X:(x_a->Prop)) (Y:(x_a->Prop)), ((or (((fequal_a_o X) Y)->False)) (((eq (x_a->Prop)) X) Y))). % Axiom help_fequal_2_1_fequal_000_062_It__a_M_Eo_J_T:(forall (X:(x_a->Prop)) (Y:(x_a->Prop)), ((or (not (((eq (x_a->Prop)) X) Y))) ((fequal_a_o X) Y))). % Axiom help_fequal_1_1_fequal_000_062_Itc__Nat__Onat_M_Eo_J_T:(forall (X:(nat->Prop)) (Y:(nat->Prop)), ((or (((fequal_nat_o X) Y)->False)) (((eq (nat->Prop)) X) Y))). % Axiom help_fequal_2_1_fequal_000_062_Itc__Nat__Onat_M_Eo_J_T:(forall (X:(nat->Prop)) (Y:(nat->Prop)), ((or (not (((eq (nat->Prop)) X) Y))) ((fequal_nat_o X) Y))). % Axiom help_fequal_1_1_fequal_000_062_Itc__Com__Opname_M_Eo_J_T:(forall (X:(pname->Prop)) (Y:(pname->Prop)), ((or (((fequal_pname_o X) Y)->False)) (((eq (pname->Prop)) X) Y))). % Axiom help_fequal_2_1_fequal_000_062_Itc__Com__Opname_M_Eo_J_T:(forall (X:(pname->Prop)) (Y:(pname->Prop)), ((or (not (((eq (pname->Prop)) X) Y))) ((fequal_pname_o X) Y))). % Axiom conj_0:(finite_finite_pname u). % Axiom conj_1:((ord_less_eq_a_o g) ((image_pname_a mgt_call) u)). % Axiom conj_2:((ord_less_eq_nat (suc na)) (finite_card_a ((image_pname_a mgt_call) u))). % Axiom conj_3:(((eq nat) (finite_card_a g)) ((minus_minus_nat (finite_card_a ((image_pname_a mgt_call) u))) (suc na))). % Axiom conj_4:((member_pname pn) u). % Axiom conj_5:(((member_a (mgt_call pn)) g)->False). % Trying to prove ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((image_pname_a mgt_call) u)) % Found conj_1:((ord_less_eq_a_o g) ((image_pname_a mgt_call) u)) % Instantiate: Y_8:=g:(x_a->Prop) % Found conj_1 as proof of ((ord_less_eq_a_o Y_8) ((image_pname_a mgt_call) u)) % Found fact_370_order__refl0:=(fact_370_order__refl ((insert_a (mgt_call pn)) g)):((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((insert_a (mgt_call pn)) g)) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_8) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_8) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_8) % Found conj_1:((ord_less_eq_a_o g) ((image_pname_a mgt_call) u)) % Found conj_1 as proof of ((ord_less_eq_a_o g) ((image_pname_a mgt_call) u)) % Found conj_1:((ord_less_eq_a_o g) ((image_pname_a mgt_call) u)) % Found conj_1 as proof of ((ord_less_eq_a_o g) ((image_pname_a mgt_call) u)) % Found conj_1:((ord_less_eq_a_o g) ((image_pname_a mgt_call) u)) % Instantiate: B_4:=g:(x_a->Prop) % Found conj_1 as proof of ((ord_less_eq_a_o B_4) ((image_pname_a mgt_call) u)) % Found eq_ref00:=(eq_ref0 B_4):(((eq (x_a->Prop)) B_4) B_4) % Found (eq_ref0 B_4) as proof of (((eq (x_a->Prop)) B_4) ((insert_a (mgt_call pn)) g)) % Found ((eq_ref (x_a->Prop)) B_4) as proof of (((eq (x_a->Prop)) B_4) ((insert_a (mgt_call pn)) g)) % Found ((eq_ref (x_a->Prop)) B_4) as proof of (((eq (x_a->Prop)) B_4) ((insert_a (mgt_call pn)) g)) % Found ((eq_ref (x_a->Prop)) B_4) as proof of (((eq (x_a->Prop)) B_4) ((insert_a (mgt_call pn)) g)) % Found fact_268_subset__refl0:=(fact_268_subset__refl ((insert_a (mgt_call pn)) g)):((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((insert_a (mgt_call pn)) g)) % Found (fact_268_subset__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_2) % Found (fact_268_subset__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_2) % Found (fact_268_subset__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_2) % Found eta_expansion000:=(eta_expansion00 ((image_pname_a mgt_call) u)):(((eq (x_a->Prop)) ((image_pname_a mgt_call) u)) (fun (x:x_a)=> (((image_pname_a mgt_call) u) x))) % Found (eta_expansion00 ((image_pname_a mgt_call) u)) as proof of (((eq (x_a->Prop)) ((image_pname_a mgt_call) u)) B_2) % Found ((eta_expansion0 Prop) ((image_pname_a mgt_call) u)) as proof of (((eq (x_a->Prop)) ((image_pname_a mgt_call) u)) B_2) % Found (((eta_expansion x_a) Prop) ((image_pname_a mgt_call) u)) as proof of (((eq (x_a->Prop)) ((image_pname_a mgt_call) u)) B_2) % Found (((eta_expansion x_a) Prop) ((image_pname_a mgt_call) u)) as proof of (((eq (x_a->Prop)) ((image_pname_a mgt_call) u)) B_2) % Found (((eta_expansion x_a) Prop) ((image_pname_a mgt_call) u)) as proof of (((eq (x_a->Prop)) ((image_pname_a mgt_call) u)) B_2) % Found fact_370_order__refl0:=(fact_370_order__refl ((insert_a (mgt_call pn)) g)):((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((insert_a (mgt_call pn)) g)) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_3) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_3) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_3) % Found eq_ref00:=(eq_ref0 B_3):(((eq (x_a->Prop)) B_3) B_3) % Found (eq_ref0 B_3) as proof of (((eq (x_a->Prop)) B_3) ((image_pname_a mgt_call) u)) % Found ((eq_ref (x_a->Prop)) B_3) as proof of (((eq (x_a->Prop)) B_3) ((image_pname_a mgt_call) u)) % Found ((eq_ref (x_a->Prop)) B_3) as proof of (((eq (x_a->Prop)) B_3) ((image_pname_a mgt_call) u)) % Found ((eq_ref (x_a->Prop)) B_3) as proof of (((eq (x_a->Prop)) B_3) ((image_pname_a mgt_call) u)) % Found conj_1:((ord_less_eq_a_o g) ((image_pname_a mgt_call) u)) % Found conj_1 as proof of ((ord_less_eq_a_o g) ((image_pname_a mgt_call) u)) % Found fact_370_order__refl0:=(fact_370_order__refl Y_6):((ord_less_eq_a_o Y_6) Y_6) % Found (fact_370_order__refl Y_6) as proof of ((ord_less_eq_a_o Y_6) g) % Found (fact_370_order__refl Y_6) as proof of ((ord_less_eq_a_o Y_6) g) % Found (fact_370_order__refl Y_6) as proof of ((ord_less_eq_a_o Y_6) g) % Found fact_370_order__refl0:=(fact_370_order__refl ((insert_a (mgt_call pn)) g)):((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((insert_a (mgt_call pn)) g)) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_6) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_6) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_6) % Found fact_370_order__refl0:=(fact_370_order__refl B_22):((ord_less_eq_a_o B_22) B_22) % Found (fact_370_order__refl B_22) as proof of ((ord_less_eq_a_o B_22) g) % Found (fact_370_order__refl B_22) as proof of ((ord_less_eq_a_o B_22) g) % Found (fact_370_order__refl B_22) as proof of ((ord_less_eq_a_o B_22) g) % Found fact_370_order__refl0:=(fact_370_order__refl ((insert_a (mgt_call pn)) g)):((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((insert_a (mgt_call pn)) g)) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_22) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_22) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_22) % Found fact_370_order__refl0:=(fact_370_order__refl Y_6):((ord_less_eq_a_o Y_6) Y_6) % Found (fact_370_order__refl Y_6) as proof of ((ord_less_eq_a_o Y_6) g) % Found (fact_370_order__refl Y_6) as proof of ((ord_less_eq_a_o Y_6) g) % Found (fact_370_order__refl Y_6) as proof of ((ord_less_eq_a_o Y_6) g) % Found fact_370_order__refl0:=(fact_370_order__refl ((insert_a (mgt_call pn)) g)):((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((insert_a (mgt_call pn)) g)) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_6) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_6) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_6) % Found fact_370_order__refl0:=(fact_370_order__refl B_22):((ord_less_eq_a_o B_22) B_22) % Found (fact_370_order__refl B_22) as proof of ((ord_less_eq_a_o B_22) g) % Found (fact_370_order__refl B_22) as proof of ((ord_less_eq_a_o B_22) g) % Found (fact_370_order__refl B_22) as proof of ((ord_less_eq_a_o B_22) g) % Found fact_370_order__refl0:=(fact_370_order__refl ((insert_a (mgt_call pn)) g)):((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((insert_a (mgt_call pn)) g)) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_22) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_22) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_22) % Found fact_444_bot__least0:=(fact_444_bot__least g):((ord_less_eq_a_o bot_bot_a_o) g) % Found (fact_444_bot__least g) as proof of ((ord_less_eq_a_o Y_8) g) % Found (fact_444_bot__least g) as proof of ((ord_less_eq_a_o Y_8) g) % Found (fact_444_bot__least g) as proof of ((ord_less_eq_a_o Y_8) g) % Found fact_370_order__refl0:=(fact_370_order__refl ((insert_a (mgt_call pn)) g)):((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((insert_a (mgt_call pn)) g)) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_8) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_8) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_8) % Found fact_444_bot__least0:=(fact_444_bot__least g):((ord_less_eq_a_o bot_bot_a_o) g) % Found (fact_444_bot__least g) as proof of ((ord_less_eq_a_o Y_8) g) % Found (fact_444_bot__least g) as proof of ((ord_less_eq_a_o Y_8) g) % Found (fact_444_bot__least g) as proof of ((ord_less_eq_a_o Y_8) g) % Found fact_370_order__refl0:=(fact_370_order__refl ((insert_a (mgt_call pn)) g)):((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((insert_a (mgt_call pn)) g)) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_8) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_8) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_8) % Found fact_423_empty__def0:=(fact_423_empty__def (fun (x:(x_a->Prop))=> (P ((insert_a (mgt_call pn)) g)))):((P ((insert_a (mgt_call pn)) g))->(P ((insert_a (mgt_call pn)) g))) % Found (fact_423_empty__def (fun (x:(x_a->Prop))=> (P ((insert_a (mgt_call pn)) g)))) as proof of (P0 ((insert_a (mgt_call pn)) g)) % Found (fact_423_empty__def (fun (x:(x_a->Prop))=> (P ((insert_a (mgt_call pn)) g)))) as proof of (P0 ((insert_a (mgt_call pn)) g)) % Found fact_423_empty__def0:=(fact_423_empty__def (fun (x:(x_a->Prop))=> (P ((insert_a (mgt_call pn)) g)))):((P ((insert_a (mgt_call pn)) g))->(P ((insert_a (mgt_call pn)) g))) % Found (fact_423_empty__def (fun (x:(x_a->Prop))=> (P ((insert_a (mgt_call pn)) g)))) as proof of (P0 ((insert_a (mgt_call pn)) g)) % Found (fact_423_empty__def (fun (x:(x_a->Prop))=> (P ((insert_a (mgt_call pn)) g)))) as proof of (P0 ((insert_a (mgt_call pn)) g)) % Found conj_30:=(conj_3 (fun (x:nat)=> (P ((image_pname_a mgt_call) u)))):((P ((image_pname_a mgt_call) u))->(P ((image_pname_a mgt_call) u))) % Found (conj_3 (fun (x:nat)=> (P ((image_pname_a mgt_call) u)))) as proof of (P0 ((image_pname_a mgt_call) u)) % Found (conj_3 (fun (x:nat)=> (P ((image_pname_a mgt_call) u)))) as proof of (P0 ((image_pname_a mgt_call) u)) % Found conj_1:((ord_less_eq_a_o g) ((image_pname_a mgt_call) u)) % Instantiate: A_74:=g:(x_a->Prop) % Found conj_1 as proof of ((ord_less_eq_a_o A_74) ((image_pname_a mgt_call) u)) % Found x:((member_a X_1) ((insert_a (mgt_call pn)) g)) % Instantiate: A_74:=((insert_a (mgt_call pn)) g):(x_a->Prop) % Found x as proof of ((member_a X_1) A_74) % Found conj_1:((ord_less_eq_a_o g) ((image_pname_a mgt_call) u)) % Instantiate: A_52:=g:(x_a->Prop) % Found conj_1 as proof of ((ord_less_eq_a_o A_52) ((image_pname_a mgt_call) u)) % Found x:((member_a X_1) ((insert_a (mgt_call pn)) g)) % Instantiate: A_52:=((insert_a (mgt_call pn)) g):(x_a->Prop) % Found x as proof of ((member_a X_1) A_52) % Found x:((member_a X_1) ((insert_a (mgt_call pn)) g)) % Instantiate: A_54:=((insert_a (mgt_call pn)) g):(x_a->Prop) % Found x as proof of ((member_a X_1) A_54) % Found conj_1:((ord_less_eq_a_o g) ((image_pname_a mgt_call) u)) % Instantiate: A_54:=g:(x_a->Prop) % Found conj_1 as proof of ((ord_less_eq_a_o A_54) ((image_pname_a mgt_call) u)) % Found fact_268_subset__refl0:=(fact_268_subset__refl B_22):((ord_less_eq_a_o B_22) B_22) % Found (fact_268_subset__refl B_22) as proof of ((ord_less_eq_a_o B_22) g) % Found (fact_268_subset__refl B_22) as proof of ((ord_less_eq_a_o B_22) g) % Found (fact_268_subset__refl B_22) as proof of ((ord_less_eq_a_o B_22) g) % Found fact_370_order__refl0:=(fact_370_order__refl ((insert_a (mgt_call pn)) g)):((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((insert_a (mgt_call pn)) g)) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_22) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_22) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_22) % Found fact_370_order__refl0:=(fact_370_order__refl Y_6):((ord_less_eq_a_o Y_6) Y_6) % Found (fact_370_order__refl Y_6) as proof of ((ord_less_eq_a_o Y_6) g) % Found (fact_370_order__refl Y_6) as proof of ((ord_less_eq_a_o Y_6) g) % Found (fact_370_order__refl Y_6) as proof of ((ord_less_eq_a_o Y_6) g) % Found fact_370_order__refl0:=(fact_370_order__refl ((insert_a (mgt_call pn)) g)):((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((insert_a (mgt_call pn)) g)) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_6) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_6) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_6) % Found fact_444_bot__least0:=(fact_444_bot__least g):((ord_less_eq_a_o bot_bot_a_o) g) % Found (fact_444_bot__least g) as proof of ((ord_less_eq_a_o B_22) g) % Found (fact_444_bot__least g) as proof of ((ord_less_eq_a_o B_22) g) % Found (fact_444_bot__least g) as proof of ((ord_less_eq_a_o B_22) g) % Found fact_370_order__refl0:=(fact_370_order__refl ((insert_a (mgt_call pn)) g)):((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((insert_a (mgt_call pn)) g)) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_22) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_22) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) B_22) % Found fact_370_order__refl0:=(fact_370_order__refl Y_6):((ord_less_eq_a_o Y_6) Y_6) % Found (fact_370_order__refl Y_6) as proof of ((ord_less_eq_a_o Y_6) g) % Found (fact_370_order__refl Y_6) as proof of ((ord_less_eq_a_o Y_6) g) % Found (fact_370_order__refl Y_6) as proof of ((ord_less_eq_a_o Y_6) g) % Found fact_370_order__refl0:=(fact_370_order__refl ((insert_a (mgt_call pn)) g)):((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((insert_a (mgt_call pn)) g)) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_6) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_6) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_6) % Found fact_444_bot__least0:=(fact_444_bot__least g):((ord_less_eq_a_o bot_bot_a_o) g) % Found (fact_444_bot__least g) as proof of ((ord_less_eq_a_o Y_8) g) % Found (fact_444_bot__least g) as proof of ((ord_less_eq_a_o Y_8) g) % Found (fact_444_bot__least g) as proof of ((ord_less_eq_a_o Y_8) g) % Found fact_370_order__refl0:=(fact_370_order__refl ((insert_a (mgt_call pn)) g)):((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((insert_a (mgt_call pn)) g)) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_8) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_8) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_8) % Found fact_370_order__refl0:=(fact_370_order__refl Y_8):((ord_less_eq_a_o Y_8) Y_8) % Found (fact_370_order__refl Y_8) as proof of ((ord_less_eq_a_o Y_8) g) % Found (fact_370_order__refl Y_8) as proof of ((ord_less_eq_a_o Y_8) g) % Found (fact_370_order__refl Y_8) as proof of ((ord_less_eq_a_o Y_8) g) % Found fact_370_order__refl0:=(fact_370_order__refl ((insert_a (mgt_call pn)) g)):((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) ((insert_a (mgt_call pn)) g)) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_8) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_8) % Found (fact_370_order__refl ((insert_a (mgt_call pn)) g)) as proof of ((ord_less_eq_a_o ((insert_a (mgt_call pn)) g)) Y_8) % Found fact_422_empty__def0:=(fact_422_empty__def (fun (x:((x_a->Prop)->Prop))=> (P ((insert_a (mgt_call pn)) g)))):((P ((insert_a (mgt_call pn)) g))->(P ((insert_a (mgt_call pn)) g))) % Found (fact_422_empty__def (fun (x:((x_a->Prop)->Prop))=> (P ((insert_a (mgt_call pn)) g)))) as proof of (P0 ((insert_a (mgt_call pn)) g)) % Found (fact_422_empty__def (fun (x:((x_a->Prop)->Prop))=> (P ((insert_a (mgt_call pn)) g)))) as proof of (P0 ((insert_a (mgt_call pn)) g)) % Found conj_30:=(conj_3 (fun (x:nat)=> (P ((image_pname_a mgt_call) u)))):((P ((image_pname_a mgt_call) u))->(P ((image_pna % EOF %------------------------------------------------------------------------------