%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT144^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:44:55 AM UTC 2026
% Result : Theorem 2.90s 1.59s
% Output : Refutation 2.90s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : DAT144^1 : TPTP v9.2.1. Released v7.0.0.
% 0.12/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n003.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 06:08:56 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.29/0.46 ---- Original DTF file ---
% 0.29/0.46 thf(spec,logic,$$dhol).
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 % File : DAT144^1 : TPTP v9.2.1. Released v7.0.0.
% 0.29/0.46 % Domain : Data Structures
% 0.29/0.46 % Problem : Coinductive stream 14
% 0.29/0.46 % Version : [Bla16] axioms : Especial.
% 0.29/0.46 % English :
% 0.29/0.46
% 0.29/0.46 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.29/0.46 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.29/0.46 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.29/0.46 % Source : [Bla16]
% 0.29/0.46 % Names : coinductive_stream__14.p [Bla16]
% 0.29/0.46
% 0.29/0.46 % Status : Theorem
% 0.29/0.46 % Rating : 0.33 v9.1.0, 0.00 v8.1.0, 0.25 v7.5.0, 0.00 v7.2.0, 0.25 v7.1.0
% 0.29/0.46 % Syntax : Number of formulae : 378 ( 141 unt; 59 typ; 0 def)
% 0.29/0.46 % Number of atoms : 828 ( 201 equ; 0 cnn)
% 0.29/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.29/0.46 % Number of connectives : 3312 ( 31 ~; 3 |; 42 &;2856 @)
% 0.29/0.46 % ( 0 <=>; 380 =>; 0 <=; 0 <~>)
% 0.29/0.46 % Maximal formula depth : 19 ( 7 avg)
% 0.29/0.46 % Number of types : 7 ( 6 usr)
% 0.29/0.46 % Number of type conns : 402 ( 402 >; 0 *; 0 +; 0 <<)
% 0.29/0.46 % Number of symbols : 55 ( 53 usr; 2 con; 0-6 aty)
% 0.29/0.46 % Number of variables : 1016 ( 68 ^; 875 !; 17 ?;1016 :)
% 0.29/0.46 % ( 56 !>; 0 ?*; 0 @-; 0 @+)
% 0.29/0.46 % SPC : TH1_THM_EQU_NAR
% 0.29/0.46
% 0.29/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.29/0.46 % 2016-07-13 15:12:48.657
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 %----Could-be-implicit typings (8)
% 0.29/0.46 thf(ty_t_Extended__Nat_Oenat,type,
% 0.29/0.46 extended_enat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_Real_Oreal,type,
% 0.29/0.46 real: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_Set_Oset,type,
% 0.29/0.46 set: $tType > $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_Num_Onum,type,
% 0.29/0.46 num: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_Nat_Onat,type,
% 0.29/0.46 nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_Int_Oint,type,
% 0.29/0.46 int: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_itself,type,
% 0.29/0.46 itself: $tType > $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_tf_a,type,
% 0.29/0.46 a: $tType ).
% 0.29/0.46
% 0.29/0.46 %----Explicit typings (51)
% 0.29/0.46 thf(sy_cl_HOL_Otype,type,
% 0.29/0.46 type:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Rings_Oring__1,type,
% 0.29/0.46 ring_1:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.29/0.46 ord:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Otop,type,
% 0.29/0.46 top:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.29/0.46 order:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.29/0.46 linorder:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.29/0.46 preorder:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Oorder__top,type,
% 0.29/0.46 order_top:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.29/0.46 linordered_idom:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 0.29/0.46 boolean_algebra:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.29/0.46 linordered_semidom:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.29/0.46 ordered_ab_group_add:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.29/0.46 linord219039673up_add:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
% 0.29/0.46 archim1727834104eiling:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Topological__Spaces_Ouniform__space,type,
% 0.29/0.46 topolo47006728_space:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.29/0.46 comple187826305attice:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 0.29/0.46 archim1804426504_field:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.29/0.46 canoni770627133id_add:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Archimedean__Field_Oceiling,type,
% 0.29/0.46 archimedean_ceiling:
% 0.29/0.46 !>[A: $tType] : ( A > int ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
% 0.29/0.46 archim2085082626_floor:
% 0.29/0.46 !>[A: $tType] : ( A > int ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Archimedean__Field_Oround,type,
% 0.29/0.46 archimedean_round:
% 0.29/0.46 !>[A: $tType] : ( A > int ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_BNF__Composition_ODEADID_Opred__DEADID,type,
% 0.29/0.46 bNF_pred_DEADID:
% 0.29/0.46 !>[A: $tType] : ( A > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_BNF__Def_OGrp,type,
% 0.29/0.46 bNF_Grp:
% 0.29/0.46 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_BNF__Def_Oeq__onp,type,
% 0.29/0.46 bNF_eq_onp:
% 0.29/0.46 !>[A: $tType] : ( ( A > $o ) > A > A > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Extended__Nat_Oenat,type,
% 0.29/0.46 extended_enat2: nat > extended_enat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.29/0.46 minus_minus:
% 0.29/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.29/0.46 uminus_uminus:
% 0.29/0.46 !>[A: $tType] : ( A > A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.29/0.46 zero_zero:
% 0.29/0.46 !>[A: $tType] : A ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_HOL_Oundefined,type,
% 0.29/0.46 undefined:
% 0.29/0.46 !>[A: $tType] : A ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Hilbert__Choice_OGreatestM,type,
% 0.29/0.46 hilbert_GreatestM:
% 0.29/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Hilbert__Choice_OLeastM,type,
% 0.29/0.46 hilbert_LeastM:
% 0.29/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Int_Onat,type,
% 0.29/0.46 nat2: int > nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Lifting_ONEG,type,
% 0.29/0.46 neg:
% 0.29/0.46 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( A > B > $o ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Lifting_OPOS,type,
% 0.29/0.46 pos:
% 0.29/0.46 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( A > B > $o ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Lifting_OQuotient,type,
% 0.29/0.46 quotient:
% 0.29/0.46 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.29/0.46 semiring_1_of_nat:
% 0.29/0.46 !>[A: $tType] : ( nat > A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Nat__Transfer_Otransfer__morphism,type,
% 0.29/0.46 nat_tr1645093318rphism:
% 0.29/0.46 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Nat__Transfer_Otsub,type,
% 0.29/0.46 nat_tsub: int > int > int ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Num_Onumeral__class_Onumeral,type,
% 0.29/0.46 numeral_numeral:
% 0.29/0.46 !>[A: $tType] : ( num > A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.29/0.46 ord_less_eq:
% 0.29/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 0.29/0.46 order_antimono:
% 0.29/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.29/0.46 top_top:
% 0.29/0.46 !>[A: $tType] : A ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pure_Otype,type,
% 0.29/0.46 type2:
% 0.29/0.46 !>[A: $tType] : ( itself @ A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Relation_ODomainp,type,
% 0.29/0.46 domainp:
% 0.29/0.46 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Relation_Oinv__imagep,type,
% 0.29/0.46 inv_imagep:
% 0.29/0.46 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_OCollect,type,
% 0.29/0.46 collect:
% 0.29/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Topological__Spaces_Omonoseq,type,
% 0.29/0.46 topological_monoseq:
% 0.29/0.46 !>[A: $tType] : ( ( nat > A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
% 0.29/0.46 topolo406746546ounded:
% 0.29/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_member,type,
% 0.29/0.46 member:
% 0.29/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_P,type,
% 0.29/0.46 p: a > $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_x,type,
% 0.29/0.46 x: a ).
% 0.29/0.46
% 0.29/0.46 %----Relevant facts (256)
% 0.29/0.46 thf(fact_0_eq__onp__eqD,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > A > $o,X: A] :
% 0.29/0.46 ( ( ( bNF_eq_onp @ A @ P )
% 0.29/0.46 = Q )
% 0.29/0.46 => ( ( P @ X )
% 0.29/0.46 = ( Q @ X @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_onp_eqD
% 0.29/0.46 thf(fact_1_eq__onp__mono0,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ! [X2: A] :
% 0.29/0.46 ( ( member @ A @ X2 @ A2 )
% 0.29/0.46 => ( ( P @ X2 )
% 0.29/0.46 => ( Q @ X2 ) ) )
% 0.29/0.46 => ! [X3: A] :
% 0.29/0.46 ( ( member @ A @ X3 @ A2 )
% 0.29/0.46 => ! [Xa: A] :
% 0.29/0.46 ( ( member @ A @ Xa @ A2 )
% 0.29/0.46 => ( ( bNF_eq_onp @ A @ P @ X3 @ Xa )
% 0.29/0.46 => ( bNF_eq_onp @ A @ Q @ X3 @ Xa ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_onp_mono0
% 0.29/0.46 thf(fact_2_eq__onp__to__eq,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,X: A,Y: A] :
% 0.29/0.46 ( ( bNF_eq_onp @ A @ P @ X @ Y )
% 0.29/0.46 => ( X = Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_onp_to_eq
% 0.29/0.46 thf(fact_3_eq__onp__same__args,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,X: A] :
% 0.29/0.46 ( ( bNF_eq_onp @ A @ P @ X @ X )
% 0.29/0.46 = ( P @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_onp_same_args
% 0.29/0.46 thf(fact_4_eq__onp__live__step,axiom,
% 0.29/0.46 ! [A: $tType,X: $o,Y: $o,P: A > $o,A3: A] :
% 0.29/0.46 ( ( X = Y )
% 0.29/0.46 => ( ( ( bNF_eq_onp @ A @ P @ A3 @ A3 )
% 0.29/0.46 & X )
% 0.29/0.46 = ( ( P @ A3 )
% 0.29/0.46 & Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_onp_live_step
% 0.29/0.46 thf(fact_5_DEADID_Orel__eq__onp,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.29/0.46 = ( bNF_eq_onp @ A @ ( bNF_pred_DEADID @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % DEADID.rel_eq_onp
% 0.29/0.46 thf(fact_6_eq__onp__le__eq,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o] :
% 0.29/0.46 ( ord_less_eq @ ( A > A > $o ) @ ( bNF_eq_onp @ A @ P )
% 0.29/0.46 @ ^ [Y2: A,Z: A] : ( Y2 = Z ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_onp_le_eq
% 0.29/0.46 thf(fact_7_type__definition__Quotient__not__empty,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P: A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
% 0.29/0.46 ( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ T )
% 0.29/0.46 => ? [X1: A] : ( P @ X1 ) ) ).
% 0.29/0.46
% 0.29/0.46 % type_definition_Quotient_not_empty
% 0.29/0.46 thf(fact_8_eq__onp__top__eq__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( bNF_eq_onp @ A @ ( top_top @ ( A > $o ) ) )
% 0.29/0.46 = ( ^ [Y2: A,Z: A] : ( Y2 = Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_onp_top_eq_eq
% 0.29/0.46 thf(fact_9_top__conj_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,P: $o,X: A] :
% 0.29/0.46 ( ( P
% 0.29/0.46 & ( top_top @ ( A > $o ) @ X ) )
% 0.29/0.46 = P ) ).
% 0.29/0.46
% 0.29/0.46 % top_conj(2)
% 0.29/0.46 thf(fact_10_top__conj_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,X: A,P: $o] :
% 0.29/0.46 ( ( ( top_top @ ( A > $o ) @ X )
% 0.29/0.46 & P )
% 0.29/0.46 = P ) ).
% 0.29/0.46
% 0.29/0.46 % top_conj(1)
% 0.29/0.46 thf(fact_11_ge__eq__refl,axiom,
% 0.29/0.46 ! [A: $tType,R: A > A > $o,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > A > $o )
% 0.29/0.46 @ ^ [Y2: A,Z: A] : ( Y2 = Z )
% 0.29/0.46 @ R )
% 0.29/0.46 => ( R @ X @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % ge_eq_refl
% 0.29/0.46 thf(fact_12_refl__ge__eq,axiom,
% 0.29/0.46 ! [A: $tType,R: A > A > $o] :
% 0.29/0.46 ( ! [X2: A] : ( R @ X2 @ X2 )
% 0.29/0.46 => ( ord_less_eq @ ( A > A > $o )
% 0.29/0.46 @ ^ [Y2: A,Z: A] : ( Y2 = Z )
% 0.29/0.46 @ R ) ) ).
% 0.29/0.46
% 0.29/0.46 % refl_ge_eq
% 0.29/0.46 thf(fact_13_Quotient__rel,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R2: A,S: A] :
% 0.29/0.46 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 0.29/0.46 => ( ( ( R @ R2 @ R2 )
% 0.29/0.46 & ( R @ S @ S )
% 0.29/0.46 & ( ( Abs @ R2 )
% 0.29/0.46 = ( Abs @ S ) ) )
% 0.29/0.46 = ( R @ R2 @ S ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_rel
% 0.29/0.46 thf(fact_14_Quotient__refl1,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R2: A,S: A] :
% 0.29/0.46 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 0.29/0.46 => ( ( R @ R2 @ S )
% 0.29/0.46 => ( R @ R2 @ R2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_refl1
% 0.29/0.46 thf(fact_15_Quotient__refl2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R2: A,S: A] :
% 0.29/0.46 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 0.29/0.46 => ( ( R @ R2 @ S )
% 0.29/0.46 => ( R @ S @ S ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_refl2
% 0.29/0.46 thf(fact_16_Quotient__Rep__eq,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,X4: B,X: A] :
% 0.29/0.46 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 0.29/0.46 => ( ( X4
% 0.29/0.46 = ( Abs @ X ) )
% 0.29/0.46 => ( ( Rep @ X4 )
% 0.29/0.46 = ( Rep @ X4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_Rep_eq
% 0.29/0.46 thf(fact_17_Quotient__abs__rep,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,A3: B] :
% 0.29/0.46 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 0.29/0.46 => ( ( Abs @ ( Rep @ A3 ) )
% 0.29/0.46 = A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_abs_rep
% 0.29/0.46 thf(fact_18_Quotient__alt__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( quotient @ A @ B )
% 0.29/0.46 = ( ^ [R3: A > A > $o,Abs2: A > B,Rep2: B > A,T2: A > B > $o] :
% 0.29/0.46 ( ! [A4: A,B2: B] :
% 0.29/0.46 ( ( T2 @ A4 @ B2 )
% 0.29/0.46 => ( ( Abs2 @ A4 )
% 0.29/0.46 = B2 ) )
% 0.29/0.46 & ! [B2: B] : ( T2 @ ( Rep2 @ B2 ) @ B2 )
% 0.29/0.46 & ! [X5: A,Y3: A] :
% 0.29/0.46 ( ( R3 @ X5 @ Y3 )
% 0.29/0.46 = ( ( T2 @ X5 @ ( Abs2 @ X5 ) )
% 0.29/0.46 & ( T2 @ Y3 @ ( Abs2 @ Y3 ) )
% 0.29/0.46 & ( ( Abs2 @ X5 )
% 0.29/0.46 = ( Abs2 @ Y3 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_alt_def
% 0.29/0.46 thf(fact_19_Quotient__rel__abs,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R2: A,S: A] :
% 0.29/0.46 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 0.29/0.46 => ( ( R @ R2 @ S )
% 0.29/0.46 => ( ( Abs @ R2 )
% 0.29/0.46 = ( Abs @ S ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_rel_abs
% 0.29/0.46 thf(fact_20_Quotient__rel__rep,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,A3: B,B3: B] :
% 0.29/0.46 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 0.29/0.46 => ( ( R @ ( Rep @ A3 ) @ ( Rep @ B3 ) )
% 0.29/0.46 = ( A3 = B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_rel_rep
% 0.29/0.46 thf(fact_21_Quotient__rep__abs,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R2: A] :
% 0.29/0.46 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 0.29/0.46 => ( ( R @ R2 @ R2 )
% 0.29/0.46 => ( R @ ( Rep @ ( Abs @ R2 ) ) @ R2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_rep_abs
% 0.29/0.46 thf(fact_22_Quotient__alt__def2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( quotient @ A @ B )
% 0.29/0.46 = ( ^ [R3: A > A > $o,Abs2: A > B,Rep2: B > A,T2: A > B > $o] :
% 0.29/0.46 ( ! [A4: A,B2: B] :
% 0.29/0.46 ( ( T2 @ A4 @ B2 )
% 0.29/0.46 => ( ( Abs2 @ A4 )
% 0.29/0.46 = B2 ) )
% 0.29/0.46 & ! [B2: B] : ( T2 @ ( Rep2 @ B2 ) @ B2 )
% 0.29/0.46 & ! [X5: A,Y3: A] :
% 0.29/0.46 ( ( R3 @ X5 @ Y3 )
% 0.29/0.46 = ( ( T2 @ X5 @ ( Abs2 @ Y3 ) )
% 0.29/0.46 & ( T2 @ Y3 @ ( Abs2 @ X5 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_alt_def2
% 0.29/0.46 thf(fact_23_Quotient__alt__def3,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( quotient @ A @ B )
% 0.29/0.46 = ( ^ [R3: A > A > $o,Abs2: A > B,Rep2: B > A,T2: A > B > $o] :
% 0.29/0.46 ( ! [A4: A,B2: B] :
% 0.29/0.46 ( ( T2 @ A4 @ B2 )
% 0.29/0.46 => ( ( Abs2 @ A4 )
% 0.29/0.46 = B2 ) )
% 0.29/0.46 & ! [B2: B] : ( T2 @ ( Rep2 @ B2 ) @ B2 )
% 0.29/0.46 & ! [X5: A,Y3: A] :
% 0.29/0.46 ( ( R3 @ X5 @ Y3 )
% 0.29/0.46 = ( ? [Z2: B] :
% 0.29/0.46 ( ( T2 @ X5 @ Z2 )
% 0.29/0.46 & ( T2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_alt_def3
% 0.29/0.46 thf(fact_24_Quotient__rel__abs2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,X: B,Y: A] :
% 0.29/0.46 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 0.29/0.46 => ( ( R @ ( Rep @ X ) @ Y )
% 0.29/0.46 => ( X
% 0.29/0.46 = ( Abs @ Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_rel_abs2
% 0.29/0.46 thf(fact_25_Quotient__rep__reflp,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,A3: B] :
% 0.29/0.46 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 0.29/0.46 => ( R @ ( Rep @ A3 ) @ ( Rep @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_rep_reflp
% 0.29/0.46 thf(fact_26_Quotient__abs__induct,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,P: B > $o,X: B] :
% 0.29/0.46 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 0.29/0.46 => ( ! [Y4: A] :
% 0.29/0.46 ( ( R @ Y4 @ Y4 )
% 0.29/0.46 => ( P @ ( Abs @ Y4 ) ) )
% 0.29/0.46 => ( P @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_abs_induct
% 0.29/0.46 thf(fact_27_Quotient__rep__abs__eq,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,T3: A] :
% 0.29/0.46 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 0.29/0.46 => ( ( R @ T3 @ T3 )
% 0.29/0.46 => ( ( ord_less_eq @ ( A > A > $o ) @ R
% 0.29/0.46 @ ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.29/0.46 => ( ( Rep @ ( Abs @ T3 ) )
% 0.29/0.46 = T3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_rep_abs_eq
% 0.29/0.46 thf(fact_28_Quotient__to__transfer,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,C: A,C2: B] :
% 0.29/0.46 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 0.29/0.46 => ( ( R @ C @ C )
% 0.29/0.46 => ( ( C2
% 0.29/0.46 = ( Abs @ C ) )
% 0.29/0.46 => ( T @ C @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_to_transfer
% 0.29/0.46 thf(fact_29_DEADID_Opred__rel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( bNF_pred_DEADID @ A )
% 0.29/0.46 = ( ^ [X5: A] : ( X5 = X5 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % DEADID.pred_rel
% 0.29/0.46 thf(fact_30_DEADID_Opred__True,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( bNF_pred_DEADID @ A )
% 0.29/0.46 = ( ^ [Uu: A] : $true ) ) ).
% 0.29/0.46
% 0.29/0.46 % DEADID.pred_True
% 0.29/0.46 thf(fact_31_predicate2D__conj,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X: A,Y: B] :
% 0.29/0.46 ( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.29/0.46 & R )
% 0.29/0.46 => ( R
% 0.29/0.46 & ( ( P @ X @ Y )
% 0.29/0.46 => ( Q @ X @ Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % predicate2D_conj
% 0.29/0.46 thf(fact_32_Quotient__rep__abs__fold__unmap,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,X4: B,X: A,Rep3: B > A] :
% 0.29/0.46 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 0.29/0.46 => ( ( X4
% 0.29/0.46 = ( Abs @ X ) )
% 0.29/0.46 => ( ( R @ X @ X )
% 0.29/0.46 => ( ( ( Rep @ X4 )
% 0.29/0.46 = ( Rep3 @ X4 ) )
% 0.29/0.46 => ( R @ ( Rep3 @ X4 ) @ X ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_rep_abs_fold_unmap
% 0.29/0.46 thf(fact_33_DEADID_Opred__cong__simp,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Ya: A] :
% 0.29/0.46 ( ( X = Ya )
% 0.29/0.46 => ( ( bNF_pred_DEADID @ A @ X )
% 0.29/0.46 = ( bNF_pred_DEADID @ A @ Ya ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % DEADID.pred_cong_simp
% 0.29/0.46 thf(fact_34_DEADID_Opred__mono__strong,axiom,
% 0.29/0.46 ! [A: $tType,X: A] :
% 0.29/0.46 ( ( bNF_pred_DEADID @ A @ X )
% 0.29/0.46 => ( bNF_pred_DEADID @ A @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % DEADID.pred_mono_strong
% 0.29/0.46 thf(fact_35_top__apply,axiom,
% 0.29/0.46 ! [C3: $tType,D: $tType] :
% 0.29/0.46 ( ( top @ C3 @ ( type2 @ C3 ) )
% 0.29/0.46 => ( ( top_top @ ( D > C3 ) )
% 0.29/0.46 = ( ^ [X5: D] : ( top_top @ C3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_apply
% 0.29/0.46 thf(fact_36_top1I,axiom,
% 0.29/0.46 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 0.29/0.46
% 0.29/0.46 % top1I
% 0.29/0.46 thf(fact_37_predicate2I,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 0.29/0.46 ( ! [X2: A,Y4: B] :
% 0.29/0.46 ( ( P @ X2 @ Y4 )
% 0.29/0.46 => ( Q @ X2 @ Y4 ) )
% 0.29/0.46 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 0.29/0.46
% 0.29/0.46 % predicate2I
% 0.29/0.46 thf(fact_38_order__refl,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_refl
% 0.29/0.46 thf(fact_39_le__funD,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.46 => ! [F: A > B,G: A > B,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_funD
% 0.29/0.46 thf(fact_40_le__funE,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.46 => ! [F: A > B,G: A > B,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_funE
% 0.29/0.46 thf(fact_41_le__funI,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.46 => ! [F: A > B,G: A > B] :
% 0.29/0.46 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.29/0.46 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_funI
% 0.29/0.46 thf(fact_42_le__fun__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.29/0.46 = ( ^ [F2: A > B,G2: A > B] :
% 0.29/0.46 ! [X5: A] : ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_fun_def
% 0.29/0.46 thf(fact_43_top__greatest,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_top @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_greatest
% 0.29/0.46 thf(fact_44_top_Oextremum__unique,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_top @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.29/0.46 = ( A3
% 0.29/0.46 = ( top_top @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % top.extremum_unique
% 0.29/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.29/0.46 ! [A: $tType,A3: A,P: A > $o] :
% 0.29/0.46 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.29/0.46 = ( P @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % mem_Collect_eq
% 0.29/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ( collect @ A
% 0.29/0.46 @ ^ [X5: A] : ( member @ A @ X5 @ A2 ) )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mem_eq
% 0.29/0.46 thf(fact_47_Collect__cong,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ! [X2: A] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 = ( Q @ X2 ) )
% 0.29/0.46 => ( ( collect @ A @ P )
% 0.29/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_cong
% 0.29/0.46 thf(fact_48_ext,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.29/0.46 ( ! [X2: A] :
% 0.29/0.46 ( ( F @ X2 )
% 0.29/0.46 = ( G @ X2 ) )
% 0.29/0.46 => ( F = G ) ) ).
% 0.29/0.46
% 0.29/0.46 % ext
% 0.29/0.46 thf(fact_49_top_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_top @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.29/0.46 => ( A3
% 0.29/0.46 = ( top_top @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % top.extremum_uniqueI
% 0.29/0.46 thf(fact_50_dual__order_Oantisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [B3: A,A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( A3 = B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.antisym
% 0.29/0.46 thf(fact_51_dual__order_Otrans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [B3: A,A3: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ C @ B3 )
% 0.29/0.46 => ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.trans
% 0.29/0.46 thf(fact_52_linorder__wlog,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [P: A > A > $o,A3: A,B3: A] :
% 0.29/0.46 ( ! [A5: A,B4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.29/0.46 => ( P @ A5 @ B4 ) )
% 0.29/0.46 => ( ! [A5: A,B4: A] :
% 0.29/0.46 ( ( P @ B4 @ A5 )
% 0.29/0.46 => ( P @ A5 @ B4 ) )
% 0.29/0.46 => ( P @ A3 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_wlog
% 0.29/0.46 thf(fact_53_dual__order_Orefl,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.refl
% 0.29/0.46 thf(fact_54_order__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,Z3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_trans
% 0.29/0.46 thf(fact_55_order__class_Oorder_Oantisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.29/0.46 => ( A3 = B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.antisym
% 0.29/0.46 thf(fact_56_ord__le__eq__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A,B3: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( B3 = C )
% 0.29/0.46 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_trans
% 0.29/0.46 thf(fact_57_ord__eq__le__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A,B3: A,C: A] :
% 0.29/0.46 ( ( A3 = B3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.29/0.46 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_trans
% 0.29/0.46 thf(fact_58_antisym__conv,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Y: A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 = ( X = Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv
% 0.29/0.46 thf(fact_59_le__cases3,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,Z3: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ X @ Z3 ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ X @ Z3 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Z3 @ X ) )
% 0.29/0.46 => ~ ( ( ord_less_eq @ A @ Z3 @ X )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_cases3
% 0.29/0.46 thf(fact_60_order_Otrans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A,B3: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.29/0.46 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.trans
% 0.29/0.46 thf(fact_61_le__cases,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_cases
% 0.29/0.46 thf(fact_62_eq__refl,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( X = Y )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_refl
% 0.29/0.46 thf(fact_63_linear,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linear
% 0.29/0.46 thf(fact_64_antisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ( X = Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym
% 0.29/0.46 thf(fact_65_eq__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.29/0.46 = ( ^ [X5: A,Y3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 0.29/0.46 & ( ord_less_eq @ A @ Y3 @ X5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_iff
% 0.29/0.46 thf(fact_66_ord__le__eq__subst,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.46 & ( ord @ A @ ( type2 @ A ) ) )
% 0.29/0.46 => ! [A3: A,B3: A,F: A > B,C: B] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( ( F @ B3 )
% 0.29/0.46 = C )
% 0.29/0.46 => ( ! [X2: A,Y4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_subst
% 0.29/0.46 thf(fact_67_ord__eq__le__subst,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.46 & ( ord @ A @ ( type2 @ A ) ) )
% 0.29/0.46 => ! [A3: A,F: B > A,B3: B,C: B] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 = ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.29/0.46 => ( ! [X2: B,Y4: B] :
% 0.29/0.46 ( ( ord_less_eq @ B @ X2 @ Y4 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_subst
% 0.29/0.46 thf(fact_68_order__subst2,axiom,
% 0.29/0.46 ! [A: $tType,C3: $tType] :
% 0.29/0.46 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.29/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.46 => ! [A3: A,B3: A,F: A > C3,C: C3] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq @ C3 @ ( F @ B3 ) @ C )
% 0.29/0.46 => ( ! [X2: A,Y4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 0.29/0.46 => ( ord_less_eq @ C3 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst2
% 0.29/0.46 thf(fact_69_order__subst1,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( ( order @ B @ ( type2 @ B ) )
% 0.29/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.46 => ! [A3: A,F: B > A,B3: B,C: B] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.29/0.46 => ( ! [X2: B,Y4: B] :
% 0.29/0.46 ( ( ord_less_eq @ B @ X2 @ Y4 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst1
% 0.29/0.46 thf(fact_70_rev__predicate2D,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
% 0.29/0.46 ( ( P @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.29/0.46 => ( Q @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev_predicate2D
% 0.29/0.46 thf(fact_71_predicate2D,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.29/0.46 => ( ( P @ X @ Y )
% 0.29/0.46 => ( Q @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % predicate2D
% 0.29/0.46 thf(fact_72_top__empty__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( top_top @ ( A > $o ) )
% 0.29/0.46 = ( ^ [X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_empty_eq
% 0.29/0.46 thf(fact_73_top__set__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( top_top @ ( set @ A ) )
% 0.29/0.46 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_set_def
% 0.29/0.46 thf(fact_74_lfp_Oleq__antisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ( X = Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfp.leq_antisym
% 0.29/0.46 thf(fact_75_gfp_Oleq__antisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Y: A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( X = Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % gfp.leq_antisym
% 0.29/0.46 thf(fact_76_lfp_Oleq__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,Z3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfp.leq_trans
% 0.29/0.46 thf(fact_77_gfp_Oleq__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Y: A,X: A,Z3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Z3 @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ Z3 @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % gfp.leq_trans
% 0.29/0.46 thf(fact_78_lfp_Oleq__refl,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfp.leq_refl
% 0.29/0.46 thf(fact_79_wlog__linorder__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [P: A > A > $o,B3: A,A3: A] :
% 0.29/0.46 ( ! [A5: A,B4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.29/0.46 => ( P @ A5 @ B4 ) )
% 0.29/0.46 => ( ( ( P @ B3 @ A3 )
% 0.29/0.46 => ( P @ A3 @ B3 ) )
% 0.29/0.46 => ( P @ A3 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % wlog_linorder_le
% 0.29/0.46 thf(fact_80_UNIV__I,axiom,
% 0.29/0.46 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % UNIV_I
% 0.29/0.46 thf(fact_81_UNIV__eq__I,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ! [X2: A] : ( member @ A @ X2 @ A2 )
% 0.29/0.46 => ( ( top_top @ ( set @ A ) )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % UNIV_eq_I
% 0.29/0.46 thf(fact_82_subset__UNIV,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_UNIV
% 0.29/0.46 thf(fact_83_UNIV__witness,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % UNIV_witness
% 0.29/0.46 thf(fact_84_iso__tuple__UNIV__I,axiom,
% 0.29/0.46 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % iso_tuple_UNIV_I
% 0.29/0.46 thf(fact_85_in__inv__imagep,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( inv_imagep @ A @ B )
% 0.29/0.46 = ( ^ [R4: A > A > $o,F2: B > A,X5: B,Y3: B] : ( R4 @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % in_inv_imagep
% 0.29/0.46 thf(fact_86_eq__onp__to__Domainp,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P: A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
% 0.29/0.46 ( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ T )
% 0.29/0.46 => ( ( domainp @ A @ B @ T )
% 0.29/0.46 = P ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_onp_to_Domainp
% 0.29/0.46 thf(fact_87_type__definition__Quotient__not__empty__witness,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,P: A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
% 0.29/0.46 ( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ T )
% 0.29/0.46 => ( P @ ( Rep @ ( undefined @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % type_definition_Quotient_not_empty_witness
% 0.29/0.46 thf(fact_88_NEG__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( neg @ A @ B )
% 0.29/0.46 = ( ^ [A6: A > B > $o,B5: A > B > $o] : ( ord_less_eq @ ( A > B > $o ) @ B5 @ A6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % NEG_def
% 0.29/0.46 thf(fact_89_antimonoD,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 & ( order @ B @ ( type2 @ B ) ) )
% 0.29/0.46 => ! [F: A > B,X: A,Y: A] :
% 0.29/0.46 ( ( order_antimono @ A @ B @ F )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antimonoD
% 0.29/0.46 thf(fact_90_subsetI,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B6: set @ A] :
% 0.29/0.46 ( ! [X2: A] :
% 0.29/0.46 ( ( member @ A @ X2 @ A2 )
% 0.29/0.46 => ( member @ A @ X2 @ B6 ) )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B6 ) ) ).
% 0.29/0.46
% 0.29/0.46 % subsetI
% 0.29/0.46 thf(fact_91_subset__antisym,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B6: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A2 )
% 0.29/0.46 => ( A2 = B6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_antisym
% 0.29/0.46 thf(fact_92_set__mp,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B6: set @ A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 0.29/0.46 => ( ( member @ A @ X @ A2 )
% 0.29/0.46 => ( member @ A @ X @ B6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % set_mp
% 0.29/0.46 thf(fact_93_in__mono,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B6: set @ A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 0.29/0.46 => ( ( member @ A @ X @ A2 )
% 0.29/0.46 => ( member @ A @ X @ B6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % in_mono
% 0.29/0.46 thf(fact_94_subsetD,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B6: set @ A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 0.29/0.46 => ( ( member @ A @ C @ A2 )
% 0.29/0.46 => ( member @ A @ C @ B6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subsetD
% 0.29/0.46 thf(fact_95_subsetCE,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B6: set @ A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 0.29/0.46 => ( ( member @ A @ C @ A2 )
% 0.29/0.46 => ( member @ A @ C @ B6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subsetCE
% 0.29/0.46 thf(fact_96_equalityE,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B6: set @ A] :
% 0.29/0.46 ( ( A2 = B6 )
% 0.29/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 0.29/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityE
% 0.29/0.46 thf(fact_97_subset__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( member @ A @ X5 @ A6 )
% 0.29/0.46 => ( member @ A @ X5 @ B5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_eq
% 0.29/0.46 thf(fact_98_equalityD1,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B6: set @ A] :
% 0.29/0.46 ( ( A2 = B6 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B6 ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityD1
% 0.29/0.46 thf(fact_99_equalityD2,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B6: set @ A] :
% 0.29/0.46 ( ( A2 = B6 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ B6 @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityD2
% 0.29/0.46 thf(fact_100_set__rev__mp,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A2: set @ A,B6: set @ A] :
% 0.29/0.46 ( ( member @ A @ X @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 0.29/0.46 => ( member @ A @ X @ B6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % set_rev_mp
% 0.29/0.46 thf(fact_101_subset__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.29/0.46 ! [T4: A] :
% 0.29/0.46 ( ( member @ A @ T4 @ A6 )
% 0.29/0.46 => ( member @ A @ T4 @ B5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_iff
% 0.29/0.46 thf(fact_102_rev__subsetD,axiom,
% 0.29/0.46 ! [A: $tType,C: A,A2: set @ A,B6: set @ A] :
% 0.29/0.46 ( ( member @ A @ C @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 0.29/0.46 => ( member @ A @ C @ B6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev_subsetD
% 0.29/0.46 thf(fact_103_subset__refl,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.29/0.46
% 0.29/0.46 % subset_refl
% 0.29/0.46 thf(fact_104_Collect__mono,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ! [X2: A] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 => ( Q @ X2 ) )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mono
% 0.29/0.46 thf(fact_105_subset__trans,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B6: set @ A,C4: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_trans
% 0.29/0.46 thf(fact_106_set__eq__subset,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ^ [Y2: set @ A,Z: set @ A] : ( Y2 = Z ) )
% 0.29/0.46 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
% 0.29/0.46 & ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % set_eq_subset
% 0.29/0.46 thf(fact_107_contra__subsetD,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B6: set @ A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 0.29/0.46 => ( ~ ( member @ A @ C @ B6 )
% 0.29/0.46 => ~ ( member @ A @ C @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % contra_subsetD
% 0.29/0.46 thf(fact_108_Collect__mono__iff,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.29/0.46 = ( ! [X5: A] :
% 0.29/0.46 ( ( P @ X5 )
% 0.29/0.46 => ( Q @ X5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mono_iff
% 0.29/0.46 thf(fact_109_DomainpE,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,R2: A > B > $o,A3: A] :
% 0.29/0.46 ( ( domainp @ A @ B @ R2 @ A3 )
% 0.29/0.46 => ~ ! [B4: B] :
% 0.29/0.46 ~ ( R2 @ A3 @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % DomainpE
% 0.29/0.46 thf(fact_110_Domainp_Ocases,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,R2: A > B > $o,A3: A] :
% 0.29/0.46 ( ( domainp @ A @ B @ R2 @ A3 )
% 0.29/0.46 => ~ ! [B4: B] :
% 0.29/0.46 ~ ( R2 @ A3 @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Domainp.cases
% 0.29/0.46 thf(fact_111_Domainp_Osimps,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( domainp @ A @ B )
% 0.29/0.46 = ( ^ [R4: A > B > $o,A4: A] :
% 0.29/0.46 ? [B2: A,C5: B] :
% 0.29/0.46 ( ( A4 = B2 )
% 0.29/0.46 & ( R4 @ B2 @ C5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Domainp.simps
% 0.29/0.46 thf(fact_112_Domainp_ODomainI,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R2: A > B > $o,A3: A,B3: B] :
% 0.29/0.46 ( ( R2 @ A3 @ B3 )
% 0.29/0.46 => ( domainp @ A @ B @ R2 @ A3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Domainp.DomainI
% 0.29/0.46 thf(fact_113_Domainp_Oinducts,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R2: A > B > $o,X: A,P: A > $o] :
% 0.29/0.46 ( ( domainp @ A @ B @ R2 @ X )
% 0.29/0.46 => ( ! [A5: A,B4: B] :
% 0.29/0.46 ( ( R2 @ A5 @ B4 )
% 0.29/0.46 => ( P @ A5 ) )
% 0.29/0.46 => ( P @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Domainp.inducts
% 0.29/0.46 thf(fact_114_NEG__trans,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A2: A > B > $o,B6: A > B > $o,C4: A > B > $o] :
% 0.29/0.46 ( ( neg @ A @ B @ A2 @ B6 )
% 0.29/0.46 => ( ( neg @ A @ B @ B6 @ C4 )
% 0.29/0.46 => ( neg @ A @ B @ A2 @ C4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % NEG_trans
% 0.29/0.46 thf(fact_115_Quotient__to__Domainp,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
% 0.29/0.46 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 0.29/0.46 => ( ( domainp @ A @ B @ T )
% 0.29/0.46 = ( ^ [X5: A] : ( R @ X5 @ X5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Quotient_to_Domainp
% 0.29/0.46 thf(fact_116_antimono__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 & ( order @ B @ ( type2 @ B ) ) )
% 0.29/0.46 => ( ( order_antimono @ A @ B )
% 0.29/0.46 = ( ^ [F2: A > B] :
% 0.29/0.46 ! [X5: A,Y3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X5 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antimono_def
% 0.29/0.46 thf(fact_117_antimonoI,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 & ( order @ B @ ( type2 @ B ) ) )
% 0.29/0.46 => ! [F: A > B] :
% 0.29/0.46 ( ! [X2: A,Y4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X2 ) ) )
% 0.29/0.46 => ( order_antimono @ A @ B @ F ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antimonoI
% 0.29/0.46 thf(fact_118_antimonoE,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 & ( order @ B @ ( type2 @ B ) ) )
% 0.29/0.46 => ! [F: A > B,X: A,Y: A] :
% 0.29/0.46 ( ( order_antimono @ A @ B @ F )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antimonoE
% 0.29/0.46 thf(fact_119_Domain__eq__top,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( domainp @ A @ A
% 0.29/0.46 @ ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.29/0.46 = ( top_top @ ( A > $o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Domain_eq_top
% 0.29/0.46 thf(fact_120_Domainp__refl,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( domainp @ A @ B )
% 0.29/0.46 = ( domainp @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % Domainp_refl
% 0.29/0.46 thf(fact_121_Domainp__iff,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( domainp @ A @ B )
% 0.29/0.46 = ( ^ [T2: A > B > $o,X5: A] :
% 0.29/0.46 ( ^ [P2: B > $o] :
% 0.29/0.46 ? [X6: B] : ( P2 @ X6 )
% 0.29/0.46 @ ( T2 @ X5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Domainp_iff
% 0.29/0.46 thf(fact_122_decseqD,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [F: nat > A,I: nat,J: nat] :
% 0.29/0.46 ( ( order_antimono @ nat @ A @ F )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ I @ J )
% 0.29/0.46 => ( ord_less_eq @ A @ ( F @ J ) @ ( F @ I ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % decseqD
% 0.29/0.46 thf(fact_123_decseq__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( order_antimono @ nat @ A )
% 0.29/0.46 = ( ^ [X7: nat > A] :
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 => ( ord_less_eq @ A @ ( X7 @ N ) @ ( X7 @ M ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % decseq_def
% 0.29/0.46 thf(fact_124_NEG__POS,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( neg @ A @ B )
% 0.29/0.46 = ( ^ [A6: A > B > $o,B5: A > B > $o] : ( pos @ A @ B @ B5 @ A6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % NEG_POS
% 0.29/0.46 thf(fact_125_POS__trans,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A2: A > B > $o,B6: A > B > $o,C4: A > B > $o] :
% 0.29/0.46 ( ( pos @ A @ B @ A2 @ B6 )
% 0.29/0.46 => ( ( pos @ A @ B @ B6 @ C4 )
% 0.29/0.46 => ( pos @ A @ B @ A2 @ C4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % POS_trans
% 0.29/0.46 thf(fact_126_POS__apply,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,R: A > B > $o,R5: A > B > $o,F: A,G: B] :
% 0.29/0.46 ( ( pos @ A @ B @ R @ R5 )
% 0.29/0.46 => ( ( R @ F @ G )
% 0.29/0.46 => ( R5 @ F @ G ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % POS_apply
% 0.29/0.46 thf(fact_127_POS__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( pos @ A @ B )
% 0.29/0.46 = ( ord_less_eq @ ( A > B > $o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % POS_def
% 0.29/0.46 thf(fact_128_POS__NEG,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( pos @ A @ B )
% 0.29/0.46 = ( ^ [A6: A > B > $o,B5: A > B > $o] : ( neg @ A @ B @ B5 @ A6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % POS_NEG
% 0.29/0.46 thf(fact_129_ex__has__least__nat,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 0.29/0.46 ( ( P @ K )
% 0.29/0.46 => ? [X2: A] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 & ! [Y5: A] :
% 0.29/0.46 ( ( P @ Y5 )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( M2 @ X2 ) @ ( M2 @ Y5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ex_has_least_nat
% 0.29/0.46 thf(fact_130_nat__le__linear,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.29/0.46 | ( ord_less_eq @ nat @ N2 @ M2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_le_linear
% 0.29/0.46 thf(fact_131_le__antisym,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ N2 @ M2 )
% 0.29/0.46 => ( M2 = N2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_antisym
% 0.29/0.46 thf(fact_132_eq__imp__le,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat] :
% 0.29/0.46 ( ( M2 = N2 )
% 0.29/0.46 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_imp_le
% 0.29/0.46 thf(fact_133_le__refl,axiom,
% 0.29/0.46 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% 0.29/0.46
% 0.29/0.46 % le_refl
% 0.29/0.46 thf(fact_134_le__trans,axiom,
% 0.29/0.46 ! [I: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I @ J )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ J @ K )
% 0.29/0.46 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_trans
% 0.29/0.46 thf(fact_135_LeastMI2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.46 => ! [P: A > $o,X: A,M2: A > B,Q: A > $o] :
% 0.29/0.46 ( ( P @ X )
% 0.29/0.46 => ( ! [Y4: A] :
% 0.29/0.46 ( ( P @ Y4 )
% 0.29/0.46 => ( ord_less_eq @ B @ ( M2 @ X ) @ ( M2 @ Y4 ) ) )
% 0.29/0.46 => ( ! [X2: A] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 => ( ! [Y5: A] :
% 0.29/0.46 ( ( P @ Y5 )
% 0.29/0.46 => ( ord_less_eq @ B @ ( M2 @ X2 ) @ ( M2 @ Y5 ) ) )
% 0.29/0.46 => ( Q @ X2 ) ) )
% 0.29/0.46 => ( Q @ ( hilbert_LeastM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % LeastMI2
% 0.29/0.46 thf(fact_136_monoseq__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( topological_monoseq @ A )
% 0.29/0.46 = ( ^ [X7: nat > A] :
% 0.29/0.46 ( ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 => ( ord_less_eq @ A @ ( X7 @ M ) @ ( X7 @ N ) ) )
% 0.29/0.46 | ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 => ( ord_less_eq @ A @ ( X7 @ N ) @ ( X7 @ M ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % monoseq_def
% 0.29/0.46 thf(fact_137_monoI2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X8: nat > A] :
% 0.29/0.46 ( ! [M3: nat,N3: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M3 @ N3 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ M3 ) ) )
% 0.29/0.46 => ( topological_monoseq @ A @ X8 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % monoI2
% 0.29/0.46 thf(fact_138_monoI1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X8: nat > A] :
% 0.29/0.46 ( ! [M3: nat,N3: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M3 @ N3 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) )
% 0.29/0.46 => ( topological_monoseq @ A @ X8 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % monoI1
% 0.29/0.46 thf(fact_139_decseq__imp__monoseq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X8: nat > A] :
% 0.29/0.46 ( ( order_antimono @ nat @ A @ X8 )
% 0.29/0.46 => ( topological_monoseq @ A @ X8 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % decseq_imp_monoseq
% 0.29/0.46 thf(fact_140_LeastM__nat__le,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,X: A,M2: A > nat] :
% 0.29/0.46 ( ( P @ X )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % LeastM_nat_le
% 0.29/0.46 thf(fact_141_LeastM__nat__lemma,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 0.29/0.46 ( ( P @ K )
% 0.29/0.46 => ( ( P @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) )
% 0.29/0.46 & ! [Y5: A] :
% 0.29/0.46 ( ( P @ Y5 )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ Y5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % LeastM_nat_lemma
% 0.29/0.46 thf(fact_142_GreatestMI2,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.46 => ! [P: A > $o,X: A,M2: A > B,Q: A > $o] :
% 0.29/0.46 ( ( P @ X )
% 0.29/0.46 => ( ! [Y4: A] :
% 0.29/0.46 ( ( P @ Y4 )
% 0.29/0.46 => ( ord_less_eq @ B @ ( M2 @ Y4 ) @ ( M2 @ X ) ) )
% 0.29/0.46 => ( ! [X2: A] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 => ( ! [Y5: A] :
% 0.29/0.46 ( ( P @ Y5 )
% 0.29/0.46 => ( ord_less_eq @ B @ ( M2 @ Y5 ) @ ( M2 @ X2 ) ) )
% 0.29/0.46 => ( Q @ X2 ) ) )
% 0.29/0.46 => ( Q @ ( hilbert_GreatestM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % GreatestMI2
% 0.29/0.46 thf(fact_143_totally__bounded__subset,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( topolo47006728_space @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [S2: set @ A,T: set @ A] :
% 0.29/0.46 ( ( topolo406746546ounded @ A @ S2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ T @ S2 )
% 0.29/0.46 => ( topolo406746546ounded @ A @ T ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % totally_bounded_subset
% 0.29/0.46 thf(fact_144_Grp__mono,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A2: set @ A,B6: set @ A,F: A > B] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 0.29/0.46 => ( ord_less_eq @ ( A > B > $o ) @ ( bNF_Grp @ A @ B @ A2 @ F ) @ ( bNF_Grp @ A @ B @ B6 @ F ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Grp_mono
% 0.29/0.46 thf(fact_145_of__nat__le__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % of_nat_le_iff
% 0.29/0.46 thf(fact_146_GrpE,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,X: A,Y: B] :
% 0.29/0.46 ( ( bNF_Grp @ A @ B @ A2 @ F @ X @ Y )
% 0.29/0.46 => ~ ( ( ( F @ X )
% 0.29/0.46 = Y )
% 0.29/0.46 => ~ ( member @ A @ X @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % GrpE
% 0.29/0.46 thf(fact_147_GrpI,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,F: B > A,X: B,Y: A,A2: set @ B] :
% 0.29/0.46 ( ( ( F @ X )
% 0.29/0.46 = Y )
% 0.29/0.46 => ( ( member @ B @ X @ A2 )
% 0.29/0.46 => ( bNF_Grp @ B @ A @ A2 @ F @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % GrpI
% 0.29/0.46 thf(fact_148_real__arch__simple,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( archim1804426504_field @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A] :
% 0.29/0.46 ? [N3: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % real_arch_simple
% 0.29/0.46 thf(fact_149_zle__int,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N2 ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % zle_int
% 0.29/0.46 thf(fact_150_transfer__int__nat__relations_I3_J,axiom,
% 0.29/0.46 ! [X: nat,Y: nat] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ X @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % transfer_int_nat_relations(3)
% 0.29/0.46 thf(fact_151_round__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % round_mono
% 0.29/0.46 thf(fact_152_nat__le__iff,axiom,
% 0.29/0.46 ! [X: int,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N2 )
% 0.29/0.46 = ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_le_iff
% 0.29/0.46 thf(fact_153_negative__zle,axiom,
% 0.29/0.46 ! [N2: nat,M2: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( semiring_1_of_nat @ int @ M2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % negative_zle
% 0.29/0.46 thf(fact_154_nat__mono,axiom,
% 0.29/0.46 ! [X: int,Y: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ X @ Y )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_mono
% 0.29/0.46 thf(fact_155_compl__le__compl__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
% 0.29/0.46 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % compl_le_compl_iff
% 0.29/0.46 thf(fact_156_neg__le__iff__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [B3: A,A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_le_iff_le
% 0.29/0.46 thf(fact_157_le__imp__neg__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_imp_neg_le
% 0.29/0.46 thf(fact_158_Compl__anti__mono,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B6: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Compl_anti_mono
% 0.29/0.46 thf(fact_159_Compl__subset__Compl__iff,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B6: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) )
% 0.29/0.46 = ( ord_less_eq @ ( set @ A ) @ B6 @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Compl_subset_Compl_iff
% 0.29/0.46 thf(fact_160_compl__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % compl_mono
% 0.29/0.46 thf(fact_161_compl__le__swap1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Y: A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % compl_le_swap1
% 0.29/0.46 thf(fact_162_compl__le__swap2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Y: A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
% 0.29/0.46 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % compl_le_swap2
% 0.29/0.46 thf(fact_163_le__minus__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ B3 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_minus_iff
% 0.29/0.46 thf(fact_164_minus__le__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
% 0.29/0.46 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minus_le_iff
% 0.29/0.46 thf(fact_165_le__nat__iff,axiom,
% 0.29/0.46 ! [K: int,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ N2 @ ( nat2 @ K ) )
% 0.29/0.46 = ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_nat_iff
% 0.29/0.46 thf(fact_166_int__nat__eq,axiom,
% 0.29/0.46 ! [Z3: int] :
% 0.29/0.46 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.29/0.46 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
% 0.29/0.46 = Z3 ) )
% 0.29/0.46 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.29/0.46 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
% 0.29/0.46 = ( zero_zero @ int ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % int_nat_eq
% 0.29/0.46 thf(fact_167_int__zle__neg,axiom,
% 0.29/0.46 ! [N2: nat,M2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M2 ) ) )
% 0.29/0.46 = ( ( N2
% 0.29/0.46 = ( zero_zero @ nat ) )
% 0.29/0.46 & ( M2
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % int_zle_neg
% 0.29/0.46 thf(fact_168_ComplI,axiom,
% 0.29/0.46 ! [A: $tType,C: A,A2: set @ A] :
% 0.29/0.46 ( ~ ( member @ A @ C @ A2 )
% 0.29/0.46 => ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ComplI
% 0.29/0.46 thf(fact_169_Compl__iff,axiom,
% 0.29/0.46 ! [A: $tType,C: A,A2: set @ A] :
% 0.29/0.46 ( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.29/0.46 = ( ~ ( member @ A @ C @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Compl_iff
% 0.29/0.46 thf(fact_170_Compl__eq__Compl__iff,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B6: set @ A] :
% 0.29/0.46 ( ( ( uminus_uminus @ ( set @ A ) @ A2 )
% 0.29/0.46 = ( uminus_uminus @ ( set @ A ) @ B6 ) )
% 0.29/0.46 = ( A2 = B6 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Compl_eq_Compl_iff
% 0.29/0.46 thf(fact_171_le__zero__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [N2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
% 0.29/0.46 = ( N2
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_zero_eq
% 0.29/0.46 thf(fact_172_le0,axiom,
% 0.29/0.46 ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% 0.29/0.46
% 0.29/0.46 % le0
% 0.29/0.46 thf(fact_173_neg__less__eq__nonneg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 0.29/0.46 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_less_eq_nonneg
% 0.29/0.46 thf(fact_174_less__eq__neg__nonpos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_eq_neg_nonpos
% 0.29/0.46 thf(fact_175_neg__le__0__iff__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
% 0.29/0.46 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_le_0_iff_le
% 0.29/0.46 thf(fact_176_neg__0__le__iff__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_0_le_iff_le
% 0.29/0.46 thf(fact_177_of__nat__le__0__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [M2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( zero_zero @ A ) )
% 0.29/0.46 = ( M2
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % of_nat_le_0_iff
% 0.29/0.46 thf(fact_178_nat__0__iff,axiom,
% 0.29/0.46 ! [I: int] :
% 0.29/0.46 ( ( ( nat2 @ I )
% 0.29/0.46 = ( zero_zero @ nat ) )
% 0.29/0.46 = ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_0_iff
% 0.29/0.46 thf(fact_179_nat__le__0,axiom,
% 0.29/0.46 ! [Z3: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) )
% 0.29/0.46 => ( ( nat2 @ Z3 )
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_le_0
% 0.29/0.46 thf(fact_180_ComplD,axiom,
% 0.29/0.46 ! [A: $tType,C: A,A2: set @ A] :
% 0.29/0.46 ( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.29/0.46 => ~ ( member @ A @ C @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % ComplD
% 0.29/0.46 thf(fact_181_double__complement,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % double_complement
% 0.29/0.46 thf(fact_182_zero__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le
% 0.29/0.46 thf(fact_183_le__0__eq,axiom,
% 0.29/0.46 ! [N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) )
% 0.29/0.46 = ( N2
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_0_eq
% 0.29/0.46 thf(fact_184_less__eq__nat_Osimps_I1_J,axiom,
% 0.29/0.46 ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% 0.29/0.46
% 0.29/0.46 % less_eq_nat.simps(1)
% 0.29/0.46 thf(fact_185_less__eq__int__code_I1_J,axiom,
% 0.29/0.46 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 0.29/0.46
% 0.29/0.46 % less_eq_int_code(1)
% 0.29/0.46 thf(fact_186_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
% 0.29/0.46 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 0.29/0.46
% 0.29/0.46 % Nat_Transfer.transfer_nat_int_function_closures(5)
% 0.29/0.46 thf(fact_187_nat__eq__iff,axiom,
% 0.29/0.46 ! [W: int,M2: nat] :
% 0.29/0.46 ( ( ( nat2 @ W )
% 0.29/0.46 = M2 )
% 0.29/0.46 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 0.29/0.46 => ( W
% 0.29/0.46 = ( semiring_1_of_nat @ int @ M2 ) ) )
% 0.29/0.46 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 0.29/0.46 => ( M2
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_eq_iff
% 0.29/0.46 thf(fact_188_nat__eq__iff2,axiom,
% 0.29/0.46 ! [M2: nat,W: int] :
% 0.29/0.46 ( ( M2
% 0.29/0.46 = ( nat2 @ W ) )
% 0.29/0.46 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 0.29/0.46 => ( W
% 0.29/0.46 = ( semiring_1_of_nat @ int @ M2 ) ) )
% 0.29/0.46 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 0.29/0.46 => ( M2
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_eq_iff2
% 0.29/0.46 thf(fact_189_of__nat__0__le__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % of_nat_0_le_iff
% 0.29/0.46 thf(fact_190_transfer__nat__int__relations_I1_J,axiom,
% 0.29/0.46 ! [X: int,Y: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.29/0.46 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
% 0.29/0.46 => ( ( ( nat2 @ X )
% 0.29/0.46 = ( nat2 @ Y ) )
% 0.29/0.46 = ( X = Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % transfer_nat_int_relations(1)
% 0.29/0.46 thf(fact_191_eq__nat__nat__iff,axiom,
% 0.29/0.46 ! [Z3: int,Z4: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.29/0.46 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z4 )
% 0.29/0.46 => ( ( ( nat2 @ Z3 )
% 0.29/0.46 = ( nat2 @ Z4 ) )
% 0.29/0.46 = ( Z3 = Z4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_nat_nat_iff
% 0.29/0.46 thf(fact_192_ex__nat,axiom,
% 0.29/0.46 ( ( ^ [P2: nat > $o] :
% 0.29/0.46 ? [X6: nat] : ( P2 @ X6 ) )
% 0.29/0.46 = ( ^ [P3: nat > $o] :
% 0.29/0.46 ? [X5: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.29/0.46 & ( P3 @ ( nat2 @ X5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ex_nat
% 0.29/0.46 thf(fact_193_all__nat,axiom,
% 0.29/0.46 ( ( ^ [P2: nat > $o] :
% 0.29/0.46 ! [X6: nat] : ( P2 @ X6 ) )
% 0.29/0.46 = ( ^ [P3: nat > $o] :
% 0.29/0.46 ! [X5: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.29/0.46 => ( P3 @ ( nat2 @ X5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % all_nat
% 0.29/0.46 thf(fact_194_transfer__int__nat__quantifiers_I2_J,axiom,
% 0.29/0.46 ! [P: int > $o] :
% 0.29/0.46 ( ( ? [X5: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.29/0.46 & ( P @ X5 ) ) )
% 0.29/0.46 = ( ? [X5: nat] : ( P @ ( semiring_1_of_nat @ int @ X5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % transfer_int_nat_quantifiers(2)
% 0.29/0.46 thf(fact_195_transfer__int__nat__quantifiers_I1_J,axiom,
% 0.29/0.46 ! [P: int > $o] :
% 0.29/0.46 ( ( ! [X5: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.29/0.46 => ( P @ X5 ) ) )
% 0.29/0.46 = ( ! [X5: nat] : ( P @ ( semiring_1_of_nat @ int @ X5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % transfer_int_nat_quantifiers(1)
% 0.29/0.46 thf(fact_196_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
% 0.29/0.46 ! [Z3: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ Z3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat_Transfer.transfer_nat_int_function_closures(9)
% 0.29/0.46 thf(fact_197_nonneg__eq__int,axiom,
% 0.29/0.46 ! [Z3: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.29/0.46 => ~ ! [M3: nat] :
% 0.29/0.46 ( Z3
% 0.29/0.46 != ( semiring_1_of_nat @ int @ M3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nonneg_eq_int
% 0.29/0.46 thf(fact_198_nonneg__int__cases,axiom,
% 0.29/0.46 ! [K: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.29/0.46 => ~ ! [N3: nat] :
% 0.29/0.46 ( K
% 0.29/0.46 != ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nonneg_int_cases
% 0.29/0.46 thf(fact_199_zero__le__imp__eq__int,axiom,
% 0.29/0.46 ! [K: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.29/0.46 => ? [N3: nat] :
% 0.29/0.46 ( K
% 0.29/0.46 = ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le_imp_eq_int
% 0.29/0.46 thf(fact_200_transfer__nat__int__relations_I3_J,axiom,
% 0.29/0.46 ! [X: int,Y: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.29/0.46 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) )
% 0.29/0.46 = ( ord_less_eq @ int @ X @ Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % transfer_nat_int_relations(3)
% 0.29/0.46 thf(fact_201_int__eq__iff,axiom,
% 0.29/0.46 ! [M2: nat,Z3: int] :
% 0.29/0.46 ( ( ( semiring_1_of_nat @ int @ M2 )
% 0.29/0.46 = Z3 )
% 0.29/0.46 = ( ( M2
% 0.29/0.46 = ( nat2 @ Z3 ) )
% 0.29/0.46 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % int_eq_iff
% 0.29/0.46 thf(fact_202_nat__0__le,axiom,
% 0.29/0.46 ! [Z3: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 0.29/0.46 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
% 0.29/0.46 = Z3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_0_le
% 0.29/0.46 thf(fact_203_negative__zle__0,axiom,
% 0.29/0.46 ! [N2: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( zero_zero @ int ) ) ).
% 0.29/0.46
% 0.29/0.46 % negative_zle_0
% 0.29/0.46 thf(fact_204_transfer__morphism__nat__int,axiom,
% 0.29/0.46 nat_tr1645093318rphism @ int @ nat @ nat2 @ ( ord_less_eq @ int @ ( zero_zero @ int ) ) ).
% 0.29/0.46
% 0.29/0.46 % transfer_morphism_nat_int
% 0.29/0.46 thf(fact_205_imp__le__cong,axiom,
% 0.29/0.46 ! [X: int,X4: int,P: $o,P4: $o] :
% 0.29/0.46 ( ( X = X4 )
% 0.29/0.46 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 0.29/0.46 => ( P = P4 ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.29/0.46 => P )
% 0.29/0.46 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 0.29/0.46 => P4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % imp_le_cong
% 0.29/0.46 thf(fact_206_conj__le__cong,axiom,
% 0.29/0.46 ! [X: int,X4: int,P: $o,P4: $o] :
% 0.29/0.46 ( ( X = X4 )
% 0.29/0.46 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 0.29/0.46 => ( P = P4 ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.29/0.46 & P )
% 0.29/0.46 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 0.29/0.46 & P4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % conj_le_cong
% 0.29/0.46 thf(fact_207_le__numeral__extra_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_numeral_extra(3)
% 0.29/0.46 thf(fact_208_int__eq__iff__numeral,axiom,
% 0.29/0.46 ! [M2: nat,V: num] :
% 0.29/0.46 ( ( ( semiring_1_of_nat @ int @ M2 )
% 0.29/0.46 = ( numeral_numeral @ int @ V ) )
% 0.29/0.46 = ( ( M2
% 0.29/0.46 = ( nat2 @ ( numeral_numeral @ int @ V ) ) )
% 0.29/0.46 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % int_eq_iff_numeral
% 0.29/0.46 thf(fact_209_numeral__le__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [M2: num,N2: num] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N2 ) )
% 0.29/0.46 = ( ord_less_eq @ num @ M2 @ N2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % numeral_le_iff
% 0.29/0.46 thf(fact_210_neg__numeral__le__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [M2: num,N2: num] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
% 0.29/0.46 = ( ord_less_eq @ num @ N2 @ M2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_numeral_le_iff
% 0.29/0.46 thf(fact_211_zero__le__numeral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [N2: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le_numeral
% 0.29/0.46 thf(fact_212_not__numeral__le__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [N2: num] :
% 0.29/0.46 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_numeral_le_zero
% 0.29/0.46 thf(fact_213_not__zero__le__neg__numeral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [N2: num] :
% 0.29/0.46 ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_zero_le_neg_numeral
% 0.29/0.46 thf(fact_214_neg__numeral__le__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_numeral_le_zero
% 0.29/0.46 thf(fact_215_not__numeral__le__neg__numeral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [M2: num,N2: num] :
% 0.29/0.46 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_numeral_le_neg_numeral
% 0.29/0.46 thf(fact_216_neg__numeral__le__numeral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [M2: num,N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_numeral_le_numeral
% 0.29/0.46 thf(fact_217_ceiling__le__neg__numeral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,V: num] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
% 0.29/0.46 = ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ceiling_le_neg_numeral
% 0.29/0.46 thf(fact_218_neg__numeral__le__floor,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [V: num,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim2085082626_floor @ A @ X ) )
% 0.29/0.46 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_numeral_le_floor
% 0.29/0.46 thf(fact_219_zero__le__floor,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim2085082626_floor @ A @ X ) )
% 0.29/0.46 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le_floor
% 0.29/0.46 thf(fact_220_ceiling__le__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
% 0.29/0.46 = ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ceiling_le_zero
% 0.29/0.46 thf(fact_221_numeral__le__floor,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [V: num,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archim2085082626_floor @ A @ X ) )
% 0.29/0.46 = ( ord_less_eq @ A @ ( numeral_numeral @ A @ V ) @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % numeral_le_floor
% 0.29/0.46 thf(fact_222_ceiling__le__numeral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,V: num] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
% 0.29/0.46 = ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ceiling_le_numeral
% 0.29/0.46 thf(fact_223_floor__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ord_less_eq @ int @ ( archim2085082626_floor @ A @ X ) @ ( archim2085082626_floor @ A @ Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % floor_mono
% 0.29/0.46 thf(fact_224_floor__le__ceiling,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A] : ( ord_less_eq @ int @ ( archim2085082626_floor @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % floor_le_ceiling
% 0.29/0.46 thf(fact_225_ceiling__ge__round,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A] : ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ceiling_ge_round
% 0.29/0.46 thf(fact_226_floor__le__round,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A] : ( ord_less_eq @ int @ ( archim2085082626_floor @ A @ X ) @ ( archimedean_round @ A @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % floor_le_round
% 0.29/0.46 thf(fact_227_ceiling__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Y: A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ceiling_mono
% 0.29/0.46 thf(fact_228_nat__ceiling__le__eq,axiom,
% 0.29/0.46 ! [X: real,A3: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A3 )
% 0.29/0.46 = ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_ceiling_le_eq
% 0.29/0.46 thf(fact_229_numeral__le__real__of__nat__iff,axiom,
% 0.29/0.46 ! [N2: num,M2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N2 ) @ ( semiring_1_of_nat @ real @ M2 ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N2 ) @ M2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % numeral_le_real_of_nat_iff
% 0.29/0.46 thf(fact_230_complete__real,axiom,
% 0.29/0.46 ! [S2: set @ real] :
% 0.29/0.46 ( ? [X3: real] : ( member @ real @ X3 @ S2 )
% 0.29/0.46 => ( ? [Z5: real] :
% 0.29/0.46 ! [X2: real] :
% 0.29/0.46 ( ( member @ real @ X2 @ S2 )
% 0.29/0.46 => ( ord_less_eq @ real @ X2 @ Z5 ) )
% 0.29/0.46 => ? [Y4: real] :
% 0.29/0.46 ( ! [X3: real] :
% 0.29/0.46 ( ( member @ real @ X3 @ S2 )
% 0.29/0.46 => ( ord_less_eq @ real @ X3 @ Y4 ) )
% 0.29/0.46 & ! [Z5: real] :
% 0.29/0.46 ( ! [X2: real] :
% 0.29/0.46 ( ( member @ real @ X2 @ S2 )
% 0.29/0.46 => ( ord_less_eq @ real @ X2 @ Z5 ) )
% 0.29/0.46 => ( ord_less_eq @ real @ Y4 @ Z5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % complete_real
% 0.29/0.46 thf(fact_231_real__nat__ceiling__ge,axiom,
% 0.29/0.46 ! [X: real] : ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % real_nat_ceiling_ge
% 0.29/0.46 thf(fact_232_nat__floor__neg,axiom,
% 0.29/0.46 ! [X: real] :
% 0.29/0.46 ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
% 0.29/0.46 => ( ( nat2 @ ( archim2085082626_floor @ real @ X ) )
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_floor_neg
% 0.29/0.46 thf(fact_233_le__nat__floor,axiom,
% 0.29/0.46 ! [X: nat,A3: real] :
% 0.29/0.46 ( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X ) @ A3 )
% 0.29/0.46 => ( ord_less_eq @ nat @ X @ ( nat2 @ ( archim2085082626_floor @ real @ A3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_nat_floor
% 0.29/0.46 thf(fact_234_enat__ord__number_I1_J,axiom,
% 0.29/0.46 ! [M2: num,N2: num] :
% 0.29/0.46 ( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( numeral_numeral @ extended_enat @ N2 ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % enat_ord_number(1)
% 0.29/0.46 thf(fact_235_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
% 0.29/0.46 ! [X: int,Y: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.29/0.46 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
% 0.29/0.46 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( nat_tsub @ X @ Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat_Transfer.transfer_nat_int_function_closures(3)
% 0.29/0.46 thf(fact_236_ile0__eq,axiom,
% 0.29/0.46 ! [N2: extended_enat] :
% 0.29/0.46 ( ( ord_less_eq @ extended_enat @ N2 @ ( zero_zero @ extended_enat ) )
% 0.29/0.46 = ( N2
% 0.29/0.46 = ( zero_zero @ extended_enat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ile0_eq
% 0.29/0.46 thf(fact_237_i0__lb,axiom,
% 0.29/0.46 ! [N2: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 ) ).
% 0.29/0.46
% 0.29/0.46 % i0_lb
% 0.29/0.46 thf(fact_238_numeral__le__enat__iff,axiom,
% 0.29/0.46 ! [M2: num,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( extended_enat2 @ N2 ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M2 ) @ N2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % numeral_le_enat_iff
% 0.29/0.46 thf(fact_239_Nat__Transfer_Otransfer__nat__int__functions_I3_J,axiom,
% 0.29/0.46 ! [X: int,Y: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.29/0.46 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
% 0.29/0.46 => ( ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) )
% 0.29/0.46 = ( nat2 @ ( nat_tsub @ X @ Y ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat_Transfer.transfer_nat_int_functions(3)
% 0.29/0.46 thf(fact_240_diff__diff__cancel,axiom,
% 0.29/0.46 ! [I: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I @ N2 )
% 0.29/0.46 => ( ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ N2 @ I ) )
% 0.29/0.46 = I ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_diff_cancel
% 0.29/0.46 thf(fact_241_diff__ge__0__iff__ge,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B3 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_ge_0_iff_ge
% 0.29/0.46 thf(fact_242_diff__is__0__eq,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ( minus_minus @ nat @ M2 @ N2 )
% 0.29/0.46 = ( zero_zero @ nat ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_is_0_eq
% 0.29/0.46 thf(fact_243_diff__is__0__eq_H,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.29/0.46 => ( ( minus_minus @ nat @ M2 @ N2 )
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_is_0_eq'
% 0.29/0.46 thf(fact_244_enat__ord__simps_I1_J,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N2 ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % enat_ord_simps(1)
% 0.29/0.46 thf(fact_245_enat__ile,axiom,
% 0.29/0.46 ! [N2: extended_enat,M2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ extended_enat @ N2 @ ( extended_enat2 @ M2 ) )
% 0.29/0.46 => ? [K2: nat] :
% 0.29/0.46 ( N2
% 0.29/0.46 = ( extended_enat2 @ K2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % enat_ile
% 0.29/0.46 thf(fact_246_eq__diff__iff,axiom,
% 0.29/0.46 ! [K: nat,M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ M2 )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ K @ N2 )
% 0.29/0.46 => ( ( ( minus_minus @ nat @ M2 @ K )
% 0.29/0.46 = ( minus_minus @ nat @ N2 @ K ) )
% 0.29/0.46 = ( M2 = N2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_diff_iff
% 0.29/0.46 thf(fact_247_le__diff__iff,axiom,
% 0.29/0.46 ! [K: nat,M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ M2 )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ K @ N2 )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_diff_iff
% 0.29/0.46 thf(fact_248_Nat_Odiff__diff__eq,axiom,
% 0.29/0.46 ! [K: nat,M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ M2 )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ K @ N2 )
% 0.29/0.46 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
% 0.29/0.46 = ( minus_minus @ nat @ M2 @ N2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.diff_diff_eq
% 0.29/0.46 thf(fact_249_diff__le__mono,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat,L: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ L ) @ ( minus_minus @ nat @ N2 @ L ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_le_mono
% 0.29/0.46 thf(fact_250_diff__le__self,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ M2 ) ).
% 0.29/0.46
% 0.29/0.46 % diff_le_self
% 0.29/0.46 thf(fact_251_le__diff__iff_H,axiom,
% 0.29/0.46 ! [A3: nat,C: nat,B3: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ A3 @ C )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ B3 @ C )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A3 ) @ ( minus_minus @ nat @ C @ B3 ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ B3 @ A3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_diff_iff'
% 0.29/0.46 thf(fact_252_diff__le__mono2,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat,L: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_le_mono2
% 0.29/0.46 thf(fact_253_of__nat__diff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ring_1 @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [N2: nat,M2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ N2 @ M2 )
% 0.29/0.46 => ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M2 @ N2 ) )
% 0.29/0.46 = ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % of_nat_diff
% 0.29/0.46 thf(fact_254_diff__eq__diff__less__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A,B3: A,C: A,D2: A] :
% 0.29/0.46 ( ( ( minus_minus @ A @ A3 @ B3 )
% 0.29/0.46 = ( minus_minus @ A @ C @ D2 ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 = ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_eq_diff_less_eq
% 0.29/0.46 thf(fact_255_diff__right__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A3: A,B3: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_right_mono
% 0.29/0.46
% 0.29/0.46 %----Type constructors (61)
% 0.29/0.46 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.29/0.46 ! [A7: $tType,A8: $tType] :
% 0.29/0.46 ( ( comple187826305attice @ A8 @ ( type2 @ A8 ) )
% 0.29/0.46 => ( comple187826305attice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 0.29/0.46 ! [A7: $tType,A8: $tType] :
% 0.29/0.46 ( ( boolean_algebra @ A8 @ ( type2 @ A8 ) )
% 0.29/0.46 => ( boolean_algebra @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.29/0.46 ! [A7: $tType,A8: $tType] :
% 0.29/0.46 ( ( order_top @ A8 @ ( type2 @ A8 ) )
% 0.29/0.46 => ( order_top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.29/0.46 ! [A7: $tType,A8: $tType] :
% 0.29/0.46 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 0.29/0.46 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.29/0.46 ! [A7: $tType,A8: $tType] :
% 0.29/0.46 ( ( order @ A8 @ ( type2 @ A8 ) )
% 0.29/0.46 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Otop,axiom,
% 0.29/0.46 ! [A7: $tType,A8: $tType] :
% 0.29/0.46 ( ( top @ A8 @ ( type2 @ A8 ) )
% 0.29/0.46 => ( top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.29/0.46 ! [A7: $tType,A8: $tType] :
% 0.29/0.46 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 0.29/0.46 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
% 0.29/0.46 linord219039673up_add @ int @ ( type2 @ int ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 0.29/0.46 ordered_ab_group_add @ int @ ( type2 @ int ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 0.29/0.46 linordered_semidom @ int @ ( type2 @ int ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 0.29/0.46 linordered_idom @ int @ ( type2 @ int ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 0.29/0.46 preorder @ int @ ( type2 @ int ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.29/0.46 linorder @ int @ ( type2 @ int ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 0.29/0.46 order @ int @ ( type2 @ int ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 0.29/0.46 ord @ int @ ( type2 @ int ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Rings_Oring__1,axiom,
% 0.29/0.46 ring_1 @ int @ ( type2 @ int ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.29/0.46 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_4,axiom,
% 0.29/0.46 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Opreorder_5,axiom,
% 0.29/0.46 preorder @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Olinorder_6,axiom,
% 0.29/0.46 linorder @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Oorder_7,axiom,
% 0.29/0.46 order @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Oord_8,axiom,
% 0.29/0.46 ord @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Num_Onum___Orderings_Opreorder_9,axiom,
% 0.29/0.46 preorder @ num @ ( type2 @ num ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Num_Onum___Orderings_Olinorder_10,axiom,
% 0.29/0.46 linorder @ num @ ( type2 @ num ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Num_Onum___Orderings_Oorder_11,axiom,
% 0.29/0.46 order @ num @ ( type2 @ num ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Num_Onum___Orderings_Oord_12,axiom,
% 0.29/0.46 ord @ num @ ( type2 @ num ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_13,axiom,
% 0.29/0.46 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_14,axiom,
% 0.29/0.46 ! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oorder__top_15,axiom,
% 0.29/0.46 ! [A7: $tType] : ( order_top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_16,axiom,
% 0.29/0.46 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oorder_17,axiom,
% 0.29/0.46 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Otop_18,axiom,
% 0.29/0.46 ! [A7: $tType] : ( top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oord_19,axiom,
% 0.29/0.46 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_20,axiom,
% 0.29/0.46 comple187826305attice @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_21,axiom,
% 0.29/0.46 boolean_algebra @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__top_22,axiom,
% 0.29/0.46 order_top @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_23,axiom,
% 0.29/0.46 preorder @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder_24,axiom,
% 0.29/0.46 linorder @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_25,axiom,
% 0.29/0.46 order @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Otop_26,axiom,
% 0.29/0.46 top @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oord_27,axiom,
% 0.29/0.46 ord @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
% 0.29/0.46 archim1804426504_field @ real @ ( type2 @ real ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
% 0.29/0.46 topolo47006728_space @ real @ ( type2 @ real ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling,axiom,
% 0.29/0.46 archim1727834104eiling @ real @ ( type2 @ real ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_28,axiom,
% 0.29/0.46 linord219039673up_add @ real @ ( type2 @ real ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_29,axiom,
% 0.29/0.46 ordered_ab_group_add @ real @ ( type2 @ real ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_30,axiom,
% 0.29/0.46 linordered_semidom @ real @ ( type2 @ real ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Olinordered__idom_31,axiom,
% 0.29/0.46 linordered_idom @ real @ ( type2 @ real ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Orderings_Opreorder_32,axiom,
% 0.29/0.46 preorder @ real @ ( type2 @ real ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Orderings_Olinorder_33,axiom,
% 0.29/0.46 linorder @ real @ ( type2 @ real ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Orderings_Oorder_34,axiom,
% 0.29/0.46 order @ real @ ( type2 @ real ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Orderings_Oord_35,axiom,
% 0.29/0.46 ord @ real @ ( type2 @ real ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Oring__1_36,axiom,
% 0.29/0.46 ring_1 @ real @ ( type2 @ real ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_37,axiom,
% 0.29/0.46 canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_38,axiom,
% 0.29/0.46 comple187826305attice @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_39,axiom,
% 0.29/0.46 order_top @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_40,axiom,
% 0.29/0.46 preorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_41,axiom,
% 0.29/0.46 linorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_42,axiom,
% 0.29/0.46 order @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Extended__Nat_Oenat___Orderings_Otop_43,axiom,
% 0.29/0.46 top @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_44,axiom,
% 0.29/0.46 ord @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.29/0.46
% 0.29/0.46 %----Conjectures (2)
% 0.29/0.46 thf(conj_0,hypothesis,
% 0.29/0.46 p @ x ).
% 0.29/0.46
% 0.29/0.46 thf(conj_1,conjecture,
% 0.29/0.46 bNF_eq_onp @ a @ p @ x @ x ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 2.90/1.52 ---- Embedded in THF ---
% 2.90/1.52 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.90/1.52 %%% Generated on Tue Feb 24 06:08:57 EST 2026
% 2.90/1.52 %%% using '$$dhol' embedding, version 1.3.0.
% 2.90/1.52 %%% Logic specification used:
% 2.90/1.52 %%% thf(spec, logic, $$dhol).
% 2.90/1.52
% 2.90/1.52 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.7l7JJvTzmF/DTF2DTF19250.p
% 2.90/1.52 thf(ty_t_Extended__Nat_Oenat, type, extended_enat: $tType).
% 2.90/1.52 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.90/1.52 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.90/1.52 thf(ty_t_Num_Onum, type, num: $tType).
% 2.90/1.52 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.90/1.52 thf(ty_t_Int_Oint, type, int: $tType).
% 2.90/1.52 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.90/1.52 thf(ty_tf_a, type, a: $tType).
% 2.90/1.52 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Rings_Oring__1, type, ring_1: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Lattices_Oboolean__algebra, type, boolean_algebra: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Archimedean__Field_Ofloor__ceiling, type, archim1727834104eiling: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Topological__Spaces_Ouniform__space, type, topolo47006728_space: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple187826305attice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Archimedean__Field_Oarchimedean__field, type, archim1804426504_field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.52 thf(sy_c_Archimedean__Field_Oceiling, type, archimedean_ceiling: (!> [A:$tType]: ((A > int)))).
% 2.90/1.52 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor, type, archim2085082626_floor: (!> [A:$tType]: ((A > int)))).
% 2.90/1.52 thf(sy_c_Archimedean__Field_Oround, type, archimedean_round: (!> [A:$tType]: ((A > int)))).
% 2.90/1.52 thf(sy_c_BNF__Composition_ODEADID_Opred__DEADID, type, bNF_pred_DEADID: (!> [A:$tType]: ((A > $o)))).
% 2.90/1.52 thf(sy_c_BNF__Def_OGrp, type, bNF_Grp: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (A > (B > $o))))))).
% 2.90/1.52 thf(sy_c_BNF__Def_Oeq__onp, type, bNF_eq_onp: (!> [A:$tType]: (((A > $o) > (A > (A > $o)))))).
% 2.90/1.52 thf(sy_c_Extended__Nat_Oenat, type, extended_enat2: (nat > extended_enat)).
% 2.90/1.52 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.90/1.52 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.90/1.52 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.90/1.52 thf(sy_c_HOL_Oundefined, type, undefined: (!> [A:$tType]: (A))).
% 2.90/1.52 thf(sy_c_Hilbert__Choice_OGreatestM, type, hilbert_GreatestM: (!> [A:$tType,B:$tType]: (((A > B) > ((A > $o) > A))))).
% 2.90/1.52 thf(sy_c_Hilbert__Choice_OLeastM, type, hilbert_LeastM: (!> [A:$tType,B:$tType]: (((A > B) > ((A > $o) > A))))).
% 2.90/1.52 thf(sy_c_Int_Onat, type, nat2: (int > nat)).
% 2.90/1.52 thf(sy_c_Lifting_ONEG, type, neg: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((A > (B > $o)) > $o))))).
% 2.90/1.52 thf(sy_c_Lifting_OPOS, type, pos: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((A > (B > $o)) > $o))))).
% 2.90/1.52 thf(sy_c_Lifting_OQuotient, type, quotient: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > ((B > A) > ((A > (B > $o)) > $o))))))).
% 2.90/1.52 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))).
% 2.90/1.52 thf(sy_c_Nat__Transfer_Otransfer__morphism, type, nat_tr1645093318rphism: (!> [B:$tType,A:$tType]: (((B > A) > ((B > $o) > $o))))).
% 2.90/1.52 thf(sy_c_Nat__Transfer_Otsub, type, nat_tsub: (int > (int > int))).
% 2.90/1.52 thf(sy_c_Num_Onumeral__class_Onumeral, type, numeral_numeral: (!> [A:$tType]: ((num > A)))).
% 2.90/1.52 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.90/1.52 thf(sy_c_Orderings_Oorder__class_Oantimono, type, order_antimono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.90/1.52 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.90/1.52 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.90/1.52 thf(sy_c_Relation_ODomainp, type, domainp: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > (A > $o))))).
% 2.90/1.52 thf(sy_c_Relation_Oinv__imagep, type, inv_imagep: (!> [B:$tType,A:$tType]: (((B > (B > $o)) > ((A > B) > (A > (A > $o))))))).
% 2.90/1.52 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.90/1.52 thf(sy_c_Topological__Spaces_Omonoseq, type, topological_monoseq: (!> [A:$tType]: (((nat > A) > $o)))).
% 2.90/1.52 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded, type, topolo406746546ounded: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.90/1.52 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.90/1.52 thf(sy_v_P, type, p: (a > $o)).
% 2.90/1.52 thf(sy_v_x, type, x: a).
% 2.90/1.52 thf(fact_0_eq__onp__eqD, axiom, (! [A:$tType,P:(A > $o),Q:(A > (A > $o)),X:A]: (((((bNF_eq_onp @ A) @ P) = Q) => ((P @ X) = ((Q @ X) @ X)))))).
% 2.90/1.52 thf(fact_1_eq__onp__mono0, axiom, (! [A:$tType,A2:(set @ A),P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ A2) => ((P @ X2) => (Q @ X2))))) => (! [X3:A]: (((((member @ A) @ X3) @ A2) => (! [Xa:A]: (((((member @ A) @ Xa) @ A2) => (((((bNF_eq_onp @ A) @ P) @ X3) @ Xa) => ((((bNF_eq_onp @ A) @ Q) @ X3) @ Xa)))))))))))).
% 2.90/1.52 thf(fact_2_eq__onp__to__eq, axiom, (! [A:$tType,P:(A > $o),X:A,Y:A]: ((((((bNF_eq_onp @ A) @ P) @ X) @ Y) => (X = Y))))).
% 2.90/1.52 thf(fact_3_eq__onp__same__args, axiom, (! [A:$tType,P:(A > $o),X:A]: ((((((bNF_eq_onp @ A) @ P) @ X) @ X) = (P @ X))))).
% 2.90/1.52 thf(fact_4_eq__onp__live__step, axiom, (! [A:$tType,X:$o,Y:$o,P:(A > $o),A3:A]: (((X = Y) => ((((((bNF_eq_onp @ A) @ P) @ A3) @ A3) & X) = ((P @ A3) & Y)))))).
% 2.90/1.52 thf(fact_5_DEADID_Orel__eq__onp, axiom, (! [A:$tType]: (((^ [Y2:A,Z:A]: ((Y2 = Z))) = ((bNF_eq_onp @ A) @ (bNF_pred_DEADID @ A)))))).
% 2.90/1.52 thf(fact_6_eq__onp__le__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((ord_less_eq @ (A > (A > $o))) @ ((bNF_eq_onp @ A) @ P)) @ (^ [Y2:A,Z:A]: ((Y2 = Z))))))).
% 2.90/1.52 thf(fact_7_type__definition__Quotient__not__empty, axiom, (! [B:$tType,A:$tType,P:(A > $o),Abs:(A > B),Rep:(B > A),T:(A > (B > $o))]: ((((((((quotient @ A) @ B) @ ((bNF_eq_onp @ A) @ P)) @ Abs) @ Rep) @ T) => (? [X1:A]: ((P @ X1))))))).
% 2.90/1.52 thf(fact_8_eq__onp__top__eq__eq, axiom, (! [A:$tType]: ((((bNF_eq_onp @ A) @ (top_top @ (A > $o))) = (^ [Y2:A,Z:A]: ((Y2 = Z))))))).
% 2.90/1.52 thf(fact_9_top__conj_I2_J, axiom, (! [A:$tType,P:$o,X:A]: (((P & ((top_top @ (A > $o)) @ X)) = P)))).
% 2.90/1.52 thf(fact_10_top__conj_I1_J, axiom, (! [A:$tType,X:A,P:$o]: (((((top_top @ (A > $o)) @ X) & P) = P)))).
% 2.90/1.52 thf(fact_11_ge__eq__refl, axiom, (! [A:$tType,R:(A > (A > $o)),X:A]: (((((ord_less_eq @ (A > (A > $o))) @ (^ [Y2:A,Z:A]: ((Y2 = Z)))) @ R) => ((R @ X) @ X))))).
% 2.90/1.52 thf(fact_12_refl__ge__eq, axiom, (! [A:$tType,R:(A > (A > $o))]: (((! [X2:A]: (((R @ X2) @ X2))) => (((ord_less_eq @ (A > (A > $o))) @ (^ [Y2:A,Z:A]: ((Y2 = Z)))) @ R))))).
% 2.90/1.52 thf(fact_13_Quotient__rel, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T:(A > (B > $o)),R2:A,S:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T) => ((((R @ R2) @ R2) & (((R @ S) @ S) & ((Abs @ R2) = (Abs @ S)))) = ((R @ R2) @ S)))))).
% 2.90/1.52 thf(fact_14_Quotient__refl1, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T:(A > (B > $o)),R2:A,S:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T) => (((R @ R2) @ S) => ((R @ R2) @ R2)))))).
% 2.90/1.52 thf(fact_15_Quotient__refl2, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T:(A > (B > $o)),R2:A,S:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T) => (((R @ R2) @ S) => ((R @ S) @ S)))))).
% 2.90/1.52 thf(fact_16_Quotient__Rep__eq, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T:(A > (B > $o)),X4:B,X:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T) => ((X4 = (Abs @ X)) => ((Rep @ X4) = (Rep @ X4))))))).
% 2.90/1.52 thf(fact_17_Quotient__abs__rep, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T:(A > (B > $o)),A3:B]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T) => ((Abs @ (Rep @ A3)) = A3))))).
% 2.90/1.52 thf(fact_18_Quotient__alt__def, axiom, (! [B:$tType,A:$tType]: ((((quotient @ A) @ B) = (^ [R3:(A > (A > $o)),Abs2:(A > B),Rep2:(B > A),T2:(A > (B > $o))]: (((! [A4:A,B2:B]: ((((T2 @ A4) @ B2) => ((Abs2 @ A4) = B2)))) & ((! [B2:B]: (((T2 @ (Rep2 @ B2)) @ B2))) & (! [X5:A,Y3:A]: ((((R3 @ X5) @ Y3) = (((T2 @ X5) @ (Abs2 @ X5)) & (((T2 @ Y3) @ (Abs2 @ Y3)) & ((Abs2 @ X5) = (Abs2 @ Y3))))))))))))))).
% 2.90/1.52 thf(fact_19_Quotient__rel__abs, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T:(A > (B > $o)),R2:A,S:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T) => (((R @ R2) @ S) => ((Abs @ R2) = (Abs @ S))))))).
% 2.90/1.52 thf(fact_20_Quotient__rel__rep, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T:(A > (B > $o)),A3:B,B3:B]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T) => (((R @ (Rep @ A3)) @ (Rep @ B3)) = (A3 = B3)))))).
% 2.90/1.52 thf(fact_21_Quotient__rep__abs, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T:(A > (B > $o)),R2:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T) => (((R @ R2) @ R2) => ((R @ (Rep @ (Abs @ R2))) @ R2)))))).
% 2.90/1.52 thf(fact_22_Quotient__alt__def2, axiom, (! [B:$tType,A:$tType]: ((((quotient @ A) @ B) = (^ [R3:(A > (A > $o)),Abs2:(A > B),Rep2:(B > A),T2:(A > (B > $o))]: (((! [A4:A,B2:B]: ((((T2 @ A4) @ B2) => ((Abs2 @ A4) = B2)))) & ((! [B2:B]: (((T2 @ (Rep2 @ B2)) @ B2))) & (! [X5:A,Y3:A]: ((((R3 @ X5) @ Y3) = (((T2 @ X5) @ (Abs2 @ Y3)) & ((T2 @ Y3) @ (Abs2 @ X5)))))))))))))).
% 2.90/1.52 thf(fact_23_Quotient__alt__def3, axiom, (! [B:$tType,A:$tType]: ((((quotient @ A) @ B) = (^ [R3:(A > (A > $o)),Abs2:(A > B),Rep2:(B > A),T2:(A > (B > $o))]: (((! [A4:A,B2:B]: ((((T2 @ A4) @ B2) => ((Abs2 @ A4) = B2)))) & ((! [B2:B]: (((T2 @ (Rep2 @ B2)) @ B2))) & (! [X5:A,Y3:A]: ((((R3 @ X5) @ Y3) = (? [Z2:B]: ((((T2 @ X5) @ Z2) & ((T2 @ Y3) @ Z2))))))))))))))).
% 2.90/1.52 thf(fact_24_Quotient__rel__abs2, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T:(A > (B > $o)),X:B,Y:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T) => (((R @ (Rep @ X)) @ Y) => (X = (Abs @ Y))))))).
% 2.90/1.52 thf(fact_25_Quotient__rep__reflp, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T:(A > (B > $o)),A3:B]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T) => ((R @ (Rep @ A3)) @ (Rep @ A3)))))).
% 2.90/1.52 thf(fact_26_Quotient__abs__induct, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T:(A > (B > $o)),P:(B > $o),X:B]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T) => ((! [Y4:A]: ((((R @ Y4) @ Y4) => (P @ (Abs @ Y4))))) => (P @ X)))))).
% 2.90/1.52 thf(fact_27_Quotient__rep__abs__eq, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T:(A > (B > $o)),T3:A]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T) => (((R @ T3) @ T3) => ((((ord_less_eq @ (A > (A > $o))) @ R) @ (^ [Y2:A,Z:A]: ((Y2 = Z)))) => ((Rep @ (Abs @ T3)) = T3))))))).
% 2.90/1.52 thf(fact_28_Quotient__to__transfer, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T:(A > (B > $o)),C:A,C2:B]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T) => (((R @ C) @ C) => ((C2 = (Abs @ C)) => ((T @ C) @ C2))))))).
% 2.90/1.52 thf(fact_29_DEADID_Opred__rel, axiom, (! [A:$tType]: (((bNF_pred_DEADID @ A) = (^ [X5:A]: ((X5 = X5))))))).
% 2.90/1.52 thf(fact_30_DEADID_Opred__True, axiom, (! [A:$tType]: (((bNF_pred_DEADID @ A) = (^ [Uu:A]: ($true)))))).
% 2.90/1.52 thf(fact_31_predicate2D__conj, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Q:(A > (B > $o)),R:$o,X:A,Y:B]: ((((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) & R) => (R & (((P @ X) @ Y) => ((Q @ X) @ Y))))))).
% 2.90/1.52 thf(fact_32_Quotient__rep__abs__fold__unmap, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T:(A > (B > $o)),X4:B,X:A,Rep3:(B > A)]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T) => ((X4 = (Abs @ X)) => (((R @ X) @ X) => (((Rep @ X4) = (Rep3 @ X4)) => ((R @ (Rep3 @ X4)) @ X)))))))).
% 2.90/1.52 thf(fact_33_DEADID_Opred__cong__simp, axiom, (! [A:$tType,X:A,Ya:A]: (((X = Ya) => (((bNF_pred_DEADID @ A) @ X) = ((bNF_pred_DEADID @ A) @ Ya)))))).
% 2.90/1.52 thf(fact_34_DEADID_Opred__mono__strong, axiom, (! [A:$tType,X:A]: ((((bNF_pred_DEADID @ A) @ X) => ((bNF_pred_DEADID @ A) @ X))))).
% 2.90/1.52 thf(fact_35_top__apply, axiom, (! [C3:$tType,D:$tType]: ((((top @ C3) @ (type2 @ C3)) => ((top_top @ (D > C3)) = (^ [X5:D]: ((top_top @ C3)))))))).
% 2.90/1.52 thf(fact_36_top1I, axiom, (! [A:$tType,X:A]: (((top_top @ (A > $o)) @ X)))).
% 2.90/1.52 thf(fact_37_predicate2I, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Q:(A > (B > $o))]: (((! [X2:A,Y4:B]: ((((P @ X2) @ Y4) => ((Q @ X2) @ Y4)))) => (((ord_less_eq @ (A > (B > $o))) @ P) @ Q))))).
% 2.90/1.52 thf(fact_38_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.90/1.52 thf(fact_39_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.90/1.52 thf(fact_40_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.90/1.52 thf(fact_41_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B)]: (((! [X2:A]: ((((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.90/1.52 thf(fact_42_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X5:A]: ((((ord_less_eq @ B) @ (F2 @ X5)) @ (G2 @ X5))))))))))).
% 2.90/1.52 thf(fact_43_top__greatest, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ (top_top @ A)))))))).
% 2.90/1.52 thf(fact_44_top_Oextremum__unique, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) = (A3 = (top_top @ A))))))))).
% 2.90/1.52 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 2.90/1.52 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X5:A]: ((((member @ A) @ X5) @ A2)))) = A2)))).
% 2.90/1.52 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.90/1.52 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 2.90/1.52 thf(fact_49_top_Oextremum__uniqueI, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) => (A3 = (top_top @ A))))))))).
% 2.90/1.52 thf(fact_50_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ A3) @ B3) => (A3 = B3))))))))).
% 2.90/1.52 thf(fact_51_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A3:A,C:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C) @ B3) => (((ord_less_eq @ A) @ C) @ A3))))))))).
% 2.90/1.52 thf(fact_52_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A3:A,B3:A]: (((! [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ B4) => ((P @ A5) @ B4)))) => ((! [A5:A,B4:A]: ((((P @ B4) @ A5) => ((P @ A5) @ B4)))) => ((P @ A3) @ B3))))))))).
% 2.90/1.52 thf(fact_53_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 2.90/1.52 thf(fact_54_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z3) => (((ord_less_eq @ A) @ X) @ Z3))))))))).
% 2.90/1.52 thf(fact_55_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ A3) => (A3 = B3))))))))).
% 2.90/1.52 thf(fact_56_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((B3 = C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 2.90/1.52 thf(fact_57_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 2.90/1.52 thf(fact_58_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.90/1.52 thf(fact_59_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z3)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z3)))) => (((((ord_less_eq @ A) @ X) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ Y)))) => (((((ord_less_eq @ A) @ Z3) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ X)))) => (~ (((((ord_less_eq @ A) @ Z3) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 2.90/1.52 thf(fact_60_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 2.90/1.52 thf(fact_61_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.90/1.52 thf(fact_62_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.90/1.52 thf(fact_63_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.90/1.52 thf(fact_64_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.90/1.52 thf(fact_65_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [X5:A,Y3:A]: (((((ord_less_eq @ A) @ X5) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X5))))))))).
% 2.90/1.52 thf(fact_66_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A3:A,B3:A,F:(A > B),C:B]: (((((ord_less_eq @ A) @ A3) @ B3) => (((F @ B3) = C) => ((! [X2:A,Y4:A]: (((((ord_less_eq @ A) @ X2) @ Y4) => (((ord_less_eq @ B) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less_eq @ B) @ (F @ A3)) @ C)))))))))).
% 2.90/1.52 thf(fact_67_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A3:A,F:(B > A),B3:B,C:B]: (((A3 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X2:B,Y4:B]: (((((ord_less_eq @ B) @ X2) @ Y4) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A3) @ (F @ C))))))))))).
% 2.90/1.52 thf(fact_68_order__subst2, axiom, (! [A:$tType,C3:$tType]: (((((order @ C3) @ (type2 @ C3)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,B3:A,F:(A > C3),C:C3]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ C3) @ (F @ B3)) @ C) => ((! [X2:A,Y4:A]: (((((ord_less_eq @ A) @ X2) @ Y4) => (((ord_less_eq @ C3) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less_eq @ C3) @ (F @ A3)) @ C)))))))))).
% 2.90/1.52 thf(fact_69_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,F:(B > A),B3:B,C:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X2:B,Y4:B]: (((((ord_less_eq @ B) @ X2) @ Y4) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A3) @ (F @ C))))))))))).
% 2.90/1.52 thf(fact_70_rev__predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),X:A,Y:B,Q:(A > (B > $o))]: ((((P @ X) @ Y) => ((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => ((Q @ X) @ Y)))))).
% 2.90/1.52 thf(fact_71_predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Q:(A > (B > $o)),X:A,Y:B]: (((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => (((P @ X) @ Y) => ((Q @ X) @ Y)))))).
% 2.90/1.52 thf(fact_72_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X5:A]: ((((member @ A) @ X5) @ (top_top @ (set @ A))))))))).
% 2.90/1.52 thf(fact_73_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.90/1.52 thf(fact_74_lfp_Oleq__antisym, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.90/1.52 thf(fact_75_gfp_Oleq__antisym, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) => (X = Y))))))))).
% 2.90/1.52 thf(fact_76_lfp_Oleq__trans, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z3) => (((ord_less_eq @ A) @ X) @ Z3))))))))).
% 2.90/1.52 thf(fact_77_gfp_Oleq__trans, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [Y:A,X:A,Z3:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ Z3) @ Y) => (((ord_less_eq @ A) @ Z3) @ X))))))))).
% 2.90/1.52 thf(fact_78_lfp_Oleq__refl, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.90/1.52 thf(fact_79_wlog__linorder__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),B3:A,A3:A]: (((! [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ B4) => ((P @ A5) @ B4)))) => ((((P @ B3) @ A3) => ((P @ A3) @ B3)) => ((P @ A3) @ B3))))))))).
% 2.90/1.52 thf(fact_80_UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.90/1.52 thf(fact_81_UNIV__eq__I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X2:A]: ((((member @ A) @ X2) @ A2))) => ((top_top @ (set @ A)) = A2))))).
% 2.90/1.52 thf(fact_82_subset__UNIV, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ (top_top @ (set @ A)))))).
% 2.90/1.52 thf(fact_83_UNIV__witness, axiom, (! [A:$tType]: ((? [X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))))).
% 2.90/1.52 thf(fact_84_iso__tuple__UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.90/1.52 thf(fact_85_in__inv__imagep, axiom, (! [B:$tType,A:$tType]: ((((inv_imagep @ A) @ B) = (^ [R4:(A > (A > $o)),F2:(B > A),X5:B,Y3:B]: (((R4 @ (F2 @ X5)) @ (F2 @ Y3)))))))).
% 2.90/1.52 thf(fact_86_eq__onp__to__Domainp, axiom, (! [B:$tType,A:$tType,P:(A > $o),Abs:(A > B),Rep:(B > A),T:(A > (B > $o))]: ((((((((quotient @ A) @ B) @ ((bNF_eq_onp @ A) @ P)) @ Abs) @ Rep) @ T) => ((((domainp @ A) @ B) @ T) = P))))).
% 2.90/1.52 thf(fact_87_type__definition__Quotient__not__empty__witness, axiom, (! [A:$tType,B:$tType,P:(A > $o),Abs:(A > B),Rep:(B > A),T:(A > (B > $o))]: ((((((((quotient @ A) @ B) @ ((bNF_eq_onp @ A) @ P)) @ Abs) @ Rep) @ T) => (P @ (Rep @ (undefined @ B))))))).
% 2.90/1.52 thf(fact_88_NEG__def, axiom, (! [B:$tType,A:$tType]: ((((neg @ A) @ B) = (^ [A6:(A > (B > $o)),B5:(A > (B > $o))]: ((((ord_less_eq @ (A > (B > $o))) @ B5) @ A6))))))).
% 2.90/1.52 thf(fact_89_antimonoD, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y:A]: (((((order_antimono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ B) @ (F @ Y)) @ (F @ X)))))))))).
% 2.90/1.52 thf(fact_90_subsetI, axiom, (! [A:$tType,A2:(set @ A),B6:(set @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ A2) => (((member @ A) @ X2) @ B6)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B6))))).
% 2.90/1.52 thf(fact_91_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ A2) => (A2 = B6)))))).
% 2.90/1.52 thf(fact_92_set__mp, axiom, (! [A:$tType,A2:(set @ A),B6:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B6) => ((((member @ A) @ X) @ A2) => (((member @ A) @ X) @ B6)))))).
% 2.90/1.52 thf(fact_93_in__mono, axiom, (! [A:$tType,A2:(set @ A),B6:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B6) => ((((member @ A) @ X) @ A2) => (((member @ A) @ X) @ B6)))))).
% 2.90/1.52 thf(fact_94_subsetD, axiom, (! [A:$tType,A2:(set @ A),B6:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B6) => ((((member @ A) @ C) @ A2) => (((member @ A) @ C) @ B6)))))).
% 2.90/1.52 thf(fact_95_subsetCE, axiom, (! [A:$tType,A2:(set @ A),B6:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B6) => ((((member @ A) @ C) @ A2) => (((member @ A) @ C) @ B6)))))).
% 2.90/1.52 thf(fact_96_equalityE, axiom, (! [A:$tType,A2:(set @ A),B6:(set @ A)]: (((A2 = B6) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B6) => (~ ((((ord_less_eq @ (set @ A)) @ B6) @ A2)))))))))).
% 2.90/1.52 thf(fact_97_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: ((! [X5:A]: (((((member @ A) @ X5) @ A6) => (((member @ A) @ X5) @ B5)))))))))).
% 2.90/1.52 thf(fact_98_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B6:(set @ A)]: (((A2 = B6) => (((ord_less_eq @ (set @ A)) @ A2) @ B6))))).
% 2.90/1.52 thf(fact_99_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B6:(set @ A)]: (((A2 = B6) => (((ord_less_eq @ (set @ A)) @ B6) @ A2))))).
% 2.90/1.52 thf(fact_100_set__rev__mp, axiom, (! [A:$tType,X:A,A2:(set @ A),B6:(set @ A)]: (((((member @ A) @ X) @ A2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B6) => (((member @ A) @ X) @ B6)))))).
% 2.90/1.52 thf(fact_101_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: ((! [T4:A]: (((((member @ A) @ T4) @ A6) => (((member @ A) @ T4) @ B5)))))))))).
% 2.90/1.52 thf(fact_102_rev__subsetD, axiom, (! [A:$tType,C:A,A2:(set @ A),B6:(set @ A)]: (((((member @ A) @ C) @ A2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B6) => (((member @ A) @ C) @ B6)))))).
% 2.90/1.52 thf(fact_103_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 2.90/1.52 thf(fact_104_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) => (Q @ X2)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.90/1.52 thf(fact_105_subset__trans, axiom, (! [A:$tType,A2:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ C4) => (((ord_less_eq @ (set @ A)) @ A2) @ C4)))))).
% 2.90/1.52 thf(fact_106_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y2:(set @ A),Z:(set @ A)]: ((Y2 = Z))) = (^ [A6:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B5) & (((ord_less_eq @ (set @ A)) @ B5) @ A6)))))))).
% 2.90/1.52 thf(fact_107_contra__subsetD, axiom, (! [A:$tType,A2:(set @ A),B6:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B6) => ((~ ((((member @ A) @ C) @ B6))) => (~ ((((member @ A) @ C) @ A2)))))))).
% 2.90/1.52 thf(fact_108_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X5:A]: (((P @ X5) => (Q @ X5)))))))).
% 2.90/1.52 thf(fact_109_DomainpE, axiom, (! [A:$tType,B:$tType,R2:(A > (B > $o)),A3:A]: ((((((domainp @ A) @ B) @ R2) @ A3) => (~ ((! [B4:B]: ((~ (((R2 @ A3) @ B4))))))))))).
% 2.90/1.52 thf(fact_110_Domainp_Ocases, axiom, (! [A:$tType,B:$tType,R2:(A > (B > $o)),A3:A]: ((((((domainp @ A) @ B) @ R2) @ A3) => (~ ((! [B4:B]: ((~ (((R2 @ A3) @ B4))))))))))).
% 2.90/1.52 thf(fact_111_Domainp_Osimps, axiom, (! [B:$tType,A:$tType]: ((((domainp @ A) @ B) = (^ [R4:(A > (B > $o)),A4:A]: ((? [B2:A,C5:B]: (((A4 = B2) & ((R4 @ B2) @ C5)))))))))).
% 2.90/1.52 thf(fact_112_Domainp_ODomainI, axiom, (! [B:$tType,A:$tType,R2:(A > (B > $o)),A3:A,B3:B]: ((((R2 @ A3) @ B3) => ((((domainp @ A) @ B) @ R2) @ A3))))).
% 2.90/1.52 thf(fact_113_Domainp_Oinducts, axiom, (! [B:$tType,A:$tType,R2:(A > (B > $o)),X:A,P:(A > $o)]: ((((((domainp @ A) @ B) @ R2) @ X) => ((! [A5:A,B4:B]: ((((R2 @ A5) @ B4) => (P @ A5)))) => (P @ X)))))).
% 2.90/1.52 thf(fact_114_NEG__trans, axiom, (! [B:$tType,A:$tType,A2:(A > (B > $o)),B6:(A > (B > $o)),C4:(A > (B > $o))]: ((((((neg @ A) @ B) @ A2) @ B6) => (((((neg @ A) @ B) @ B6) @ C4) => ((((neg @ A) @ B) @ A2) @ C4)))))).
% 2.90/1.52 thf(fact_115_Quotient__to__Domainp, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T:(A > (B > $o))]: ((((((((quotient @ A) @ B) @ R) @ Abs) @ Rep) @ T) => ((((domainp @ A) @ B) @ T) = (^ [X5:A]: (((R @ X5) @ X5)))))))).
% 2.90/1.52 thf(fact_116_antimono__def, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (((order_antimono @ A) @ B) = (^ [F2:(A > B)]: ((! [X5:A,Y3:A]: (((((ord_less_eq @ A) @ X5) @ Y3) => (((ord_less_eq @ B) @ (F2 @ Y3)) @ (F2 @ X5)))))))))))).
% 2.90/1.52 thf(fact_117_antimonoI, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B)]: (((! [X2:A,Y4:A]: (((((ord_less_eq @ A) @ X2) @ Y4) => (((ord_less_eq @ B) @ (F @ Y4)) @ (F @ X2))))) => (((order_antimono @ A) @ B) @ F)))))))).
% 2.90/1.52 thf(fact_118_antimonoE, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y:A]: (((((order_antimono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ B) @ (F @ Y)) @ (F @ X)))))))))).
% 2.90/1.52 thf(fact_119_Domain__eq__top, axiom, (! [A:$tType]: (((((domainp @ A) @ A) @ (^ [Y2:A,Z:A]: ((Y2 = Z)))) = (top_top @ (A > $o)))))).
% 2.90/1.52 thf(fact_120_Domainp__refl, axiom, (! [B:$tType,A:$tType]: ((((domainp @ A) @ B) = ((domainp @ A) @ B))))).
% 2.90/1.52 thf(fact_121_Domainp__iff, axiom, (! [B:$tType,A:$tType]: ((((domainp @ A) @ B) = (^ [T2:(A > (B > $o)),X5:A]: (((^ [P2:(B > $o)]: ((? [X6:B]: ((P2 @ X6))))) @ (T2 @ X5)))))))).
% 2.90/1.52 thf(fact_122_decseqD, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [F:(nat > A),I:nat,J:nat]: (((((order_antimono @ nat) @ A) @ F) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ A) @ (F @ J)) @ (F @ I)))))))))).
% 2.90/1.52 thf(fact_123_decseq__def, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (((order_antimono @ nat) @ A) = (^ [X7:(nat > A)]: ((! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ A) @ (X7 @ N)) @ (X7 @ M)))))))))))).
% 2.90/1.52 thf(fact_124_NEG__POS, axiom, (! [B:$tType,A:$tType]: ((((neg @ A) @ B) = (^ [A6:(A > (B > $o)),B5:(A > (B > $o))]: (((((pos @ A) @ B) @ B5) @ A6))))))).
% 2.90/1.52 thf(fact_125_POS__trans, axiom, (! [B:$tType,A:$tType,A2:(A > (B > $o)),B6:(A > (B > $o)),C4:(A > (B > $o))]: ((((((pos @ A) @ B) @ A2) @ B6) => (((((pos @ A) @ B) @ B6) @ C4) => ((((pos @ A) @ B) @ A2) @ C4)))))).
% 2.90/1.52 thf(fact_126_POS__apply, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),R5:(A > (B > $o)),F:A,G:B]: ((((((pos @ A) @ B) @ R) @ R5) => (((R @ F) @ G) => ((R5 @ F) @ G)))))).
% 2.90/1.52 thf(fact_127_POS__def, axiom, (! [B:$tType,A:$tType]: ((((pos @ A) @ B) = (ord_less_eq @ (A > (B > $o))))))).
% 2.90/1.52 thf(fact_128_POS__NEG, axiom, (! [B:$tType,A:$tType]: ((((pos @ A) @ B) = (^ [A6:(A > (B > $o)),B5:(A > (B > $o))]: (((((neg @ A) @ B) @ B5) @ A6))))))).
% 2.90/1.52 thf(fact_129_ex__has__least__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat)]: (((P @ K) => (? [X2:A]: (((P @ X2) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M2 @ X2)) @ (M2 @ Y5)))))))))))).
% 2.90/1.52 thf(fact_130_nat__le__linear, axiom, (! [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) | (((ord_less_eq @ nat) @ N2) @ M2))))).
% 2.90/1.52 thf(fact_131_le__antisym, axiom, (! [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) => ((((ord_less_eq @ nat) @ N2) @ M2) => (M2 = N2)))))).
% 2.90/1.52 thf(fact_132_eq__imp__le, axiom, (! [M2:nat,N2:nat]: (((M2 = N2) => (((ord_less_eq @ nat) @ M2) @ N2))))).
% 2.90/1.52 thf(fact_133_le__refl, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ N2) @ N2)))).
% 2.90/1.52 thf(fact_134_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 2.90/1.52 thf(fact_135_LeastMI2, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [P:(A > $o),X:A,M2:(A > B),Q:(A > $o)]: (((P @ X) => ((! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ B) @ (M2 @ X)) @ (M2 @ Y4))))) => ((! [X2:A]: (((P @ X2) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ B) @ (M2 @ X2)) @ (M2 @ Y5))))) => (Q @ X2))))) => (Q @ ((((hilbert_LeastM @ A) @ B) @ M2) @ P))))))))))).
% 2.90/1.52 thf(fact_136_monoseq__def, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((topological_monoseq @ A) = (^ [X7:(nat > A)]: (((! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ A) @ (X7 @ M)) @ (X7 @ N))))) | (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ A) @ (X7 @ N)) @ (X7 @ M))))))))))))).
% 2.90/1.52 thf(fact_137_monoI2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X8:(nat > A)]: (((! [M3:nat,N3:nat]: (((((ord_less_eq @ nat) @ M3) @ N3) => (((ord_less_eq @ A) @ (X8 @ N3)) @ (X8 @ M3))))) => ((topological_monoseq @ A) @ X8)))))))).
% 2.90/1.52 thf(fact_138_monoI1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X8:(nat > A)]: (((! [M3:nat,N3:nat]: (((((ord_less_eq @ nat) @ M3) @ N3) => (((ord_less_eq @ A) @ (X8 @ M3)) @ (X8 @ N3))))) => ((topological_monoseq @ A) @ X8)))))))).
% 2.90/1.52 thf(fact_139_decseq__imp__monoseq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X8:(nat > A)]: (((((order_antimono @ nat) @ A) @ X8) => ((topological_monoseq @ A) @ X8)))))))).
% 2.90/1.52 thf(fact_140_LeastM__nat__le, axiom, (! [A:$tType,P:(A > $o),X:A,M2:(A > nat)]: (((P @ X) => (((ord_less_eq @ nat) @ (M2 @ ((((hilbert_LeastM @ A) @ nat) @ M2) @ P))) @ (M2 @ X)))))).
% 2.90/1.52 thf(fact_141_LeastM__nat__lemma, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat)]: (((P @ K) => ((P @ ((((hilbert_LeastM @ A) @ nat) @ M2) @ P)) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M2 @ ((((hilbert_LeastM @ A) @ nat) @ M2) @ P))) @ (M2 @ Y5)))))))))).
% 2.90/1.52 thf(fact_142_GreatestMI2, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [P:(A > $o),X:A,M2:(A > B),Q:(A > $o)]: (((P @ X) => ((! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ B) @ (M2 @ Y4)) @ (M2 @ X))))) => ((! [X2:A]: (((P @ X2) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ B) @ (M2 @ Y5)) @ (M2 @ X2))))) => (Q @ X2))))) => (Q @ ((((hilbert_GreatestM @ A) @ B) @ M2) @ P))))))))))).
% 2.90/1.52 thf(fact_143_totally__bounded__subset, axiom, (! [A:$tType]: ((((topolo47006728_space @ A) @ (type2 @ A)) => (! [S2:(set @ A),T:(set @ A)]: ((((topolo406746546ounded @ A) @ S2) => ((((ord_less_eq @ (set @ A)) @ T) @ S2) => ((topolo406746546ounded @ A) @ T))))))))).
% 2.90/1.52 thf(fact_144_Grp__mono, axiom, (! [B:$tType,A:$tType,A2:(set @ A),B6:(set @ A),F:(A > B)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B6) => (((ord_less_eq @ (A > (B > $o))) @ ((((bNF_Grp @ A) @ B) @ A2) @ F)) @ ((((bNF_Grp @ A) @ B) @ B6) @ F)))))).
% 2.90/1.52 thf(fact_145_of__nat__le__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [M2:nat,N2:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M2)) @ ((semiring_1_of_nat @ A) @ N2)) = (((ord_less_eq @ nat) @ M2) @ N2)))))))).
% 2.90/1.52 thf(fact_146_GrpE, axiom, (! [B:$tType,A:$tType,A2:(set @ A),F:(A > B),X:A,Y:B]: ((((((((bNF_Grp @ A) @ B) @ A2) @ F) @ X) @ Y) => (~ ((((F @ X) = Y) => (~ ((((member @ A) @ X) @ A2)))))))))).
% 2.90/1.52 thf(fact_147_GrpI, axiom, (! [B:$tType,A:$tType,F:(B > A),X:B,Y:A,A2:(set @ B)]: ((((F @ X) = Y) => ((((member @ B) @ X) @ A2) => ((((((bNF_Grp @ B) @ A) @ A2) @ F) @ X) @ Y)))))).
% 2.90/1.52 thf(fact_148_real__arch__simple, axiom, (! [A:$tType]: ((((archim1804426504_field @ A) @ (type2 @ A)) => (! [X:A]: ((? [N3:nat]: ((((ord_less_eq @ A) @ X) @ ((semiring_1_of_nat @ A) @ N3)))))))))).
% 2.90/1.52 thf(fact_149_zle__int, axiom, (! [M2:nat,N2:nat]: (((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ M2)) @ ((semiring_1_of_nat @ int) @ N2)) = (((ord_less_eq @ nat) @ M2) @ N2))))).
% 2.90/1.52 thf(fact_150_transfer__int__nat__relations_I3_J, axiom, (! [X:nat,Y:nat]: (((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ X)) @ ((semiring_1_of_nat @ int) @ Y)) = (((ord_less_eq @ nat) @ X) @ Y))))).
% 2.90/1.52 thf(fact_151_round__mono, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ int) @ ((archimedean_round @ A) @ X)) @ ((archimedean_round @ A) @ Y))))))))).
% 2.90/1.52 thf(fact_152_nat__le__iff, axiom, (! [X:int,N2:nat]: (((((ord_less_eq @ nat) @ (nat2 @ X)) @ N2) = (((ord_less_eq @ int) @ X) @ ((semiring_1_of_nat @ int) @ N2)))))).
% 2.90/1.52 thf(fact_153_negative__zle, axiom, (! [N2:nat,M2:nat]: ((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N2))) @ ((semiring_1_of_nat @ int) @ M2))))).
% 2.90/1.52 thf(fact_154_nat__mono, axiom, (! [X:int,Y:int]: (((((ord_less_eq @ int) @ X) @ Y) => (((ord_less_eq @ nat) @ (nat2 @ X)) @ (nat2 @ Y)))))).
% 2.90/1.52 thf(fact_155_compl__le__compl__iff, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X)) @ ((uminus_uminus @ A) @ Y)) = (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.90/1.52 thf(fact_156_neg__le__iff__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B3)) @ ((uminus_uminus @ A) @ A3)) = (((ord_less_eq @ A) @ A3) @ B3)))))))).
% 2.90/1.52 thf(fact_157_le__imp__neg__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B3)) @ ((uminus_uminus @ A) @ A3))))))))).
% 2.90/1.52 thf(fact_158_Compl__anti__mono, axiom, (! [A:$tType,A2:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B6) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ B6)) @ ((uminus_uminus @ (set @ A)) @ A2)))))).
% 2.90/1.52 thf(fact_159_Compl__subset__Compl__iff, axiom, (! [A:$tType,A2:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A2)) @ ((uminus_uminus @ (set @ A)) @ B6)) = (((ord_less_eq @ (set @ A)) @ B6) @ A2))))).
% 2.90/1.52 thf(fact_160_compl__mono, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y)) @ ((uminus_uminus @ A) @ X))))))))).
% 2.90/1.52 thf(fact_161_compl__le__swap1, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ ((uminus_uminus @ A) @ X)) => (((ord_less_eq @ A) @ X) @ ((uminus_uminus @ A) @ Y))))))))).
% 2.90/1.52 thf(fact_162_compl__le__swap2, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y)) @ X) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X)) @ Y)))))))).
% 2.90/1.52 thf(fact_163_le__minus__iff, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ ((uminus_uminus @ A) @ B3)) = (((ord_less_eq @ A) @ B3) @ ((uminus_uminus @ A) @ A3))))))))).
% 2.90/1.52 thf(fact_164_minus__le__iff, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ B3) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B3)) @ A3)))))))).
% 2.90/1.52 thf(fact_165_le__nat__iff, axiom, (! [K:int,N2:nat]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((((ord_less_eq @ nat) @ N2) @ (nat2 @ K)) = (((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ N2)) @ K)))))).
% 2.90/1.52 thf(fact_166_int__nat__eq, axiom, (! [Z3:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3) => (((semiring_1_of_nat @ int) @ (nat2 @ Z3)) = Z3)) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3))) => (((semiring_1_of_nat @ int) @ (nat2 @ Z3)) = (zero_zero @ int))))))).
% 2.90/1.52 thf(fact_167_int__zle__neg, axiom, (! [N2:nat,M2:nat]: (((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ N2)) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ M2))) = ((N2 = (zero_zero @ nat)) & (M2 = (zero_zero @ nat))))))).
% 2.90/1.52 thf(fact_168_ComplI, axiom, (! [A:$tType,C:A,A2:(set @ A)]: (((~ ((((member @ A) @ C) @ A2))) => (((member @ A) @ C) @ ((uminus_uminus @ (set @ A)) @ A2)))))).
% 2.90/1.52 thf(fact_169_Compl__iff, axiom, (! [A:$tType,C:A,A2:(set @ A)]: (((((member @ A) @ C) @ ((uminus_uminus @ (set @ A)) @ A2)) = (~ ((((member @ A) @ C) @ A2))))))).
% 2.90/1.52 thf(fact_170_Compl__eq__Compl__iff, axiom, (! [A:$tType,A2:(set @ A),B6:(set @ A)]: (((((uminus_uminus @ (set @ A)) @ A2) = ((uminus_uminus @ (set @ A)) @ B6)) = (A2 = B6))))).
% 2.90/1.52 thf(fact_171_le__zero__eq, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N2:A]: (((((ord_less_eq @ A) @ N2) @ (zero_zero @ A)) = (N2 = (zero_zero @ A))))))))).
% 2.90/1.52 thf(fact_172_le0, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N2)))).
% 2.90/1.52 thf(fact_173_neg__less__eq__nonneg, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ A3) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))).
% 2.90/1.52 thf(fact_174_less__eq__neg__nonpos, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ ((uminus_uminus @ A) @ A3)) = (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))).
% 2.90/1.52 thf(fact_175_neg__le__0__iff__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))).
% 2.90/1.52 thf(fact_176_neg__0__le__iff__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A3)) = (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))).
% 2.90/1.52 thf(fact_177_of__nat__le__0__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [M2:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M2)) @ (zero_zero @ A)) = (M2 = (zero_zero @ nat))))))))).
% 2.90/1.52 thf(fact_178_nat__0__iff, axiom, (! [I:int]: ((((nat2 @ I) = (zero_zero @ nat)) = (((ord_less_eq @ int) @ I) @ (zero_zero @ int)))))).
% 2.90/1.52 thf(fact_179_nat__le__0, axiom, (! [Z3:int]: (((((ord_less_eq @ int) @ Z3) @ (zero_zero @ int)) => ((nat2 @ Z3) = (zero_zero @ nat)))))).
% 2.90/1.52 thf(fact_180_ComplD, axiom, (! [A:$tType,C:A,A2:(set @ A)]: (((((member @ A) @ C) @ ((uminus_uminus @ (set @ A)) @ A2)) => (~ ((((member @ A) @ C) @ A2))))))).
% 2.90/1.52 thf(fact_181_double__complement, axiom, (! [A:$tType,A2:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A2)) = A2)))).
% 2.90/1.52 thf(fact_182_zero__le, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.90/1.52 thf(fact_183_le__0__eq, axiom, (! [N2:nat]: (((((ord_less_eq @ nat) @ N2) @ (zero_zero @ nat)) = (N2 = (zero_zero @ nat)))))).
% 2.90/1.52 thf(fact_184_less__eq__nat_Osimps_I1_J, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N2)))).
% 2.90/1.52 thf(fact_185_less__eq__int__code_I1_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ (zero_zero @ int))).
% 2.90/1.52 thf(fact_186_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ (zero_zero @ int))).
% 2.90/1.52 thf(fact_187_nat__eq__iff, axiom, (! [W:int,M2:nat]: ((((nat2 @ W) = M2) = (((((ord_less_eq @ int) @ (zero_zero @ int)) @ W) => (W = ((semiring_1_of_nat @ int) @ M2))) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ W))) => (M2 = (zero_zero @ nat)))))))).
% 2.90/1.52 thf(fact_188_nat__eq__iff2, axiom, (! [M2:nat,W:int]: (((M2 = (nat2 @ W)) = (((((ord_less_eq @ int) @ (zero_zero @ int)) @ W) => (W = ((semiring_1_of_nat @ int) @ M2))) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ W))) => (M2 = (zero_zero @ nat)))))))).
% 2.90/1.52 thf(fact_189_of__nat__0__le__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [N2:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N2)))))))).
% 2.90/1.52 thf(fact_190_transfer__nat__int__relations_I1_J, axiom, (! [X:int,Y:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y) => (((nat2 @ X) = (nat2 @ Y)) = (X = Y))))))).
% 2.90/1.52 thf(fact_191_eq__nat__nat__iff, axiom, (! [Z3:int,Z4:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z4) => (((nat2 @ Z3) = (nat2 @ Z4)) = (Z3 = Z4))))))).
% 2.90/1.52 thf(fact_192_ex__nat, axiom, ((^ [P2:(nat > $o)]: ((? [X6:nat]: ((P2 @ X6))))) = (^ [P3:(nat > $o)]: ((? [X5:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X5) & (P3 @ (nat2 @ X5))))))))).
% 2.90/1.52 thf(fact_193_all__nat, axiom, ((^ [P2:(nat > $o)]: ((! [X6:nat]: ((P2 @ X6))))) = (^ [P3:(nat > $o)]: ((! [X5:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X5) => (P3 @ (nat2 @ X5))))))))).
% 2.90/1.52 thf(fact_194_transfer__int__nat__quantifiers_I2_J, axiom, (! [P:(int > $o)]: (((? [X5:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X5) & (P @ X5)))) = (? [X5:nat]: ((P @ ((semiring_1_of_nat @ int) @ X5)))))))).
% 2.90/1.52 thf(fact_195_transfer__int__nat__quantifiers_I1_J, axiom, (! [P:(int > $o)]: (((! [X5:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X5) => (P @ X5)))) = (! [X5:nat]: ((P @ ((semiring_1_of_nat @ int) @ X5)))))))).
% 2.90/1.52 thf(fact_196_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J, axiom, (! [Z3:nat]: ((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((semiring_1_of_nat @ int) @ Z3))))).
% 2.90/1.52 thf(fact_197_nonneg__eq__int, axiom, (! [Z3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3) => (~ ((! [M3:nat]: ((Z3 != ((semiring_1_of_nat @ int) @ M3)))))))))).
% 2.90/1.52 thf(fact_198_nonneg__int__cases, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (~ ((! [N3:nat]: ((K != ((semiring_1_of_nat @ int) @ N3)))))))))).
% 2.90/1.52 thf(fact_199_zero__le__imp__eq__int, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (? [N3:nat]: ((K = ((semiring_1_of_nat @ int) @ N3)))))))).
% 2.90/1.52 thf(fact_200_transfer__nat__int__relations_I3_J, axiom, (! [X:int,Y:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y) => ((((ord_less_eq @ nat) @ (nat2 @ X)) @ (nat2 @ Y)) = (((ord_less_eq @ int) @ X) @ Y))))))).
% 2.90/1.52 thf(fact_201_int__eq__iff, axiom, (! [M2:nat,Z3:int]: (((((semiring_1_of_nat @ int) @ M2) = Z3) = ((M2 = (nat2 @ Z3)) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3)))))).
% 2.90/1.52 thf(fact_202_nat__0__le, axiom, (! [Z3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z3) => (((semiring_1_of_nat @ int) @ (nat2 @ Z3)) = Z3))))).
% 2.90/1.52 thf(fact_203_negative__zle__0, axiom, (! [N2:nat]: ((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N2))) @ (zero_zero @ int))))).
% 2.90/1.52 thf(fact_204_transfer__morphism__nat__int, axiom, ((((nat_tr1645093318rphism @ int) @ nat) @ nat2) @ ((ord_less_eq @ int) @ (zero_zero @ int)))).
% 2.90/1.52 thf(fact_205_imp__le__cong, axiom, (! [X:int,X4:int,P:$o,P4:$o]: (((X = X4) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X4) => (P = P4)) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => P) = ((((ord_less_eq @ int) @ (zero_zero @ int)) @ X4) => P4))))))).
% 2.90/1.52 thf(fact_206_conj__le__cong, axiom, (! [X:int,X4:int,P:$o,P4:$o]: (((X = X4) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X4) => (P = P4)) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) & P) = ((((ord_less_eq @ int) @ (zero_zero @ int)) @ X4) & P4))))))).
% 2.90/1.52 thf(fact_207_le__numeral__extra_I3_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.90/1.52 thf(fact_208_int__eq__iff__numeral, axiom, (! [M2:nat,V:num]: (((((semiring_1_of_nat @ int) @ M2) = ((numeral_numeral @ int) @ V)) = ((M2 = (nat2 @ ((numeral_numeral @ int) @ V))) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ ((numeral_numeral @ int) @ V))))))).
% 2.90/1.52 thf(fact_209_numeral__le__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [M2:num,N2:num]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ M2)) @ ((numeral_numeral @ A) @ N2)) = (((ord_less_eq @ num) @ M2) @ N2)))))))).
% 2.90/1.52 thf(fact_210_neg__numeral__le__iff, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [M2:num,N2:num]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))) = (((ord_less_eq @ num) @ N2) @ M2)))))))).
% 2.90/1.52 thf(fact_211_zero__le__numeral, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [N2:num]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ N2)))))))).
% 2.90/1.52 thf(fact_212_not__numeral__le__zero, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [N2:num]: ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ N2)) @ (zero_zero @ A)))))))))).
% 2.90/1.52 thf(fact_213_not__zero__le__neg__numeral, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [N2:num]: ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))))))))))).
% 2.90/1.52 thf(fact_214_neg__numeral__le__zero, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [N2:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))) @ (zero_zero @ A)))))))).
% 2.90/1.52 thf(fact_215_not__numeral__le__neg__numeral, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [M2:num,N2:num]: ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ M2)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))))))))))).
% 2.90/1.52 thf(fact_216_neg__numeral__le__numeral, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [M2:num,N2:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))) @ ((numeral_numeral @ A) @ N2)))))))).
% 2.90/1.52 thf(fact_217_ceiling__le__neg__numeral, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [X:A,V:num]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V))) = (((ord_less_eq @ A) @ X) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V)))))))))).
% 2.90/1.52 thf(fact_218_neg__numeral__le__floor, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [V:num,X:A]: (((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V))) @ ((archim2085082626_floor @ A) @ X)) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) @ X)))))))).
% 2.90/1.52 thf(fact_219_zero__le__floor, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [X:A]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((archim2085082626_floor @ A) @ X)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ X)))))))).
% 2.90/1.52 thf(fact_220_ceiling__le__zero, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [X:A]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ (zero_zero @ int)) = (((ord_less_eq @ A) @ X) @ (zero_zero @ A))))))))).
% 2.90/1.52 thf(fact_221_numeral__le__floor, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [V:num,X:A]: (((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ V)) @ ((archim2085082626_floor @ A) @ X)) = (((ord_less_eq @ A) @ ((numeral_numeral @ A) @ V)) @ X)))))))).
% 2.90/1.52 thf(fact_222_ceiling__le__numeral, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [X:A,V:num]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ ((numeral_numeral @ int) @ V)) = (((ord_less_eq @ A) @ X) @ ((numeral_numeral @ A) @ V))))))))).
% 2.90/1.52 thf(fact_223_floor__mono, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ int) @ ((archim2085082626_floor @ A) @ X)) @ ((archim2085082626_floor @ A) @ Y))))))))).
% 2.90/1.52 thf(fact_224_floor__le__ceiling, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ int) @ ((archim2085082626_floor @ A) @ X)) @ ((archimedean_ceiling @ A) @ X)))))))).
% 2.90/1.52 thf(fact_225_ceiling__ge__round, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ int) @ ((archimedean_round @ A) @ X)) @ ((archimedean_ceiling @ A) @ X)))))))).
% 2.90/1.52 thf(fact_226_floor__le__round, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ int) @ ((archim2085082626_floor @ A) @ X)) @ ((archimedean_round @ A) @ X)))))))).
% 2.90/1.52 thf(fact_227_ceiling__mono, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => (((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ Y)) @ ((archimedean_ceiling @ A) @ X))))))))).
% 2.90/1.52 thf(fact_228_nat__ceiling__le__eq, axiom, (! [X:real,A3:nat]: (((((ord_less_eq @ nat) @ (nat2 @ ((archimedean_ceiling @ real) @ X))) @ A3) = (((ord_less_eq @ real) @ X) @ ((semiring_1_of_nat @ real) @ A3)))))).
% 2.90/1.52 thf(fact_229_numeral__le__real__of__nat__iff, axiom, (! [N2:num,M2:nat]: (((((ord_less_eq @ real) @ ((numeral_numeral @ real) @ N2)) @ ((semiring_1_of_nat @ real) @ M2)) = (((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ N2)) @ M2))))).
% 2.90/1.52 thf(fact_230_complete__real, axiom, (! [S2:(set @ real)]: (((? [X3:real]: ((((member @ real) @ X3) @ S2))) => ((? [Z5:real]: ((! [X2:real]: (((((member @ real) @ X2) @ S2) => (((ord_less_eq @ real) @ X2) @ Z5)))))) => (? [Y4:real]: (((! [X3:real]: (((((member @ real) @ X3) @ S2) => (((ord_less_eq @ real) @ X3) @ Y4)))) & (! [Z5:real]: (((! [X2:real]: (((((member @ real) @ X2) @ S2) => (((ord_less_eq @ real) @ X2) @ Z5)))) => (((ord_less_eq @ real) @ Y4) @ Z5)))))))))))).
% 2.90/1.52 thf(fact_231_real__nat__ceiling__ge, axiom, (! [X:real]: ((((ord_less_eq @ real) @ X) @ ((semiring_1_of_nat @ real) @ (nat2 @ ((archimedean_ceiling @ real) @ X))))))).
% 2.90/1.52 thf(fact_232_nat__floor__neg, axiom, (! [X:real]: (((((ord_less_eq @ real) @ X) @ (zero_zero @ real)) => ((nat2 @ ((archim2085082626_floor @ real) @ X)) = (zero_zero @ nat)))))).
% 2.90/1.52 thf(fact_233_le__nat__floor, axiom, (! [X:nat,A3:real]: (((((ord_less_eq @ real) @ ((semiring_1_of_nat @ real) @ X)) @ A3) => (((ord_less_eq @ nat) @ X) @ (nat2 @ ((archim2085082626_floor @ real) @ A3))))))).
% 2.90/1.52 thf(fact_234_enat__ord__number_I1_J, axiom, (! [M2:num,N2:num]: (((((ord_less_eq @ extended_enat) @ ((numeral_numeral @ extended_enat) @ M2)) @ ((numeral_numeral @ extended_enat) @ N2)) = (((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ M2)) @ ((numeral_numeral @ nat) @ N2)))))).
% 2.90/1.52 thf(fact_235_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J, axiom, (! [X:int,Y:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y) => (((ord_less_eq @ int) @ (zero_zero @ int)) @ ((nat_tsub @ X) @ Y))))))).
% 2.90/1.52 thf(fact_236_ile0__eq, axiom, (! [N2:extended_enat]: (((((ord_less_eq @ extended_enat) @ N2) @ (zero_zero @ extended_enat)) = (N2 = (zero_zero @ extended_enat)))))).
% 2.90/1.52 thf(fact_237_i0__lb, axiom, (! [N2:extended_enat]: ((((ord_less_eq @ extended_enat) @ (zero_zero @ extended_enat)) @ N2)))).
% 2.90/1.52 thf(fact_238_numeral__le__enat__iff, axiom, (! [M2:num,N2:nat]: (((((ord_less_eq @ extended_enat) @ ((numeral_numeral @ extended_enat) @ M2)) @ (extended_enat2 @ N2)) = (((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ M2)) @ N2))))).
% 2.90/1.52 thf(fact_239_Nat__Transfer_Otransfer__nat__int__functions_I3_J, axiom, (! [X:int,Y:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y) => ((((minus_minus @ nat) @ (nat2 @ X)) @ (nat2 @ Y)) = (nat2 @ ((nat_tsub @ X) @ Y)))))))).
% 2.90/1.52 thf(fact_240_diff__diff__cancel, axiom, (! [I:nat,N2:nat]: (((((ord_less_eq @ nat) @ I) @ N2) => ((((minus_minus @ nat) @ N2) @ (((minus_minus @ nat) @ N2) @ I)) = I))))).
% 2.90/1.52 thf(fact_241_diff__ge__0__iff__ge, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A3) @ B3)) = (((ord_less_eq @ A) @ B3) @ A3)))))))).
% 2.90/1.52 thf(fact_242_diff__is__0__eq, axiom, (! [M2:nat,N2:nat]: ((((((minus_minus @ nat) @ M2) @ N2) = (zero_zero @ nat)) = (((ord_less_eq @ nat) @ M2) @ N2))))).
% 2.90/1.52 thf(fact_243_diff__is__0__eq_H, axiom, (! [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) => ((((minus_minus @ nat) @ M2) @ N2) = (zero_zero @ nat)))))).
% 2.90/1.52 thf(fact_244_enat__ord__simps_I1_J, axiom, (! [M2:nat,N2:nat]: (((((ord_less_eq @ extended_enat) @ (extended_enat2 @ M2)) @ (extended_enat2 @ N2)) = (((ord_less_eq @ nat) @ M2) @ N2))))).
% 2.90/1.52 thf(fact_245_enat__ile, axiom, (! [N2:extended_enat,M2:nat]: (((((ord_less_eq @ extended_enat) @ N2) @ (extended_enat2 @ M2)) => (? [K2:nat]: ((N2 = (extended_enat2 @ K2)))))))).
% 2.90/1.52 thf(fact_246_eq__diff__iff, axiom, (! [K:nat,M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ M2) => ((((ord_less_eq @ nat) @ K) @ N2) => (((((minus_minus @ nat) @ M2) @ K) = (((minus_minus @ nat) @ N2) @ K)) = (M2 = N2))))))).
% 2.90/1.52 thf(fact_247_le__diff__iff, axiom, (! [K:nat,M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ M2) => ((((ord_less_eq @ nat) @ K) @ N2) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M2) @ K)) @ (((minus_minus @ nat) @ N2) @ K)) = (((ord_less_eq @ nat) @ M2) @ N2))))))).
% 2.90/1.52 thf(fact_248_Nat_Odiff__diff__eq, axiom, (! [K:nat,M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ M2) => ((((ord_less_eq @ nat) @ K) @ N2) => ((((minus_minus @ nat) @ (((minus_minus @ nat) @ M2) @ K)) @ (((minus_minus @ nat) @ N2) @ K)) = (((minus_minus @ nat) @ M2) @ N2))))))).
% 2.90/1.52 thf(fact_249_diff__le__mono, axiom, (! [M2:nat,N2:nat,L:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M2) @ L)) @ (((minus_minus @ nat) @ N2) @ L)))))).
% 2.90/1.52 thf(fact_250_diff__le__self, axiom, (! [M2:nat,N2:nat]: ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M2) @ N2)) @ M2)))).
% 2.90/1.52 thf(fact_251_le__diff__iff_H, axiom, (! [A3:nat,C:nat,B3:nat]: (((((ord_less_eq @ nat) @ A3) @ C) => ((((ord_less_eq @ nat) @ B3) @ C) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ C) @ A3)) @ (((minus_minus @ nat) @ C) @ B3)) = (((ord_less_eq @ nat) @ B3) @ A3))))))).
% 2.90/1.52 thf(fact_252_diff__le__mono2, axiom, (! [M2:nat,N2:nat,L:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ L) @ N2)) @ (((minus_minus @ nat) @ L) @ M2)))))).
% 2.90/1.52 thf(fact_253_of__nat__diff, axiom, (! [A:$tType]: ((((ring_1 @ A) @ (type2 @ A)) => (! [N2:nat,M2:nat]: (((((ord_less_eq @ nat) @ N2) @ M2) => (((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ M2) @ N2)) = (((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ M2)) @ ((semiring_1_of_nat @ A) @ N2)))))))))).
% 2.90/1.52 thf(fact_254_diff__eq__diff__less__eq, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A,D2:A]: ((((((minus_minus @ A) @ A3) @ B3) = (((minus_minus @ A) @ C) @ D2)) => ((((ord_less_eq @ A) @ A3) @ B3) = (((ord_less_eq @ A) @ C) @ D2))))))))).
% 2.90/1.52 thf(fact_255_diff__right__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ C)) @ (((minus_minus @ A) @ B3) @ C))))))))).
% 2.90/1.52 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A7:$tType,A8:$tType]: ((((comple187826305attice @ A8) @ (type2 @ A8)) => ((comple187826305attice @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.52 thf(tcon_fun___Lattices_Oboolean__algebra, axiom, (! [A7:$tType,A8:$tType]: ((((boolean_algebra @ A8) @ (type2 @ A8)) => ((boolean_algebra @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.52 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A7:$tType,A8:$tType]: ((((order_top @ A8) @ (type2 @ A8)) => ((order_top @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.52 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: ((((preorder @ A8) @ (type2 @ A8)) => ((preorder @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.52 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: ((((order @ A8) @ (type2 @ A8)) => ((order @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.52 thf(tcon_fun___Orderings_Otop, axiom, (! [A7:$tType,A8:$tType]: ((((top @ A8) @ (type2 @ A8)) => ((top @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.52 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: ((((ord @ A8) @ (type2 @ A8)) => ((ord @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.52 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add, axiom, ((linord219039673up_add @ int) @ (type2 @ int))).
% 2.90/1.52 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add, axiom, ((ordered_ab_group_add @ int) @ (type2 @ int))).
% 2.90/1.52 thf(tcon_Int_Oint___Rings_Olinordered__semidom, axiom, ((linordered_semidom @ int) @ (type2 @ int))).
% 2.90/1.52 thf(tcon_Int_Oint___Rings_Olinordered__idom, axiom, ((linordered_idom @ int) @ (type2 @ int))).
% 2.90/1.52 thf(tcon_Int_Oint___Orderings_Opreorder_1, axiom, ((preorder @ int) @ (type2 @ int))).
% 2.90/1.52 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, ((linorder @ int) @ (type2 @ int))).
% 2.90/1.52 thf(tcon_Int_Oint___Orderings_Oorder_2, axiom, ((order @ int) @ (type2 @ int))).
% 2.90/1.52 thf(tcon_Int_Oint___Orderings_Oord_3, axiom, ((ord @ int) @ (type2 @ int))).
% 2.90/1.52 thf(tcon_Int_Oint___Rings_Oring__1, axiom, ((ring_1 @ int) @ (type2 @ int))).
% 2.90/1.52 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, ((canoni770627133id_add @ nat) @ (type2 @ nat))).
% 2.90/1.52 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_4, axiom, ((linordered_semidom @ nat) @ (type2 @ nat))).
% 2.90/1.52 thf(tcon_Nat_Onat___Orderings_Opreorder_5, axiom, ((preorder @ nat) @ (type2 @ nat))).
% 2.90/1.52 thf(tcon_Nat_Onat___Orderings_Olinorder_6, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 2.90/1.52 thf(tcon_Nat_Onat___Orderings_Oorder_7, axiom, ((order @ nat) @ (type2 @ nat))).
% 2.90/1.52 thf(tcon_Nat_Onat___Orderings_Oord_8, axiom, ((ord @ nat) @ (type2 @ nat))).
% 2.90/1.52 thf(tcon_Num_Onum___Orderings_Opreorder_9, axiom, ((preorder @ num) @ (type2 @ num))).
% 2.90/1.52 thf(tcon_Num_Onum___Orderings_Olinorder_10, axiom, ((linorder @ num) @ (type2 @ num))).
% 2.90/1.52 thf(tcon_Num_Onum___Orderings_Oorder_11, axiom, ((order @ num) @ (type2 @ num))).
% 2.90/1.52 thf(tcon_Num_Onum___Orderings_Oord_12, axiom, ((ord @ num) @ (type2 @ num))).
% 2.90/1.52 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_13, axiom, (! [A7:$tType]: (((comple187826305attice @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.52 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_14, axiom, (! [A7:$tType]: (((boolean_algebra @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.52 thf(tcon_Set_Oset___Orderings_Oorder__top_15, axiom, (! [A7:$tType]: (((order_top @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.52 thf(tcon_Set_Oset___Orderings_Opreorder_16, axiom, (! [A7:$tType]: (((preorder @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.52 thf(tcon_Set_Oset___Orderings_Oorder_17, axiom, (! [A7:$tType]: (((order @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.52 thf(tcon_Set_Oset___Orderings_Otop_18, axiom, (! [A7:$tType]: (((top @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.52 thf(tcon_Set_Oset___Orderings_Oord_19, axiom, (! [A7:$tType]: (((ord @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.52 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_20, axiom, ((comple187826305attice @ $o) @ (type2 @ $o))).
% 2.90/1.52 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_21, axiom, ((boolean_algebra @ $o) @ (type2 @ $o))).
% 2.90/1.52 thf(tcon_HOL_Obool___Orderings_Oorder__top_22, axiom, ((order_top @ $o) @ (type2 @ $o))).
% 2.90/1.52 thf(tcon_HOL_Obool___Orderings_Opreorder_23, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.90/1.52 thf(tcon_HOL_Obool___Orderings_Olinorder_24, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.90/1.52 thf(tcon_HOL_Obool___Orderings_Oorder_25, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.90/1.52 thf(tcon_HOL_Obool___Orderings_Otop_26, axiom, ((top @ $o) @ (type2 @ $o))).
% 2.90/1.52 thf(tcon_HOL_Obool___Orderings_Oord_27, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.90/1.52 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field, axiom, ((archim1804426504_field @ real) @ (type2 @ real))).
% 2.90/1.52 thf(tcon_Real_Oreal___Topological__Spaces_Ouniform__space, axiom, ((topolo47006728_space @ real) @ (type2 @ real))).
% 2.90/1.52 thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling, axiom, ((archim1727834104eiling @ real) @ (type2 @ real))).
% 2.90/1.52 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_28, axiom, ((linord219039673up_add @ real) @ (type2 @ real))).
% 2.90/1.52 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_29, axiom, ((ordered_ab_group_add @ real) @ (type2 @ real))).
% 2.90/1.52 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_30, axiom, ((linordered_semidom @ real) @ (type2 @ real))).
% 2.90/1.52 thf(tcon_Real_Oreal___Rings_Olinordered__idom_31, axiom, ((linordered_idom @ real) @ (type2 @ real))).
% 2.90/1.52 thf(tcon_Real_Oreal___Orderings_Opreorder_32, axiom, ((preorder @ real) @ (type2 @ real))).
% 2.90/1.52 thf(tcon_Real_Oreal___Orderings_Olinorder_33, axiom, ((linorder @ real) @ (type2 @ real))).
% 2.90/1.52 thf(tcon_Real_Oreal___Orderings_Oorder_34, axiom, ((order @ real) @ (type2 @ real))).
% 2.90/1.52 thf(tcon_Real_Oreal___Orderings_Oord_35, axiom, ((ord @ real) @ (type2 @ real))).
% 2.90/1.52 thf(tcon_Real_Oreal___Rings_Oring__1_36, axiom, ((ring_1 @ real) @ (type2 @ real))).
% 2.90/1.52 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_37, axiom, ((canoni770627133id_add @ extended_enat) @ (type2 @ extended_enat))).
% 2.90/1.52 thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_38, axiom, ((comple187826305attice @ extended_enat) @ (type2 @ extended_enat))).
% 2.90/1.52 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_39, axiom, ((order_top @ extended_enat) @ (type2 @ extended_enat))).
% 2.90/1.52 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_40, axiom, ((preorder @ extended_enat) @ (type2 @ extended_enat))).
% 2.90/1.52 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_41, axiom, ((linorder @ extended_enat) @ (type2 @ extended_enat))).
% 2.90/1.52 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_42, axiom, ((order @ extended_enat) @ (type2 @ extended_enat))).
% 2.90/1.52 thf(tcon_Extended__Nat_Oenat___Orderings_Otop_43, axiom, ((top @ extended_enat) @ (type2 @ extended_enat))).
% 2.90/1.52 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_44, axiom, ((ord @ extended_enat) @ (type2 @ extended_enat))).
% 2.90/1.52 thf(conj_0, hypothesis, (p @ x)).
% 2.90/1.52 thf(conj_1, conjecture, ((((bNF_eq_onp @ a) @ p) @ x) @ x)).
% 2.90/1.52 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.7l7JJvTzmF/DTF2DTF19250.p
% 2.90/1.52 ------------------------
% 2.90/1.55 ---- Cleaned THF ---
% 2.90/1.55 thf(ty_t_Extended__Nat_Oenat,type,
% 2.90/1.55 extended_enat: $tType ).
% 2.90/1.55
% 2.90/1.55 thf(ty_t_Real_Oreal,type,
% 2.90/1.55 real: $tType ).
% 2.90/1.55
% 2.90/1.55 thf(ty_t_Set_Oset,type,
% 2.90/1.55 set: $tType > $tType ).
% 2.90/1.55
% 2.90/1.55 thf(ty_t_Num_Onum,type,
% 2.90/1.55 num: $tType ).
% 2.90/1.55
% 2.90/1.55 thf(ty_t_Nat_Onat,type,
% 2.90/1.55 nat: $tType ).
% 2.90/1.55
% 2.90/1.55 thf(ty_t_Int_Oint,type,
% 2.90/1.55 int: $tType ).
% 2.90/1.55
% 2.90/1.55 thf(ty_t_itself,type,
% 2.90/1.55 itself: $tType > $tType ).
% 2.90/1.55
% 2.90/1.55 thf(ty_tf_a,type,
% 2.90/1.55 a: $tType ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_HOL_Otype,type,
% 2.90/1.55 type:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Rings_Oring__1,type,
% 2.90/1.55 ring_1:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Orderings_Oord,type,
% 2.90/1.55 ord:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Orderings_Otop,type,
% 2.90/1.55 top:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Orderings_Oorder,type,
% 2.90/1.55 order:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Orderings_Olinorder,type,
% 2.90/1.55 linorder:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Orderings_Opreorder,type,
% 2.90/1.55 preorder:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Orderings_Oorder__top,type,
% 2.90/1.55 order_top:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.90/1.55 linordered_idom:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 2.90/1.55 boolean_algebra:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.90/1.55 linordered_semidom:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.90/1.55 ordered_ab_group_add:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 2.90/1.55 linord219039673up_add:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
% 2.90/1.55 archim1727834104eiling:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Topological__Spaces_Ouniform__space,type,
% 2.90/1.55 topolo47006728_space:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 2.90/1.55 comple187826305attice:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 2.90/1.55 archim1804426504_field:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.90/1.55 canoni770627133id_add:
% 2.90/1.55 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Archimedean__Field_Oceiling,type,
% 2.90/1.55 archimedean_ceiling:
% 2.90/1.55 !>[A: $tType] : ( A > int ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
% 2.90/1.55 archim2085082626_floor:
% 2.90/1.55 !>[A: $tType] : ( A > int ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Archimedean__Field_Oround,type,
% 2.90/1.55 archimedean_round:
% 2.90/1.55 !>[A: $tType] : ( A > int ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_BNF__Composition_ODEADID_Opred__DEADID,type,
% 2.90/1.55 bNF_pred_DEADID:
% 2.90/1.55 !>[A: $tType] : ( A > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_BNF__Def_OGrp,type,
% 2.90/1.55 bNF_Grp:
% 2.90/1.55 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_BNF__Def_Oeq__onp,type,
% 2.90/1.55 bNF_eq_onp:
% 2.90/1.55 !>[A: $tType] : ( ( A > $o ) > A > A > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Extended__Nat_Oenat,type,
% 2.90/1.55 extended_enat2: nat > extended_enat ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.90/1.55 minus_minus:
% 2.90/1.55 !>[A: $tType] : ( A > A > A ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 2.90/1.55 uminus_uminus:
% 2.90/1.55 !>[A: $tType] : ( A > A ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.90/1.55 zero_zero:
% 2.90/1.55 !>[A: $tType] : A ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_HOL_Oundefined,type,
% 2.90/1.55 undefined:
% 2.90/1.55 !>[A: $tType] : A ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Hilbert__Choice_OGreatestM,type,
% 2.90/1.55 hilbert_GreatestM:
% 2.90/1.55 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Hilbert__Choice_OLeastM,type,
% 2.90/1.55 hilbert_LeastM:
% 2.90/1.55 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Int_Onat,type,
% 2.90/1.55 nat2: int > nat ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Lifting_ONEG,type,
% 2.90/1.55 neg:
% 2.90/1.55 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( A > B > $o ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Lifting_OPOS,type,
% 2.90/1.55 pos:
% 2.90/1.55 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( A > B > $o ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Lifting_OQuotient,type,
% 2.90/1.55 quotient:
% 2.90/1.55 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 2.90/1.55 semiring_1_of_nat:
% 2.90/1.55 !>[A: $tType] : ( nat > A ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Nat__Transfer_Otransfer__morphism,type,
% 2.90/1.55 nat_tr1645093318rphism:
% 2.90/1.55 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Nat__Transfer_Otsub,type,
% 2.90/1.55 nat_tsub: int > int > int ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Num_Onumeral__class_Onumeral,type,
% 2.90/1.55 numeral_numeral:
% 2.90/1.55 !>[A: $tType] : ( num > A ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.90/1.55 ord_less_eq:
% 2.90/1.55 !>[A: $tType] : ( A > A > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 2.90/1.55 order_antimono:
% 2.90/1.55 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.90/1.55 top_top:
% 2.90/1.55 !>[A: $tType] : A ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Pure_Otype,type,
% 2.90/1.55 type2:
% 2.90/1.55 !>[A: $tType] : ( itself @ A ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Relation_ODomainp,type,
% 2.90/1.55 domainp:
% 2.90/1.55 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Relation_Oinv__imagep,type,
% 2.90/1.55 inv_imagep:
% 2.90/1.55 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Set_OCollect,type,
% 2.90/1.55 collect:
% 2.90/1.55 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Topological__Spaces_Omonoseq,type,
% 2.90/1.55 topological_monoseq:
% 2.90/1.55 !>[A: $tType] : ( ( nat > A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
% 2.90/1.55 topolo406746546ounded:
% 2.90/1.55 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_c_member,type,
% 2.90/1.55 member:
% 2.90/1.55 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(sy_v_P,type,
% 2.90/1.55 p: a > $o ).
% 2.90/1.55
% 2.90/1.55 thf(sy_v_x,type,
% 2.90/1.55 x: a ).
% 2.90/1.55
% 2.90/1.55 thf(fact_0_eq__onp__eqD,axiom,
% 2.90/1.55 ! [A: $tType,P: A > $o,Q: A > A > $o,X: A] :
% 2.90/1.55 ( ( ( bNF_eq_onp @ A @ P )
% 2.90/1.55 = Q )
% 2.90/1.55 => ( ( P @ X )
% 2.90/1.55 = ( Q @ X @ X ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_1_eq__onp__mono0,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A,P: A > $o,Q: A > $o] :
% 2.90/1.55 ( ! [X2: A] :
% 2.90/1.55 ( ( member @ A @ X2 @ A2 )
% 2.90/1.55 => ( ( P @ X2 )
% 2.90/1.55 => ( Q @ X2 ) ) )
% 2.90/1.55 => ! [X3: A] :
% 2.90/1.55 ( ( member @ A @ X3 @ A2 )
% 2.90/1.55 => ! [Xa: A] :
% 2.90/1.55 ( ( member @ A @ Xa @ A2 )
% 2.90/1.55 => ( ( bNF_eq_onp @ A @ P @ X3 @ Xa )
% 2.90/1.55 => ( bNF_eq_onp @ A @ Q @ X3 @ Xa ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_2_eq__onp__to__eq,axiom,
% 2.90/1.55 ! [A: $tType,P: A > $o,X: A,Y: A] :
% 2.90/1.55 ( ( bNF_eq_onp @ A @ P @ X @ Y )
% 2.90/1.55 => ( X = Y ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_3_eq__onp__same__args,axiom,
% 2.90/1.55 ! [A: $tType,P: A > $o,X: A] :
% 2.90/1.55 ( ( bNF_eq_onp @ A @ P @ X @ X )
% 2.90/1.55 = ( P @ X ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_4_eq__onp__live__step,axiom,
% 2.90/1.55 ! [A: $tType,X: $o,Y: $o,P: A > $o,A3: A] :
% 2.90/1.55 ( ( X = Y )
% 2.90/1.55 => ( ( ( bNF_eq_onp @ A @ P @ A3 @ A3 )
% 2.90/1.55 & X )
% 2.90/1.55 = ( ( P @ A3 )
% 2.90/1.55 & Y ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_5_DEADID_Orel__eq__onp,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.90/1.55 = ( bNF_eq_onp @ A @ ( bNF_pred_DEADID @ A ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_6_eq__onp__le__eq,axiom,
% 2.90/1.55 ! [A: $tType,P: A > $o] :
% 2.90/1.55 ( ord_less_eq @ ( A > A > $o ) @ ( bNF_eq_onp @ A @ P )
% 2.90/1.55 @ ^ [Y2: A,Z: A] : ( Y2 = Z ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_7_type__definition__Quotient__not__empty,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,P: A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
% 2.90/1.55 ( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ T )
% 2.90/1.55 => ? [X1: A] : ( P @ X1 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_8_eq__onp__top__eq__eq,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( bNF_eq_onp @ A @ ( top_top @ ( A > $o ) ) )
% 2.90/1.55 = ( ^ [Y2: A,Z: A] : ( Y2 = Z ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_9_top__conj_I2_J,axiom,
% 2.90/1.55 ! [A: $tType,P: $o,X: A] :
% 2.90/1.55 ( ( P
% 2.90/1.55 & ( top_top @ ( A > $o ) @ X ) )
% 2.90/1.55 = P ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_10_top__conj_I1_J,axiom,
% 2.90/1.55 ! [A: $tType,X: A,P: $o] :
% 2.90/1.55 ( ( ( top_top @ ( A > $o ) @ X )
% 2.90/1.55 & P )
% 2.90/1.55 = P ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_11_ge__eq__refl,axiom,
% 2.90/1.55 ! [A: $tType,R: A > A > $o,X: A] :
% 2.90/1.55 ( ( ord_less_eq @ ( A > A > $o )
% 2.90/1.55 @ ^ [Y2: A,Z: A] : ( Y2 = Z )
% 2.90/1.55 @ R )
% 2.90/1.55 => ( R @ X @ X ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_12_refl__ge__eq,axiom,
% 2.90/1.55 ! [A: $tType,R: A > A > $o] :
% 2.90/1.55 ( ! [X2: A] : ( R @ X2 @ X2 )
% 2.90/1.55 => ( ord_less_eq @ ( A > A > $o )
% 2.90/1.55 @ ^ [Y2: A,Z: A] : ( Y2 = Z )
% 2.90/1.55 @ R ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_13_Quotient__rel,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R2: A,S: A] :
% 2.90/1.55 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 2.90/1.55 => ( ( ( R @ R2 @ R2 )
% 2.90/1.55 & ( R @ S @ S )
% 2.90/1.55 & ( ( Abs @ R2 )
% 2.90/1.55 = ( Abs @ S ) ) )
% 2.90/1.55 = ( R @ R2 @ S ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_14_Quotient__refl1,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R2: A,S: A] :
% 2.90/1.55 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 2.90/1.55 => ( ( R @ R2 @ S )
% 2.90/1.55 => ( R @ R2 @ R2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_15_Quotient__refl2,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R2: A,S: A] :
% 2.90/1.55 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 2.90/1.55 => ( ( R @ R2 @ S )
% 2.90/1.55 => ( R @ S @ S ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_16_Quotient__Rep__eq,axiom,
% 2.90/1.55 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,X4: B,X: A] :
% 2.90/1.55 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 2.90/1.55 => ( ( X4
% 2.90/1.55 = ( Abs @ X ) )
% 2.90/1.55 => ( ( Rep @ X4 )
% 2.90/1.55 = ( Rep @ X4 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_17_Quotient__abs__rep,axiom,
% 2.90/1.55 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,A3: B] :
% 2.90/1.55 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 2.90/1.55 => ( ( Abs @ ( Rep @ A3 ) )
% 2.90/1.55 = A3 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_18_Quotient__alt__def,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( quotient @ A @ B )
% 2.90/1.55 = ( ^ [R3: A > A > $o,Abs2: A > B,Rep2: B > A,T2: A > B > $o] :
% 2.90/1.55 ( ! [A4: A,B2: B] :
% 2.90/1.55 ( ( T2 @ A4 @ B2 )
% 2.90/1.55 => ( ( Abs2 @ A4 )
% 2.90/1.55 = B2 ) )
% 2.90/1.55 & ! [B2: B] : ( T2 @ ( Rep2 @ B2 ) @ B2 )
% 2.90/1.55 & ! [X5: A,Y3: A] :
% 2.90/1.55 ( ( R3 @ X5 @ Y3 )
% 2.90/1.55 = ( ( T2 @ X5 @ ( Abs2 @ X5 ) )
% 2.90/1.55 & ( T2 @ Y3 @ ( Abs2 @ Y3 ) )
% 2.90/1.55 & ( ( Abs2 @ X5 )
% 2.90/1.55 = ( Abs2 @ Y3 ) ) ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_19_Quotient__rel__abs,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R2: A,S: A] :
% 2.90/1.55 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 2.90/1.55 => ( ( R @ R2 @ S )
% 2.90/1.55 => ( ( Abs @ R2 )
% 2.90/1.55 = ( Abs @ S ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_20_Quotient__rel__rep,axiom,
% 2.90/1.55 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,A3: B,B3: B] :
% 2.90/1.55 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 2.90/1.55 => ( ( R @ ( Rep @ A3 ) @ ( Rep @ B3 ) )
% 2.90/1.55 = ( A3 = B3 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_21_Quotient__rep__abs,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R2: A] :
% 2.90/1.55 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 2.90/1.55 => ( ( R @ R2 @ R2 )
% 2.90/1.55 => ( R @ ( Rep @ ( Abs @ R2 ) ) @ R2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_22_Quotient__alt__def2,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( quotient @ A @ B )
% 2.90/1.55 = ( ^ [R3: A > A > $o,Abs2: A > B,Rep2: B > A,T2: A > B > $o] :
% 2.90/1.55 ( ! [A4: A,B2: B] :
% 2.90/1.55 ( ( T2 @ A4 @ B2 )
% 2.90/1.55 => ( ( Abs2 @ A4 )
% 2.90/1.55 = B2 ) )
% 2.90/1.55 & ! [B2: B] : ( T2 @ ( Rep2 @ B2 ) @ B2 )
% 2.90/1.55 & ! [X5: A,Y3: A] :
% 2.90/1.55 ( ( R3 @ X5 @ Y3 )
% 2.90/1.55 = ( ( T2 @ X5 @ ( Abs2 @ Y3 ) )
% 2.90/1.55 & ( T2 @ Y3 @ ( Abs2 @ X5 ) ) ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_23_Quotient__alt__def3,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( quotient @ A @ B )
% 2.90/1.55 = ( ^ [R3: A > A > $o,Abs2: A > B,Rep2: B > A,T2: A > B > $o] :
% 2.90/1.55 ( ! [A4: A,B2: B] :
% 2.90/1.55 ( ( T2 @ A4 @ B2 )
% 2.90/1.55 => ( ( Abs2 @ A4 )
% 2.90/1.55 = B2 ) )
% 2.90/1.55 & ! [B2: B] : ( T2 @ ( Rep2 @ B2 ) @ B2 )
% 2.90/1.55 & ! [X5: A,Y3: A] :
% 2.90/1.55 ( ( R3 @ X5 @ Y3 )
% 2.90/1.55 = ( ? [Z2: B] :
% 2.90/1.55 ( ( T2 @ X5 @ Z2 )
% 2.90/1.55 & ( T2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_24_Quotient__rel__abs2,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,X: B,Y: A] :
% 2.90/1.55 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 2.90/1.55 => ( ( R @ ( Rep @ X ) @ Y )
% 2.90/1.55 => ( X
% 2.90/1.55 = ( Abs @ Y ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_25_Quotient__rep__reflp,axiom,
% 2.90/1.55 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,A3: B] :
% 2.90/1.55 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 2.90/1.55 => ( R @ ( Rep @ A3 ) @ ( Rep @ A3 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_26_Quotient__abs__induct,axiom,
% 2.90/1.55 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,P: B > $o,X: B] :
% 2.90/1.55 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 2.90/1.55 => ( ! [Y4: A] :
% 2.90/1.55 ( ( R @ Y4 @ Y4 )
% 2.90/1.55 => ( P @ ( Abs @ Y4 ) ) )
% 2.90/1.55 => ( P @ X ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_27_Quotient__rep__abs__eq,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,T3: A] :
% 2.90/1.55 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 2.90/1.55 => ( ( R @ T3 @ T3 )
% 2.90/1.55 => ( ( ord_less_eq @ ( A > A > $o ) @ R
% 2.90/1.55 @ ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.90/1.55 => ( ( Rep @ ( Abs @ T3 ) )
% 2.90/1.55 = T3 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_28_Quotient__to__transfer,axiom,
% 2.90/1.55 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,C: A,C2: B] :
% 2.90/1.55 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 2.90/1.55 => ( ( R @ C @ C )
% 2.90/1.55 => ( ( C2
% 2.90/1.55 = ( Abs @ C ) )
% 2.90/1.55 => ( T @ C @ C2 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_29_DEADID_Opred__rel,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( bNF_pred_DEADID @ A )
% 2.90/1.55 = ( ^ [X5: A] : ( X5 = X5 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_30_DEADID_Opred__True,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( bNF_pred_DEADID @ A )
% 2.90/1.55 = ( ^ [Uu: A] : $true ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_31_predicate2D__conj,axiom,
% 2.90/1.55 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X: A,Y: B] :
% 2.90/1.55 ( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 2.90/1.55 & R )
% 2.90/1.55 => ( R
% 2.90/1.55 & ( ( P @ X @ Y )
% 2.90/1.55 => ( Q @ X @ Y ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_32_Quotient__rep__abs__fold__unmap,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,X4: B,X: A,Rep3: B > A] :
% 2.90/1.55 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 2.90/1.55 => ( ( X4
% 2.90/1.55 = ( Abs @ X ) )
% 2.90/1.55 => ( ( R @ X @ X )
% 2.90/1.55 => ( ( ( Rep @ X4 )
% 2.90/1.55 = ( Rep3 @ X4 ) )
% 2.90/1.55 => ( R @ ( Rep3 @ X4 ) @ X ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_33_DEADID_Opred__cong__simp,axiom,
% 2.90/1.55 ! [A: $tType,X: A,Ya: A] :
% 2.90/1.55 ( ( X = Ya )
% 2.90/1.55 => ( ( bNF_pred_DEADID @ A @ X )
% 2.90/1.55 = ( bNF_pred_DEADID @ A @ Ya ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_34_DEADID_Opred__mono__strong,axiom,
% 2.90/1.55 ! [A: $tType,X: A] :
% 2.90/1.55 ( ( bNF_pred_DEADID @ A @ X )
% 2.90/1.55 => ( bNF_pred_DEADID @ A @ X ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_35_top__apply,axiom,
% 2.90/1.55 ! [C3: $tType,D: $tType] :
% 2.90/1.55 ( ( top @ C3 @ ( type2 @ C3 ) )
% 2.90/1.55 => ( ( top_top @ ( D > C3 ) )
% 2.90/1.55 = ( ^ [X5: D] : ( top_top @ C3 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_36_top1I,axiom,
% 2.90/1.55 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_37_predicate2I,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 2.90/1.55 ( ! [X2: A,Y4: B] :
% 2.90/1.55 ( ( P @ X2 @ Y4 )
% 2.90/1.55 => ( Q @ X2 @ Y4 ) )
% 2.90/1.55 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_38_order__refl,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( preorder @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_39_le__funD,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.55 => ! [F: A > B,G: A > B,X: A] :
% 2.90/1.55 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.90/1.55 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_40_le__funE,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.55 => ! [F: A > B,G: A > B,X: A] :
% 2.90/1.55 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.90/1.55 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_41_le__funI,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.55 => ! [F: A > B,G: A > B] :
% 2.90/1.55 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 2.90/1.55 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_42_le__fun__def,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.55 => ( ( ord_less_eq @ ( A > B ) )
% 2.90/1.55 = ( ^ [F2: A > B,G2: A > B] :
% 2.90/1.55 ! [X5: A] : ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_43_top__greatest,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order_top @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_44_top_Oextremum__unique,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order_top @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 2.90/1.55 = ( A3
% 2.90/1.55 = ( top_top @ A ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_45_mem__Collect__eq,axiom,
% 2.90/1.55 ! [A: $tType,A3: A,P: A > $o] :
% 2.90/1.55 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 2.90/1.55 = ( P @ A3 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_46_Collect__mem__eq,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A] :
% 2.90/1.55 ( ( collect @ A
% 2.90/1.55 @ ^ [X5: A] : ( member @ A @ X5 @ A2 ) )
% 2.90/1.55 = A2 ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_47_Collect__cong,axiom,
% 2.90/1.55 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.90/1.55 ( ! [X2: A] :
% 2.90/1.55 ( ( P @ X2 )
% 2.90/1.55 = ( Q @ X2 ) )
% 2.90/1.55 => ( ( collect @ A @ P )
% 2.90/1.55 = ( collect @ A @ Q ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_48_ext,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.90/1.55 ( ! [X2: A] :
% 2.90/1.55 ( ( F @ X2 )
% 2.90/1.55 = ( G @ X2 ) )
% 2.90/1.55 => ( F = G ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_49_top_Oextremum__uniqueI,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order_top @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 2.90/1.55 => ( A3
% 2.90/1.55 = ( top_top @ A ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_50_dual__order_Oantisym,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [B3: A,A3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.90/1.55 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.55 => ( A3 = B3 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_51_dual__order_Otrans,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [B3: A,A3: A,C: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.90/1.55 => ( ( ord_less_eq @ A @ C @ B3 )
% 2.90/1.55 => ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_52_linorder__wlog,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linorder @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [P: A > A > $o,A3: A,B3: A] :
% 2.90/1.55 ( ! [A5: A,B4: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.90/1.55 => ( P @ A5 @ B4 ) )
% 2.90/1.55 => ( ! [A5: A,B4: A] :
% 2.90/1.55 ( ( P @ B4 @ A5 )
% 2.90/1.55 => ( P @ A5 @ B4 ) )
% 2.90/1.55 => ( P @ A3 @ B3 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_53_dual__order_Orefl,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_54_order__trans,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( preorder @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A,Y: A,Z3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.55 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 2.90/1.55 => ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_55_order__class_Oorder_Oantisym,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A,B3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.55 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.90/1.55 => ( A3 = B3 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_56_ord__le__eq__trans,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( ord @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A,B3: A,C: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.55 => ( ( B3 = C )
% 2.90/1.55 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_57_ord__eq__le__trans,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( ord @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A,B3: A,C: A] :
% 2.90/1.55 ( ( A3 = B3 )
% 2.90/1.55 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.90/1.55 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_58_antisym__conv,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [Y: A,X: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.55 => ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.55 = ( X = Y ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_59_le__cases3,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linorder @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A,Y: A,Z3: A] :
% 2.90/1.55 ( ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.55 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 2.90/1.55 => ( ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.55 => ~ ( ord_less_eq @ A @ X @ Z3 ) )
% 2.90/1.55 => ( ( ( ord_less_eq @ A @ X @ Z3 )
% 2.90/1.55 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 2.90/1.55 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 2.90/1.55 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 2.90/1.55 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 2.90/1.55 => ~ ( ord_less_eq @ A @ Z3 @ X ) )
% 2.90/1.55 => ~ ( ( ord_less_eq @ A @ Z3 @ X )
% 2.90/1.55 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_60_order_Otrans,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A,B3: A,C: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.55 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.90/1.55 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_61_le__cases,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linorder @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A,Y: A] :
% 2.90/1.55 ( ~ ( ord_less_eq @ A @ X @ Y )
% 2.90/1.55 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_62_eq__refl,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( preorder @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A,Y: A] :
% 2.90/1.55 ( ( X = Y )
% 2.90/1.55 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_63_linear,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linorder @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A,Y: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.55 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_64_antisym,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A,Y: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.55 => ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.55 => ( X = Y ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_65_eq__iff,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.90/1.55 = ( ^ [X5: A,Y3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 2.90/1.55 & ( ord_less_eq @ A @ Y3 @ X5 ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_66_ord__le__eq__subst,axiom,
% 2.90/1.55 ! [A: $tType,B: $tType] :
% 2.90/1.55 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.55 & ( ord @ A @ ( type2 @ A ) ) )
% 2.90/1.55 => ! [A3: A,B3: A,F: A > B,C: B] :
% 2.90/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.55 => ( ( ( F @ B3 )
% 2.90/1.55 = C )
% 2.90/1.55 => ( ! [X2: A,Y4: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 2.90/1.55 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.90/1.55 => ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_67_ord__eq__le__subst,axiom,
% 2.90/1.55 ! [A: $tType,B: $tType] :
% 2.90/1.55 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.55 & ( ord @ A @ ( type2 @ A ) ) )
% 2.90/1.55 => ! [A3: A,F: B > A,B3: B,C: B] :
% 2.90/1.55 ( ( A3
% 2.90/1.55 = ( F @ B3 ) )
% 2.90/1.55 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.90/1.55 => ( ! [X2: B,Y4: B] :
% 2.90/1.55 ( ( ord_less_eq @ B @ X2 @ Y4 )
% 2.90/1.55 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.90/1.55 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_68_order__subst2,axiom,
% 2.90/1.55 ! [A: $tType,C3: $tType] :
% 2.90/1.55 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 2.90/1.55 & ( order @ A @ ( type2 @ A ) ) )
% 2.90/1.55 => ! [A3: A,B3: A,F: A > C3,C: C3] :
% 2.90/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.55 => ( ( ord_less_eq @ C3 @ ( F @ B3 ) @ C )
% 2.90/1.55 => ( ! [X2: A,Y4: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 2.90/1.55 => ( ord_less_eq @ C3 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.90/1.55 => ( ord_less_eq @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_69_order__subst1,axiom,
% 2.90/1.55 ! [A: $tType,B: $tType] :
% 2.90/1.55 ( ( ( order @ B @ ( type2 @ B ) )
% 2.90/1.55 & ( order @ A @ ( type2 @ A ) ) )
% 2.90/1.55 => ! [A3: A,F: B > A,B3: B,C: B] :
% 2.90/1.55 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 2.90/1.55 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.90/1.55 => ( ! [X2: B,Y4: B] :
% 2.90/1.55 ( ( ord_less_eq @ B @ X2 @ Y4 )
% 2.90/1.55 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.90/1.55 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_70_rev__predicate2D,axiom,
% 2.90/1.55 ! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
% 2.90/1.55 ( ( P @ X @ Y )
% 2.90/1.55 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 2.90/1.55 => ( Q @ X @ Y ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_71_predicate2D,axiom,
% 2.90/1.55 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
% 2.90/1.55 ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 2.90/1.55 => ( ( P @ X @ Y )
% 2.90/1.55 => ( Q @ X @ Y ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_72_top__empty__eq,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( top_top @ ( A > $o ) )
% 2.90/1.55 = ( ^ [X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_73_top__set__def,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( top_top @ ( set @ A ) )
% 2.90/1.55 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_74_lfp_Oleq__antisym,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A,Y: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.55 => ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.55 => ( X = Y ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_75_gfp_Oleq__antisym,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [Y: A,X: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.55 => ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.55 => ( X = Y ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_76_lfp_Oleq__trans,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A,Y: A,Z3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.55 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 2.90/1.55 => ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_77_gfp_Oleq__trans,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [Y: A,X: A,Z3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.55 => ( ( ord_less_eq @ A @ Z3 @ Y )
% 2.90/1.55 => ( ord_less_eq @ A @ Z3 @ X ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_78_lfp_Oleq__refl,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_79_wlog__linorder__le,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linorder @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [P: A > A > $o,B3: A,A3: A] :
% 2.90/1.55 ( ! [A5: A,B4: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.90/1.55 => ( P @ A5 @ B4 ) )
% 2.90/1.55 => ( ( ( P @ B3 @ A3 )
% 2.90/1.55 => ( P @ A3 @ B3 ) )
% 2.90/1.55 => ( P @ A3 @ B3 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_80_UNIV__I,axiom,
% 2.90/1.55 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_81_UNIV__eq__I,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A] :
% 2.90/1.55 ( ! [X2: A] : ( member @ A @ X2 @ A2 )
% 2.90/1.55 => ( ( top_top @ ( set @ A ) )
% 2.90/1.55 = A2 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_82_subset__UNIV,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_83_UNIV__witness,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_84_iso__tuple__UNIV__I,axiom,
% 2.90/1.55 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_85_in__inv__imagep,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( inv_imagep @ A @ B )
% 2.90/1.55 = ( ^ [R4: A > A > $o,F2: B > A,X5: B,Y3: B] : ( R4 @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_86_eq__onp__to__Domainp,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,P: A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
% 2.90/1.55 ( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ T )
% 2.90/1.55 => ( ( domainp @ A @ B @ T )
% 2.90/1.55 = P ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_87_type__definition__Quotient__not__empty__witness,axiom,
% 2.90/1.55 ! [A: $tType,B: $tType,P: A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
% 2.90/1.55 ( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ T )
% 2.90/1.55 => ( P @ ( Rep @ ( undefined @ B ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_88_NEG__def,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( neg @ A @ B )
% 2.90/1.55 = ( ^ [A6: A > B > $o,B5: A > B > $o] : ( ord_less_eq @ ( A > B > $o ) @ B5 @ A6 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_89_antimonoD,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 & ( order @ B @ ( type2 @ B ) ) )
% 2.90/1.55 => ! [F: A > B,X: A,Y: A] :
% 2.90/1.55 ( ( order_antimono @ A @ B @ F )
% 2.90/1.55 => ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.55 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_90_subsetI,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A,B6: set @ A] :
% 2.90/1.55 ( ! [X2: A] :
% 2.90/1.55 ( ( member @ A @ X2 @ A2 )
% 2.90/1.55 => ( member @ A @ X2 @ B6 ) )
% 2.90/1.55 => ( ord_less_eq @ ( set @ A ) @ A2 @ B6 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_91_subset__antisym,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A,B6: set @ A] :
% 2.90/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 2.90/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A2 )
% 2.90/1.55 => ( A2 = B6 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_92_set__mp,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A,B6: set @ A,X: A] :
% 2.90/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 2.90/1.55 => ( ( member @ A @ X @ A2 )
% 2.90/1.55 => ( member @ A @ X @ B6 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_93_in__mono,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A,B6: set @ A,X: A] :
% 2.90/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 2.90/1.55 => ( ( member @ A @ X @ A2 )
% 2.90/1.55 => ( member @ A @ X @ B6 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_94_subsetD,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A,B6: set @ A,C: A] :
% 2.90/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 2.90/1.55 => ( ( member @ A @ C @ A2 )
% 2.90/1.55 => ( member @ A @ C @ B6 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_95_subsetCE,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A,B6: set @ A,C: A] :
% 2.90/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 2.90/1.55 => ( ( member @ A @ C @ A2 )
% 2.90/1.55 => ( member @ A @ C @ B6 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_96_equalityE,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A,B6: set @ A] :
% 2.90/1.55 ( ( A2 = B6 )
% 2.90/1.55 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 2.90/1.55 => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_97_subset__eq,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 2.90/1.55 = ( ^ [A6: set @ A,B5: set @ A] :
% 2.90/1.55 ! [X5: A] :
% 2.90/1.55 ( ( member @ A @ X5 @ A6 )
% 2.90/1.55 => ( member @ A @ X5 @ B5 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_98_equalityD1,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A,B6: set @ A] :
% 2.90/1.55 ( ( A2 = B6 )
% 2.90/1.55 => ( ord_less_eq @ ( set @ A ) @ A2 @ B6 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_99_equalityD2,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A,B6: set @ A] :
% 2.90/1.55 ( ( A2 = B6 )
% 2.90/1.55 => ( ord_less_eq @ ( set @ A ) @ B6 @ A2 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_100_set__rev__mp,axiom,
% 2.90/1.55 ! [A: $tType,X: A,A2: set @ A,B6: set @ A] :
% 2.90/1.55 ( ( member @ A @ X @ A2 )
% 2.90/1.55 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 2.90/1.55 => ( member @ A @ X @ B6 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_101_subset__iff,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 2.90/1.55 = ( ^ [A6: set @ A,B5: set @ A] :
% 2.90/1.55 ! [T4: A] :
% 2.90/1.55 ( ( member @ A @ T4 @ A6 )
% 2.90/1.55 => ( member @ A @ T4 @ B5 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_102_rev__subsetD,axiom,
% 2.90/1.55 ! [A: $tType,C: A,A2: set @ A,B6: set @ A] :
% 2.90/1.55 ( ( member @ A @ C @ A2 )
% 2.90/1.55 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 2.90/1.55 => ( member @ A @ C @ B6 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_103_subset__refl,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_104_Collect__mono,axiom,
% 2.90/1.55 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.90/1.55 ( ! [X2: A] :
% 2.90/1.55 ( ( P @ X2 )
% 2.90/1.55 => ( Q @ X2 ) )
% 2.90/1.55 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_105_subset__trans,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A,B6: set @ A,C4: set @ A] :
% 2.90/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 2.90/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 2.90/1.55 => ( ord_less_eq @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_106_set__eq__subset,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( ^ [Y2: set @ A,Z: set @ A] : ( Y2 = Z ) )
% 2.90/1.55 = ( ^ [A6: set @ A,B5: set @ A] :
% 2.90/1.55 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
% 2.90/1.55 & ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_107_contra__subsetD,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A,B6: set @ A,C: A] :
% 2.90/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 2.90/1.55 => ( ~ ( member @ A @ C @ B6 )
% 2.90/1.55 => ~ ( member @ A @ C @ A2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_108_Collect__mono__iff,axiom,
% 2.90/1.55 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.90/1.55 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.90/1.55 = ( ! [X5: A] :
% 2.90/1.55 ( ( P @ X5 )
% 2.90/1.55 => ( Q @ X5 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_109_DomainpE,axiom,
% 2.90/1.55 ! [A: $tType,B: $tType,R2: A > B > $o,A3: A] :
% 2.90/1.55 ( ( domainp @ A @ B @ R2 @ A3 )
% 2.90/1.55 => ~ ! [B4: B] :
% 2.90/1.55 ~ ( R2 @ A3 @ B4 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_110_Domainp_Ocases,axiom,
% 2.90/1.55 ! [A: $tType,B: $tType,R2: A > B > $o,A3: A] :
% 2.90/1.55 ( ( domainp @ A @ B @ R2 @ A3 )
% 2.90/1.55 => ~ ! [B4: B] :
% 2.90/1.55 ~ ( R2 @ A3 @ B4 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_111_Domainp_Osimps,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( domainp @ A @ B )
% 2.90/1.55 = ( ^ [R4: A > B > $o,A4: A] :
% 2.90/1.55 ? [B2: A,C5: B] :
% 2.90/1.55 ( ( A4 = B2 )
% 2.90/1.55 & ( R4 @ B2 @ C5 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_112_Domainp_ODomainI,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,R2: A > B > $o,A3: A,B3: B] :
% 2.90/1.55 ( ( R2 @ A3 @ B3 )
% 2.90/1.55 => ( domainp @ A @ B @ R2 @ A3 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_113_Domainp_Oinducts,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,R2: A > B > $o,X: A,P: A > $o] :
% 2.90/1.55 ( ( domainp @ A @ B @ R2 @ X )
% 2.90/1.55 => ( ! [A5: A,B4: B] :
% 2.90/1.55 ( ( R2 @ A5 @ B4 )
% 2.90/1.55 => ( P @ A5 ) )
% 2.90/1.55 => ( P @ X ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_114_NEG__trans,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,A2: A > B > $o,B6: A > B > $o,C4: A > B > $o] :
% 2.90/1.55 ( ( neg @ A @ B @ A2 @ B6 )
% 2.90/1.55 => ( ( neg @ A @ B @ B6 @ C4 )
% 2.90/1.55 => ( neg @ A @ B @ A2 @ C4 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_115_Quotient__to__Domainp,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
% 2.90/1.55 ( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
% 2.90/1.55 => ( ( domainp @ A @ B @ T )
% 2.90/1.55 = ( ^ [X5: A] : ( R @ X5 @ X5 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_116_antimono__def,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 & ( order @ B @ ( type2 @ B ) ) )
% 2.90/1.55 => ( ( order_antimono @ A @ B )
% 2.90/1.55 = ( ^ [F2: A > B] :
% 2.90/1.55 ! [X5: A,Y3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 2.90/1.55 => ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X5 ) ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_117_antimonoI,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 & ( order @ B @ ( type2 @ B ) ) )
% 2.90/1.55 => ! [F: A > B] :
% 2.90/1.55 ( ! [X2: A,Y4: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 2.90/1.55 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X2 ) ) )
% 2.90/1.55 => ( order_antimono @ A @ B @ F ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_118_antimonoE,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 & ( order @ B @ ( type2 @ B ) ) )
% 2.90/1.55 => ! [F: A > B,X: A,Y: A] :
% 2.90/1.55 ( ( order_antimono @ A @ B @ F )
% 2.90/1.55 => ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.55 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_119_Domain__eq__top,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( domainp @ A @ A
% 2.90/1.55 @ ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.90/1.55 = ( top_top @ ( A > $o ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_120_Domainp__refl,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( domainp @ A @ B )
% 2.90/1.55 = ( domainp @ A @ B ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_121_Domainp__iff,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( domainp @ A @ B )
% 2.90/1.55 = ( ^ [T2: A > B > $o,X5: A] :
% 2.90/1.55 ( ^ [P2: B > $o] :
% 2.90/1.55 ? [X6: B] : ( P2 @ X6 )
% 2.90/1.55 @ ( T2 @ X5 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_122_decseqD,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [F: nat > A,I: nat,J: nat] :
% 2.90/1.55 ( ( order_antimono @ nat @ A @ F )
% 2.90/1.55 => ( ( ord_less_eq @ nat @ I @ J )
% 2.90/1.55 => ( ord_less_eq @ A @ ( F @ J ) @ ( F @ I ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_123_decseq__def,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 => ( ( order_antimono @ nat @ A )
% 2.90/1.55 = ( ^ [X7: nat > A] :
% 2.90/1.55 ! [M: nat,N: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ M @ N )
% 2.90/1.55 => ( ord_less_eq @ A @ ( X7 @ N ) @ ( X7 @ M ) ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_124_NEG__POS,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( neg @ A @ B )
% 2.90/1.55 = ( ^ [A6: A > B > $o,B5: A > B > $o] : ( pos @ A @ B @ B5 @ A6 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_125_POS__trans,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,A2: A > B > $o,B6: A > B > $o,C4: A > B > $o] :
% 2.90/1.55 ( ( pos @ A @ B @ A2 @ B6 )
% 2.90/1.55 => ( ( pos @ A @ B @ B6 @ C4 )
% 2.90/1.55 => ( pos @ A @ B @ A2 @ C4 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_126_POS__apply,axiom,
% 2.90/1.55 ! [A: $tType,B: $tType,R: A > B > $o,R5: A > B > $o,F: A,G: B] :
% 2.90/1.55 ( ( pos @ A @ B @ R @ R5 )
% 2.90/1.55 => ( ( R @ F @ G )
% 2.90/1.55 => ( R5 @ F @ G ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_127_POS__def,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( pos @ A @ B )
% 2.90/1.55 = ( ord_less_eq @ ( A > B > $o ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_128_POS__NEG,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( pos @ A @ B )
% 2.90/1.55 = ( ^ [A6: A > B > $o,B5: A > B > $o] : ( neg @ A @ B @ B5 @ A6 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_129_ex__has__least__nat,axiom,
% 2.90/1.55 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 2.90/1.55 ( ( P @ K )
% 2.90/1.55 => ? [X2: A] :
% 2.90/1.55 ( ( P @ X2 )
% 2.90/1.55 & ! [Y5: A] :
% 2.90/1.55 ( ( P @ Y5 )
% 2.90/1.55 => ( ord_less_eq @ nat @ ( M2 @ X2 ) @ ( M2 @ Y5 ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_130_nat__le__linear,axiom,
% 2.90/1.55 ! [M2: nat,N2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.90/1.55 | ( ord_less_eq @ nat @ N2 @ M2 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_131_le__antisym,axiom,
% 2.90/1.55 ! [M2: nat,N2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.90/1.55 => ( ( ord_less_eq @ nat @ N2 @ M2 )
% 2.90/1.55 => ( M2 = N2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_132_eq__imp__le,axiom,
% 2.90/1.55 ! [M2: nat,N2: nat] :
% 2.90/1.55 ( ( M2 = N2 )
% 2.90/1.55 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_133_le__refl,axiom,
% 2.90/1.55 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_134_le__trans,axiom,
% 2.90/1.55 ! [I: nat,J: nat,K: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ I @ J )
% 2.90/1.55 => ( ( ord_less_eq @ nat @ J @ K )
% 2.90/1.55 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_135_LeastMI2,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.55 => ! [P: A > $o,X: A,M2: A > B,Q: A > $o] :
% 2.90/1.55 ( ( P @ X )
% 2.90/1.55 => ( ! [Y4: A] :
% 2.90/1.55 ( ( P @ Y4 )
% 2.90/1.55 => ( ord_less_eq @ B @ ( M2 @ X ) @ ( M2 @ Y4 ) ) )
% 2.90/1.55 => ( ! [X2: A] :
% 2.90/1.55 ( ( P @ X2 )
% 2.90/1.55 => ( ! [Y5: A] :
% 2.90/1.55 ( ( P @ Y5 )
% 2.90/1.55 => ( ord_less_eq @ B @ ( M2 @ X2 ) @ ( M2 @ Y5 ) ) )
% 2.90/1.55 => ( Q @ X2 ) ) )
% 2.90/1.55 => ( Q @ ( hilbert_LeastM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_136_monoseq__def,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 => ( ( topological_monoseq @ A )
% 2.90/1.55 = ( ^ [X7: nat > A] :
% 2.90/1.55 ( ! [M: nat,N: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ M @ N )
% 2.90/1.55 => ( ord_less_eq @ A @ ( X7 @ M ) @ ( X7 @ N ) ) )
% 2.90/1.55 | ! [M: nat,N: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ M @ N )
% 2.90/1.55 => ( ord_less_eq @ A @ ( X7 @ N ) @ ( X7 @ M ) ) ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_137_monoI2,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X8: nat > A] :
% 2.90/1.55 ( ! [M3: nat,N3: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ M3 @ N3 )
% 2.90/1.55 => ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ M3 ) ) )
% 2.90/1.55 => ( topological_monoseq @ A @ X8 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_138_monoI1,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X8: nat > A] :
% 2.90/1.55 ( ! [M3: nat,N3: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ M3 @ N3 )
% 2.90/1.55 => ( ord_less_eq @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) )
% 2.90/1.55 => ( topological_monoseq @ A @ X8 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_139_decseq__imp__monoseq,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X8: nat > A] :
% 2.90/1.55 ( ( order_antimono @ nat @ A @ X8 )
% 2.90/1.55 => ( topological_monoseq @ A @ X8 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_140_LeastM__nat__le,axiom,
% 2.90/1.55 ! [A: $tType,P: A > $o,X: A,M2: A > nat] :
% 2.90/1.55 ( ( P @ X )
% 2.90/1.55 => ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ X ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_141_LeastM__nat__lemma,axiom,
% 2.90/1.55 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 2.90/1.55 ( ( P @ K )
% 2.90/1.55 => ( ( P @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) )
% 2.90/1.55 & ! [Y5: A] :
% 2.90/1.55 ( ( P @ Y5 )
% 2.90/1.55 => ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ Y5 ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_142_GreatestMI2,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType] :
% 2.90/1.55 ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.55 => ! [P: A > $o,X: A,M2: A > B,Q: A > $o] :
% 2.90/1.55 ( ( P @ X )
% 2.90/1.55 => ( ! [Y4: A] :
% 2.90/1.55 ( ( P @ Y4 )
% 2.90/1.55 => ( ord_less_eq @ B @ ( M2 @ Y4 ) @ ( M2 @ X ) ) )
% 2.90/1.55 => ( ! [X2: A] :
% 2.90/1.55 ( ( P @ X2 )
% 2.90/1.55 => ( ! [Y5: A] :
% 2.90/1.55 ( ( P @ Y5 )
% 2.90/1.55 => ( ord_less_eq @ B @ ( M2 @ Y5 ) @ ( M2 @ X2 ) ) )
% 2.90/1.55 => ( Q @ X2 ) ) )
% 2.90/1.55 => ( Q @ ( hilbert_GreatestM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_143_totally__bounded__subset,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( topolo47006728_space @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [S2: set @ A,T: set @ A] :
% 2.90/1.55 ( ( topolo406746546ounded @ A @ S2 )
% 2.90/1.55 => ( ( ord_less_eq @ ( set @ A ) @ T @ S2 )
% 2.90/1.55 => ( topolo406746546ounded @ A @ T ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_144_Grp__mono,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,A2: set @ A,B6: set @ A,F: A > B] :
% 2.90/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 2.90/1.55 => ( ord_less_eq @ ( A > B > $o ) @ ( bNF_Grp @ A @ B @ A2 @ F ) @ ( bNF_Grp @ A @ B @ B6 @ F ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_145_of__nat__le__iff,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [M2: nat,N2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) )
% 2.90/1.55 = ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_146_GrpE,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,X: A,Y: B] :
% 2.90/1.55 ( ( bNF_Grp @ A @ B @ A2 @ F @ X @ Y )
% 2.90/1.55 => ~ ( ( ( F @ X )
% 2.90/1.55 = Y )
% 2.90/1.55 => ~ ( member @ A @ X @ A2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_147_GrpI,axiom,
% 2.90/1.55 ! [B: $tType,A: $tType,F: B > A,X: B,Y: A,A2: set @ B] :
% 2.90/1.55 ( ( ( F @ X )
% 2.90/1.55 = Y )
% 2.90/1.55 => ( ( member @ B @ X @ A2 )
% 2.90/1.55 => ( bNF_Grp @ B @ A @ A2 @ F @ X @ Y ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_148_real__arch__simple,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( archim1804426504_field @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A] :
% 2.90/1.55 ? [N3: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_149_zle__int,axiom,
% 2.90/1.55 ! [M2: nat,N2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N2 ) )
% 2.90/1.55 = ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_150_transfer__int__nat__relations_I3_J,axiom,
% 2.90/1.55 ! [X: nat,Y: nat] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y ) )
% 2.90/1.55 = ( ord_less_eq @ nat @ X @ Y ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_151_round__mono,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A,Y: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.55 => ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_152_nat__le__iff,axiom,
% 2.90/1.55 ! [X: int,N2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N2 )
% 2.90/1.55 = ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_153_negative__zle,axiom,
% 2.90/1.55 ! [N2: nat,M2: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( semiring_1_of_nat @ int @ M2 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_154_nat__mono,axiom,
% 2.90/1.55 ! [X: int,Y: int] :
% 2.90/1.55 ( ( ord_less_eq @ int @ X @ Y )
% 2.90/1.55 => ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_155_compl__le__compl__iff,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A,Y: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
% 2.90/1.55 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_156_neg__le__iff__le,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [B3: A,A3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) )
% 2.90/1.55 = ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_157_le__imp__neg__le,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A,B3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.55 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_158_Compl__anti__mono,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A,B6: set @ A] :
% 2.90/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
% 2.90/1.55 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_159_Compl__subset__Compl__iff,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A,B6: set @ A] :
% 2.90/1.55 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) )
% 2.90/1.55 = ( ord_less_eq @ ( set @ A ) @ B6 @ A2 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_160_compl__mono,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A,Y: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.55 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_161_compl__le__swap1,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [Y: A,X: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
% 2.90/1.55 => ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_162_compl__le__swap2,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [Y: A,X: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
% 2.90/1.55 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_163_le__minus__iff,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A,B3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
% 2.90/1.55 = ( ord_less_eq @ A @ B3 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_164_minus__le__iff,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A,B3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
% 2.90/1.55 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ A3 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_165_le__nat__iff,axiom,
% 2.90/1.55 ! [K: int,N2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.90/1.55 => ( ( ord_less_eq @ nat @ N2 @ ( nat2 @ K ) )
% 2.90/1.55 = ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_166_int__nat__eq,axiom,
% 2.90/1.55 ! [Z3: int] :
% 2.90/1.55 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 2.90/1.55 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
% 2.90/1.55 = Z3 ) )
% 2.90/1.55 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 2.90/1.55 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
% 2.90/1.55 = ( zero_zero @ int ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_167_int__zle__neg,axiom,
% 2.90/1.55 ! [N2: nat,M2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M2 ) ) )
% 2.90/1.55 = ( ( N2
% 2.90/1.55 = ( zero_zero @ nat ) )
% 2.90/1.55 & ( M2
% 2.90/1.55 = ( zero_zero @ nat ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_168_ComplI,axiom,
% 2.90/1.55 ! [A: $tType,C: A,A2: set @ A] :
% 2.90/1.55 ( ~ ( member @ A @ C @ A2 )
% 2.90/1.55 => ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_169_Compl__iff,axiom,
% 2.90/1.55 ! [A: $tType,C: A,A2: set @ A] :
% 2.90/1.55 ( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.90/1.55 = ( ~ ( member @ A @ C @ A2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_170_Compl__eq__Compl__iff,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A,B6: set @ A] :
% 2.90/1.55 ( ( ( uminus_uminus @ ( set @ A ) @ A2 )
% 2.90/1.55 = ( uminus_uminus @ ( set @ A ) @ B6 ) )
% 2.90/1.55 = ( A2 = B6 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_171_le__zero__eq,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [N2: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
% 2.90/1.55 = ( N2
% 2.90/1.55 = ( zero_zero @ A ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_172_le0,axiom,
% 2.90/1.55 ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_173_neg__less__eq__nonneg,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 2.90/1.55 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_174_less__eq__neg__nonpos,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
% 2.90/1.55 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_175_neg__le__0__iff__le,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
% 2.90/1.55 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_176_neg__0__le__iff__le,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
% 2.90/1.55 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_177_of__nat__le__0__iff,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [M2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( zero_zero @ A ) )
% 2.90/1.55 = ( M2
% 2.90/1.55 = ( zero_zero @ nat ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_178_nat__0__iff,axiom,
% 2.90/1.55 ! [I: int] :
% 2.90/1.55 ( ( ( nat2 @ I )
% 2.90/1.55 = ( zero_zero @ nat ) )
% 2.90/1.55 = ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_179_nat__le__0,axiom,
% 2.90/1.55 ! [Z3: int] :
% 2.90/1.55 ( ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) )
% 2.90/1.55 => ( ( nat2 @ Z3 )
% 2.90/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_180_ComplD,axiom,
% 2.90/1.55 ! [A: $tType,C: A,A2: set @ A] :
% 2.90/1.55 ( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.90/1.55 => ~ ( member @ A @ C @ A2 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_181_double__complement,axiom,
% 2.90/1.55 ! [A: $tType,A2: set @ A] :
% 2.90/1.55 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.90/1.55 = A2 ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_182_zero__le,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_183_le__0__eq,axiom,
% 2.90/1.55 ! [N2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) )
% 2.90/1.55 = ( N2
% 2.90/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_184_less__eq__nat_Osimps_I1_J,axiom,
% 2.90/1.55 ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_185_less__eq__int__code_I1_J,axiom,
% 2.90/1.55 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_186_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
% 2.90/1.55 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_187_nat__eq__iff,axiom,
% 2.90/1.55 ! [W: int,M2: nat] :
% 2.90/1.55 ( ( ( nat2 @ W )
% 2.90/1.55 = M2 )
% 2.90/1.55 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 2.90/1.55 => ( W
% 2.90/1.55 = ( semiring_1_of_nat @ int @ M2 ) ) )
% 2.90/1.55 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 2.90/1.55 => ( M2
% 2.90/1.55 = ( zero_zero @ nat ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_188_nat__eq__iff2,axiom,
% 2.90/1.55 ! [M2: nat,W: int] :
% 2.90/1.55 ( ( M2
% 2.90/1.55 = ( nat2 @ W ) )
% 2.90/1.55 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 2.90/1.55 => ( W
% 2.90/1.55 = ( semiring_1_of_nat @ int @ M2 ) ) )
% 2.90/1.55 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
% 2.90/1.55 => ( M2
% 2.90/1.55 = ( zero_zero @ nat ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_189_of__nat__0__le__iff,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_190_transfer__nat__int__relations_I1_J,axiom,
% 2.90/1.55 ! [X: int,Y: int] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.90/1.55 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
% 2.90/1.55 => ( ( ( nat2 @ X )
% 2.90/1.55 = ( nat2 @ Y ) )
% 2.90/1.55 = ( X = Y ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_191_eq__nat__nat__iff,axiom,
% 2.90/1.55 ! [Z3: int,Z4: int] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 2.90/1.55 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z4 )
% 2.90/1.55 => ( ( ( nat2 @ Z3 )
% 2.90/1.55 = ( nat2 @ Z4 ) )
% 2.90/1.55 = ( Z3 = Z4 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_192_ex__nat,axiom,
% 2.90/1.55 ( ( ^ [P2: nat > $o] :
% 2.90/1.55 ? [X6: nat] : ( P2 @ X6 ) )
% 2.90/1.55 = ( ^ [P3: nat > $o] :
% 2.90/1.55 ? [X5: int] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 2.90/1.55 & ( P3 @ ( nat2 @ X5 ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_193_all__nat,axiom,
% 2.90/1.55 ( ( ^ [P2: nat > $o] :
% 2.90/1.55 ! [X6: nat] : ( P2 @ X6 ) )
% 2.90/1.55 = ( ^ [P3: nat > $o] :
% 2.90/1.55 ! [X5: int] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 2.90/1.55 => ( P3 @ ( nat2 @ X5 ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_194_transfer__int__nat__quantifiers_I2_J,axiom,
% 2.90/1.55 ! [P: int > $o] :
% 2.90/1.55 ( ( ? [X5: int] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 2.90/1.55 & ( P @ X5 ) ) )
% 2.90/1.55 = ( ? [X5: nat] : ( P @ ( semiring_1_of_nat @ int @ X5 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_195_transfer__int__nat__quantifiers_I1_J,axiom,
% 2.90/1.55 ! [P: int > $o] :
% 2.90/1.55 ( ( ! [X5: int] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 2.90/1.55 => ( P @ X5 ) ) )
% 2.90/1.55 = ( ! [X5: nat] : ( P @ ( semiring_1_of_nat @ int @ X5 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_196_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
% 2.90/1.55 ! [Z3: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ Z3 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_197_nonneg__eq__int,axiom,
% 2.90/1.55 ! [Z3: int] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 2.90/1.55 => ~ ! [M3: nat] :
% 2.90/1.55 ( Z3
% 2.90/1.55 != ( semiring_1_of_nat @ int @ M3 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_198_nonneg__int__cases,axiom,
% 2.90/1.55 ! [K: int] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.90/1.55 => ~ ! [N3: nat] :
% 2.90/1.55 ( K
% 2.90/1.55 != ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_199_zero__le__imp__eq__int,axiom,
% 2.90/1.55 ! [K: int] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.90/1.55 => ? [N3: nat] :
% 2.90/1.55 ( K
% 2.90/1.55 = ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_200_transfer__nat__int__relations_I3_J,axiom,
% 2.90/1.55 ! [X: int,Y: int] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.90/1.55 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
% 2.90/1.55 => ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) )
% 2.90/1.55 = ( ord_less_eq @ int @ X @ Y ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_201_int__eq__iff,axiom,
% 2.90/1.55 ! [M2: nat,Z3: int] :
% 2.90/1.55 ( ( ( semiring_1_of_nat @ int @ M2 )
% 2.90/1.55 = Z3 )
% 2.90/1.55 = ( ( M2
% 2.90/1.55 = ( nat2 @ Z3 ) )
% 2.90/1.55 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_202_nat__0__le,axiom,
% 2.90/1.55 ! [Z3: int] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
% 2.90/1.55 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
% 2.90/1.55 = Z3 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_203_negative__zle__0,axiom,
% 2.90/1.55 ! [N2: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( zero_zero @ int ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_204_transfer__morphism__nat__int,axiom,
% 2.90/1.55 nat_tr1645093318rphism @ int @ nat @ nat2 @ ( ord_less_eq @ int @ ( zero_zero @ int ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_205_imp__le__cong,axiom,
% 2.90/1.55 ! [X: int,X4: int,P: $o,P4: $o] :
% 2.90/1.55 ( ( X = X4 )
% 2.90/1.55 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 2.90/1.55 => ( P = P4 ) )
% 2.90/1.55 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.90/1.55 => P )
% 2.90/1.55 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 2.90/1.55 => P4 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_206_conj__le__cong,axiom,
% 2.90/1.55 ! [X: int,X4: int,P: $o,P4: $o] :
% 2.90/1.55 ( ( X = X4 )
% 2.90/1.55 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 2.90/1.55 => ( P = P4 ) )
% 2.90/1.55 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.90/1.55 & P )
% 2.90/1.55 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 2.90/1.55 & P4 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_207_le__numeral__extra_I3_J,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.90/1.55 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_208_int__eq__iff__numeral,axiom,
% 2.90/1.55 ! [M2: nat,V: num] :
% 2.90/1.55 ( ( ( semiring_1_of_nat @ int @ M2 )
% 2.90/1.55 = ( numeral_numeral @ int @ V ) )
% 2.90/1.55 = ( ( M2
% 2.90/1.55 = ( nat2 @ ( numeral_numeral @ int @ V ) ) )
% 2.90/1.55 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_209_numeral__le__iff,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [M2: num,N2: num] :
% 2.90/1.55 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N2 ) )
% 2.90/1.55 = ( ord_less_eq @ num @ M2 @ N2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_210_neg__numeral__le__iff,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [M2: num,N2: num] :
% 2.90/1.55 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
% 2.90/1.55 = ( ord_less_eq @ num @ N2 @ M2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_211_zero__le__numeral,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [N2: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_212_not__numeral__le__zero,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [N2: num] :
% 2.90/1.55 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( zero_zero @ A ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_213_not__zero__le__neg__numeral,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [N2: num] :
% 2.90/1.55 ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_214_neg__numeral__le__zero,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ ( zero_zero @ A ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_215_not__numeral__le__neg__numeral,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [M2: num,N2: num] :
% 2.90/1.55 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_216_neg__numeral__le__numeral,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [M2: num,N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_217_ceiling__le__neg__numeral,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A,V: num] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
% 2.90/1.55 = ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_218_neg__numeral__le__floor,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [V: num,X: A] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim2085082626_floor @ A @ X ) )
% 2.90/1.55 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_219_zero__le__floor,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim2085082626_floor @ A @ X ) )
% 2.90/1.55 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_220_ceiling__le__zero,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
% 2.90/1.55 = ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_221_numeral__le__floor,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [V: num,X: A] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archim2085082626_floor @ A @ X ) )
% 2.90/1.55 = ( ord_less_eq @ A @ ( numeral_numeral @ A @ V ) @ X ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_222_ceiling__le__numeral,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A,V: num] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
% 2.90/1.55 = ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_223_floor__mono,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A,Y: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.55 => ( ord_less_eq @ int @ ( archim2085082626_floor @ A @ X ) @ ( archim2085082626_floor @ A @ Y ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_224_floor__le__ceiling,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A] : ( ord_less_eq @ int @ ( archim2085082626_floor @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_225_ceiling__ge__round,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A] : ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_226_floor__le__round,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [X: A] : ( ord_less_eq @ int @ ( archim2085082626_floor @ A @ X ) @ ( archimedean_round @ A @ X ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_227_ceiling__mono,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( archim1727834104eiling @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [Y: A,X: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.55 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_228_nat__ceiling__le__eq,axiom,
% 2.90/1.55 ! [X: real,A3: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A3 )
% 2.90/1.55 = ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A3 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_229_numeral__le__real__of__nat__iff,axiom,
% 2.90/1.55 ! [N2: num,M2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N2 ) @ ( semiring_1_of_nat @ real @ M2 ) )
% 2.90/1.55 = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N2 ) @ M2 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_230_complete__real,axiom,
% 2.90/1.55 ! [S2: set @ real] :
% 2.90/1.55 ( ? [X3: real] : ( member @ real @ X3 @ S2 )
% 2.90/1.55 => ( ? [Z5: real] :
% 2.90/1.55 ! [X2: real] :
% 2.90/1.55 ( ( member @ real @ X2 @ S2 )
% 2.90/1.55 => ( ord_less_eq @ real @ X2 @ Z5 ) )
% 2.90/1.55 => ? [Y4: real] :
% 2.90/1.55 ( ! [X3: real] :
% 2.90/1.55 ( ( member @ real @ X3 @ S2 )
% 2.90/1.55 => ( ord_less_eq @ real @ X3 @ Y4 ) )
% 2.90/1.55 & ! [Z5: real] :
% 2.90/1.55 ( ! [X2: real] :
% 2.90/1.55 ( ( member @ real @ X2 @ S2 )
% 2.90/1.55 => ( ord_less_eq @ real @ X2 @ Z5 ) )
% 2.90/1.55 => ( ord_less_eq @ real @ Y4 @ Z5 ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_231_real__nat__ceiling__ge,axiom,
% 2.90/1.55 ! [X: real] : ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_232_nat__floor__neg,axiom,
% 2.90/1.55 ! [X: real] :
% 2.90/1.55 ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
% 2.90/1.55 => ( ( nat2 @ ( archim2085082626_floor @ real @ X ) )
% 2.90/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_233_le__nat__floor,axiom,
% 2.90/1.55 ! [X: nat,A3: real] :
% 2.90/1.55 ( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X ) @ A3 )
% 2.90/1.55 => ( ord_less_eq @ nat @ X @ ( nat2 @ ( archim2085082626_floor @ real @ A3 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_234_enat__ord__number_I1_J,axiom,
% 2.90/1.55 ! [M2: num,N2: num] :
% 2.90/1.55 ( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( numeral_numeral @ extended_enat @ N2 ) )
% 2.90/1.55 = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_235_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
% 2.90/1.55 ! [X: int,Y: int] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.90/1.55 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
% 2.90/1.55 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( nat_tsub @ X @ Y ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_236_ile0__eq,axiom,
% 2.90/1.55 ! [N2: extended_enat] :
% 2.90/1.55 ( ( ord_less_eq @ extended_enat @ N2 @ ( zero_zero @ extended_enat ) )
% 2.90/1.55 = ( N2
% 2.90/1.55 = ( zero_zero @ extended_enat ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_237_i0__lb,axiom,
% 2.90/1.55 ! [N2: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_238_numeral__le__enat__iff,axiom,
% 2.90/1.55 ! [M2: num,N2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( extended_enat2 @ N2 ) )
% 2.90/1.55 = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M2 ) @ N2 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_239_Nat__Transfer_Otransfer__nat__int__functions_I3_J,axiom,
% 2.90/1.55 ! [X: int,Y: int] :
% 2.90/1.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.90/1.55 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
% 2.90/1.55 => ( ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) )
% 2.90/1.55 = ( nat2 @ ( nat_tsub @ X @ Y ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_240_diff__diff__cancel,axiom,
% 2.90/1.55 ! [I: nat,N2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ I @ N2 )
% 2.90/1.55 => ( ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ N2 @ I ) )
% 2.90/1.55 = I ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_241_diff__ge__0__iff__ge,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A,B3: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B3 ) )
% 2.90/1.55 = ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_242_diff__is__0__eq,axiom,
% 2.90/1.55 ! [M2: nat,N2: nat] :
% 2.90/1.55 ( ( ( minus_minus @ nat @ M2 @ N2 )
% 2.90/1.55 = ( zero_zero @ nat ) )
% 2.90/1.55 = ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_243_diff__is__0__eq_H,axiom,
% 2.90/1.55 ! [M2: nat,N2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.90/1.55 => ( ( minus_minus @ nat @ M2 @ N2 )
% 2.90/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_244_enat__ord__simps_I1_J,axiom,
% 2.90/1.55 ! [M2: nat,N2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N2 ) )
% 2.90/1.55 = ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_245_enat__ile,axiom,
% 2.90/1.55 ! [N2: extended_enat,M2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ extended_enat @ N2 @ ( extended_enat2 @ M2 ) )
% 2.90/1.55 => ? [K2: nat] :
% 2.90/1.55 ( N2
% 2.90/1.55 = ( extended_enat2 @ K2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_246_eq__diff__iff,axiom,
% 2.90/1.55 ! [K: nat,M2: nat,N2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ K @ M2 )
% 2.90/1.55 => ( ( ord_less_eq @ nat @ K @ N2 )
% 2.90/1.55 => ( ( ( minus_minus @ nat @ M2 @ K )
% 2.90/1.55 = ( minus_minus @ nat @ N2 @ K ) )
% 2.90/1.55 = ( M2 = N2 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_247_le__diff__iff,axiom,
% 2.90/1.55 ! [K: nat,M2: nat,N2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ K @ M2 )
% 2.90/1.55 => ( ( ord_less_eq @ nat @ K @ N2 )
% 2.90/1.55 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
% 2.90/1.55 = ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_248_Nat_Odiff__diff__eq,axiom,
% 2.90/1.55 ! [K: nat,M2: nat,N2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ K @ M2 )
% 2.90/1.55 => ( ( ord_less_eq @ nat @ K @ N2 )
% 2.90/1.55 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
% 2.90/1.55 = ( minus_minus @ nat @ M2 @ N2 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_249_diff__le__mono,axiom,
% 2.90/1.55 ! [M2: nat,N2: nat,L: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.90/1.55 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ L ) @ ( minus_minus @ nat @ N2 @ L ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_250_diff__le__self,axiom,
% 2.90/1.55 ! [M2: nat,N2: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ M2 ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_251_le__diff__iff_H,axiom,
% 2.90/1.55 ! [A3: nat,C: nat,B3: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ A3 @ C )
% 2.90/1.55 => ( ( ord_less_eq @ nat @ B3 @ C )
% 2.90/1.55 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A3 ) @ ( minus_minus @ nat @ C @ B3 ) )
% 2.90/1.55 = ( ord_less_eq @ nat @ B3 @ A3 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_252_diff__le__mono2,axiom,
% 2.90/1.55 ! [M2: nat,N2: nat,L: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.90/1.55 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_253_of__nat__diff,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( ring_1 @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [N2: nat,M2: nat] :
% 2.90/1.55 ( ( ord_less_eq @ nat @ N2 @ M2 )
% 2.90/1.55 => ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M2 @ N2 ) )
% 2.90/1.55 = ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_254_diff__eq__diff__less__eq,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A,B3: A,C: A,D2: A] :
% 2.90/1.55 ( ( ( minus_minus @ A @ A3 @ B3 )
% 2.90/1.55 = ( minus_minus @ A @ C @ D2 ) )
% 2.90/1.55 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.55 = ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(fact_255_diff__right__mono,axiom,
% 2.90/1.55 ! [A: $tType] :
% 2.90/1.55 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.55 => ! [A3: A,B3: A,C: A] :
% 2.90/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.55 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 2.90/1.55 ! [A7: $tType,A8: $tType] :
% 2.90/1.55 ( ( comple187826305attice @ A8 @ ( type2 @ A8 ) )
% 2.90/1.55 => ( comple187826305attice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 2.90/1.55 ! [A7: $tType,A8: $tType] :
% 2.90/1.55 ( ( boolean_algebra @ A8 @ ( type2 @ A8 ) )
% 2.90/1.55 => ( boolean_algebra @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 2.90/1.55 ! [A7: $tType,A8: $tType] :
% 2.90/1.55 ( ( order_top @ A8 @ ( type2 @ A8 ) )
% 2.90/1.55 => ( order_top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.90/1.55 ! [A7: $tType,A8: $tType] :
% 2.90/1.55 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 2.90/1.55 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.90/1.55 ! [A7: $tType,A8: $tType] :
% 2.90/1.55 ( ( order @ A8 @ ( type2 @ A8 ) )
% 2.90/1.55 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_fun___Orderings_Otop,axiom,
% 2.90/1.55 ! [A7: $tType,A8: $tType] :
% 2.90/1.55 ( ( top @ A8 @ ( type2 @ A8 ) )
% 2.90/1.55 => ( top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_fun___Orderings_Oord,axiom,
% 2.90/1.55 ! [A7: $tType,A8: $tType] :
% 2.90/1.55 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 2.90/1.55 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
% 2.90/1.55 linord219039673up_add @ int @ ( type2 @ int ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 2.90/1.55 ordered_ab_group_add @ int @ ( type2 @ int ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 2.90/1.55 linordered_semidom @ int @ ( type2 @ int ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 2.90/1.55 linordered_idom @ int @ ( type2 @ int ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 2.90/1.55 preorder @ int @ ( type2 @ int ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 2.90/1.55 linorder @ int @ ( type2 @ int ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 2.90/1.55 order @ int @ ( type2 @ int ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 2.90/1.55 ord @ int @ ( type2 @ int ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Int_Oint___Rings_Oring__1,axiom,
% 2.90/1.55 ring_1 @ int @ ( type2 @ int ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.90/1.55 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_4,axiom,
% 2.90/1.55 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Nat_Onat___Orderings_Opreorder_5,axiom,
% 2.90/1.55 preorder @ nat @ ( type2 @ nat ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Nat_Onat___Orderings_Olinorder_6,axiom,
% 2.90/1.55 linorder @ nat @ ( type2 @ nat ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Nat_Onat___Orderings_Oorder_7,axiom,
% 2.90/1.55 order @ nat @ ( type2 @ nat ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Nat_Onat___Orderings_Oord_8,axiom,
% 2.90/1.55 ord @ nat @ ( type2 @ nat ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Num_Onum___Orderings_Opreorder_9,axiom,
% 2.90/1.55 preorder @ num @ ( type2 @ num ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Num_Onum___Orderings_Olinorder_10,axiom,
% 2.90/1.55 linorder @ num @ ( type2 @ num ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Num_Onum___Orderings_Oorder_11,axiom,
% 2.90/1.55 order @ num @ ( type2 @ num ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Num_Onum___Orderings_Oord_12,axiom,
% 2.90/1.55 ord @ num @ ( type2 @ num ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_13,axiom,
% 2.90/1.55 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_14,axiom,
% 2.90/1.55 ! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Set_Oset___Orderings_Oorder__top_15,axiom,
% 2.90/1.55 ! [A7: $tType] : ( order_top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Set_Oset___Orderings_Opreorder_16,axiom,
% 2.90/1.55 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Set_Oset___Orderings_Oorder_17,axiom,
% 2.90/1.55 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Set_Oset___Orderings_Otop_18,axiom,
% 2.90/1.55 ! [A7: $tType] : ( top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Set_Oset___Orderings_Oord_19,axiom,
% 2.90/1.55 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_20,axiom,
% 2.90/1.55 comple187826305attice @ $o @ ( type2 @ $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_21,axiom,
% 2.90/1.55 boolean_algebra @ $o @ ( type2 @ $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_HOL_Obool___Orderings_Oorder__top_22,axiom,
% 2.90/1.55 order_top @ $o @ ( type2 @ $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_HOL_Obool___Orderings_Opreorder_23,axiom,
% 2.90/1.55 preorder @ $o @ ( type2 @ $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_HOL_Obool___Orderings_Olinorder_24,axiom,
% 2.90/1.55 linorder @ $o @ ( type2 @ $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_HOL_Obool___Orderings_Oorder_25,axiom,
% 2.90/1.55 order @ $o @ ( type2 @ $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_HOL_Obool___Orderings_Otop_26,axiom,
% 2.90/1.55 top @ $o @ ( type2 @ $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_HOL_Obool___Orderings_Oord_27,axiom,
% 2.90/1.55 ord @ $o @ ( type2 @ $o ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
% 2.90/1.55 archim1804426504_field @ real @ ( type2 @ real ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
% 2.90/1.55 topolo47006728_space @ real @ ( type2 @ real ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling,axiom,
% 2.90/1.55 archim1727834104eiling @ real @ ( type2 @ real ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_28,axiom,
% 2.90/1.55 linord219039673up_add @ real @ ( type2 @ real ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_29,axiom,
% 2.90/1.55 ordered_ab_group_add @ real @ ( type2 @ real ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_30,axiom,
% 2.90/1.55 linordered_semidom @ real @ ( type2 @ real ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Real_Oreal___Rings_Olinordered__idom_31,axiom,
% 2.90/1.55 linordered_idom @ real @ ( type2 @ real ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Real_Oreal___Orderings_Opreorder_32,axiom,
% 2.90/1.55 preorder @ real @ ( type2 @ real ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Real_Oreal___Orderings_Olinorder_33,axiom,
% 2.90/1.55 linorder @ real @ ( type2 @ real ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Real_Oreal___Orderings_Oorder_34,axiom,
% 2.90/1.55 order @ real @ ( type2 @ real ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Real_Oreal___Orderings_Oord_35,axiom,
% 2.90/1.55 ord @ real @ ( type2 @ real ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Real_Oreal___Rings_Oring__1_36,axiom,
% 2.90/1.55 ring_1 @ real @ ( type2 @ real ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_37,axiom,
% 2.90/1.55 canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_38,axiom,
% 2.90/1.55 comple187826305attice @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_39,axiom,
% 2.90/1.55 order_top @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_40,axiom,
% 2.90/1.55 preorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_41,axiom,
% 2.90/1.55 linorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_42,axiom,
% 2.90/1.55 order @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Extended__Nat_Oenat___Orderings_Otop_43,axiom,
% 2.90/1.55 top @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.90/1.55
% 2.90/1.55 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_44,axiom,
% 2.90/1.55 ord @ extended_enat @ ( type2 @ extended_enat ) ).
% 2.90/1.55
% 2.90/1.55 thf(conj_0,hypothesis,
% 2.90/1.55 p @ x ).
% 2.90/1.55
% 2.90/1.55 thf(conj_1,conjecture,
% 2.90/1.55 bNF_eq_onp @ a @ p @ x @ x ).
% 2.90/1.55 ------------------------
% 2.90/1.57 % (19460)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_19250 for (2999ds/275Mi)
% 2.90/1.57 % (19461)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_19250 for (2999ds/18Mi)
% 2.90/1.57 % (19456)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_19250 for (2999ds/4Mi)
% 2.90/1.57 % (19456)Instruction limit reached!
% 2.90/1.57 % (19456)------------------------------
% 2.90/1.57 % (19456)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.57 % (19456)Termination reason: Unknown
% 2.90/1.57 % (19456)Termination phase: shuffling
% 2.90/1.57
% 2.90/1.57 % (19456)Memory used [KB]: 1407
% 2.90/1.57 % (19456)Time elapsed: 0.003 s
% 2.90/1.57 % (19456)Instructions burned: 5 (million)
% 2.90/1.57 % (19456)------------------------------
% 2.90/1.57 % (19456)------------------------------
% 2.90/1.58 % (19461)Instruction limit reached!
% 2.90/1.58 % (19461)------------------------------
% 2.90/1.58 % (19461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.58 % (19461)Termination reason: Unknown
% 2.90/1.58 % (19461)Termination phase: shuffling
% 2.90/1.58
% 2.90/1.58 % (19461)Memory used [KB]: 1791
% 2.90/1.58 % (19461)Time elapsed: 0.008 s
% 2.90/1.58 % (19461)Instructions burned: 19 (million)
% 2.90/1.58 % (19461)------------------------------
% 2.90/1.58 % (19461)------------------------------
% 2.90/1.58 % (19455)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_19250 for (2999ds/183Mi)
% 2.90/1.58 % (19457)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_19250 for (2999ds/27Mi)
% 2.90/1.58 % (19459)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_19250 for (2999ds/2Mi)
% 2.90/1.58 % (19458)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_19250 for (2999ds/2Mi)
% 2.90/1.58 % (19458)Instruction limit reached!
% 2.90/1.58 % (19458)------------------------------
% 2.90/1.58 % (19458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.58 % (19458)Termination reason: Unknown
% 2.90/1.58 % (19458)Termination phase: shuffling
% 2.90/1.58
% 2.90/1.58 % (19458)Memory used [KB]: 1407
% 2.90/1.58 % (19458)Time elapsed: 0.004 s
% 2.90/1.58 % (19458)Instructions burned: 3 (million)
% 2.90/1.58 % (19458)------------------------------
% 2.90/1.58 % (19458)------------------------------
% 2.90/1.58 % (19459)Instruction limit reached!
% 2.90/1.58 % (19459)------------------------------
% 2.90/1.58 % (19459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.58 % (19459)Termination reason: Unknown
% 2.90/1.58 % (19459)Termination phase: shuffling
% 2.90/1.58
% 2.90/1.58 % (19459)Memory used [KB]: 1407
% 2.90/1.58 % (19459)Time elapsed: 0.004 s
% 2.90/1.58 % (19459)Instructions burned: 3 (million)
% 2.90/1.58 % (19459)------------------------------
% 2.90/1.58 % (19459)------------------------------
% 2.90/1.58 % (19462)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_19250 for (2999ds/3Mi)
% 2.90/1.58 % (19462)Instruction limit reached!
% 2.90/1.58 % (19462)------------------------------
% 2.90/1.58 % (19462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.58 % (19462)Termination reason: Unknown
% 2.90/1.58 % (19462)Termination phase: shuffling
% 2.90/1.58
% 2.90/1.58 % (19462)Memory used [KB]: 1407
% 2.90/1.58 % (19462)Time elapsed: 0.003 s
% 2.90/1.58 % (19462)Instructions burned: 4 (million)
% 2.90/1.58 % (19462)------------------------------
% 2.90/1.58 % (19462)------------------------------
% 2.90/1.59 % (19460)First to succeed.
% 2.90/1.59 % (19463)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_19250 for (2999ds/37Mi)
% 2.90/1.59 % (19460)Refutation found. Thanks to Tanya!
% 2.90/1.59 % SZS status Theorem for DTF2THF_19250
% 2.90/1.59 % SZS output start Proof for DTF2THF_19250
% 2.90/1.59 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.90/1.59 thf(type_def_7, type, int: $tType).
% 2.90/1.59 thf(type_def_8, type, set: ($tType) > $tType).
% 2.90/1.59 thf(type_def_9, type, nat: $tType).
% 2.90/1.59 thf(type_def_10, type, extended_enat: $tType).
% 2.90/1.59 thf(type_def_11, type, num: $tType).
% 2.90/1.59 thf(type_def_12, type, a: $tType).
% 2.90/1.59 thf(type_def_13, type, real: $tType).
% 2.90/1.59 thf(func_def_0, type, extended_enat: $tType).
% 2.90/1.59 thf(func_def_1, type, real: $tType).
% 2.90/1.59 thf(func_def_2, type, set: ($tType) > $tType).
% 2.90/1.59 thf(func_def_3, type, num: $tType).
% 2.90/1.59 thf(func_def_4, type, nat: $tType).
% 2.90/1.59 thf(func_def_5, type, int: $tType).
% 2.90/1.59 thf(func_def_6, type, itself: ($tType) > $tType).
% 2.90/1.59 thf(func_def_7, type, a: $tType).
% 2.90/1.59 thf(func_def_8, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_9, type, ring_1: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_10, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_11, type, top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_12, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_13, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_14, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_15, type, order_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_16, type, linordered_idom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_17, type, boolean_algebra: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_18, type, linordered_semidom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_19, type, ordered_ab_group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_20, type, linord219039673up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_21, type, archim1727834104eiling: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_22, type, topolo47006728_space: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_23, type, comple187826305attice: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_24, type, archim1804426504_field: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_25, type, canoni770627133id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.59 thf(func_def_26, type, archimedean_ceiling: !>[X0: $tType]:(X0 > int)).
% 2.90/1.59 thf(func_def_27, type, archim2085082626_floor: !>[X0: $tType]:(X0 > int)).
% 2.90/1.59 thf(func_def_28, type, archimedean_round: !>[X0: $tType]:(X0 > int)).
% 2.90/1.59 thf(func_def_29, type, bNF_pred_DEADID: !>[X0: $tType]:(X0 > $o)).
% 2.90/1.59 thf(func_def_30, type, bNF_Grp: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X0 > X1 > $o)).
% 2.90/1.59 thf(func_def_31, type, bNF_eq_onp: !>[X0: $tType]:((X0 > $o) > X0 > X0 > $o)).
% 2.90/1.59 thf(func_def_32, type, extended_enat2: nat > extended_enat).
% 2.90/1.59 thf(func_def_33, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.90/1.59 thf(func_def_34, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 2.90/1.59 thf(func_def_35, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.90/1.59 thf(func_def_36, type, undefined: !>[X0: $tType]:(X0)).
% 2.90/1.59 thf(func_def_37, type, hilbert_GreatestM: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > X0)).
% 2.90/1.59 thf(func_def_38, type, hilbert_LeastM: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > X0)).
% 2.90/1.59 thf(func_def_39, type, nat2: int > nat).
% 2.90/1.59 thf(func_def_40, type, neg: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1 > $o) > $o)).
% 2.90/1.59 thf(func_def_41, type, pos: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > X1 > $o) > $o)).
% 2.90/1.59 thf(func_def_42, type, quotient: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > (X1 > X0) > (X0 > X1 > $o) > $o)).
% 2.90/1.59 thf(func_def_43, type, semiring_1_of_nat: !>[X0: $tType]:(nat > X0)).
% 2.90/1.59 thf(func_def_44, type, nat_tr1645093318rphism: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > $o)).
% 2.90/1.59 thf(func_def_45, type, nat_tsub: int > int > int).
% 2.90/1.59 thf(func_def_46, type, numeral_numeral: !>[X0: $tType]:(num > X0)).
% 2.90/1.59 thf(func_def_47, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.90/1.59 thf(func_def_48, type, order_antimono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 2.90/1.59 thf(func_def_49, type, top_top: !>[X0: $tType]:(X0)).
% 2.90/1.59 thf(func_def_50, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.90/1.59 thf(func_def_51, type, domainp: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X0 > $o)).
% 2.90/1.59 thf(func_def_52, type, inv_imagep: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > X1 > X1 > $o)).
% 2.90/1.59 thf(func_def_53, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.90/1.59 thf(func_def_54, type, topological_monoseq: !>[X0: $tType]:((nat > X0) > $o)).
% 2.90/1.59 thf(func_def_55, type, topolo406746546ounded: !>[X0: $tType]:(set @ X0 > $o)).
% 2.90/1.59 thf(func_def_56, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.90/1.59 thf(func_def_57, type, p: a > $o).
% 2.90/1.59 thf(func_def_58, type, x: a).
% 2.90/1.59 thf(func_def_75, type, sK0: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > set @ X0 > X0)).
% 2.90/1.59 thf(func_def_76, type, sK1: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.90/1.59 thf(func_def_77, type, sK2: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.90/1.59 thf(func_def_78, type, sK3: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0 > $o) > X1)).
% 2.90/1.59 thf(func_def_79, type, sK4: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0 > $o) > X0)).
% 2.90/1.59 thf(func_def_80, type, sK5: !>[X0: $tType]:(set @ X0 > X0)).
% 2.90/1.59 thf(func_def_81, type, sK6: !>[X0: $tType]:(X0)).
% 2.90/1.59 thf(func_def_82, type, sK7: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.90/1.59 thf(func_def_83, type, sK8: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.90/1.59 thf(func_def_84, type, sK9: !>[X0: $tType, X1: $tType]:((X1 > $o) > (X0 > X1) > (X0 > X0 > $o) > X0)).
% 2.90/1.59 thf(func_def_86, type, ph11: !>[X0: $tType]:(X0)).
% 2.90/1.59 thf(f1135,plain,(
% 2.90/1.59 $false),
% 2.90/1.59 inference(subsumption_resolution,[status(thm)],[f1134,f1027])).
% 2.90/1.59 thf(f1027,plain,(
% 2.90/1.59 ((p @ x) = $true)),
% 2.90/1.59 inference(cnf_transformation,[status(thm)],[f699])).
% 2.90/1.59 thf(f699,plain,(
% 2.90/1.59 ((p @ x) = $true)),
% 2.90/1.59 inference(fool_elimination,[status(thm)],[f698])).
% 2.90/1.59 thf(f698,plain,(
% 2.90/1.59 (p @ x)),
% 2.90/1.59 inference(rectify,[status(thm)],[f318])).
% 2.90/1.59 thf(f318,axiom,(
% 2.90/1.59 (p @ x)),
% 2.90/1.59 file('/export/starexec/sandbox2/tmp/tmp.7l7JJvTzmF/DTF2THF_19250.p',conj_0)).
% 2.90/1.59 thf(f1134,plain,(
% 2.90/1.59 ((p @ x) != $true)),
% 2.90/1.59 inference(trivial_inequality_removal,[status(thm)],[f1131])).
% 2.90/1.59 thf(f1131,plain,(
% 2.90/1.59 ((p @ x) != $true) | ($true != $true)),
% 2.90/1.59 inference(superposition,[status(thm)],[f1084,f1116])).
% 2.90/1.59 thf(f1116,plain,(
% 2.90/1.59 ( ! [X0 : $tType,X3 : X0 > $o,X4 : X0] : (($true = (bNF_eq_onp @ X0 @ X3 @ X4 @ X4)) | ((X3 @ X4) != $true)) )),
% 2.90/1.59 inference(equality_resolution,[status(thm)],[f1115])).
% 2.90/1.59 thf(f1115,plain,(
% 2.90/1.59 ( ! [X0 : $tType,X3 : X0 > $o,X1 : $o,X4 : X0] : (($true = (bNF_eq_onp @ X0 @ X3 @ X4 @ X4)) | ((X3 @ X4) != $true) | ($true != X1)) )),
% 2.90/1.59 inference(equality_resolution,[status(thm)],[f1061])).
% 2.90/1.59 thf(f1061,plain,(
% 2.90/1.59 ( ! [X0 : $tType,X2 : $o,X3 : X0 > $o,X1 : $o,X4 : X0] : (($true = (bNF_eq_onp @ X0 @ X3 @ X4 @ X4)) | ($true != X2) | ((X3 @ X4) != $true) | (X1 != X2)) )),
% 2.90/1.59 inference(cnf_transformation,[status(thm)],[f1008])).
% 2.90/1.59 thf(f1008,plain,(
% 2.90/1.59 ! [X0 : $tType,X1 : $o,X2 : $o,X3 : X0 > $o,X4 : X0] : ((((($true = X2) & ((X3 @ X4) = $true)) | ($true != X1) | ($true != (bNF_eq_onp @ X0 @ X3 @ X4 @ X4))) & ((($true = X1) & ($true = (bNF_eq_onp @ X0 @ X3 @ X4 @ X4))) | ($true != X2) | ((X3 @ X4) != $true))) | (X1 != X2))),
% 2.90/1.59 inference(flattening,[status(thm)],[f1007])).
% 2.90/1.59 thf(f1007,plain,(
% 2.90/1.59 ! [X0 : $tType,X1 : $o,X2 : $o,X3 : X0 > $o,X4 : X0] : ((((($true = X2) & ((X3 @ X4) = $true)) | (($true != X1) | ($true != (bNF_eq_onp @ X0 @ X3 @ X4 @ X4)))) & ((($true = X1) & ($true = (bNF_eq_onp @ X0 @ X3 @ X4 @ X4))) | (($true != X2) | ((X3 @ X4) != $true)))) | (X1 != X2))),
% 2.90/1.59 inference(nnf_transformation,[status(thm)],[f979])).
% 2.90/1.59 thf(f979,plain,(
% 2.90/1.59 ! [X0 : $tType,X1 : $o,X2 : $o,X3 : X0 > $o,X4 : X0] : (((($true = X2) & ((X3 @ X4) = $true)) <=> (($true = X1) & ($true = (bNF_eq_onp @ X0 @ X3 @ X4 @ X4)))) | (X1 != X2))),
% 2.90/1.59 inference(ennf_transformation,[status(thm)],[f379])).
% 2.90/1.59 thf(f379,plain,(
% 2.90/1.59 ! [X0 : $tType,X1 : $o,X2 : $o,X3 : X0 > $o,X4 : X0] : ((X1 = X2) => ((($true = X2) & ((X3 @ X4) = $true)) <=> (($true = X1) & ($true = (bNF_eq_onp @ X0 @ X3 @ X4 @ X4)))))),
% 2.90/1.59 inference(fool_elimination,[status(thm)],[f378])).
% 2.90/1.59 thf(f378,plain,(
% 2.90/1.59 ! [X0 : $tType,X1 : $o,X2 : $o,X3 : X0 > $o,X4 : X0] : ((X1 = X2) => ((X3 @ X4) & X2 = X1 & (bNF_eq_onp @ X0 @ X3 @ X4 @ X4)))),
% 2.90/1.59 inference(rectify,[status(thm)],[f5])).
% 2.90/1.59 thf(f5,axiom,(
% 2.90/1.59 ! [X0 : $tType,X4 : $o,X9 : $o,X2 : X0 > $o,X10 : X0] : ((X4 = X9) => ((X2 @ X10) & X9 = X4 & (bNF_eq_onp @ X0 @ X2 @ X10 @ X10)))),
% 2.90/1.59 file('/export/starexec/sandbox2/tmp/tmp.7l7JJvTzmF/DTF2THF_19250.p',fact_4_eq__onp__live__step)).
% 2.90/1.59 thf(f1084,plain,(
% 2.90/1.59 ((bNF_eq_onp @ a @ p @ x @ x) != $true)),
% 2.90/1.59 inference(cnf_transformation,[status(thm)],[f922])).
% 2.90/1.59 thf(f922,plain,(
% 2.90/1.59 ((bNF_eq_onp @ a @ p @ x @ x) != $true)),
% 2.90/1.59 inference(flattening,[status(thm)],[f693])).
% 2.90/1.59 thf(f693,plain,(
% 2.90/1.59 ~((bNF_eq_onp @ a @ p @ x @ x) = $true)),
% 2.90/1.59 inference(fool_elimination,[status(thm)],[f692])).
% 2.90/1.59 thf(f692,plain,(
% 2.90/1.59 ~(bNF_eq_onp @ a @ p @ x @ x)),
% 2.90/1.59 inference(rectify,[status(thm)],[f320])).
% 2.90/1.59 thf(f320,negated_conjecture,(
% 2.90/1.59 ~(bNF_eq_onp @ a @ p @ x @ x)),
% 2.90/1.59 inference(negated_conjecture,[status(cth)],[f319])).
% 2.90/1.59 thf(f319,conjecture,(
% 2.90/1.59 (bNF_eq_onp @ a @ p @ x @ x)),
% 2.90/1.59 file('/export/starexec/sandbox2/tmp/tmp.7l7JJvTzmF/DTF2THF_19250.p',conj_1)).
% 2.90/1.59 % SZS output end Proof for DTF2THF_19250
% 2.90/1.59 % (19460)------------------------------
% 2.90/1.59 % (19460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.59 % (19460)Termination reason: Refutation
% 2.90/1.59
% 2.90/1.59 % (19460)Memory used [KB]: 6524
% 2.90/1.59 % (19460)Time elapsed: 0.019 s
% 2.90/1.59 % (19460)Instructions burned: 50 (million)
% 2.90/1.59 % (19460)------------------------------
% 2.90/1.59 % (19460)------------------------------
% 2.90/1.59 % (19454)Success in time 0.035 s
%------------------------------------------------------------------------------