%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP072^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n020.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 08:55:19 AM UTC 2026
% Result : Theorem 2.71s 1.67s
% Output : Refutation 2.71s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP072^2 : TPTP v9.2.1. Released v7.5.0.
% 0.13/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n020.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:25:12 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.47 thf(spec,logic,$$dhol).
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 % File : ITP072^2 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.47 % Domain : Interactive Theorem Proving
% 0.19/0.47 % Problem : Sledgehammer HF problem prob_113__5324706_1
% 0.19/0.47 % Version : Especial.
% 0.19/0.47 % English :
% 0.19/0.47
% 0.19/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.47 % Source : [Des21]
% 0.19/0.47 % Names : HF/prob_113__5324706_1 [Des21]
% 0.19/0.47
% 0.19/0.47 % Status : Theorem
% 0.19/0.47 % Rating : 0.33 v9.1.0, 0.00 v7.5.0
% 0.19/0.47 % Syntax : Number of formulae : 340 ( 117 unt; 50 typ; 0 def)
% 0.19/0.47 % Number of atoms : 835 ( 258 equ; 0 cnn)
% 0.19/0.47 % Maximal formula atoms : 13 ( 2 avg)
% 0.19/0.47 % Number of connectives : 3500 ( 86 ~; 9 |; 46 &;2951 @)
% 0.19/0.47 % ( 0 <=>; 408 =>; 0 <=; 0 <~>)
% 0.19/0.47 % Maximal formula depth : 20 ( 8 avg)
% 0.19/0.47 % Number of types : 3 ( 2 usr)
% 0.19/0.47 % Number of type conns : 155 ( 155 >; 0 *; 0 +; 0 <<)
% 0.19/0.47 % Number of symbols : 50 ( 48 usr; 2 con; 0-6 aty)
% 0.19/0.47 % Number of variables : 991 ( 48 ^; 875 !; 23 ?; 991 :)
% 0.19/0.47 % ( 45 !>; 0 ?*; 0 @-; 0 @+)
% 0.19/0.47 % SPC : TH1_THM_EQU_NAR
% 0.19/0.47
% 0.19/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:21:13.623
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 % Could-be-implicit typings (3)
% 0.19/0.47 thf(ty_t_HF__Mirabelle__fsbjehakzm_Ohf,type,
% 0.19/0.47 hF_Mirabelle_hf: $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_Set_Oset,type,
% 0.19/0.47 set: $tType > $tType ).
% 0.19/0.47
% 0.19/0.47 thf(ty_t_Nat_Onat,type,
% 0.19/0.47 nat: $tType ).
% 0.19/0.47
% 0.19/0.47 % Explicit typings (47)
% 0.19/0.47 thf(sy_cl_HOL_Otype,type,
% 0.19/0.47 type:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Groups_Ozero,type,
% 0.19/0.47 zero:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Orderings_Obot,type,
% 0.19/0.47 bot:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.19/0.47 ord:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Num_Oneg__numeral,type,
% 0.19/0.47 neg_numeral:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.19/0.47 order:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Groups_Ogroup__add,type,
% 0.19/0.47 group_add:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Rings_Osemiring__1,type,
% 0.19/0.47 semiring_1:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.19/0.47 finite_finite:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.19/0.47 linorder:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.19/0.47 preorder:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.19/0.47 order_bot:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.19/0.47 comm_monoid_diff:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Parity_Osemiring__bit__shifts,type,
% 0.19/0.47 semiring_bit_shifts:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.19/0.47 ordered_ab_group_add:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.19/0.47 cancel1352612707id_add:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.19/0.47 cancel146912293up_add:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.19/0.47 linord1659791738miring:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.19/0.47 canoni770627133id_add:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_cl_Parity_Ounique__euclidean__semiring__with__bit__shifts,type,
% 0.19/0.47 unique788075200shifts:
% 0.19/0.47 !>[A: $tType] : $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Finite__Set_Ocard,type,
% 0.19/0.47 finite_card:
% 0.19/0.47 !>[B: $tType] : ( ( set @ B ) > nat ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Finite__Set_Ofinite,type,
% 0.19/0.47 finite_finite2:
% 0.19/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Fun_Oinj__on,type,
% 0.19/0.47 inj_on:
% 0.19/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.19/0.47 minus_minus:
% 0.19/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.19/0.47 zero_zero:
% 0.19/0.47 !>[A: $tType] : A ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHF,type,
% 0.19/0.47 hF_Mirabelle_HF: ( set @ hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohf_OAbs__hf,type,
% 0.19/0.47 hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfset,type,
% 0.19/0.47 hF_Mirabelle_hfset: hF_Mirabelle_hf > ( set @ hF_Mirabelle_hf ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohinsert,type,
% 0.19/0.47 hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohmem,type,
% 0.19/0.47 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
% 0.19/0.47 lattic1704895705min_on:
% 0.19/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Nat_Osemiring__1__class_ONats,type,
% 0.19/0.47 semiring_1_Nats:
% 0.19/0.47 !>[A: $tType] : ( set @ A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
% 0.19/0.47 semiri532925092at_aux:
% 0.19/0.47 !>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
% 0.19/0.47 neg_numeral_dbl:
% 0.19/0.47 !>[A: $tType] : ( A > A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.19/0.47 bot_bot:
% 0.19/0.47 !>[A: $tType] : A ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.19/0.47 ord_less_eq:
% 0.19/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Parity_Osemiring__bit__shifts__class_Odrop__bit,type,
% 0.19/0.47 semiri2097166173op_bit:
% 0.19/0.47 !>[A: $tType] : ( nat > A > A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Parity_Osemiring__bit__shifts__class_Opush__bit,type,
% 0.19/0.47 semiri1924326578sh_bit:
% 0.19/0.47 !>[A: $tType] : ( nat > A > A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Relation_Oinv__imagep,type,
% 0.19/0.47 inv_imagep:
% 0.19/0.47 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Set_OCollect,type,
% 0.19/0.47 collect:
% 0.19/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Set_Oimage,type,
% 0.19/0.47 image:
% 0.19/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Set_Oinsert,type,
% 0.19/0.47 insert:
% 0.19/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Set_Ois__empty,type,
% 0.19/0.47 is_empty:
% 0.19/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Set_Ois__singleton,type,
% 0.19/0.47 is_singleton:
% 0.19/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_Set_Othe__elem,type,
% 0.19/0.47 the_elem:
% 0.19/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_c_member,type,
% 0.19/0.47 member:
% 0.19/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(sy_v_z,type,
% 0.19/0.47 z: hF_Mirabelle_hf ).
% 0.19/0.47
% 0.19/0.47 % Relevant facts (254)
% 0.19/0.47 thf(fact_0_hf__ext,axiom,
% 0.19/0.47 ( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z ) )
% 0.19/0.47 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.19/0.47 ! [X: hF_Mirabelle_hf] :
% 0.19/0.47 ( ( hF_Mirabelle_hmem @ X @ A2 )
% 0.19/0.47 = ( hF_Mirabelle_hmem @ X @ B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % hf_ext
% 0.19/0.47 thf(fact_1_hemptyE,axiom,
% 0.19/0.47 ! [A3: hF_Mirabelle_hf] :
% 0.19/0.47 ~ ( hF_Mirabelle_hmem @ A3 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.19/0.47
% 0.19/0.47 % hemptyE
% 0.19/0.47 thf(fact_2_hmem__hempty,axiom,
% 0.19/0.47 ! [A3: hF_Mirabelle_hf] :
% 0.19/0.47 ~ ( hF_Mirabelle_hmem @ A3 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.19/0.47
% 0.19/0.47 % hmem_hempty
% 0.19/0.47 thf(fact_3_hf__cases,axiom,
% 0.19/0.47 ! [Y2: hF_Mirabelle_hf] :
% 0.19/0.47 ( ( Y2
% 0.19/0.47 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.19/0.47 => ~ ! [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.19/0.47 ( ( Y2
% 0.19/0.47 = ( hF_Mirabelle_hinsert @ A4 @ B3 ) )
% 0.19/0.47 => ( hF_Mirabelle_hmem @ A4 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % hf_cases
% 0.19/0.47 thf(fact_4_hmem__hinsert,axiom,
% 0.19/0.47 ! [A3: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.19/0.47 ( ( hF_Mirabelle_hmem @ A3 @ ( hF_Mirabelle_hinsert @ B4 @ C ) )
% 0.19/0.47 = ( ( A3 = B4 )
% 0.19/0.47 | ( hF_Mirabelle_hmem @ A3 @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % hmem_hinsert
% 0.19/0.47 thf(fact_5_hmem__def,axiom,
% 0.19/0.47 ( hF_Mirabelle_hmem
% 0.19/0.47 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( member @ hF_Mirabelle_hf @ A2 @ ( hF_Mirabelle_hfset @ B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % hmem_def
% 0.19/0.47 thf(fact_6_zero__reorient,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( zero @ A )
% 0.19/0.47 => ! [X2: A] :
% 0.19/0.47 ( ( ( zero_zero @ A )
% 0.19/0.47 = X2 )
% 0.19/0.47 = ( X2
% 0.19/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % zero_reorient
% 0.19/0.47 thf(fact_7_Abs__hf__0,axiom,
% 0.19/0.47 ( ( hF_Mirabelle_Abs_hf @ ( zero_zero @ nat ) )
% 0.19/0.47 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.19/0.47
% 0.19/0.47 % Abs_hf_0
% 0.19/0.47 thf(fact_8_dbl__simps_I2_J,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( neg_numeral @ A )
% 0.19/0.47 => ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
% 0.19/0.47 = ( zero_zero @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dbl_simps(2)
% 0.19/0.47 thf(fact_9_Zero__hf__def,axiom,
% 0.19/0.47 ( ( zero_zero @ hF_Mirabelle_hf )
% 0.19/0.47 = ( hF_Mirabelle_HF @ ( bot_bot @ ( set @ hF_Mirabelle_hf ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Zero_hf_def
% 0.19/0.47 thf(fact_10_push__bit__of__0,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( semiring_bit_shifts @ A )
% 0.19/0.47 => ! [N: nat] :
% 0.19/0.47 ( ( semiri1924326578sh_bit @ A @ N @ ( zero_zero @ A ) )
% 0.19/0.47 = ( zero_zero @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % push_bit_of_0
% 0.19/0.47 thf(fact_11_push__bit__eq__0__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( unique788075200shifts @ A )
% 0.19/0.47 => ! [N: nat,A3: A] :
% 0.19/0.47 ( ( ( semiri1924326578sh_bit @ A @ N @ A3 )
% 0.19/0.47 = ( zero_zero @ A ) )
% 0.19/0.47 = ( A3
% 0.19/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % push_bit_eq_0_iff
% 0.19/0.47 thf(fact_12_drop__bit__of__0,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( semiring_bit_shifts @ A )
% 0.19/0.47 => ! [N: nat] :
% 0.19/0.47 ( ( semiri2097166173op_bit @ A @ N @ ( zero_zero @ A ) )
% 0.19/0.47 = ( zero_zero @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % drop_bit_of_0
% 0.19/0.47 thf(fact_13_Nats__0,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( semiring_1 @ A )
% 0.19/0.47 => ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Nats_0
% 0.19/0.47 thf(fact_14_HF__hfset,axiom,
% 0.19/0.47 ! [A3: hF_Mirabelle_hf] :
% 0.19/0.47 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A3 ) )
% 0.19/0.47 = A3 ) ).
% 0.19/0.47
% 0.19/0.47 % HF_hfset
% 0.19/0.47 thf(fact_15_empty__iff,axiom,
% 0.19/0.47 ! [A: $tType,C: A] :
% 0.19/0.47 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % empty_iff
% 0.19/0.47 thf(fact_16_all__not__in__conv,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( ( ! [X: A] :
% 0.19/0.47 ~ ( member @ A @ X @ A5 ) )
% 0.19/0.47 = ( A5
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % all_not_in_conv
% 0.19/0.47 thf(fact_17_Collect__empty__eq,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o] :
% 0.19/0.47 ( ( ( collect @ A @ P )
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = ( ! [X: A] :
% 0.19/0.47 ~ ( P @ X ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_empty_eq
% 0.19/0.47 thf(fact_18_empty__Collect__eq,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o] :
% 0.19/0.47 ( ( ( bot_bot @ ( set @ A ) )
% 0.19/0.47 = ( collect @ A @ P ) )
% 0.19/0.47 = ( ! [X: A] :
% 0.19/0.47 ~ ( P @ X ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % empty_Collect_eq
% 0.19/0.47 thf(fact_19_bot__apply,axiom,
% 0.19/0.47 ! [C2: $tType,D: $tType] :
% 0.19/0.47 ( ( bot @ C2 )
% 0.19/0.47 => ( ( bot_bot @ ( D > C2 ) )
% 0.19/0.47 = ( ^ [X: D] : ( bot_bot @ C2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_apply
% 0.19/0.47 thf(fact_20_hinsert__def,axiom,
% 0.19/0.47 ( hF_Mirabelle_hinsert
% 0.19/0.47 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( hF_Mirabelle_HF @ ( insert @ hF_Mirabelle_hf @ A2 @ ( hF_Mirabelle_hfset @ B2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % hinsert_def
% 0.19/0.47 thf(fact_21_hfset__HF,axiom,
% 0.19/0.47 ! [A5: set @ hF_Mirabelle_hf] :
% 0.19/0.47 ( ( finite_finite2 @ hF_Mirabelle_hf @ A5 )
% 0.19/0.47 => ( ( hF_Mirabelle_hfset @ ( hF_Mirabelle_HF @ A5 ) )
% 0.19/0.47 = A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % hfset_HF
% 0.19/0.47 thf(fact_22_zero__natural_Orsp,axiom,
% 0.19/0.47 ( ( zero_zero @ nat )
% 0.19/0.47 = ( zero_zero @ nat ) ) ).
% 0.19/0.47
% 0.19/0.47 % zero_natural.rsp
% 0.19/0.47 thf(fact_23_emptyE,axiom,
% 0.19/0.47 ! [A: $tType,A3: A] :
% 0.19/0.47 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % emptyE
% 0.19/0.47 thf(fact_24_equals0D,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,A3: A] :
% 0.19/0.47 ( ( A5
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ~ ( member @ A @ A3 @ A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % equals0D
% 0.19/0.47 thf(fact_25_insert__absorb2,axiom,
% 0.19/0.47 ! [A: $tType,X2: A,A5: set @ A] :
% 0.19/0.47 ( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A5 ) )
% 0.19/0.47 = ( insert @ A @ X2 @ A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_absorb2
% 0.19/0.47 thf(fact_26_insert__iff,axiom,
% 0.19/0.47 ! [A: $tType,A3: A,B4: A,A5: set @ A] :
% 0.19/0.47 ( ( member @ A @ A3 @ ( insert @ A @ B4 @ A5 ) )
% 0.19/0.47 = ( ( A3 = B4 )
% 0.19/0.47 | ( member @ A @ A3 @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_iff
% 0.19/0.47 thf(fact_27_insertCI,axiom,
% 0.19/0.47 ! [A: $tType,A3: A,B5: set @ A,B4: A] :
% 0.19/0.47 ( ( ~ ( member @ A @ A3 @ B5 )
% 0.19/0.47 => ( A3 = B4 ) )
% 0.19/0.47 => ( member @ A @ A3 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertCI
% 0.19/0.47 thf(fact_28_singletonI,axiom,
% 0.19/0.47 ! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % singletonI
% 0.19/0.47 thf(fact_29_mk__disjoint__insert,axiom,
% 0.19/0.47 ! [A: $tType,A3: A,A5: set @ A] :
% 0.19/0.47 ( ( member @ A @ A3 @ A5 )
% 0.19/0.47 => ? [B6: set @ A] :
% 0.19/0.47 ( ( A5
% 0.19/0.47 = ( insert @ A @ A3 @ B6 ) )
% 0.19/0.47 & ~ ( member @ A @ A3 @ B6 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % mk_disjoint_insert
% 0.19/0.47 thf(fact_30_insert__commute,axiom,
% 0.19/0.47 ! [A: $tType,X2: A,Y2: A,A5: set @ A] :
% 0.19/0.47 ( ( insert @ A @ X2 @ ( insert @ A @ Y2 @ A5 ) )
% 0.19/0.47 = ( insert @ A @ Y2 @ ( insert @ A @ X2 @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_commute
% 0.19/0.47 thf(fact_31_insert__eq__iff,axiom,
% 0.19/0.47 ! [A: $tType,A3: A,A5: set @ A,B4: A,B5: set @ A] :
% 0.19/0.47 ( ~ ( member @ A @ A3 @ A5 )
% 0.19/0.47 => ( ~ ( member @ A @ B4 @ B5 )
% 0.19/0.47 => ( ( ( insert @ A @ A3 @ A5 )
% 0.19/0.47 = ( insert @ A @ B4 @ B5 ) )
% 0.19/0.47 = ( ( ( A3 = B4 )
% 0.19/0.47 => ( A5 = B5 ) )
% 0.19/0.47 & ( ( A3 != B4 )
% 0.19/0.47 => ? [C3: set @ A] :
% 0.19/0.47 ( ( A5
% 0.19/0.47 = ( insert @ A @ B4 @ C3 ) )
% 0.19/0.47 & ~ ( member @ A @ B4 @ C3 )
% 0.19/0.47 & ( B5
% 0.19/0.47 = ( insert @ A @ A3 @ C3 ) )
% 0.19/0.47 & ~ ( member @ A @ A3 @ C3 ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_eq_iff
% 0.19/0.47 thf(fact_32_insert__absorb,axiom,
% 0.19/0.47 ! [A: $tType,A3: A,A5: set @ A] :
% 0.19/0.47 ( ( member @ A @ A3 @ A5 )
% 0.19/0.47 => ( ( insert @ A @ A3 @ A5 )
% 0.19/0.47 = A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_absorb
% 0.19/0.47 thf(fact_33_insert__ident,axiom,
% 0.19/0.47 ! [A: $tType,X2: A,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ~ ( member @ A @ X2 @ A5 )
% 0.19/0.47 => ( ~ ( member @ A @ X2 @ B5 )
% 0.19/0.47 => ( ( ( insert @ A @ X2 @ A5 )
% 0.19/0.47 = ( insert @ A @ X2 @ B5 ) )
% 0.19/0.47 = ( A5 = B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_ident
% 0.19/0.47 thf(fact_34_Set_Oset__insert,axiom,
% 0.19/0.47 ! [A: $tType,X2: A,A5: set @ A] :
% 0.19/0.47 ( ( member @ A @ X2 @ A5 )
% 0.19/0.47 => ~ ! [B6: set @ A] :
% 0.19/0.47 ( ( A5
% 0.19/0.47 = ( insert @ A @ X2 @ B6 ) )
% 0.19/0.47 => ( member @ A @ X2 @ B6 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Set.set_insert
% 0.19/0.47 thf(fact_35_insertI2,axiom,
% 0.19/0.47 ! [A: $tType,A3: A,B5: set @ A,B4: A] :
% 0.19/0.47 ( ( member @ A @ A3 @ B5 )
% 0.19/0.47 => ( member @ A @ A3 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertI2
% 0.19/0.47 thf(fact_36_insertI1,axiom,
% 0.19/0.47 ! [A: $tType,A3: A,B5: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertI1
% 0.19/0.47 thf(fact_37_insertE,axiom,
% 0.19/0.47 ! [A: $tType,A3: A,B4: A,A5: set @ A] :
% 0.19/0.47 ( ( member @ A @ A3 @ ( insert @ A @ B4 @ A5 ) )
% 0.19/0.47 => ( ( A3 != B4 )
% 0.19/0.47 => ( member @ A @ A3 @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertE
% 0.19/0.47 thf(fact_38_finite__cases,axiom,
% 0.19/0.47 ! [A: $tType,F: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F )
% 0.19/0.47 => ( ( F
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ~ ! [A6: set @ A,X3: A] :
% 0.19/0.47 ( ( F
% 0.19/0.47 = ( insert @ A @ X3 @ A6 ) )
% 0.19/0.47 => ( ~ ( member @ A @ X3 @ A6 )
% 0.19/0.47 => ~ ( finite_finite2 @ A @ A6 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_cases
% 0.19/0.47 thf(fact_39_bot__set__def,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( bot_bot @ ( set @ A ) )
% 0.19/0.47 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_set_def
% 0.19/0.47 thf(fact_40_singleton__inject,axiom,
% 0.19/0.47 ! [A: $tType,A3: A,B4: A] :
% 0.19/0.47 ( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 => ( A3 = B4 ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_inject
% 0.19/0.47 thf(fact_41_insert__not__empty,axiom,
% 0.19/0.47 ! [A: $tType,A3: A,A5: set @ A] :
% 0.19/0.47 ( ( insert @ A @ A3 @ A5 )
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_not_empty
% 0.19/0.47 thf(fact_42_doubleton__eq__iff,axiom,
% 0.19/0.47 ! [A: $tType,A3: A,B4: A,C: A,D2: A] :
% 0.19/0.47 ( ( ( insert @ A @ A3 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 = ( insert @ A @ C @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.19/0.47 = ( ( ( A3 = C )
% 0.19/0.47 & ( B4 = D2 ) )
% 0.19/0.47 | ( ( A3 = D2 )
% 0.19/0.47 & ( B4 = C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % doubleton_eq_iff
% 0.19/0.47 thf(fact_43_mem__Collect__eq,axiom,
% 0.19/0.47 ! [A: $tType,A3: A,P: A > $o] :
% 0.19/0.47 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.19/0.47 = ( P @ A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % mem_Collect_eq
% 0.19/0.47 thf(fact_44_Collect__mem__eq,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( ( collect @ A
% 0.19/0.47 @ ^ [X: A] : ( member @ A @ X @ A5 ) )
% 0.19/0.47 = A5 ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_mem_eq
% 0.19/0.47 thf(fact_45_Collect__cong,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.47 ( ! [X3: A] :
% 0.19/0.47 ( ( P @ X3 )
% 0.19/0.47 = ( Q @ X3 ) )
% 0.19/0.47 => ( ( collect @ A @ P )
% 0.19/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_cong
% 0.19/0.47 thf(fact_46_ext,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F2: A > B,G: A > B] :
% 0.19/0.47 ( ! [X3: A] :
% 0.19/0.47 ( ( F2 @ X3 )
% 0.19/0.47 = ( G @ X3 ) )
% 0.19/0.47 => ( F2 = G ) ) ).
% 0.19/0.47
% 0.19/0.47 % ext
% 0.19/0.47 thf(fact_47_singleton__iff,axiom,
% 0.19/0.47 ! [A: $tType,B4: A,A3: A] :
% 0.19/0.47 ( ( member @ A @ B4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 = ( B4 = A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_iff
% 0.19/0.47 thf(fact_48_singletonD,axiom,
% 0.19/0.47 ! [A: $tType,B4: A,A3: A] :
% 0.19/0.47 ( ( member @ A @ B4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 => ( B4 = A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % singletonD
% 0.19/0.47 thf(fact_49_bot__nat__def,axiom,
% 0.19/0.47 ( ( bot_bot @ nat )
% 0.19/0.47 = ( zero_zero @ nat ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_nat_def
% 0.19/0.47 thf(fact_50_finite__hfset,axiom,
% 0.19/0.47 ! [A3: hF_Mirabelle_hf] : ( finite_finite2 @ hF_Mirabelle_hf @ ( hF_Mirabelle_hfset @ A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_hfset
% 0.19/0.47 thf(fact_51_bot__fun__def,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( bot @ B )
% 0.19/0.47 => ( ( bot_bot @ ( A > B ) )
% 0.19/0.47 = ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_fun_def
% 0.19/0.47 thf(fact_52_ex__in__conv,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( ( ? [X: A] : ( member @ A @ X @ A5 ) )
% 0.19/0.47 = ( A5
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ex_in_conv
% 0.19/0.47 thf(fact_53_equals0I,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( ! [Y3: A] :
% 0.19/0.47 ~ ( member @ A @ Y3 @ A5 )
% 0.19/0.47 => ( A5
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equals0I
% 0.19/0.47 thf(fact_54_finite__insert,axiom,
% 0.19/0.47 ! [A: $tType,A3: A,A5: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ ( insert @ A @ A3 @ A5 ) )
% 0.19/0.47 = ( finite_finite2 @ A @ A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_insert
% 0.19/0.47 thf(fact_55_finite_Ocases,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A3 )
% 0.19/0.47 => ( ( A3
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ~ ! [A6: set @ A] :
% 0.19/0.47 ( ? [A4: A] :
% 0.19/0.47 ( A3
% 0.19/0.47 = ( insert @ A @ A4 @ A6 ) )
% 0.19/0.47 => ~ ( finite_finite2 @ A @ A6 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite.cases
% 0.19/0.47 thf(fact_56_finite_Osimps,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_finite2 @ A )
% 0.19/0.47 = ( ^ [A2: set @ A] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 | ? [A7: set @ A,B2: A] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 = ( insert @ A @ B2 @ A7 ) )
% 0.19/0.47 & ( finite_finite2 @ A @ A7 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite.simps
% 0.19/0.47 thf(fact_57_finite__induct,axiom,
% 0.19/0.47 ! [A: $tType,F: set @ A,P: ( set @ A ) > $o] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F )
% 0.19/0.47 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ! [X3: A,F3: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F3 )
% 0.19/0.47 => ( ~ ( member @ A @ X3 @ F3 )
% 0.19/0.47 => ( ( P @ F3 )
% 0.19/0.47 => ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) )
% 0.19/0.47 => ( P @ F ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_induct
% 0.19/0.47 thf(fact_58_finite_Oinducts,axiom,
% 0.19/0.47 ! [A: $tType,X2: set @ A,P: ( set @ A ) > $o] :
% 0.19/0.47 ( ( finite_finite2 @ A @ X2 )
% 0.19/0.47 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ! [A6: set @ A,A4: A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A6 )
% 0.19/0.47 => ( ( P @ A6 )
% 0.19/0.47 => ( P @ ( insert @ A @ A4 @ A6 ) ) ) )
% 0.19/0.47 => ( P @ X2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite.inducts
% 0.19/0.47 thf(fact_59_finite__ne__induct,axiom,
% 0.19/0.47 ! [A: $tType,F: set @ A,P: ( set @ A ) > $o] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F )
% 0.19/0.47 => ( ( F
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ! [X3: A] : ( P @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 => ( ! [X3: A,F3: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F3 )
% 0.19/0.47 => ( ( F3
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ~ ( member @ A @ X3 @ F3 )
% 0.19/0.47 => ( ( P @ F3 )
% 0.19/0.47 => ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) ) )
% 0.19/0.47 => ( P @ F ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_ne_induct
% 0.19/0.47 thf(fact_60_infinite__finite__induct,axiom,
% 0.19/0.47 ! [A: $tType,P: ( set @ A ) > $o,A5: set @ A] :
% 0.19/0.47 ( ! [A6: set @ A] :
% 0.19/0.47 ( ~ ( finite_finite2 @ A @ A6 )
% 0.19/0.47 => ( P @ A6 ) )
% 0.19/0.47 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ! [X3: A,F3: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F3 )
% 0.19/0.47 => ( ~ ( member @ A @ X3 @ F3 )
% 0.19/0.47 => ( ( P @ F3 )
% 0.19/0.47 => ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) )
% 0.19/0.47 => ( P @ A5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_finite_induct
% 0.19/0.47 thf(fact_61_the__elem__eq,axiom,
% 0.19/0.47 ! [A: $tType,X2: A] :
% 0.19/0.47 ( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 = X2 ) ).
% 0.19/0.47
% 0.19/0.47 % the_elem_eq
% 0.19/0.47 thf(fact_62_finite__code,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_finite @ A )
% 0.19/0.47 => ( ( finite_finite2 @ A )
% 0.19/0.47 = ( ^ [A7: set @ A] : $true ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_code
% 0.19/0.47 thf(fact_63_finite__set__choice,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,A5: set @ A,P: A > B > $o] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A5 )
% 0.19/0.47 => ( ! [X3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ A5 )
% 0.19/0.47 => ? [X_1: B] : ( P @ X3 @ X_1 ) )
% 0.19/0.47 => ? [F4: A > B] :
% 0.19/0.47 ! [X4: A] :
% 0.19/0.47 ( ( member @ A @ X4 @ A5 )
% 0.19/0.47 => ( P @ X4 @ ( F4 @ X4 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_set_choice
% 0.19/0.47 thf(fact_64_finite,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_finite @ A )
% 0.19/0.47 => ! [A5: set @ A] : ( finite_finite2 @ A @ A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite
% 0.19/0.47 thf(fact_65_finite_OemptyI,axiom,
% 0.19/0.47 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite.emptyI
% 0.19/0.47 thf(fact_66_infinite__imp__nonempty,axiom,
% 0.19/0.47 ! [A: $tType,S: set @ A] :
% 0.19/0.47 ( ~ ( finite_finite2 @ A @ S )
% 0.19/0.47 => ( S
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_imp_nonempty
% 0.19/0.47 thf(fact_67_finite_OinsertI,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,A3: A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A5 )
% 0.19/0.47 => ( finite_finite2 @ A @ ( insert @ A @ A3 @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite.insertI
% 0.19/0.47 thf(fact_68_bot__empty__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( bot_bot @ ( A > $o ) )
% 0.19/0.47 = ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_empty_eq
% 0.19/0.47 thf(fact_69_Collect__empty__eq__bot,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o] :
% 0.19/0.47 ( ( ( collect @ A @ P )
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = ( P
% 0.19/0.47 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_empty_eq_bot
% 0.19/0.47 thf(fact_70_is__singleton__the__elem,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( is_singleton @ A )
% 0.19/0.47 = ( ^ [A7: set @ A] :
% 0.19/0.47 ( A7
% 0.19/0.47 = ( insert @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singleton_the_elem
% 0.19/0.47 thf(fact_71_is__singletonI,axiom,
% 0.19/0.47 ! [A: $tType,X2: A] : ( is_singleton @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singletonI
% 0.19/0.47 thf(fact_72_Set_Ois__empty__def,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( is_empty @ A )
% 0.19/0.47 = ( ^ [A7: set @ A] :
% 0.19/0.47 ( A7
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Set.is_empty_def
% 0.19/0.47 thf(fact_73_is__singleton__def,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( is_singleton @ A )
% 0.19/0.47 = ( ^ [A7: set @ A] :
% 0.19/0.47 ? [X: A] :
% 0.19/0.47 ( A7
% 0.19/0.47 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singleton_def
% 0.19/0.47 thf(fact_74_is__singletonE,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( ( is_singleton @ A @ A5 )
% 0.19/0.47 => ~ ! [X3: A] :
% 0.19/0.47 ( A5
% 0.19/0.47 != ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singletonE
% 0.19/0.47 thf(fact_75_is__singletonI_H,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( ( A5
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ! [X3: A,Y3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ A5 )
% 0.19/0.47 => ( ( member @ A @ Y3 @ A5 )
% 0.19/0.47 => ( X3 = Y3 ) ) )
% 0.19/0.47 => ( is_singleton @ A @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singletonI'
% 0.19/0.47 thf(fact_76_card__0__eq,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A5 )
% 0.19/0.47 => ( ( ( finite_card @ A @ A5 )
% 0.19/0.47 = ( zero_zero @ nat ) )
% 0.19/0.47 = ( A5
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_0_eq
% 0.19/0.47 thf(fact_77_finite__subset__induct,axiom,
% 0.19/0.47 ! [A: $tType,F: set @ A,A5: set @ A,P: ( set @ A ) > $o] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ F @ A5 )
% 0.19/0.47 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ! [A4: A,F3: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F3 )
% 0.19/0.47 => ( ( member @ A @ A4 @ A5 )
% 0.19/0.47 => ( ~ ( member @ A @ A4 @ F3 )
% 0.19/0.47 => ( ( P @ F3 )
% 0.19/0.47 => ( P @ ( insert @ A @ A4 @ F3 ) ) ) ) ) )
% 0.19/0.47 => ( P @ F ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_subset_induct
% 0.19/0.47 thf(fact_78_finite__subset__induct_H,axiom,
% 0.19/0.47 ! [A: $tType,F: set @ A,A5: set @ A,P: ( set @ A ) > $o] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ F @ A5 )
% 0.19/0.47 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ! [A4: A,F3: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F3 )
% 0.19/0.47 => ( ( member @ A @ A4 @ A5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ F3 @ A5 )
% 0.19/0.47 => ( ~ ( member @ A @ A4 @ F3 )
% 0.19/0.47 => ( ( P @ F3 )
% 0.19/0.47 => ( P @ ( insert @ A @ A4 @ F3 ) ) ) ) ) ) )
% 0.19/0.47 => ( P @ F ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_subset_induct'
% 0.19/0.47 thf(fact_79_in__inv__imagep,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( inv_imagep @ A @ B )
% 0.19/0.47 = ( ^ [R: A > A > $o,F5: B > A,X: B,Y4: B] : ( R @ ( F5 @ X ) @ ( F5 @ Y4 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % in_inv_imagep
% 0.19/0.47 thf(fact_80_inj__on__HF,axiom,
% 0.19/0.47 inj_on @ ( set @ hF_Mirabelle_hf ) @ hF_Mirabelle_hf @ hF_Mirabelle_HF @ ( collect @ ( set @ hF_Mirabelle_hf ) @ ( finite_finite2 @ hF_Mirabelle_hf ) ) ).
% 0.19/0.47
% 0.19/0.47 % inj_on_HF
% 0.19/0.47 thf(fact_81_of__nat__aux_Osimps_I1_J,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( semiring_1 @ A )
% 0.19/0.47 => ! [Inc: A > A,I: A] :
% 0.19/0.47 ( ( semiri532925092at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
% 0.19/0.47 = I ) ) ).
% 0.19/0.47
% 0.19/0.47 % of_nat_aux.simps(1)
% 0.19/0.47 thf(fact_82_order__refl,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( preorder @ A )
% 0.19/0.47 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_refl
% 0.19/0.47 thf(fact_83_subset__antisym,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
% 0.19/0.47 => ( A5 = B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_antisym
% 0.19/0.47 thf(fact_84_subsetI,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ! [X3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ A5 )
% 0.19/0.47 => ( member @ A @ X3 @ B5 ) )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % subsetI
% 0.19/0.47 thf(fact_85_le__zero__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( canoni770627133id_add @ A )
% 0.19/0.47 => ! [N: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.19/0.47 = ( N
% 0.19/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_zero_eq
% 0.19/0.47 thf(fact_86_subset__empty,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = ( A5
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_empty
% 0.19/0.47 thf(fact_87_empty__subsetI,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% 0.19/0.47
% 0.19/0.47 % empty_subsetI
% 0.19/0.47 thf(fact_88_insert__subset,axiom,
% 0.19/0.47 ! [A: $tType,X2: A,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ A5 ) @ B5 )
% 0.19/0.47 = ( ( member @ A @ X2 @ B5 )
% 0.19/0.47 & ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_subset
% 0.19/0.47 thf(fact_89_singleton__insert__inj__eq_H,axiom,
% 0.19/0.47 ! [A: $tType,A3: A,A5: set @ A,B4: A] :
% 0.19/0.47 ( ( ( insert @ A @ A3 @ A5 )
% 0.19/0.47 = ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 = ( ( A3 = B4 )
% 0.19/0.47 & ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_insert_inj_eq'
% 0.19/0.47 thf(fact_90_singleton__insert__inj__eq,axiom,
% 0.19/0.47 ! [A: $tType,B4: A,A3: A,A5: set @ A] :
% 0.19/0.47 ( ( ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = ( insert @ A @ A3 @ A5 ) )
% 0.19/0.47 = ( ( A3 = B4 )
% 0.19/0.47 & ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_insert_inj_eq
% 0.19/0.47 thf(fact_91_card_Oempty,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = ( zero_zero @ nat ) ) ).
% 0.19/0.47
% 0.19/0.47 % card.empty
% 0.19/0.47 thf(fact_92_card_Oinfinite,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( ~ ( finite_finite2 @ A @ A5 )
% 0.19/0.47 => ( ( finite_card @ A @ A5 )
% 0.19/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card.infinite
% 0.19/0.47 thf(fact_93_dual__order_Oantisym,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [B4: A,A3: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.19/0.47 => ( A3 = B4 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.antisym
% 0.19/0.47 thf(fact_94_dual__order_Oeq__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.19/0.47 = ( ^ [A2: A,B2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.19/0.47 & ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.eq_iff
% 0.19/0.47 thf(fact_95_dual__order_Otrans,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [B4: A,A3: A,C: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ C @ B4 )
% 0.19/0.47 => ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.trans
% 0.19/0.47 thf(fact_96_linorder__wlog,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A )
% 0.19/0.47 => ! [P: A > A > $o,A3: A,B4: A] :
% 0.19/0.47 ( ! [A4: A,B3: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.47 => ( P @ A4 @ B3 ) )
% 0.19/0.47 => ( ! [A4: A,B3: A] :
% 0.19/0.47 ( ( P @ B3 @ A4 )
% 0.19/0.47 => ( P @ A4 @ B3 ) )
% 0.19/0.47 => ( P @ A3 @ B4 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % linorder_wlog
% 0.19/0.47 thf(fact_97_dual__order_Orefl,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.refl
% 0.19/0.47 thf(fact_98_order__trans,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( preorder @ A )
% 0.19/0.47 => ! [X2: A,Y2: A,Z2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.19/0.47 => ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_trans
% 0.19/0.47 thf(fact_99_order__class_Oorder_Oantisym,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [A3: A,B4: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.19/0.47 => ( A3 = B4 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_class.order.antisym
% 0.19/0.47 thf(fact_100_ord__le__eq__trans,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord @ A )
% 0.19/0.47 => ! [A3: A,B4: A,C: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.19/0.47 => ( ( B4 = C )
% 0.19/0.47 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_le_eq_trans
% 0.19/0.47 thf(fact_101_ord__eq__le__trans,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord @ A )
% 0.19/0.47 => ! [A3: A,B4: A,C: A] :
% 0.19/0.47 ( ( A3 = B4 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ B4 @ C )
% 0.19/0.47 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_eq_le_trans
% 0.19/0.47 thf(fact_102_order__class_Oorder_Oeq__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.19/0.47 = ( ^ [A2: A,B2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.19/0.47 & ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_class.order.eq_iff
% 0.19/0.47 thf(fact_103_antisym__conv,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [Y2: A,X2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ Y2 @ X2 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.47 = ( X2 = Y2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % antisym_conv
% 0.19/0.47 thf(fact_104_le__cases3,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A )
% 0.19/0.47 => ! [X2: A,Y2: A,Z2: A] :
% 0.19/0.47 ( ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.47 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 0.19/0.47 => ( ( ( ord_less_eq @ A @ Y2 @ X2 )
% 0.19/0.47 => ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
% 0.19/0.47 => ( ( ( ord_less_eq @ A @ X2 @ Z2 )
% 0.19/0.47 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 0.19/0.47 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 0.19/0.47 => ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
% 0.19/0.47 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.19/0.47 => ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
% 0.19/0.47 => ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
% 0.19/0.47 => ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_cases3
% 0.19/0.47 thf(fact_105_order_Otrans,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [A3: A,B4: A,C: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ B4 @ C )
% 0.19/0.47 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order.trans
% 0.19/0.47 thf(fact_106_le__cases,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A )
% 0.19/0.47 => ! [X2: A,Y2: A] :
% 0.19/0.47 ( ~ ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_cases
% 0.19/0.47 thf(fact_107_eq__refl,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( preorder @ A )
% 0.19/0.47 => ! [X2: A,Y2: A] :
% 0.19/0.47 ( ( X2 = Y2 )
% 0.19/0.47 => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_refl
% 0.19/0.47 thf(fact_108_linear,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A )
% 0.19/0.47 => ! [X2: A,Y2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.47 | ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % linear
% 0.19/0.47 thf(fact_109_antisym,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [X2: A,Y2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ Y2 @ X2 )
% 0.19/0.47 => ( X2 = Y2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % antisym
% 0.19/0.47 thf(fact_110_eq__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.19/0.47 = ( ^ [X: A,Y4: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.19/0.47 & ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_iff
% 0.19/0.47 thf(fact_111_ord__le__eq__subst,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( ord @ B )
% 0.19/0.47 & ( ord @ A ) )
% 0.19/0.47 => ! [A3: A,B4: A,F2: A > B,C: B] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.19/0.47 => ( ( ( F2 @ B4 )
% 0.19/0.47 = C )
% 0.19/0.47 => ( ! [X3: A,Y3: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.19/0.47 => ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.47 => ( ord_less_eq @ B @ ( F2 @ A3 ) @ C ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_le_eq_subst
% 0.19/0.47 thf(fact_112_ord__eq__le__subst,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( ord @ B )
% 0.19/0.47 & ( ord @ A ) )
% 0.19/0.47 => ! [A3: A,F2: B > A,B4: B,C: B] :
% 0.19/0.47 ( ( A3
% 0.19/0.47 = ( F2 @ B4 ) )
% 0.19/0.47 => ( ( ord_less_eq @ B @ B4 @ C )
% 0.19/0.47 => ( ! [X3: B,Y3: B] :
% 0.19/0.47 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 0.19/0.47 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.47 => ( ord_less_eq @ A @ A3 @ ( F2 @ C ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_eq_le_subst
% 0.19/0.47 thf(fact_113_order__subst2,axiom,
% 0.19/0.47 ! [A: $tType,C2: $tType] :
% 0.19/0.47 ( ( ( order @ C2 )
% 0.19/0.47 & ( order @ A ) )
% 0.19/0.47 => ! [A3: A,B4: A,F2: A > C2,C: C2] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.19/0.47 => ( ( ord_less_eq @ C2 @ ( F2 @ B4 ) @ C )
% 0.19/0.47 => ( ! [X3: A,Y3: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.19/0.47 => ( ord_less_eq @ C2 @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.47 => ( ord_less_eq @ C2 @ ( F2 @ A3 ) @ C ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_subst2
% 0.19/0.47 thf(fact_114_order__subst1,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( order @ B )
% 0.19/0.47 & ( order @ A ) )
% 0.19/0.47 => ! [A3: A,F2: B > A,B4: B,C: B] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A3 @ ( F2 @ B4 ) )
% 0.19/0.47 => ( ( ord_less_eq @ B @ B4 @ C )
% 0.19/0.47 => ( ! [X3: B,Y3: B] :
% 0.19/0.47 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 0.19/0.47 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.47 => ( ord_less_eq @ A @ A3 @ ( F2 @ C ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_subst1
% 0.19/0.47 thf(fact_115_Collect__mono__iff,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.19/0.47 = ( ! [X: A] :
% 0.19/0.47 ( ( P @ X )
% 0.19/0.47 => ( Q @ X ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_mono_iff
% 0.19/0.47 thf(fact_116_le__fun__def,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ord @ B )
% 0.19/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.19/0.47 = ( ^ [F5: A > B,G2: A > B] :
% 0.19/0.47 ! [X: A] : ( ord_less_eq @ B @ ( F5 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_fun_def
% 0.19/0.47 thf(fact_117_set__eq__subset,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ^ [Y: set @ A,Z: set @ A] : ( Y = Z ) )
% 0.19/0.47 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.19/0.47 & ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % set_eq_subset
% 0.19/0.47 thf(fact_118_le__funI,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ord @ B )
% 0.19/0.47 => ! [F2: A > B,G: A > B] :
% 0.19/0.47 ( ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
% 0.19/0.47 => ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_funI
% 0.19/0.47 thf(fact_119_le__funE,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ord @ B )
% 0.19/0.47 => ! [F2: A > B,G: A > B,X2: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( A > B ) @ F2 @ G )
% 0.19/0.47 => ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_funE
% 0.19/0.47 thf(fact_120_le__funD,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ord @ B )
% 0.19/0.47 => ! [F2: A > B,G: A > B,X2: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( A > B ) @ F2 @ G )
% 0.19/0.47 => ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_funD
% 0.19/0.47 thf(fact_121_subset__trans,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_trans
% 0.19/0.47 thf(fact_122_Collect__mono,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.47 ( ! [X3: A] :
% 0.19/0.47 ( ( P @ X3 )
% 0.19/0.47 => ( Q @ X3 ) )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_mono
% 0.19/0.47 thf(fact_123_subset__refl,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% 0.19/0.47
% 0.19/0.47 % subset_refl
% 0.19/0.47 thf(fact_124_subset__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.19/0.47 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.19/0.47 ! [T: A] :
% 0.19/0.47 ( ( member @ A @ T @ A7 )
% 0.19/0.47 => ( member @ A @ T @ B7 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_iff
% 0.19/0.47 thf(fact_125_equalityD2,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( A5 = B5 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % equalityD2
% 0.19/0.47 thf(fact_126_equalityD1,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( A5 = B5 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % equalityD1
% 0.19/0.47 thf(fact_127_subset__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.19/0.47 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.19/0.47 ! [X: A] :
% 0.19/0.47 ( ( member @ A @ X @ A7 )
% 0.19/0.47 => ( member @ A @ X @ B7 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_eq
% 0.19/0.47 thf(fact_128_equalityE,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( A5 = B5 )
% 0.19/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equalityE
% 0.19/0.47 thf(fact_129_subsetD,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A,C: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ( member @ A @ C @ A5 )
% 0.19/0.47 => ( member @ A @ C @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subsetD
% 0.19/0.47 thf(fact_130_in__mono,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A,X2: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ( member @ A @ X2 @ A5 )
% 0.19/0.47 => ( member @ A @ X2 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % in_mono
% 0.19/0.47 thf(fact_131_card__subset__eq,axiom,
% 0.19/0.47 ! [A: $tType,B5: set @ A,A5: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ B5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ( ( finite_card @ A @ A5 )
% 0.19/0.47 = ( finite_card @ A @ B5 ) )
% 0.19/0.47 => ( A5 = B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_subset_eq
% 0.19/0.47 thf(fact_132_infinite__arbitrarily__large,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,N: nat] :
% 0.19/0.47 ( ~ ( finite_finite2 @ A @ A5 )
% 0.19/0.47 => ? [B6: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ B6 )
% 0.19/0.47 & ( ( finite_card @ A @ B6 )
% 0.19/0.47 = N )
% 0.19/0.47 & ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_arbitrarily_large
% 0.19/0.47 thf(fact_133_zero__le,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( canoni770627133id_add @ A )
% 0.19/0.47 => ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % zero_le
% 0.19/0.47 thf(fact_134_le__numeral__extra_I3_J,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linord1659791738miring @ A )
% 0.19/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_numeral_extra(3)
% 0.19/0.47 thf(fact_135_bot_Oextremum,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order_bot @ A )
% 0.19/0.47 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot.extremum
% 0.19/0.47 thf(fact_136_bot_Oextremum__unique,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order_bot @ A )
% 0.19/0.47 => ! [A3: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.19/0.47 = ( A3
% 0.19/0.47 = ( bot_bot @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot.extremum_unique
% 0.19/0.47 thf(fact_137_bot_Oextremum__uniqueI,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order_bot @ A )
% 0.19/0.47 => ! [A3: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.19/0.47 => ( A3
% 0.19/0.47 = ( bot_bot @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot.extremum_uniqueI
% 0.19/0.47 thf(fact_138_finite__has__maximal2,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [A5: set @ A,A3: A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A5 )
% 0.19/0.47 => ( ( member @ A @ A3 @ A5 )
% 0.19/0.47 => ? [X3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ A5 )
% 0.19/0.47 & ( ord_less_eq @ A @ A3 @ X3 )
% 0.19/0.47 & ! [Xa: A] :
% 0.19/0.47 ( ( member @ A @ Xa @ A5 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ X3 @ Xa )
% 0.19/0.47 => ( X3 = Xa ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_has_maximal2
% 0.19/0.47 thf(fact_139_finite__has__minimal2,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [A5: set @ A,A3: A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A5 )
% 0.19/0.47 => ( ( member @ A @ A3 @ A5 )
% 0.19/0.47 => ? [X3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ A5 )
% 0.19/0.47 & ( ord_less_eq @ A @ X3 @ A3 )
% 0.19/0.47 & ! [Xa: A] :
% 0.19/0.47 ( ( member @ A @ Xa @ A5 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ Xa @ X3 )
% 0.19/0.47 => ( X3 = Xa ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_has_minimal2
% 0.19/0.47 thf(fact_140_finite__subset,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ( finite_finite2 @ A @ B5 )
% 0.19/0.47 => ( finite_finite2 @ A @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_subset
% 0.19/0.47 thf(fact_141_infinite__super,axiom,
% 0.19/0.47 ! [A: $tType,S: set @ A,T2: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
% 0.19/0.47 => ( ~ ( finite_finite2 @ A @ S )
% 0.19/0.47 => ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_super
% 0.19/0.47 thf(fact_142_rev__finite__subset,axiom,
% 0.19/0.47 ! [A: $tType,B5: set @ A,A5: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ B5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( finite_finite2 @ A @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % rev_finite_subset
% 0.19/0.47 thf(fact_143_insert__mono,axiom,
% 0.19/0.47 ! [A: $tType,C4: set @ A,D3: set @ A,A3: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C4 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_mono
% 0.19/0.47 thf(fact_144_subset__insert,axiom,
% 0.19/0.47 ! [A: $tType,X2: A,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ~ ( member @ A @ X2 @ A5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ B5 ) )
% 0.19/0.47 = ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_insert
% 0.19/0.47 thf(fact_145_subset__insertI,axiom,
% 0.19/0.47 ! [A: $tType,B5: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A3 @ B5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_insertI
% 0.19/0.47 thf(fact_146_subset__insertI2,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A,B4: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_insertI2
% 0.19/0.47 thf(fact_147_finite__has__minimal,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [A5: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A5 )
% 0.19/0.47 => ( ( A5
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ? [X3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ A5 )
% 0.19/0.47 & ! [Xa: A] :
% 0.19/0.47 ( ( member @ A @ Xa @ A5 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ Xa @ X3 )
% 0.19/0.47 => ( X3 = Xa ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_has_minimal
% 0.19/0.47 thf(fact_148_finite__has__maximal,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A )
% 0.19/0.47 => ! [A5: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A5 )
% 0.19/0.47 => ( ( A5
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ? [X3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ A5 )
% 0.19/0.47 & ! [Xa: A] :
% 0.19/0.47 ( ( member @ A @ Xa @ A5 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ X3 @ Xa )
% 0.19/0.47 => ( X3 = Xa ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_has_maximal
% 0.19/0.47 thf(fact_149_subset__singleton__iff,axiom,
% 0.19/0.47 ! [A: $tType,X5: set @ A,A3: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 = ( ( X5
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 | ( X5
% 0.19/0.47 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_singleton_iff
% 0.19/0.47 thf(fact_150_subset__singletonD,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,X2: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 => ( ( A5
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 | ( A5
% 0.19/0.47 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_singletonD
% 0.19/0.47 thf(fact_151_card__eq__0__iff,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( ( ( finite_card @ A @ A5 )
% 0.19/0.47 = ( zero_zero @ nat ) )
% 0.19/0.47 = ( ( A5
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 | ~ ( finite_finite2 @ A @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_eq_0_iff
% 0.19/0.47 thf(fact_152_inj__on__empty,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F2: A > B] : ( inj_on @ A @ B @ F2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % inj_on_empty
% 0.19/0.47 thf(fact_153_finite__ranking__induct,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( linorder @ A )
% 0.19/0.47 => ! [S: set @ B,P: ( set @ B ) > $o,F2: B > A] :
% 0.19/0.47 ( ( finite_finite2 @ B @ S )
% 0.19/0.47 => ( ( P @ ( bot_bot @ ( set @ B ) ) )
% 0.19/0.47 => ( ! [X3: B,S2: set @ B] :
% 0.19/0.47 ( ( finite_finite2 @ B @ S2 )
% 0.19/0.47 => ( ! [Y5: B] :
% 0.19/0.47 ( ( member @ B @ Y5 @ S2 )
% 0.19/0.47 => ( ord_less_eq @ A @ ( F2 @ Y5 ) @ ( F2 @ X3 ) ) )
% 0.19/0.47 => ( ( P @ S2 )
% 0.19/0.47 => ( P @ ( insert @ B @ X3 @ S2 ) ) ) ) )
% 0.19/0.47 => ( P @ S ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_ranking_induct
% 0.19/0.47 thf(fact_154_inj__on__subset,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F2: A > B,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( inj_on @ A @ B @ F2 @ A5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
% 0.19/0.47 => ( inj_on @ A @ B @ F2 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % inj_on_subset
% 0.19/0.47 thf(fact_155_subset__inj__on,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F2: A > B,B5: set @ A,A5: set @ A] :
% 0.19/0.47 ( ( inj_on @ A @ B @ F2 @ B5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( inj_on @ A @ B @ F2 @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_inj_on
% 0.19/0.47 thf(fact_156_insert__subsetI,axiom,
% 0.19/0.47 ! [A: $tType,X2: A,A5: set @ A,X5: set @ A] :
% 0.19/0.47 ( ( member @ A @ X2 @ A5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ X5 @ A5 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ X5 ) @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_subsetI
% 0.19/0.47 thf(fact_157_subset__emptyI,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( ! [X3: A] :
% 0.19/0.47 ~ ( member @ A @ X3 @ A5 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_emptyI
% 0.19/0.47 thf(fact_158_bot__nat__0_Oextremum,axiom,
% 0.19/0.47 ! [A3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A3 ) ).
% 0.19/0.47
% 0.19/0.47 % bot_nat_0.extremum
% 0.19/0.47 thf(fact_159_le0,axiom,
% 0.19/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.19/0.47
% 0.19/0.47 % le0
% 0.19/0.47 thf(fact_160_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.19/0.47 ! [A3: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
% 0.19/0.47 => ( A3
% 0.19/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_nat_0.extremum_uniqueI
% 0.19/0.47 thf(fact_161_bot__nat__0_Oextremum__unique,axiom,
% 0.19/0.47 ! [A3: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
% 0.19/0.47 = ( A3
% 0.19/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_nat_0.extremum_unique
% 0.19/0.47 thf(fact_162_le__0__eq,axiom,
% 0.19/0.47 ! [N: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.19/0.47 = ( N
% 0.19/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_0_eq
% 0.19/0.47 thf(fact_163_less__eq__nat_Osimps_I1_J,axiom,
% 0.19/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.19/0.47
% 0.19/0.47 % less_eq_nat.simps(1)
% 0.19/0.47 thf(fact_164_finite__if__finite__subsets__card__bdd,axiom,
% 0.19/0.47 ! [A: $tType,F: set @ A,C4: nat] :
% 0.19/0.47 ( ! [G3: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ G3 @ F )
% 0.19/0.47 => ( ( finite_finite2 @ A @ G3 )
% 0.19/0.47 => ( ord_less_eq @ nat @ ( finite_card @ A @ G3 ) @ C4 ) ) )
% 0.19/0.47 => ( ( finite_finite2 @ A @ F )
% 0.19/0.47 & ( ord_less_eq @ nat @ ( finite_card @ A @ F ) @ C4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_if_finite_subsets_card_bdd
% 0.19/0.47 thf(fact_165_card__seteq,axiom,
% 0.19/0.47 ! [A: $tType,B5: set @ A,A5: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ B5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ B5 ) @ ( finite_card @ A @ A5 ) )
% 0.19/0.47 => ( A5 = B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_seteq
% 0.19/0.47 thf(fact_166_card__mono,axiom,
% 0.19/0.47 ! [A: $tType,B5: set @ A,A5: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ B5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_mono
% 0.19/0.47 thf(fact_167_card__insert__le,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,X2: A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A5 )
% 0.19/0.47 => ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ ( insert @ A @ X2 @ A5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_insert_le
% 0.19/0.47 thf(fact_168_inj__on__inverseI,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,A5: set @ A,G: B > A,F2: A > B] :
% 0.19/0.47 ( ! [X3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ A5 )
% 0.19/0.47 => ( ( G @ ( F2 @ X3 ) )
% 0.19/0.47 = X3 ) )
% 0.19/0.47 => ( inj_on @ A @ B @ F2 @ A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % inj_on_inverseI
% 0.19/0.47 thf(fact_169_inj__on__contraD,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F2: A > B,A5: set @ A,X2: A,Y2: A] :
% 0.19/0.47 ( ( inj_on @ A @ B @ F2 @ A5 )
% 0.19/0.47 => ( ( X2 != Y2 )
% 0.19/0.47 => ( ( member @ A @ X2 @ A5 )
% 0.19/0.47 => ( ( member @ A @ Y2 @ A5 )
% 0.19/0.47 => ( ( F2 @ X2 )
% 0.19/0.47 != ( F2 @ Y2 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % inj_on_contraD
% 0.19/0.47 thf(fact_170_inj__on__eq__iff,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F2: A > B,A5: set @ A,X2: A,Y2: A] :
% 0.19/0.47 ( ( inj_on @ A @ B @ F2 @ A5 )
% 0.19/0.47 => ( ( member @ A @ X2 @ A5 )
% 0.19/0.47 => ( ( member @ A @ Y2 @ A5 )
% 0.19/0.47 => ( ( ( F2 @ X2 )
% 0.19/0.47 = ( F2 @ Y2 ) )
% 0.19/0.47 = ( X2 = Y2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % inj_on_eq_iff
% 0.19/0.47 thf(fact_171_inj__on__cong,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,A5: set @ A,F2: A > B,G: A > B] :
% 0.19/0.47 ( ! [A4: A] :
% 0.19/0.47 ( ( member @ A @ A4 @ A5 )
% 0.19/0.47 => ( ( F2 @ A4 )
% 0.19/0.47 = ( G @ A4 ) ) )
% 0.19/0.47 => ( ( inj_on @ A @ B @ F2 @ A5 )
% 0.19/0.47 = ( inj_on @ A @ B @ G @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % inj_on_cong
% 0.19/0.47 thf(fact_172_inj__on__def,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( inj_on @ A @ B )
% 0.19/0.47 = ( ^ [F5: A > B,A7: set @ A] :
% 0.19/0.47 ! [X: A] :
% 0.19/0.47 ( ( member @ A @ X @ A7 )
% 0.19/0.47 => ! [Y4: A] :
% 0.19/0.47 ( ( member @ A @ Y4 @ A7 )
% 0.19/0.47 => ( ( ( F5 @ X )
% 0.19/0.47 = ( F5 @ Y4 ) )
% 0.19/0.47 => ( X = Y4 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % inj_on_def
% 0.19/0.47 thf(fact_173_inj__onI,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,A5: set @ A,F2: A > B] :
% 0.19/0.47 ( ! [X3: A,Y3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ A5 )
% 0.19/0.47 => ( ( member @ A @ Y3 @ A5 )
% 0.19/0.47 => ( ( ( F2 @ X3 )
% 0.19/0.47 = ( F2 @ Y3 ) )
% 0.19/0.47 => ( X3 = Y3 ) ) ) )
% 0.19/0.47 => ( inj_on @ A @ B @ F2 @ A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % inj_onI
% 0.19/0.47 thf(fact_174_inj__onD,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F2: A > B,A5: set @ A,X2: A,Y2: A] :
% 0.19/0.47 ( ( inj_on @ A @ B @ F2 @ A5 )
% 0.19/0.47 => ( ( ( F2 @ X2 )
% 0.19/0.47 = ( F2 @ Y2 ) )
% 0.19/0.47 => ( ( member @ A @ X2 @ A5 )
% 0.19/0.47 => ( ( member @ A @ Y2 @ A5 )
% 0.19/0.47 => ( X2 = Y2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % inj_onD
% 0.19/0.47 thf(fact_175_obtain__subset__with__card__n,axiom,
% 0.19/0.47 ! [A: $tType,N: nat,S: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S ) )
% 0.19/0.47 => ~ ! [T3: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ T3 @ S )
% 0.19/0.47 => ( ( ( finite_card @ A @ T3 )
% 0.19/0.47 = N )
% 0.19/0.47 => ~ ( finite_finite2 @ A @ T3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % obtain_subset_with_card_n
% 0.19/0.47 thf(fact_176_card__le__if__inj__on__rel,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,B5: set @ A,A5: set @ B,R2: B > A > $o] :
% 0.19/0.47 ( ( finite_finite2 @ A @ B5 )
% 0.19/0.47 => ( ! [A4: B] :
% 0.19/0.47 ( ( member @ B @ A4 @ A5 )
% 0.19/0.47 => ? [B8: A] :
% 0.19/0.47 ( ( member @ A @ B8 @ B5 )
% 0.19/0.47 & ( R2 @ A4 @ B8 ) ) )
% 0.19/0.47 => ( ! [A1: B,A22: B,B3: A] :
% 0.19/0.47 ( ( member @ B @ A1 @ A5 )
% 0.19/0.47 => ( ( member @ B @ A22 @ A5 )
% 0.19/0.47 => ( ( member @ A @ B3 @ B5 )
% 0.19/0.47 => ( ( R2 @ A1 @ B3 )
% 0.19/0.47 => ( ( R2 @ A22 @ B3 )
% 0.19/0.47 => ( A1 = A22 ) ) ) ) ) )
% 0.19/0.47 => ( ord_less_eq @ nat @ ( finite_card @ B @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_le_if_inj_on_rel
% 0.19/0.47 thf(fact_177_arg__min__least,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( linorder @ B )
% 0.19/0.47 => ! [S: set @ A,Y2: A,F2: A > B] :
% 0.19/0.47 ( ( finite_finite2 @ A @ S )
% 0.19/0.47 => ( ( S
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ( member @ A @ Y2 @ S )
% 0.19/0.47 => ( ord_less_eq @ B @ ( F2 @ ( lattic1704895705min_on @ A @ B @ F2 @ S ) ) @ ( F2 @ Y2 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % arg_min_least
% 0.19/0.47 thf(fact_178_bounded__Max__nat,axiom,
% 0.19/0.47 ! [P: nat > $o,X2: nat,M: nat] :
% 0.19/0.47 ( ( P @ X2 )
% 0.19/0.47 => ( ! [X3: nat] :
% 0.19/0.47 ( ( P @ X3 )
% 0.19/0.47 => ( ord_less_eq @ nat @ X3 @ M ) )
% 0.19/0.47 => ~ ! [M2: nat] :
% 0.19/0.47 ( ( P @ M2 )
% 0.19/0.47 => ~ ! [X4: nat] :
% 0.19/0.47 ( ( P @ X4 )
% 0.19/0.47 => ( ord_less_eq @ nat @ X4 @ M2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bounded_Max_nat
% 0.19/0.47 thf(fact_179_ex__has__least__nat,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o,K: A,M3: A > nat] :
% 0.19/0.47 ( ( P @ K )
% 0.19/0.47 => ? [X3: A] :
% 0.19/0.47 ( ( P @ X3 )
% 0.19/0.47 & ! [Y5: A] :
% 0.19/0.47 ( ( P @ Y5 )
% 0.19/0.47 => ( ord_less_eq @ nat @ ( M3 @ X3 ) @ ( M3 @ Y5 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ex_has_least_nat
% 0.19/0.47 thf(fact_180_Nat_Oex__has__greatest__nat,axiom,
% 0.19/0.47 ! [P: nat > $o,K: nat,B4: nat] :
% 0.19/0.47 ( ( P @ K )
% 0.19/0.47 => ( ! [Y3: nat] :
% 0.19/0.47 ( ( P @ Y3 )
% 0.19/0.47 => ( ord_less_eq @ nat @ Y3 @ B4 ) )
% 0.19/0.47 => ? [X3: nat] :
% 0.19/0.47 ( ( P @ X3 )
% 0.19/0.47 & ! [Y5: nat] :
% 0.19/0.47 ( ( P @ Y5 )
% 0.19/0.47 => ( ord_less_eq @ nat @ Y5 @ X3 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Nat.ex_has_greatest_nat
% 0.19/0.47 thf(fact_181_nat__le__linear,axiom,
% 0.19/0.47 ! [M3: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ M3 @ N )
% 0.19/0.47 | ( ord_less_eq @ nat @ N @ M3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_le_linear
% 0.19/0.47 thf(fact_182_le__antisym,axiom,
% 0.19/0.47 ! [M3: nat,N: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ M3 @ N )
% 0.19/0.47 => ( ( ord_less_eq @ nat @ N @ M3 )
% 0.19/0.47 => ( M3 = N ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_antisym
% 0.19/0.47 thf(fact_183_eq__imp__le,axiom,
% 0.19/0.47 ! [M3: nat,N: nat] :
% 0.19/0.47 ( ( M3 = N )
% 0.19/0.47 => ( ord_less_eq @ nat @ M3 @ N ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_imp_le
% 0.19/0.47 thf(fact_184_le__trans,axiom,
% 0.19/0.47 ! [I: nat,J: nat,K: nat] :
% 0.19/0.47 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.47 => ( ( ord_less_eq @ nat @ J @ K )
% 0.19/0.47 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_trans
% 0.19/0.47 thf(fact_185_le__refl,axiom,
% 0.19/0.47 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.19/0.47
% 0.19/0.47 % le_refl
% 0.19/0.47 thf(fact_186_finite__nat__set__iff__bounded__le,axiom,
% 0.19/0.47 ( ( finite_finite2 @ nat )
% 0.19/0.47 = ( ^ [N2: set @ nat] :
% 0.19/0.47 ? [M4: nat] :
% 0.19/0.47 ! [X: nat] :
% 0.19/0.47 ( ( member @ nat @ X @ N2 )
% 0.19/0.47 => ( ord_less_eq @ nat @ X @ M4 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_nat_set_iff_bounded_le
% 0.19/0.47 thf(fact_187_arg__min__if__finite_I1_J,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( order @ B )
% 0.19/0.47 => ! [S: set @ A,F2: A > B] :
% 0.19/0.47 ( ( finite_finite2 @ A @ S )
% 0.19/0.47 => ( ( S
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( member @ A @ ( lattic1704895705min_on @ A @ B @ F2 @ S ) @ S ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % arg_min_if_finite(1)
% 0.19/0.47 thf(fact_188_card__Diff1__le,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,X2: A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A5 )
% 0.19/0.47 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_Diff1_le
% 0.19/0.47 thf(fact_189_card__le__inj,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,A5: set @ A,B5: set @ B] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A5 )
% 0.19/0.47 => ( ( finite_finite2 @ B @ B5 )
% 0.19/0.47 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B5 ) )
% 0.19/0.47 => ? [F4: A > B] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A5 ) @ B5 )
% 0.19/0.47 & ( inj_on @ A @ B @ F4 @ A5 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_le_inj
% 0.19/0.47 thf(fact_190_card__inj__on__le,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F2: A > B,A5: set @ A,B5: set @ B] :
% 0.19/0.47 ( ( inj_on @ A @ B @ F2 @ A5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A5 ) @ B5 )
% 0.19/0.47 => ( ( finite_finite2 @ B @ B5 )
% 0.19/0.47 => ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B5 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_inj_on_le
% 0.19/0.47 thf(fact_191_image__eqI,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,B4: A,F2: B > A,X2: B,A5: set @ B] :
% 0.19/0.47 ( ( B4
% 0.19/0.47 = ( F2 @ X2 ) )
% 0.19/0.47 => ( ( member @ B @ X2 @ A5 )
% 0.19/0.47 => ( member @ A @ B4 @ ( image @ B @ A @ F2 @ A5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_eqI
% 0.19/0.47 thf(fact_192_DiffI,axiom,
% 0.19/0.47 ! [A: $tType,C: A,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( member @ A @ C @ A5 )
% 0.19/0.47 => ( ~ ( member @ A @ C @ B5 )
% 0.19/0.47 => ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % DiffI
% 0.19/0.47 thf(fact_193_Diff__iff,axiom,
% 0.19/0.47 ! [A: $tType,C: A,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
% 0.19/0.47 = ( ( member @ A @ C @ A5 )
% 0.19/0.47 & ~ ( member @ A @ C @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_iff
% 0.19/0.47 thf(fact_194_Diff__idemp,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ B5 )
% 0.19/0.47 = ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_idemp
% 0.19/0.47 thf(fact_195_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( cancel1352612707id_add @ A )
% 0.19/0.47 => ! [A3: A] :
% 0.19/0.47 ( ( minus_minus @ A @ A3 @ A3 )
% 0.19/0.47 = ( zero_zero @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % cancel_comm_monoid_add_class.diff_cancel
% 0.19/0.47 thf(fact_196_diff__zero,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( cancel1352612707id_add @ A )
% 0.19/0.47 => ! [A3: A] :
% 0.19/0.47 ( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.47 = A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_zero
% 0.19/0.47 thf(fact_197_zero__diff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( comm_monoid_diff @ A )
% 0.19/0.47 => ! [A3: A] :
% 0.19/0.47 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.47 = ( zero_zero @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % zero_diff
% 0.19/0.47 thf(fact_198_diff__0__right,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( group_add @ A )
% 0.19/0.47 => ! [A3: A] :
% 0.19/0.47 ( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.47 = A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_0_right
% 0.19/0.47 thf(fact_199_diff__self,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( group_add @ A )
% 0.19/0.47 => ! [A3: A] :
% 0.19/0.47 ( ( minus_minus @ A @ A3 @ A3 )
% 0.19/0.47 = ( zero_zero @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_self
% 0.19/0.47 thf(fact_200_image__empty,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F2: B > A] :
% 0.19/0.47 ( ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) )
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_empty
% 0.19/0.47 thf(fact_201_empty__is__image,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F2: B > A,A5: set @ B] :
% 0.19/0.47 ( ( ( bot_bot @ ( set @ A ) )
% 0.19/0.47 = ( image @ B @ A @ F2 @ A5 ) )
% 0.19/0.47 = ( A5
% 0.19/0.47 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % empty_is_image
% 0.19/0.47 thf(fact_202_image__is__empty,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F2: B > A,A5: set @ B] :
% 0.19/0.47 ( ( ( image @ B @ A @ F2 @ A5 )
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = ( A5
% 0.19/0.47 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_is_empty
% 0.19/0.47 thf(fact_203_finite__imageI,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F: set @ A,H: A > B] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F )
% 0.19/0.47 => ( finite_finite2 @ B @ ( image @ A @ B @ H @ F ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_imageI
% 0.19/0.47 thf(fact_204_image__insert,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F2: B > A,A3: B,B5: set @ B] :
% 0.19/0.47 ( ( image @ B @ A @ F2 @ ( insert @ B @ A3 @ B5 ) )
% 0.19/0.47 = ( insert @ A @ ( F2 @ A3 ) @ ( image @ B @ A @ F2 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_insert
% 0.19/0.47 thf(fact_205_insert__image,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,X2: A,A5: set @ A,F2: A > B] :
% 0.19/0.47 ( ( member @ A @ X2 @ A5 )
% 0.19/0.47 => ( ( insert @ B @ ( F2 @ X2 ) @ ( image @ A @ B @ F2 @ A5 ) )
% 0.19/0.47 = ( image @ A @ B @ F2 @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_image
% 0.19/0.47 thf(fact_206_Diff__empty,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( ( minus_minus @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = A5 ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_empty
% 0.19/0.47 thf(fact_207_empty__Diff,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 )
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % empty_Diff
% 0.19/0.47 thf(fact_208_Diff__cancel,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A] :
% 0.19/0.47 ( ( minus_minus @ ( set @ A ) @ A5 @ A5 )
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_cancel
% 0.19/0.47 thf(fact_209_finite__Diff,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A5 )
% 0.19/0.47 => ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_Diff
% 0.19/0.47 thf(fact_210_finite__Diff2,axiom,
% 0.19/0.47 ! [A: $tType,B5: set @ A,A5: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ B5 )
% 0.19/0.47 => ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
% 0.19/0.47 = ( finite_finite2 @ A @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_Diff2
% 0.19/0.47 thf(fact_211_Diff__insert0,axiom,
% 0.19/0.47 ! [A: $tType,X2: A,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ~ ( member @ A @ X2 @ A5 )
% 0.19/0.47 => ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ B5 ) )
% 0.19/0.47 = ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_insert0
% 0.19/0.47 thf(fact_212_insert__Diff1,axiom,
% 0.19/0.47 ! [A: $tType,X2: A,B5: set @ A,A5: set @ A] :
% 0.19/0.47 ( ( member @ A @ X2 @ B5 )
% 0.19/0.47 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A5 ) @ B5 )
% 0.19/0.47 = ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_Diff1
% 0.19/0.47 thf(fact_213_diff__ge__0__iff__ge,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A )
% 0.19/0.47 => ! [A3: A,B4: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B4 ) )
% 0.19/0.47 = ( ord_less_eq @ A @ B4 @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_ge_0_iff_ge
% 0.19/0.47 thf(fact_214_Diff__eq__empty__iff,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( ( minus_minus @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_eq_empty_iff
% 0.19/0.47 thf(fact_215_insert__Diff__single,axiom,
% 0.19/0.47 ! [A: $tType,A3: A,A5: set @ A] :
% 0.19/0.47 ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.19/0.47 = ( insert @ A @ A3 @ A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_Diff_single
% 0.19/0.47 thf(fact_216_finite__Diff__insert,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,A3: A,B5: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ B5 ) ) )
% 0.19/0.47 = ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_Diff_insert
% 0.19/0.47 thf(fact_217_inj__on__insert,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F2: A > B,A3: A,A5: set @ A] :
% 0.19/0.47 ( ( inj_on @ A @ B @ F2 @ ( insert @ A @ A3 @ A5 ) )
% 0.19/0.47 = ( ( inj_on @ A @ B @ F2 @ A5 )
% 0.19/0.47 & ~ ( member @ B @ ( F2 @ A3 ) @ ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % inj_on_insert
% 0.19/0.47 thf(fact_218_image__diff__subset,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F2: B > A,A5: set @ B,B5: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A5 ) @ ( image @ B @ A @ F2 @ B5 ) ) @ ( image @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A5 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_diff_subset
% 0.19/0.47 thf(fact_219_inj__on__image__set__diff,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F2: A > B,C4: set @ A,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( inj_on @ A @ B @ F2 @ C4 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ C4 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 0.19/0.47 => ( ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
% 0.19/0.47 = ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A5 ) @ ( image @ A @ B @ F2 @ B5 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % inj_on_image_set_diff
% 0.19/0.47 thf(fact_220_diff__mono,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A )
% 0.19/0.47 => ! [A3: A,B4: A,D2: A,C: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ D2 @ C )
% 0.19/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B4 @ D2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_mono
% 0.19/0.47 thf(fact_221_diff__left__mono,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A )
% 0.19/0.47 => ! [B4: A,A3: A,C: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ B4 @ A3 )
% 0.19/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B4 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_left_mono
% 0.19/0.47 thf(fact_222_diff__right__mono,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A )
% 0.19/0.47 => ! [A3: A,B4: A,C: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.19/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B4 @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_right_mono
% 0.19/0.47 thf(fact_223_diff__eq__diff__less__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A )
% 0.19/0.47 => ! [A3: A,B4: A,C: A,D2: A] :
% 0.19/0.47 ( ( ( minus_minus @ A @ A3 @ B4 )
% 0.19/0.47 = ( minus_minus @ A @ C @ D2 ) )
% 0.19/0.47 => ( ( ord_less_eq @ A @ A3 @ B4 )
% 0.19/0.47 = ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_eq_diff_less_eq
% 0.19/0.47 thf(fact_224_all__subset__image,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F2: B > A,A5: set @ B,P: ( set @ A ) > $o] :
% 0.19/0.47 ( ( ! [B7: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F2 @ A5 ) )
% 0.19/0.47 => ( P @ B7 ) ) )
% 0.19/0.47 = ( ! [B7: set @ B] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ B ) @ B7 @ A5 )
% 0.19/0.47 => ( P @ ( image @ B @ A @ F2 @ B7 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % all_subset_image
% 0.19/0.47 thf(fact_225_image__mono,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,A5: set @ A,B5: set @ A,F2: A > B] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A5 ) @ ( image @ A @ B @ F2 @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_mono
% 0.19/0.47 thf(fact_226_image__subsetI,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,A5: set @ A,F2: A > B,B5: set @ B] :
% 0.19/0.47 ( ! [X3: A] :
% 0.19/0.47 ( ( member @ A @ X3 @ A5 )
% 0.19/0.47 => ( member @ B @ ( F2 @ X3 ) @ B5 ) )
% 0.19/0.47 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A5 ) @ B5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subsetI
% 0.19/0.47 thf(fact_227_subset__imageE,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,B5: set @ A,F2: B > A,A5: set @ B] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F2 @ A5 ) )
% 0.19/0.47 => ~ ! [C5: set @ B] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A5 )
% 0.19/0.47 => ( B5
% 0.19/0.47 != ( image @ B @ A @ F2 @ C5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_imageE
% 0.19/0.47 thf(fact_228_image__subset__iff,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F2: B > A,A5: set @ B,B5: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A5 ) @ B5 )
% 0.19/0.47 = ( ! [X: B] :
% 0.19/0.47 ( ( member @ B @ X @ A5 )
% 0.19/0.47 => ( member @ A @ ( F2 @ X ) @ B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subset_iff
% 0.19/0.47 thf(fact_229_subset__image__iff,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,B5: set @ A,F2: B > A,A5: set @ B] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F2 @ A5 ) )
% 0.19/0.47 = ( ? [AA: set @ B] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ B ) @ AA @ A5 )
% 0.19/0.47 & ( B5
% 0.19/0.47 = ( image @ B @ A @ F2 @ AA ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_image_iff
% 0.19/0.47 thf(fact_230_Diff__mono,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,C4: set @ A,D3: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ D3 @ B5 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_mono
% 0.19/0.47 thf(fact_231_Diff__subset,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ A5 ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_subset
% 0.19/0.47 thf(fact_232_double__diff,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 0.19/0.47 => ( ( minus_minus @ ( set @ A ) @ B5 @ ( minus_minus @ ( set @ A ) @ C4 @ A5 ) )
% 0.19/0.47 = A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % double_diff
% 0.19/0.47 thf(fact_233_le__iff__diff__le__0,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A )
% 0.19/0.47 => ( ( ord_less_eq @ A )
% 0.19/0.47 = ( ^ [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_iff_diff_le_0
% 0.19/0.47 thf(fact_234_finite__surj,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,A5: set @ A,B5: set @ B,F2: A > B] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( image @ A @ B @ F2 @ A5 ) )
% 0.19/0.47 => ( finite_finite2 @ B @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_surj
% 0.19/0.47 thf(fact_235_finite__subset__image,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,B5: set @ A,F2: B > A,A5: set @ B] :
% 0.19/0.47 ( ( finite_finite2 @ A @ B5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F2 @ A5 ) )
% 0.19/0.47 => ? [C5: set @ B] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A5 )
% 0.19/0.47 & ( finite_finite2 @ B @ C5 )
% 0.19/0.47 & ( B5
% 0.19/0.47 = ( image @ B @ A @ F2 @ C5 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_subset_image
% 0.19/0.47 thf(fact_236_ex__finite__subset__image,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F2: B > A,A5: set @ B,P: ( set @ A ) > $o] :
% 0.19/0.47 ( ( ? [B7: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ B7 )
% 0.19/0.47 & ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F2 @ A5 ) )
% 0.19/0.47 & ( P @ B7 ) ) )
% 0.19/0.47 = ( ? [B7: set @ B] :
% 0.19/0.47 ( ( finite_finite2 @ B @ B7 )
% 0.19/0.47 & ( ord_less_eq @ ( set @ B ) @ B7 @ A5 )
% 0.19/0.47 & ( P @ ( image @ B @ A @ F2 @ B7 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ex_finite_subset_image
% 0.19/0.47 thf(fact_237_all__finite__subset__image,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F2: B > A,A5: set @ B,P: ( set @ A ) > $o] :
% 0.19/0.47 ( ( ! [B7: set @ A] :
% 0.19/0.47 ( ( ( finite_finite2 @ A @ B7 )
% 0.19/0.47 & ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F2 @ A5 ) ) )
% 0.19/0.47 => ( P @ B7 ) ) )
% 0.19/0.47 = ( ! [B7: set @ B] :
% 0.19/0.47 ( ( ( finite_finite2 @ B @ B7 )
% 0.19/0.47 & ( ord_less_eq @ ( set @ B ) @ B7 @ A5 ) )
% 0.19/0.47 => ( P @ ( image @ B @ A @ F2 @ B7 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % all_finite_subset_image
% 0.19/0.47 thf(fact_238_finite__imageD,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,F2: B > A,A5: set @ B] :
% 0.19/0.47 ( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ A5 ) )
% 0.19/0.47 => ( ( inj_on @ B @ A @ F2 @ A5 )
% 0.19/0.47 => ( finite_finite2 @ B @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_imageD
% 0.19/0.47 thf(fact_239_finite__image__iff,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F2: A > B,A5: set @ A] :
% 0.19/0.47 ( ( inj_on @ A @ B @ F2 @ A5 )
% 0.19/0.47 => ( ( finite_finite2 @ B @ ( image @ A @ B @ F2 @ A5 ) )
% 0.19/0.47 = ( finite_finite2 @ A @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_image_iff
% 0.19/0.47 thf(fact_240_inj__on__image__eq__iff,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F2: A > B,C4: set @ A,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( inj_on @ A @ B @ F2 @ C4 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 0.19/0.47 => ( ( ( image @ A @ B @ F2 @ A5 )
% 0.19/0.47 = ( image @ A @ B @ F2 @ B5 ) )
% 0.19/0.47 = ( A5 = B5 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % inj_on_image_eq_iff
% 0.19/0.47 thf(fact_241_inj__on__image__mem__iff,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F2: A > B,B5: set @ A,A3: A,A5: set @ A] :
% 0.19/0.47 ( ( inj_on @ A @ B @ F2 @ B5 )
% 0.19/0.47 => ( ( member @ A @ A3 @ B5 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.19/0.47 => ( ( member @ B @ ( F2 @ A3 ) @ ( image @ A @ B @ F2 @ A5 ) )
% 0.19/0.47 = ( member @ A @ A3 @ A5 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % inj_on_image_mem_iff
% 0.19/0.47 thf(fact_242_inj__img__insertE,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F2: A > B,A5: set @ A,X2: B,B5: set @ B] :
% 0.19/0.47 ( ( inj_on @ A @ B @ F2 @ A5 )
% 0.19/0.47 => ( ~ ( member @ B @ X2 @ B5 )
% 0.19/0.47 => ( ( ( insert @ B @ X2 @ B5 )
% 0.19/0.47 = ( image @ A @ B @ F2 @ A5 ) )
% 0.19/0.47 => ~ ! [X6: A,A8: set @ A] :
% 0.19/0.47 ( ~ ( member @ A @ X6 @ A8 )
% 0.19/0.47 => ( ( A5
% 0.19/0.47 = ( insert @ A @ X6 @ A8 ) )
% 0.19/0.47 => ( ( X2
% 0.19/0.47 = ( F2 @ X6 ) )
% 0.19/0.47 => ( B5
% 0.19/0.47 != ( image @ A @ B @ F2 @ A8 ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % inj_img_insertE
% 0.19/0.47 thf(fact_243_card__image,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F2: A > B,A5: set @ A] :
% 0.19/0.47 ( ( inj_on @ A @ B @ F2 @ A5 )
% 0.19/0.47 => ( ( finite_card @ B @ ( image @ A @ B @ F2 @ A5 ) )
% 0.19/0.47 = ( finite_card @ A @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_image
% 0.19/0.47 thf(fact_244_Diff__insert__absorb,axiom,
% 0.19/0.47 ! [A: $tType,X2: A,A5: set @ A] :
% 0.19/0.47 ( ~ ( member @ A @ X2 @ A5 )
% 0.19/0.47 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A5 ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 = A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_insert_absorb
% 0.19/0.47 thf(fact_245_Diff__insert2,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,A3: A,B5: set @ A] :
% 0.19/0.47 ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ B5 ) )
% 0.19/0.47 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_insert2
% 0.19/0.47 thf(fact_246_insert__Diff,axiom,
% 0.19/0.47 ! [A: $tType,A3: A,A5: set @ A] :
% 0.19/0.47 ( ( member @ A @ A3 @ A5 )
% 0.19/0.47 => ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.19/0.47 = A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_Diff
% 0.19/0.47 thf(fact_247_Diff__insert,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,A3: A,B5: set @ A] :
% 0.19/0.47 ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ B5 ) )
% 0.19/0.47 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_insert
% 0.19/0.47 thf(fact_248_subset__Diff__insert,axiom,
% 0.19/0.47 ! [A: $tType,A5: set @ A,B5: set @ A,X2: A,C4: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B5 @ ( insert @ A @ X2 @ C4 ) ) )
% 0.19/0.47 = ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B5 @ C4 ) )
% 0.19/0.47 & ~ ( member @ A @ X2 @ A5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_Diff_insert
% 0.19/0.47 thf(fact_249_in__image__insert__iff,axiom,
% 0.19/0.47 ! [A: $tType,B5: set @ ( set @ A ),X2: A,A5: set @ A] :
% 0.19/0.47 ( ! [C5: set @ A] :
% 0.19/0.47 ( ( member @ ( set @ A ) @ C5 @ B5 )
% 0.19/0.47 => ~ ( member @ A @ X2 @ C5 ) )
% 0.19/0.47 => ( ( member @ ( set @ A ) @ A5 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X2 ) @ B5 ) )
% 0.19/0.47 = ( ( member @ A @ X2 @ A5 )
% 0.19/0.47 & ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % in_image_insert_iff
% 0.19/0.47 thf(fact_250_diff__eq__diff__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( group_add @ A )
% 0.19/0.47 => ! [A3: A,B4: A,C: A,D2: A] :
% 0.19/0.47 ( ( ( minus_minus @ A @ A3 @ B4 )
% 0.19/0.47 = ( minus_minus @ A @ C @ D2 ) )
% 0.19/0.47 => ( ( A3 = B4 )
% 0.19/0.47 = ( C = D2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_eq_diff_eq
% 0.19/0.47 thf(fact_251_diff__right__commute,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( cancel146912293up_add @ A )
% 0.19/0.47 => ! [A3: A,C: A,B4: A] :
% 0.19/0.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C ) @ B4 )
% 0.19/0.47 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B4 ) @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_right_commute
% 0.19/0.47 thf(fact_252_DiffE,axiom,
% 0.19/0.47 ! [A: $tType,C: A,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
% 0.19/0.47 => ~ ( ( member @ A @ C @ A5 )
% 0.19/0.47 => ( member @ A @ C @ B5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % DiffE
% 0.19/0.47 thf(fact_253_DiffD1,axiom,
% 0.19/0.47 ! [A: $tType,C: A,A5: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
% 0.19/0.47 => ( member @ A @ C @ A5 ) ) ).
% 0.19/0.47
% 0.19/0.47 % DiffD1
% 0.19/0.47
% 0.19/0.47 % Type constructors (35)
% 0.19/0.47 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( order_bot @ A10 )
% 0.19/0.47 => ( order_bot @ ( A9 > A10 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( preorder @ A10 )
% 0.19/0.47 => ( preorder @ ( A9 > A10 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ A9 )
% 0.19/0.47 & ( finite_finite @ A10 ) )
% 0.19/0.47 => ( finite_finite @ ( A9 > A10 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( order @ A10 )
% 0.19/0.47 => ( order @ ( A9 > A10 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( ord @ A10 )
% 0.19/0.47 => ( ord @ ( A9 > A10 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_fun___Orderings_Obot,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( bot @ A10 )
% 0.19/0.47 => ( bot @ ( A9 > A10 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Nat_Onat___Parity_Ounique__euclidean__semiring__with__bit__shifts,axiom,
% 0.19/0.47 unique788075200shifts @ nat ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.19/0.47 canoni770627133id_add @ nat ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 0.19/0.47 linord1659791738miring @ nat ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.19/0.47 cancel146912293up_add @ nat ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 0.19/0.47 cancel1352612707id_add @ nat ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Nat_Onat___Parity_Osemiring__bit__shifts,axiom,
% 0.19/0.47 semiring_bit_shifts @ nat ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 0.19/0.47 comm_monoid_diff @ nat ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
% 0.19/0.47 order_bot @ nat ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
% 0.19/0.47 preorder @ nat ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.19/0.47 linorder @ nat ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__1,axiom,
% 0.19/0.47 semiring_1 @ nat ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
% 0.19/0.47 order @ nat ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
% 0.19/0.47 ord @ nat ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
% 0.19/0.47 bot @ nat ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.19/0.47 zero @ nat ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
% 0.19/0.47 ! [A9: $tType] : ( order_bot @ ( set @ A9 ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 0.19/0.47 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Finite__Set_Ofinite_8,axiom,
% 0.19/0.47 ! [A9: $tType] :
% 0.19/0.47 ( ( finite_finite @ A9 )
% 0.19/0.47 => ( finite_finite @ ( set @ A9 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
% 0.19/0.47 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
% 0.19/0.47 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
% 0.19/0.47 ! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Orderings_Oorder__bot_12,axiom,
% 0.19/0.47 order_bot @ $o ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_13,axiom,
% 0.19/0.47 preorder @ $o ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_14,axiom,
% 0.19/0.47 linorder @ $o ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Finite__Set_Ofinite_15,axiom,
% 0.19/0.47 finite_finite @ $o ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
% 0.19/0.47 order @ $o ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Orderings_Oord_17,axiom,
% 0.19/0.47 ord @ $o ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Orderings_Obot_18,axiom,
% 0.19/0.47 bot @ $o ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ozero_19,axiom,
% 0.19/0.47 zero @ hF_Mirabelle_hf ).
% 0.19/0.47
% 0.19/0.47 % Conjectures (1)
% 0.19/0.47 thf(conj_0,conjecture,
% 0.19/0.47 ( ( z
% 0.19/0.47 = ( zero_zero @ hF_Mirabelle_hf ) )
% 0.19/0.47 = ( ! [X: hF_Mirabelle_hf] :
% 0.19/0.47 ~ ( hF_Mirabelle_hmem @ X @ z ) ) ) ).
% 0.19/0.47
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 ------------------------
% 2.63/1.52 ---- Embedded in THF ---
% 2.63/1.52 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.63/1.52 %%% Generated on Tue Feb 24 08:25:13 EST 2026
% 2.63/1.52 %%% using '$$dhol' embedding, version 1.3.0.
% 2.63/1.52 %%% Logic specification used:
% 2.63/1.52 %%% thf(spec, logic, $$dhol).
% 2.63/1.52
% 2.63/1.52 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.2mXs7qeb79/DTF2DTF32116.p
% 2.63/1.52 thf(ty_t_HF__Mirabelle__fsbjehakzm_Ohf, type, hF_Mirabelle_hf: $tType).
% 2.63/1.52 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.63/1.52 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.63/1.52 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Num_Oneg__numeral, type, neg_numeral: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Groups_Ocomm__monoid__diff, type, comm_monoid_diff: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Parity_Osemiring__bit__shifts, type, semiring_bit_shifts: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_cl_Parity_Ounique__euclidean__semiring__with__bit__shifts, type, unique788075200shifts: (!> [A:$tType]: ($o))).
% 2.63/1.52 thf(sy_c_Finite__Set_Ocard, type, finite_card: (!> [B:$tType]: (((set @ B) > nat)))).
% 2.63/1.52 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.63/1.52 thf(sy_c_Fun_Oinj__on, type, inj_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 2.63/1.52 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.63/1.52 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.63/1.52 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHF, type, hF_Mirabelle_HF: ((set @ hF_Mirabelle_hf) > hF_Mirabelle_hf)).
% 2.63/1.52 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohf_OAbs__hf, type, hF_Mirabelle_Abs_hf: (nat > hF_Mirabelle_hf)).
% 2.63/1.52 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfset, type, hF_Mirabelle_hfset: (hF_Mirabelle_hf > (set @ hF_Mirabelle_hf))).
% 2.63/1.52 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohinsert, type, hF_Mirabelle_hinsert: (hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf))).
% 2.63/1.52 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohmem, type, hF_Mirabelle_hmem: (hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))).
% 2.63/1.52 thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on, type, lattic1704895705min_on: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > B))))).
% 2.63/1.52 thf(sy_c_Nat_Osemiring__1__class_ONats, type, semiring_1_Nats: (!> [A:$tType]: ((set @ A)))).
% 2.63/1.52 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux, type, semiri532925092at_aux: (!> [A:$tType]: (((A > A) > (nat > (A > A)))))).
% 2.63/1.52 thf(sy_c_Num_Oneg__numeral__class_Odbl, type, neg_numeral_dbl: (!> [A:$tType]: ((A > A)))).
% 2.63/1.52 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.63/1.52 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.63/1.52 thf(sy_c_Parity_Osemiring__bit__shifts__class_Odrop__bit, type, semiri2097166173op_bit: (!> [A:$tType]: ((nat > (A > A))))).
% 2.63/1.52 thf(sy_c_Parity_Osemiring__bit__shifts__class_Opush__bit, type, semiri1924326578sh_bit: (!> [A:$tType]: ((nat > (A > A))))).
% 2.63/1.52 thf(sy_c_Relation_Oinv__imagep, type, inv_imagep: (!> [B:$tType,A:$tType]: (((B > (B > $o)) > ((A > B) > (A > (A > $o))))))).
% 2.63/1.52 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.63/1.52 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.63/1.52 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.63/1.52 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.63/1.52 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.63/1.52 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 2.63/1.52 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.63/1.52 thf(sy_v_z, type, z: hF_Mirabelle_hf).
% 2.63/1.52 thf(fact_0_hf__ext, axiom, ((^ [Y:hF_Mirabelle_hf,Z:hF_Mirabelle_hf]: ((Y = Z))) = (^ [A2:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: ((! [X:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ A2) = ((hF_Mirabelle_hmem @ X) @ B2)))))))).
% 2.63/1.52 thf(fact_1_hemptyE, axiom, (! [A3:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ A3) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 2.63/1.52 thf(fact_2_hmem__hempty, axiom, (! [A3:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ A3) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 2.63/1.52 thf(fact_3_hf__cases, axiom, (! [Y2:hF_Mirabelle_hf]: (((Y2 != (zero_zero @ hF_Mirabelle_hf)) => (~ ((! [A4:hF_Mirabelle_hf,B3:hF_Mirabelle_hf]: (((Y2 = ((hF_Mirabelle_hinsert @ A4) @ B3)) => ((hF_Mirabelle_hmem @ A4) @ B3)))))))))).
% 2.63/1.52 thf(fact_4_hmem__hinsert, axiom, (! [A3:hF_Mirabelle_hf,B4:hF_Mirabelle_hf,C:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ A3) @ ((hF_Mirabelle_hinsert @ B4) @ C)) = ((A3 = B4) | ((hF_Mirabelle_hmem @ A3) @ C)))))).
% 2.63/1.52 thf(fact_5_hmem__def, axiom, (hF_Mirabelle_hmem = (^ [A2:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: ((((member @ hF_Mirabelle_hf) @ A2) @ (hF_Mirabelle_hfset @ B2)))))).
% 2.63/1.52 thf(fact_6_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X2:A]: ((((zero_zero @ A) = X2) = (X2 = (zero_zero @ A))))))))).
% 2.63/1.52 thf(fact_7_Abs__hf__0, axiom, ((hF_Mirabelle_Abs_hf @ (zero_zero @ nat)) = (zero_zero @ hF_Mirabelle_hf))).
% 2.63/1.52 thf(fact_8_dbl__simps_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.63/1.52 thf(fact_9_Zero__hf__def, axiom, ((zero_zero @ hF_Mirabelle_hf) = (hF_Mirabelle_HF @ (bot_bot @ (set @ hF_Mirabelle_hf))))).
% 2.63/1.52 thf(fact_10_push__bit__of__0, axiom, (! [A:$tType]: (((semiring_bit_shifts @ A) => (! [N:nat]: (((((semiri1924326578sh_bit @ A) @ N) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.63/1.52 thf(fact_11_push__bit__eq__0__iff, axiom, (! [A:$tType]: (((unique788075200shifts @ A) => (! [N:nat,A3:A]: ((((((semiri1924326578sh_bit @ A) @ N) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 2.63/1.52 thf(fact_12_drop__bit__of__0, axiom, (! [A:$tType]: (((semiring_bit_shifts @ A) => (! [N:nat]: (((((semiri2097166173op_bit @ A) @ N) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.63/1.52 thf(fact_13_Nats__0, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((member @ A) @ (zero_zero @ A)) @ (semiring_1_Nats @ A)))))).
% 2.63/1.52 thf(fact_14_HF__hfset, axiom, (! [A3:hF_Mirabelle_hf]: (((hF_Mirabelle_HF @ (hF_Mirabelle_hfset @ A3)) = A3)))).
% 2.63/1.52 thf(fact_15_empty__iff, axiom, (! [A:$tType,C:A]: ((~ ((((member @ A) @ C) @ (bot_bot @ (set @ A)))))))).
% 2.63/1.52 thf(fact_16_all__not__in__conv, axiom, (! [A:$tType,A5:(set @ A)]: (((! [X:A]: ((~ ((((member @ A) @ X) @ A5))))) = (A5 = (bot_bot @ (set @ A))))))).
% 2.63/1.52 thf(fact_17_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X:A]: ((~ ((P @ X))))))))).
% 2.63/1.52 thf(fact_18_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X:A]: ((~ ((P @ X))))))))).
% 2.63/1.52 thf(fact_19_bot__apply, axiom, (! [C2:$tType,D:$tType]: (((bot @ C2) => ((bot_bot @ (D > C2)) = (^ [X:D]: ((bot_bot @ C2)))))))).
% 2.63/1.52 thf(fact_20_hinsert__def, axiom, (hF_Mirabelle_hinsert = (^ [A2:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: ((hF_Mirabelle_HF @ (((insert @ hF_Mirabelle_hf) @ A2) @ (hF_Mirabelle_hfset @ B2))))))).
% 2.63/1.52 thf(fact_21_hfset__HF, axiom, (! [A5:(set @ hF_Mirabelle_hf)]: ((((finite_finite2 @ hF_Mirabelle_hf) @ A5) => ((hF_Mirabelle_hfset @ (hF_Mirabelle_HF @ A5)) = A5))))).
% 2.63/1.52 thf(fact_22_zero__natural_Orsp, axiom, ((zero_zero @ nat) = (zero_zero @ nat))).
% 2.63/1.52 thf(fact_23_emptyE, axiom, (! [A:$tType,A3:A]: ((~ ((((member @ A) @ A3) @ (bot_bot @ (set @ A)))))))).
% 2.63/1.52 thf(fact_24_equals0D, axiom, (! [A:$tType,A5:(set @ A),A3:A]: (((A5 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A3) @ A5))))))).
% 2.63/1.52 thf(fact_25_insert__absorb2, axiom, (! [A:$tType,X2:A,A5:(set @ A)]: (((((insert @ A) @ X2) @ (((insert @ A) @ X2) @ A5)) = (((insert @ A) @ X2) @ A5))))).
% 2.63/1.52 thf(fact_26_insert__iff, axiom, (! [A:$tType,A3:A,B4:A,A5:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B4) @ A5)) = ((A3 = B4) | (((member @ A) @ A3) @ A5)))))).
% 2.63/1.52 thf(fact_27_insertCI, axiom, (! [A:$tType,A3:A,B5:(set @ A),B4:A]: ((((~ ((((member @ A) @ A3) @ B5))) => (A3 = B4)) => (((member @ A) @ A3) @ (((insert @ A) @ B4) @ B5)))))).
% 2.63/1.52 thf(fact_28_singletonI, axiom, (! [A:$tType,A3:A]: ((((member @ A) @ A3) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))).
% 2.63/1.52 thf(fact_29_mk__disjoint__insert, axiom, (! [A:$tType,A3:A,A5:(set @ A)]: (((((member @ A) @ A3) @ A5) => (? [B6:(set @ A)]: (((A5 = (((insert @ A) @ A3) @ B6)) & (~ ((((member @ A) @ A3) @ B6)))))))))).
% 2.63/1.52 thf(fact_30_insert__commute, axiom, (! [A:$tType,X2:A,Y2:A,A5:(set @ A)]: (((((insert @ A) @ X2) @ (((insert @ A) @ Y2) @ A5)) = (((insert @ A) @ Y2) @ (((insert @ A) @ X2) @ A5)))))).
% 2.63/1.52 thf(fact_31_insert__eq__iff, axiom, (! [A:$tType,A3:A,A5:(set @ A),B4:A,B5:(set @ A)]: (((~ ((((member @ A) @ A3) @ A5))) => ((~ ((((member @ A) @ B4) @ B5))) => (((((insert @ A) @ A3) @ A5) = (((insert @ A) @ B4) @ B5)) = (((A3 = B4) => (A5 = B5)) & ((A3 != B4) => (? [C3:(set @ A)]: (((A5 = (((insert @ A) @ B4) @ C3)) & ((~ ((((member @ A) @ B4) @ C3))) & ((B5 = (((insert @ A) @ A3) @ C3)) & (~ ((((member @ A) @ A3) @ C3)))))))))))))))).
% 2.63/1.52 thf(fact_32_insert__absorb, axiom, (! [A:$tType,A3:A,A5:(set @ A)]: (((((member @ A) @ A3) @ A5) => ((((insert @ A) @ A3) @ A5) = A5))))).
% 2.63/1.52 thf(fact_33_insert__ident, axiom, (! [A:$tType,X2:A,A5:(set @ A),B5:(set @ A)]: (((~ ((((member @ A) @ X2) @ A5))) => ((~ ((((member @ A) @ X2) @ B5))) => (((((insert @ A) @ X2) @ A5) = (((insert @ A) @ X2) @ B5)) = (A5 = B5))))))).
% 2.63/1.52 thf(fact_34_Set_Oset__insert, axiom, (! [A:$tType,X2:A,A5:(set @ A)]: (((((member @ A) @ X2) @ A5) => (~ ((! [B6:(set @ A)]: (((A5 = (((insert @ A) @ X2) @ B6)) => (((member @ A) @ X2) @ B6)))))))))).
% 2.63/1.52 thf(fact_35_insertI2, axiom, (! [A:$tType,A3:A,B5:(set @ A),B4:A]: (((((member @ A) @ A3) @ B5) => (((member @ A) @ A3) @ (((insert @ A) @ B4) @ B5)))))).
% 2.63/1.52 thf(fact_36_insertI1, axiom, (! [A:$tType,A3:A,B5:(set @ A)]: ((((member @ A) @ A3) @ (((insert @ A) @ A3) @ B5))))).
% 2.63/1.52 thf(fact_37_insertE, axiom, (! [A:$tType,A3:A,B4:A,A5:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B4) @ A5)) => ((A3 != B4) => (((member @ A) @ A3) @ A5)))))).
% 2.63/1.52 thf(fact_38_finite__cases, axiom, (! [A:$tType,F:(set @ A)]: ((((finite_finite2 @ A) @ F) => ((F != (bot_bot @ (set @ A))) => (~ ((! [A6:(set @ A),X3:A]: (((F = (((insert @ A) @ X3) @ A6)) => ((~ ((((member @ A) @ X3) @ A6))) => (~ (((finite_finite2 @ A) @ A6)))))))))))))).
% 2.63/1.52 thf(fact_39_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.63/1.52 thf(fact_40_singleton__inject, axiom, (! [A:$tType,A3:A,B4:A]: ((((((insert @ A) @ A3) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B4) @ (bot_bot @ (set @ A)))) => (A3 = B4))))).
% 2.63/1.52 thf(fact_41_insert__not__empty, axiom, (! [A:$tType,A3:A,A5:(set @ A)]: (((((insert @ A) @ A3) @ A5) != (bot_bot @ (set @ A)))))).
% 2.63/1.52 thf(fact_42_doubleton__eq__iff, axiom, (! [A:$tType,A3:A,B4:A,C:A,D2:A]: ((((((insert @ A) @ A3) @ (((insert @ A) @ B4) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C) @ (((insert @ A) @ D2) @ (bot_bot @ (set @ A))))) = (((A3 = C) & (B4 = D2)) | ((A3 = D2) & (B4 = C))))))).
% 2.63/1.52 thf(fact_43_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 2.63/1.52 thf(fact_44_Collect__mem__eq, axiom, (! [A:$tType,A5:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A5)))) = A5)))).
% 2.63/1.52 thf(fact_45_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.63/1.52 thf(fact_46_ext, axiom, (! [B:$tType,A:$tType,F2:(A > B),G:(A > B)]: (((! [X3:A]: (((F2 @ X3) = (G @ X3)))) => (F2 = G))))).
% 2.63/1.52 thf(fact_47_singleton__iff, axiom, (! [A:$tType,B4:A,A3:A]: (((((member @ A) @ B4) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) = (B4 = A3))))).
% 2.63/1.52 thf(fact_48_singletonD, axiom, (! [A:$tType,B4:A,A3:A]: (((((member @ A) @ B4) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) => (B4 = A3))))).
% 2.63/1.52 thf(fact_49_bot__nat__def, axiom, ((bot_bot @ nat) = (zero_zero @ nat))).
% 2.63/1.52 thf(fact_50_finite__hfset, axiom, (! [A3:hF_Mirabelle_hf]: (((finite_finite2 @ hF_Mirabelle_hf) @ (hF_Mirabelle_hfset @ A3))))).
% 2.63/1.52 thf(fact_51_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X:A]: ((bot_bot @ B)))))))).
% 2.63/1.52 thf(fact_52_ex__in__conv, axiom, (! [A:$tType,A5:(set @ A)]: (((? [X:A]: ((((member @ A) @ X) @ A5))) = (A5 != (bot_bot @ (set @ A))))))).
% 2.63/1.52 thf(fact_53_equals0I, axiom, (! [A:$tType,A5:(set @ A)]: (((! [Y3:A]: ((~ ((((member @ A) @ Y3) @ A5))))) => (A5 = (bot_bot @ (set @ A))))))).
% 2.63/1.52 thf(fact_54_finite__insert, axiom, (! [A:$tType,A3:A,A5:(set @ A)]: ((((finite_finite2 @ A) @ (((insert @ A) @ A3) @ A5)) = ((finite_finite2 @ A) @ A5))))).
% 2.63/1.52 thf(fact_55_finite_Ocases, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (~ ((! [A6:(set @ A)]: (((? [A4:A]: ((A3 = (((insert @ A) @ A4) @ A6)))) => (~ (((finite_finite2 @ A) @ A6))))))))))))).
% 2.63/1.52 thf(fact_56_finite_Osimps, axiom, (! [A:$tType]: (((finite_finite2 @ A) = (^ [A2:(set @ A)]: (((A2 = (bot_bot @ (set @ A))) | (? [A7:(set @ A),B2:A]: (((A2 = (((insert @ A) @ B2) @ A7)) & ((finite_finite2 @ A) @ A7))))))))))).
% 2.63/1.52 thf(fact_57_finite__induct, axiom, (! [A:$tType,F:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ F) => ((P @ (bot_bot @ (set @ A))) => ((! [X3:A,F3:(set @ A)]: ((((finite_finite2 @ A) @ F3) => ((~ ((((member @ A) @ X3) @ F3))) => ((P @ F3) => (P @ (((insert @ A) @ X3) @ F3))))))) => (P @ F))))))).
% 2.63/1.52 thf(fact_58_finite_Oinducts, axiom, (! [A:$tType,X2:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ X2) => ((P @ (bot_bot @ (set @ A))) => ((! [A6:(set @ A),A4:A]: ((((finite_finite2 @ A) @ A6) => ((P @ A6) => (P @ (((insert @ A) @ A4) @ A6)))))) => (P @ X2))))))).
% 2.63/1.52 thf(fact_59_finite__ne__induct, axiom, (! [A:$tType,F:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ F) => ((F != (bot_bot @ (set @ A))) => ((! [X3:A]: ((P @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))) => ((! [X3:A,F3:(set @ A)]: ((((finite_finite2 @ A) @ F3) => ((F3 != (bot_bot @ (set @ A))) => ((~ ((((member @ A) @ X3) @ F3))) => ((P @ F3) => (P @ (((insert @ A) @ X3) @ F3)))))))) => (P @ F)))))))).
% 2.63/1.52 thf(fact_60_infinite__finite__induct, axiom, (! [A:$tType,P:((set @ A) > $o),A5:(set @ A)]: (((! [A6:(set @ A)]: (((~ (((finite_finite2 @ A) @ A6))) => (P @ A6)))) => ((P @ (bot_bot @ (set @ A))) => ((! [X3:A,F3:(set @ A)]: ((((finite_finite2 @ A) @ F3) => ((~ ((((member @ A) @ X3) @ F3))) => ((P @ F3) => (P @ (((insert @ A) @ X3) @ F3))))))) => (P @ A5))))))).
% 2.63/1.52 thf(fact_61_the__elem__eq, axiom, (! [A:$tType,X2:A]: ((((the_elem @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) = X2)))).
% 2.63/1.52 thf(fact_62_finite__code, axiom, (! [A:$tType]: (((finite_finite @ A) => ((finite_finite2 @ A) = (^ [A7:(set @ A)]: ($true))))))).
% 2.63/1.52 thf(fact_63_finite__set__choice, axiom, (! [B:$tType,A:$tType,A5:(set @ A),P:(A > (B > $o))]: ((((finite_finite2 @ A) @ A5) => ((! [X3:A]: (((((member @ A) @ X3) @ A5) => (? [X_1:B]: (((P @ X3) @ X_1)))))) => (? [F4:(A > B)]: ((! [X4:A]: (((((member @ A) @ X4) @ A5) => ((P @ X4) @ (F4 @ X4)))))))))))).
% 2.63/1.52 thf(fact_64_finite, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A5:(set @ A)]: (((finite_finite2 @ A) @ A5))))))).
% 2.63/1.52 thf(fact_65_finite_OemptyI, axiom, (! [A:$tType]: (((finite_finite2 @ A) @ (bot_bot @ (set @ A)))))).
% 2.63/1.52 thf(fact_66_infinite__imp__nonempty, axiom, (! [A:$tType,S:(set @ A)]: (((~ (((finite_finite2 @ A) @ S))) => (S != (bot_bot @ (set @ A))))))).
% 2.63/1.52 thf(fact_67_finite_OinsertI, axiom, (! [A:$tType,A5:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A5) => ((finite_finite2 @ A) @ (((insert @ A) @ A3) @ A5)))))).
% 2.63/1.52 thf(fact_68_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X:A]: ((((member @ A) @ X) @ (bot_bot @ (set @ A))))))))).
% 2.63/1.52 thf(fact_69_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 2.63/1.52 thf(fact_70_is__singleton__the__elem, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A7:(set @ A)]: ((A7 = (((insert @ A) @ ((the_elem @ A) @ A7)) @ (bot_bot @ (set @ A)))))))))).
% 2.63/1.52 thf(fact_71_is__singletonI, axiom, (! [A:$tType,X2:A]: (((is_singleton @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))))).
% 2.63/1.52 thf(fact_72_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A7:(set @ A)]: ((A7 = (bot_bot @ (set @ A))))))))).
% 2.63/1.52 thf(fact_73_is__singleton__def, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A7:(set @ A)]: ((? [X:A]: ((A7 = (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))))).
% 2.63/1.52 thf(fact_74_is__singletonE, axiom, (! [A:$tType,A5:(set @ A)]: ((((is_singleton @ A) @ A5) => (~ ((! [X3:A]: ((A5 != (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))).
% 2.63/1.52 thf(fact_75_is__singletonI_H, axiom, (! [A:$tType,A5:(set @ A)]: (((A5 != (bot_bot @ (set @ A))) => ((! [X3:A,Y3:A]: (((((member @ A) @ X3) @ A5) => ((((member @ A) @ Y3) @ A5) => (X3 = Y3))))) => ((is_singleton @ A) @ A5)))))).
% 2.63/1.52 thf(fact_76_card__0__eq, axiom, (! [A:$tType,A5:(set @ A)]: ((((finite_finite2 @ A) @ A5) => ((((finite_card @ A) @ A5) = (zero_zero @ nat)) = (A5 = (bot_bot @ (set @ A)))))))).
% 2.63/1.52 thf(fact_77_finite__subset__induct, axiom, (! [A:$tType,F:(set @ A),A5:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ F) => ((((ord_less_eq @ (set @ A)) @ F) @ A5) => ((P @ (bot_bot @ (set @ A))) => ((! [A4:A,F3:(set @ A)]: ((((finite_finite2 @ A) @ F3) => ((((member @ A) @ A4) @ A5) => ((~ ((((member @ A) @ A4) @ F3))) => ((P @ F3) => (P @ (((insert @ A) @ A4) @ F3)))))))) => (P @ F)))))))).
% 2.63/1.52 thf(fact_78_finite__subset__induct_H, axiom, (! [A:$tType,F:(set @ A),A5:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ F) => ((((ord_less_eq @ (set @ A)) @ F) @ A5) => ((P @ (bot_bot @ (set @ A))) => ((! [A4:A,F3:(set @ A)]: ((((finite_finite2 @ A) @ F3) => ((((member @ A) @ A4) @ A5) => ((((ord_less_eq @ (set @ A)) @ F3) @ A5) => ((~ ((((member @ A) @ A4) @ F3))) => ((P @ F3) => (P @ (((insert @ A) @ A4) @ F3))))))))) => (P @ F)))))))).
% 2.63/1.52 thf(fact_79_in__inv__imagep, axiom, (! [B:$tType,A:$tType]: ((((inv_imagep @ A) @ B) = (^ [R:(A > (A > $o)),F5:(B > A),X:B,Y4:B]: (((R @ (F5 @ X)) @ (F5 @ Y4)))))))).
% 2.63/1.52 thf(fact_80_inj__on__HF, axiom, ((((inj_on @ (set @ hF_Mirabelle_hf)) @ hF_Mirabelle_hf) @ hF_Mirabelle_HF) @ ((collect @ (set @ hF_Mirabelle_hf)) @ (finite_finite2 @ hF_Mirabelle_hf)))).
% 2.63/1.52 thf(fact_81_of__nat__aux_Osimps_I1_J, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [Inc:(A > A),I:A]: ((((((semiri532925092at_aux @ A) @ Inc) @ (zero_zero @ nat)) @ I) = I))))))).
% 2.63/1.52 thf(fact_82_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 2.63/1.52 thf(fact_83_subset__antisym, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ A5) => (A5 = B5)))))).
% 2.63/1.52 thf(fact_84_subsetI, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A5) => (((member @ A) @ X3) @ B5)))) => (((ord_less_eq @ (set @ A)) @ A5) @ B5))))).
% 2.63/1.52 thf(fact_85_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.63/1.52 thf(fact_86_subset__empty, axiom, (! [A:$tType,A5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ (bot_bot @ (set @ A))) = (A5 = (bot_bot @ (set @ A))))))).
% 2.63/1.52 thf(fact_87_empty__subsetI, axiom, (! [A:$tType,A5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A5)))).
% 2.63/1.52 thf(fact_88_insert__subset, axiom, (! [A:$tType,X2:A,A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X2) @ A5)) @ B5) = ((((member @ A) @ X2) @ B5) & (((ord_less_eq @ (set @ A)) @ A5) @ B5)))))).
% 2.63/1.52 thf(fact_89_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A3:A,A5:(set @ A),B4:A]: ((((((insert @ A) @ A3) @ A5) = (((insert @ A) @ B4) @ (bot_bot @ (set @ A)))) = ((A3 = B4) & (((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ B4) @ (bot_bot @ (set @ A))))))))).
% 2.63/1.52 thf(fact_90_singleton__insert__inj__eq, axiom, (! [A:$tType,B4:A,A3:A,A5:(set @ A)]: ((((((insert @ A) @ B4) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A3) @ A5)) = ((A3 = B4) & (((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ B4) @ (bot_bot @ (set @ A))))))))).
% 2.63/1.52 thf(fact_91_card_Oempty, axiom, (! [A:$tType]: ((((finite_card @ A) @ (bot_bot @ (set @ A))) = (zero_zero @ nat))))).
% 2.63/1.52 thf(fact_92_card_Oinfinite, axiom, (! [A:$tType,A5:(set @ A)]: (((~ (((finite_finite2 @ A) @ A5))) => (((finite_card @ A) @ A5) = (zero_zero @ nat)))))).
% 2.63/1.52 thf(fact_93_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B4:A,A3:A]: (((((ord_less_eq @ A) @ B4) @ A3) => ((((ord_less_eq @ A) @ A3) @ B4) => (A3 = B4))))))))).
% 2.63/1.52 thf(fact_94_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [A2:A,B2:A]: (((((ord_less_eq @ A) @ B2) @ A2) & (((ord_less_eq @ A) @ A2) @ B2))))))))).
% 2.63/1.52 thf(fact_95_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B4:A,A3:A,C:A]: (((((ord_less_eq @ A) @ B4) @ A3) => ((((ord_less_eq @ A) @ C) @ B4) => (((ord_less_eq @ A) @ C) @ A3))))))))).
% 2.63/1.52 thf(fact_96_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B4:A]: (((! [A4:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((P @ A4) @ B3)))) => ((! [A4:A,B3:A]: ((((P @ B3) @ A4) => ((P @ A4) @ B3)))) => ((P @ A3) @ B4))))))))).
% 2.63/1.52 thf(fact_97_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 2.63/1.52 thf(fact_98_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z2) => (((ord_less_eq @ A) @ X2) @ Z2))))))))).
% 2.63/1.52 thf(fact_99_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B4:A]: (((((ord_less_eq @ A) @ A3) @ B4) => ((((ord_less_eq @ A) @ B4) @ A3) => (A3 = B4))))))))).
% 2.63/1.52 thf(fact_100_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B4:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B4) => ((B4 = C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 2.63/1.52 thf(fact_101_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B4:A,C:A]: (((A3 = B4) => ((((ord_less_eq @ A) @ B4) @ C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 2.63/1.52 thf(fact_102_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) & (((ord_less_eq @ A) @ B2) @ A2))))))))).
% 2.63/1.52 thf(fact_103_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y2:A,X2:A]: (((((ord_less_eq @ A) @ Y2) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y2) = (X2 = Y2))))))))).
% 2.63/1.52 thf(fact_104_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A,Z2:A]: ((((((ord_less_eq @ A) @ X2) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ Z2)))) => (((((ord_less_eq @ A) @ Y2) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Z2)))) => (((((ord_less_eq @ A) @ X2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y2)))) => (((((ord_less_eq @ A) @ Z2) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ X2)))) => (((((ord_less_eq @ A) @ Y2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X2)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Y2))))))))))))))))).
% 2.63/1.52 thf(fact_105_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B4:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B4) => ((((ord_less_eq @ A) @ B4) @ C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 2.63/1.52 thf(fact_106_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X2)))))))).
% 2.63/1.52 thf(fact_107_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y2:A]: (((X2 = Y2) => (((ord_less_eq @ A) @ X2) @ Y2)))))))).
% 2.63/1.52 thf(fact_108_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) | (((ord_less_eq @ A) @ Y2) @ X2)))))))).
% 2.63/1.52 thf(fact_109_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ X2) => (X2 = Y2))))))))).
% 2.63/1.52 thf(fact_110_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) & (((ord_less_eq @ A) @ Y4) @ X))))))))).
% 2.63/1.52 thf(fact_111_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B4:A,F2:(A > B),C:B]: (((((ord_less_eq @ A) @ A3) @ B4) => (((F2 @ B4) = C) => ((! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ B) @ (F2 @ X3)) @ (F2 @ Y3))))) => (((ord_less_eq @ B) @ (F2 @ A3)) @ C)))))))))).
% 2.63/1.52 thf(fact_112_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F2:(B > A),B4:B,C:B]: (((A3 = (F2 @ B4)) => ((((ord_less_eq @ B) @ B4) @ C) => ((! [X3:B,Y3:B]: (((((ord_less_eq @ B) @ X3) @ Y3) => (((ord_less_eq @ A) @ (F2 @ X3)) @ (F2 @ Y3))))) => (((ord_less_eq @ A) @ A3) @ (F2 @ C))))))))))).
% 2.63/1.52 thf(fact_113_order__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A3:A,B4:A,F2:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A3) @ B4) => ((((ord_less_eq @ C2) @ (F2 @ B4)) @ C) => ((! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ C2) @ (F2 @ X3)) @ (F2 @ Y3))))) => (((ord_less_eq @ C2) @ (F2 @ A3)) @ C)))))))))).
% 2.63/1.52 thf(fact_114_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F2:(B > A),B4:B,C:B]: (((((ord_less_eq @ A) @ A3) @ (F2 @ B4)) => ((((ord_less_eq @ B) @ B4) @ C) => ((! [X3:B,Y3:B]: (((((ord_less_eq @ B) @ X3) @ Y3) => (((ord_less_eq @ A) @ (F2 @ X3)) @ (F2 @ Y3))))) => (((ord_less_eq @ A) @ A3) @ (F2 @ C))))))))))).
% 2.63/1.52 thf(fact_115_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X:A]: (((P @ X) => (Q @ X)))))))).
% 2.63/1.52 thf(fact_116_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F5:(A > B),G2:(A > B)]: ((! [X:A]: ((((ord_less_eq @ B) @ (F5 @ X)) @ (G2 @ X))))))))))).
% 2.63/1.52 thf(fact_117_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y:(set @ A),Z:(set @ A)]: ((Y = Z))) = (^ [A7:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) & (((ord_less_eq @ (set @ A)) @ B7) @ A7)))))))).
% 2.63/1.52 thf(fact_118_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F2 @ X3)) @ (G @ X3)))) => (((ord_less_eq @ (A > B)) @ F2) @ G)))))))).
% 2.63/1.52 thf(fact_119_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F2) @ G) => (((ord_less_eq @ B) @ (F2 @ X2)) @ (G @ X2))))))))).
% 2.63/1.52 thf(fact_120_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F2) @ G) => (((ord_less_eq @ B) @ (F2 @ X2)) @ (G @ X2))))))))).
% 2.63/1.52 thf(fact_121_subset__trans, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ C4) => (((ord_less_eq @ (set @ A)) @ A5) @ C4)))))).
% 2.63/1.52 thf(fact_122_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.63/1.52 thf(fact_123_subset__refl, axiom, (! [A:$tType,A5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A5) @ A5)))).
% 2.63/1.52 thf(fact_124_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: ((! [T:A]: (((((member @ A) @ T) @ A7) => (((member @ A) @ T) @ B7)))))))))).
% 2.63/1.52 thf(fact_125_equalityD2, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((A5 = B5) => (((ord_less_eq @ (set @ A)) @ B5) @ A5))))).
% 2.63/1.52 thf(fact_126_equalityD1, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((A5 = B5) => (((ord_less_eq @ (set @ A)) @ A5) @ B5))))).
% 2.63/1.52 thf(fact_127_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ A7) => (((member @ A) @ X) @ B7)))))))))).
% 2.63/1.52 thf(fact_128_equalityE, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((A5 = B5) => (~ (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => (~ ((((ord_less_eq @ (set @ A)) @ B5) @ A5)))))))))).
% 2.63/1.52 thf(fact_129_subsetD, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((member @ A) @ C) @ A5) => (((member @ A) @ C) @ B5)))))).
% 2.63/1.52 thf(fact_130_in__mono, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A),X2:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((member @ A) @ X2) @ A5) => (((member @ A) @ X2) @ B5)))))).
% 2.63/1.52 thf(fact_131_card__subset__eq, axiom, (! [A:$tType,B5:(set @ A),A5:(set @ A)]: ((((finite_finite2 @ A) @ B5) => ((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((finite_card @ A) @ A5) = ((finite_card @ A) @ B5)) => (A5 = B5))))))).
% 2.63/1.52 thf(fact_132_infinite__arbitrarily__large, axiom, (! [A:$tType,A5:(set @ A),N:nat]: (((~ (((finite_finite2 @ A) @ A5))) => (? [B6:(set @ A)]: ((((finite_finite2 @ A) @ B6) & ((((finite_card @ A) @ B6) = N) & (((ord_less_eq @ (set @ A)) @ B6) @ A5))))))))).
% 2.63/1.52 thf(fact_133_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X2))))))).
% 2.63/1.52 thf(fact_134_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.63/1.52 thf(fact_135_bot_Oextremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A3))))))).
% 2.63/1.52 thf(fact_136_bot_Oextremum__unique, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) = (A3 = (bot_bot @ A))))))))).
% 2.63/1.52 thf(fact_137_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) => (A3 = (bot_bot @ A))))))))).
% 2.63/1.52 thf(fact_138_finite__has__maximal2, axiom, (! [A:$tType]: (((order @ A) => (! [A5:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A5) => ((((member @ A) @ A3) @ A5) => (? [X3:A]: (((((member @ A) @ X3) @ A5) & ((((ord_less_eq @ A) @ A3) @ X3) & (! [Xa:A]: (((((member @ A) @ Xa) @ A5) => ((((ord_less_eq @ A) @ X3) @ Xa) => (X3 = Xa))))))))))))))))).
% 2.63/1.52 thf(fact_139_finite__has__minimal2, axiom, (! [A:$tType]: (((order @ A) => (! [A5:(set @ A),A3:A]: ((((finite_finite2 @ A) @ A5) => ((((member @ A) @ A3) @ A5) => (? [X3:A]: (((((member @ A) @ X3) @ A5) & ((((ord_less_eq @ A) @ X3) @ A3) & (! [Xa:A]: (((((member @ A) @ Xa) @ A5) => ((((ord_less_eq @ A) @ Xa) @ X3) => (X3 = Xa))))))))))))))))).
% 2.63/1.52 thf(fact_140_finite__subset, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => (((finite_finite2 @ A) @ B5) => ((finite_finite2 @ A) @ A5)))))).
% 2.63/1.52 thf(fact_141_infinite__super, axiom, (! [A:$tType,S:(set @ A),T2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S) @ T2) => ((~ (((finite_finite2 @ A) @ S))) => (~ (((finite_finite2 @ A) @ T2)))))))).
% 2.63/1.52 thf(fact_142_rev__finite__subset, axiom, (! [A:$tType,B5:(set @ A),A5:(set @ A)]: ((((finite_finite2 @ A) @ B5) => ((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((finite_finite2 @ A) @ A5)))))).
% 2.63/1.52 thf(fact_143_insert__mono, axiom, (! [A:$tType,C4:(set @ A),D3:(set @ A),A3:A]: (((((ord_less_eq @ (set @ A)) @ C4) @ D3) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A3) @ C4)) @ (((insert @ A) @ A3) @ D3)))))).
% 2.63/1.52 thf(fact_144_subset__insert, axiom, (! [A:$tType,X2:A,A5:(set @ A),B5:(set @ A)]: (((~ ((((member @ A) @ X2) @ A5))) => ((((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ X2) @ B5)) = (((ord_less_eq @ (set @ A)) @ A5) @ B5)))))).
% 2.63/1.52 thf(fact_145_subset__insertI, axiom, (! [A:$tType,B5:(set @ A),A3:A]: ((((ord_less_eq @ (set @ A)) @ B5) @ (((insert @ A) @ A3) @ B5))))).
% 2.63/1.52 thf(fact_146_subset__insertI2, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A),B4:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => (((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ B4) @ B5)))))).
% 2.63/1.52 thf(fact_147_finite__has__minimal, axiom, (! [A:$tType]: (((order @ A) => (! [A5:(set @ A)]: ((((finite_finite2 @ A) @ A5) => ((A5 != (bot_bot @ (set @ A))) => (? [X3:A]: (((((member @ A) @ X3) @ A5) & (! [Xa:A]: (((((member @ A) @ Xa) @ A5) => ((((ord_less_eq @ A) @ Xa) @ X3) => (X3 = Xa)))))))))))))))).
% 2.63/1.52 thf(fact_148_finite__has__maximal, axiom, (! [A:$tType]: (((order @ A) => (! [A5:(set @ A)]: ((((finite_finite2 @ A) @ A5) => ((A5 != (bot_bot @ (set @ A))) => (? [X3:A]: (((((member @ A) @ X3) @ A5) & (! [Xa:A]: (((((member @ A) @ Xa) @ A5) => ((((ord_less_eq @ A) @ X3) @ Xa) => (X3 = Xa)))))))))))))))).
% 2.63/1.52 thf(fact_149_subset__singleton__iff, axiom, (! [A:$tType,X5:(set @ A),A3:A]: (((((ord_less_eq @ (set @ A)) @ X5) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) = ((X5 = (bot_bot @ (set @ A))) | (X5 = (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))))).
% 2.63/1.52 thf(fact_150_subset__singletonD, axiom, (! [A:$tType,A5:(set @ A),X2:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) => ((A5 = (bot_bot @ (set @ A))) | (A5 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))))))).
% 2.63/1.52 thf(fact_151_card__eq__0__iff, axiom, (! [A:$tType,A5:(set @ A)]: (((((finite_card @ A) @ A5) = (zero_zero @ nat)) = ((A5 = (bot_bot @ (set @ A))) | (~ (((finite_finite2 @ A) @ A5)))))))).
% 2.63/1.52 thf(fact_152_inj__on__empty, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: (((((inj_on @ A) @ B) @ F2) @ (bot_bot @ (set @ A)))))).
% 2.63/1.52 thf(fact_153_finite__ranking__induct, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [S:(set @ B),P:((set @ B) > $o),F2:(B > A)]: ((((finite_finite2 @ B) @ S) => ((P @ (bot_bot @ (set @ B))) => ((! [X3:B,S2:(set @ B)]: ((((finite_finite2 @ B) @ S2) => ((! [Y5:B]: (((((member @ B) @ Y5) @ S2) => (((ord_less_eq @ A) @ (F2 @ Y5)) @ (F2 @ X3))))) => ((P @ S2) => (P @ (((insert @ B) @ X3) @ S2))))))) => (P @ S)))))))))).
% 2.63/1.52 thf(fact_154_inj__on__subset, axiom, (! [B:$tType,A:$tType,F2:(A > B),A5:(set @ A),B5:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ A5) => ((((ord_less_eq @ (set @ A)) @ B5) @ A5) => ((((inj_on @ A) @ B) @ F2) @ B5)))))).
% 2.63/1.52 thf(fact_155_subset__inj__on, axiom, (! [B:$tType,A:$tType,F2:(A > B),B5:(set @ A),A5:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ B5) => ((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((inj_on @ A) @ B) @ F2) @ A5)))))).
% 2.63/1.52 thf(fact_156_insert__subsetI, axiom, (! [A:$tType,X2:A,A5:(set @ A),X5:(set @ A)]: (((((member @ A) @ X2) @ A5) => ((((ord_less_eq @ (set @ A)) @ X5) @ A5) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X2) @ X5)) @ A5)))))).
% 2.63/1.52 thf(fact_157_subset__emptyI, axiom, (! [A:$tType,A5:(set @ A)]: (((! [X3:A]: ((~ ((((member @ A) @ X3) @ A5))))) => (((ord_less_eq @ (set @ A)) @ A5) @ (bot_bot @ (set @ A))))))).
% 2.63/1.52 thf(fact_158_bot__nat__0_Oextremum, axiom, (! [A3:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A3)))).
% 2.63/1.52 thf(fact_159_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.63/1.52 thf(fact_160_bot__nat__0_Oextremum__uniqueI, axiom, (! [A3:nat]: (((((ord_less_eq @ nat) @ A3) @ (zero_zero @ nat)) => (A3 = (zero_zero @ nat)))))).
% 2.63/1.52 thf(fact_161_bot__nat__0_Oextremum__unique, axiom, (! [A3:nat]: (((((ord_less_eq @ nat) @ A3) @ (zero_zero @ nat)) = (A3 = (zero_zero @ nat)))))).
% 2.63/1.52 thf(fact_162_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 2.63/1.52 thf(fact_163_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.63/1.52 thf(fact_164_finite__if__finite__subsets__card__bdd, axiom, (! [A:$tType,F:(set @ A),C4:nat]: (((! [G3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ G3) @ F) => (((finite_finite2 @ A) @ G3) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ G3)) @ C4))))) => (((finite_finite2 @ A) @ F) & (((ord_less_eq @ nat) @ ((finite_card @ A) @ F)) @ C4)))))).
% 2.63/1.52 thf(fact_165_card__seteq, axiom, (! [A:$tType,B5:(set @ A),A5:(set @ A)]: ((((finite_finite2 @ A) @ B5) => ((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ B5)) @ ((finite_card @ A) @ A5)) => (A5 = B5))))))).
% 2.63/1.52 thf(fact_166_card__mono, axiom, (! [A:$tType,B5:(set @ A),A5:(set @ A)]: ((((finite_finite2 @ A) @ B5) => ((((ord_less_eq @ (set @ A)) @ A5) @ B5) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ A5)) @ ((finite_card @ A) @ B5))))))).
% 2.63/1.52 thf(fact_167_card__insert__le, axiom, (! [A:$tType,A5:(set @ A),X2:A]: ((((finite_finite2 @ A) @ A5) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ A5)) @ ((finite_card @ A) @ (((insert @ A) @ X2) @ A5))))))).
% 2.63/1.52 thf(fact_168_inj__on__inverseI, axiom, (! [B:$tType,A:$tType,A5:(set @ A),G:(B > A),F2:(A > B)]: (((! [X3:A]: (((((member @ A) @ X3) @ A5) => ((G @ (F2 @ X3)) = X3)))) => ((((inj_on @ A) @ B) @ F2) @ A5))))).
% 2.63/1.52 thf(fact_169_inj__on__contraD, axiom, (! [B:$tType,A:$tType,F2:(A > B),A5:(set @ A),X2:A,Y2:A]: ((((((inj_on @ A) @ B) @ F2) @ A5) => ((X2 != Y2) => ((((member @ A) @ X2) @ A5) => ((((member @ A) @ Y2) @ A5) => ((F2 @ X2) != (F2 @ Y2))))))))).
% 2.63/1.52 thf(fact_170_inj__on__eq__iff, axiom, (! [B:$tType,A:$tType,F2:(A > B),A5:(set @ A),X2:A,Y2:A]: ((((((inj_on @ A) @ B) @ F2) @ A5) => ((((member @ A) @ X2) @ A5) => ((((member @ A) @ Y2) @ A5) => (((F2 @ X2) = (F2 @ Y2)) = (X2 = Y2)))))))).
% 2.63/1.52 thf(fact_171_inj__on__cong, axiom, (! [B:$tType,A:$tType,A5:(set @ A),F2:(A > B),G:(A > B)]: (((! [A4:A]: (((((member @ A) @ A4) @ A5) => ((F2 @ A4) = (G @ A4))))) => (((((inj_on @ A) @ B) @ F2) @ A5) = ((((inj_on @ A) @ B) @ G) @ A5)))))).
% 2.63/1.52 thf(fact_172_inj__on__def, axiom, (! [B:$tType,A:$tType]: ((((inj_on @ A) @ B) = (^ [F5:(A > B),A7:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ A7) => (! [Y4:A]: (((((member @ A) @ Y4) @ A7) => (((F5 @ X) = (F5 @ Y4)) => (X = Y4)))))))))))))).
% 2.63/1.52 thf(fact_173_inj__onI, axiom, (! [B:$tType,A:$tType,A5:(set @ A),F2:(A > B)]: (((! [X3:A,Y3:A]: (((((member @ A) @ X3) @ A5) => ((((member @ A) @ Y3) @ A5) => (((F2 @ X3) = (F2 @ Y3)) => (X3 = Y3)))))) => ((((inj_on @ A) @ B) @ F2) @ A5))))).
% 2.63/1.52 thf(fact_174_inj__onD, axiom, (! [B:$tType,A:$tType,F2:(A > B),A5:(set @ A),X2:A,Y2:A]: ((((((inj_on @ A) @ B) @ F2) @ A5) => (((F2 @ X2) = (F2 @ Y2)) => ((((member @ A) @ X2) @ A5) => ((((member @ A) @ Y2) @ A5) => (X2 = Y2)))))))).
% 2.63/1.52 thf(fact_175_obtain__subset__with__card__n, axiom, (! [A:$tType,N:nat,S:(set @ A)]: (((((ord_less_eq @ nat) @ N) @ ((finite_card @ A) @ S)) => (~ ((! [T3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ T3) @ S) => ((((finite_card @ A) @ T3) = N) => (~ (((finite_finite2 @ A) @ T3))))))))))))).
% 2.63/1.52 thf(fact_176_card__le__if__inj__on__rel, axiom, (! [B:$tType,A:$tType,B5:(set @ A),A5:(set @ B),R2:(B > (A > $o))]: ((((finite_finite2 @ A) @ B5) => ((! [A4:B]: (((((member @ B) @ A4) @ A5) => (? [B8:A]: (((((member @ A) @ B8) @ B5) & ((R2 @ A4) @ B8))))))) => ((! [A1:B,A22:B,B3:A]: (((((member @ B) @ A1) @ A5) => ((((member @ B) @ A22) @ A5) => ((((member @ A) @ B3) @ B5) => (((R2 @ A1) @ B3) => (((R2 @ A22) @ B3) => (A1 = A22)))))))) => (((ord_less_eq @ nat) @ ((finite_card @ B) @ A5)) @ ((finite_card @ A) @ B5)))))))).
% 2.63/1.52 thf(fact_177_arg__min__least, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [S:(set @ A),Y2:A,F2:(A > B)]: ((((finite_finite2 @ A) @ S) => ((S != (bot_bot @ (set @ A))) => ((((member @ A) @ Y2) @ S) => (((ord_less_eq @ B) @ (F2 @ ((((lattic1704895705min_on @ A) @ B) @ F2) @ S))) @ (F2 @ Y2))))))))))).
% 2.63/1.52 thf(fact_178_bounded__Max__nat, axiom, (! [P:(nat > $o),X2:nat,M:nat]: (((P @ X2) => ((! [X3:nat]: (((P @ X3) => (((ord_less_eq @ nat) @ X3) @ M)))) => (~ ((! [M2:nat]: (((P @ M2) => (~ ((! [X4:nat]: (((P @ X4) => (((ord_less_eq @ nat) @ X4) @ M2)))))))))))))))).
% 2.63/1.52 thf(fact_179_ex__has__least__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M3:(A > nat)]: (((P @ K) => (? [X3:A]: (((P @ X3) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M3 @ X3)) @ (M3 @ Y5)))))))))))).
% 2.63/1.52 thf(fact_180_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K:nat,B4:nat]: (((P @ K) => ((! [Y3:nat]: (((P @ Y3) => (((ord_less_eq @ nat) @ Y3) @ B4)))) => (? [X3:nat]: (((P @ X3) & (! [Y5:nat]: (((P @ Y5) => (((ord_less_eq @ nat) @ Y5) @ X3)))))))))))).
% 2.63/1.52 thf(fact_181_nat__le__linear, axiom, (! [M3:nat,N:nat]: (((((ord_less_eq @ nat) @ M3) @ N) | (((ord_less_eq @ nat) @ N) @ M3))))).
% 2.63/1.52 thf(fact_182_le__antisym, axiom, (! [M3:nat,N:nat]: (((((ord_less_eq @ nat) @ M3) @ N) => ((((ord_less_eq @ nat) @ N) @ M3) => (M3 = N)))))).
% 2.63/1.52 thf(fact_183_eq__imp__le, axiom, (! [M3:nat,N:nat]: (((M3 = N) => (((ord_less_eq @ nat) @ M3) @ N))))).
% 2.63/1.52 thf(fact_184_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 2.63/1.52 thf(fact_185_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.63/1.52 thf(fact_186_finite__nat__set__iff__bounded__le, axiom, ((finite_finite2 @ nat) = (^ [N2:(set @ nat)]: ((? [M4:nat]: ((! [X:nat]: (((((member @ nat) @ X) @ N2) => (((ord_less_eq @ nat) @ X) @ M4)))))))))).
% 2.63/1.52 thf(fact_187_arg__min__if__finite_I1_J, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [S:(set @ A),F2:(A > B)]: ((((finite_finite2 @ A) @ S) => ((S != (bot_bot @ (set @ A))) => (((member @ A) @ ((((lattic1704895705min_on @ A) @ B) @ F2) @ S)) @ S))))))))).
% 2.63/1.52 thf(fact_188_card__Diff1__le, axiom, (! [A:$tType,A5:(set @ A),X2:A]: ((((finite_finite2 @ A) @ A5) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A5) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))) @ ((finite_card @ A) @ A5)))))).
% 2.63/1.52 thf(fact_189_card__le__inj, axiom, (! [B:$tType,A:$tType,A5:(set @ A),B5:(set @ B)]: ((((finite_finite2 @ A) @ A5) => (((finite_finite2 @ B) @ B5) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ A5)) @ ((finite_card @ B) @ B5)) => (? [F4:(A > B)]: (((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F4) @ A5)) @ B5) & ((((inj_on @ A) @ B) @ F4) @ A5)))))))))).
% 2.63/1.52 thf(fact_190_card__inj__on__le, axiom, (! [A:$tType,B:$tType,F2:(A > B),A5:(set @ A),B5:(set @ B)]: ((((((inj_on @ A) @ B) @ F2) @ A5) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A5)) @ B5) => (((finite_finite2 @ B) @ B5) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ A5)) @ ((finite_card @ B) @ B5)))))))).
% 2.63/1.52 thf(fact_191_image__eqI, axiom, (! [A:$tType,B:$tType,B4:A,F2:(B > A),X2:B,A5:(set @ B)]: (((B4 = (F2 @ X2)) => ((((member @ B) @ X2) @ A5) => (((member @ A) @ B4) @ ((((image @ B) @ A) @ F2) @ A5))))))).
% 2.63/1.52 thf(fact_192_DiffI, axiom, (! [A:$tType,C:A,A5:(set @ A),B5:(set @ A)]: (((((member @ A) @ C) @ A5) => ((~ ((((member @ A) @ C) @ B5))) => (((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A5) @ B5))))))).
% 2.63/1.52 thf(fact_193_Diff__iff, axiom, (! [A:$tType,C:A,A5:(set @ A),B5:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A5) @ B5)) = ((((member @ A) @ C) @ A5) & (~ ((((member @ A) @ C) @ B5)))))))).
% 2.63/1.52 thf(fact_194_Diff__idemp, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A5) @ B5)) @ B5) = (((minus_minus @ (set @ A)) @ A5) @ B5))))).
% 2.63/1.52 thf(fact_195_cancel__comm__monoid__add__class_Odiff__cancel, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A3:A]: (((((minus_minus @ A) @ A3) @ A3) = (zero_zero @ A)))))))).
% 2.63/1.52 thf(fact_196_diff__zero, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A3:A]: (((((minus_minus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 2.63/1.52 thf(fact_197_zero__diff, axiom, (! [A:$tType]: (((comm_monoid_diff @ A) => (! [A3:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A3) = (zero_zero @ A)))))))).
% 2.63/1.52 thf(fact_198_diff__0__right, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: (((((minus_minus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 2.63/1.52 thf(fact_199_diff__self, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: (((((minus_minus @ A) @ A3) @ A3) = (zero_zero @ A)))))))).
% 2.63/1.52 thf(fact_200_image__empty, axiom, (! [B:$tType,A:$tType,F2:(B > A)]: ((((((image @ B) @ A) @ F2) @ (bot_bot @ (set @ B))) = (bot_bot @ (set @ A)))))).
% 2.63/1.52 thf(fact_201_empty__is__image, axiom, (! [A:$tType,B:$tType,F2:(B > A),A5:(set @ B)]: ((((bot_bot @ (set @ A)) = ((((image @ B) @ A) @ F2) @ A5)) = (A5 = (bot_bot @ (set @ B))))))).
% 2.63/1.52 thf(fact_202_image__is__empty, axiom, (! [A:$tType,B:$tType,F2:(B > A),A5:(set @ B)]: (((((((image @ B) @ A) @ F2) @ A5) = (bot_bot @ (set @ A))) = (A5 = (bot_bot @ (set @ B))))))).
% 2.63/1.52 thf(fact_203_finite__imageI, axiom, (! [B:$tType,A:$tType,F:(set @ A),H:(A > B)]: ((((finite_finite2 @ A) @ F) => ((finite_finite2 @ B) @ ((((image @ A) @ B) @ H) @ F)))))).
% 2.63/1.52 thf(fact_204_image__insert, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:B,B5:(set @ B)]: ((((((image @ B) @ A) @ F2) @ (((insert @ B) @ A3) @ B5)) = (((insert @ A) @ (F2 @ A3)) @ ((((image @ B) @ A) @ F2) @ B5)))))).
% 2.63/1.52 thf(fact_205_insert__image, axiom, (! [B:$tType,A:$tType,X2:A,A5:(set @ A),F2:(A > B)]: (((((member @ A) @ X2) @ A5) => ((((insert @ B) @ (F2 @ X2)) @ ((((image @ A) @ B) @ F2) @ A5)) = ((((image @ A) @ B) @ F2) @ A5)))))).
% 2.63/1.52 thf(fact_206_Diff__empty, axiom, (! [A:$tType,A5:(set @ A)]: (((((minus_minus @ (set @ A)) @ A5) @ (bot_bot @ (set @ A))) = A5)))).
% 2.63/1.52 thf(fact_207_empty__Diff, axiom, (! [A:$tType,A5:(set @ A)]: (((((minus_minus @ (set @ A)) @ (bot_bot @ (set @ A))) @ A5) = (bot_bot @ (set @ A)))))).
% 2.63/1.52 thf(fact_208_Diff__cancel, axiom, (! [A:$tType,A5:(set @ A)]: (((((minus_minus @ (set @ A)) @ A5) @ A5) = (bot_bot @ (set @ A)))))).
% 2.63/1.52 thf(fact_209_finite__Diff, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: ((((finite_finite2 @ A) @ A5) => ((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A5) @ B5)))))).
% 2.63/1.52 thf(fact_210_finite__Diff2, axiom, (! [A:$tType,B5:(set @ A),A5:(set @ A)]: ((((finite_finite2 @ A) @ B5) => (((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A5) @ B5)) = ((finite_finite2 @ A) @ A5)))))).
% 2.63/1.52 thf(fact_211_Diff__insert0, axiom, (! [A:$tType,X2:A,A5:(set @ A),B5:(set @ A)]: (((~ ((((member @ A) @ X2) @ A5))) => ((((minus_minus @ (set @ A)) @ A5) @ (((insert @ A) @ X2) @ B5)) = (((minus_minus @ (set @ A)) @ A5) @ B5)))))).
% 2.63/1.52 thf(fact_212_insert__Diff1, axiom, (! [A:$tType,X2:A,B5:(set @ A),A5:(set @ A)]: (((((member @ A) @ X2) @ B5) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X2) @ A5)) @ B5) = (((minus_minus @ (set @ A)) @ A5) @ B5)))))).
% 2.63/1.52 thf(fact_213_diff__ge__0__iff__ge, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B4:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A3) @ B4)) = (((ord_less_eq @ A) @ B4) @ A3)))))))).
% 2.63/1.52 thf(fact_214_Diff__eq__empty__iff, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: ((((((minus_minus @ (set @ A)) @ A5) @ B5) = (bot_bot @ (set @ A))) = (((ord_less_eq @ (set @ A)) @ A5) @ B5))))).
% 2.63/1.52 thf(fact_215_insert__Diff__single, axiom, (! [A:$tType,A3:A,A5:(set @ A)]: (((((insert @ A) @ A3) @ (((minus_minus @ (set @ A)) @ A5) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) = (((insert @ A) @ A3) @ A5))))).
% 2.63/1.52 thf(fact_216_finite__Diff__insert, axiom, (! [A:$tType,A5:(set @ A),A3:A,B5:(set @ A)]: ((((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A5) @ (((insert @ A) @ A3) @ B5))) = ((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A5) @ B5)))))).
% 2.63/1.52 thf(fact_217_inj__on__insert, axiom, (! [B:$tType,A:$tType,F2:(A > B),A3:A,A5:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ (((insert @ A) @ A3) @ A5)) = (((((inj_on @ A) @ B) @ F2) @ A5) & (~ ((((member @ B) @ (F2 @ A3)) @ ((((image @ A) @ B) @ F2) @ (((minus_minus @ (set @ A)) @ A5) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))))))))).
% 2.63/1.52 thf(fact_218_image__diff__subset, axiom, (! [A:$tType,B:$tType,F2:(B > A),A5:(set @ B),B5:(set @ B)]: ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ ((((image @ B) @ A) @ F2) @ A5)) @ ((((image @ B) @ A) @ F2) @ B5))) @ ((((image @ B) @ A) @ F2) @ (((minus_minus @ (set @ B)) @ A5) @ B5)))))).
% 2.63/1.52 thf(fact_219_inj__on__image__set__diff, axiom, (! [B:$tType,A:$tType,F2:(A > B),C4:(set @ A),A5:(set @ A),B5:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ C4) => ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A5) @ B5)) @ C4) => ((((ord_less_eq @ (set @ A)) @ B5) @ C4) => (((((image @ A) @ B) @ F2) @ (((minus_minus @ (set @ A)) @ A5) @ B5)) = (((minus_minus @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A5)) @ ((((image @ A) @ B) @ F2) @ B5))))))))).
% 2.63/1.52 thf(fact_220_diff__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B4:A,D2:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B4) => ((((ord_less_eq @ A) @ D2) @ C) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ C)) @ (((minus_minus @ A) @ B4) @ D2)))))))))).
% 2.63/1.52 thf(fact_221_diff__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B4:A,A3:A,C:A]: (((((ord_less_eq @ A) @ B4) @ A3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C) @ A3)) @ (((minus_minus @ A) @ C) @ B4))))))))).
% 2.63/1.52 thf(fact_222_diff__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B4:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B4) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ C)) @ (((minus_minus @ A) @ B4) @ C))))))))).
% 2.63/1.52 thf(fact_223_diff__eq__diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B4:A,C:A,D2:A]: ((((((minus_minus @ A) @ A3) @ B4) = (((minus_minus @ A) @ C) @ D2)) => ((((ord_less_eq @ A) @ A3) @ B4) = (((ord_less_eq @ A) @ C) @ D2))))))))).
% 2.63/1.52 thf(fact_224_all__subset__image, axiom, (! [A:$tType,B:$tType,F2:(B > A),A5:(set @ B),P:((set @ A) > $o)]: (((! [B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B7) @ ((((image @ B) @ A) @ F2) @ A5)) => (P @ B7)))) = (! [B7:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ B7) @ A5) => (P @ ((((image @ B) @ A) @ F2) @ B7))))))))).
% 2.63/1.52 thf(fact_225_image__mono, axiom, (! [B:$tType,A:$tType,A5:(set @ A),B5:(set @ A),F2:(A > B)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A5)) @ ((((image @ A) @ B) @ F2) @ B5)))))).
% 2.63/1.52 thf(fact_226_image__subsetI, axiom, (! [A:$tType,B:$tType,A5:(set @ A),F2:(A > B),B5:(set @ B)]: (((! [X3:A]: (((((member @ A) @ X3) @ A5) => (((member @ B) @ (F2 @ X3)) @ B5)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A5)) @ B5))))).
% 2.63/1.52 thf(fact_227_subset__imageE, axiom, (! [A:$tType,B:$tType,B5:(set @ A),F2:(B > A),A5:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B5) @ ((((image @ B) @ A) @ F2) @ A5)) => (~ ((! [C5:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C5) @ A5) => (B5 != ((((image @ B) @ A) @ F2) @ C5))))))))))).
% 2.63/1.52 thf(fact_228_image__subset__iff, axiom, (! [A:$tType,B:$tType,F2:(B > A),A5:(set @ B),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F2) @ A5)) @ B5) = (! [X:B]: (((((member @ B) @ X) @ A5) => (((member @ A) @ (F2 @ X)) @ B5)))))))).
% 2.63/1.52 thf(fact_229_subset__image__iff, axiom, (! [A:$tType,B:$tType,B5:(set @ A),F2:(B > A),A5:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B5) @ ((((image @ B) @ A) @ F2) @ A5)) = (? [AA:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ AA) @ A5) & (B5 = ((((image @ B) @ A) @ F2) @ AA))))))))).
% 2.63/1.52 thf(fact_230_Diff__mono, axiom, (! [A:$tType,A5:(set @ A),C4:(set @ A),D3:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ C4) => ((((ord_less_eq @ (set @ A)) @ D3) @ B5) => (((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A5) @ B5)) @ (((minus_minus @ (set @ A)) @ C4) @ D3))))))).
% 2.63/1.52 thf(fact_231_Diff__subset, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A5) @ B5)) @ A5)))).
% 2.63/1.52 thf(fact_232_double__diff, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ C4) => ((((minus_minus @ (set @ A)) @ B5) @ (((minus_minus @ (set @ A)) @ C4) @ A5)) = A5)))))).
% 2.63/1.52 thf(fact_233_le__iff__diff__le__0, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => ((ord_less_eq @ A) = (^ [A2:A,B2:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ (zero_zero @ A))))))))).
% 2.63/1.52 thf(fact_234_finite__surj, axiom, (! [A:$tType,B:$tType,A5:(set @ A),B5:(set @ B),F2:(A > B)]: ((((finite_finite2 @ A) @ A5) => ((((ord_less_eq @ (set @ B)) @ B5) @ ((((image @ A) @ B) @ F2) @ A5)) => ((finite_finite2 @ B) @ B5)))))).
% 2.63/1.52 thf(fact_235_finite__subset__image, axiom, (! [A:$tType,B:$tType,B5:(set @ A),F2:(B > A),A5:(set @ B)]: ((((finite_finite2 @ A) @ B5) => ((((ord_less_eq @ (set @ A)) @ B5) @ ((((image @ B) @ A) @ F2) @ A5)) => (? [C5:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C5) @ A5) & (((finite_finite2 @ B) @ C5) & (B5 = ((((image @ B) @ A) @ F2) @ C5))))))))))).
% 2.63/1.52 thf(fact_236_ex__finite__subset__image, axiom, (! [A:$tType,B:$tType,F2:(B > A),A5:(set @ B),P:((set @ A) > $o)]: (((? [B7:(set @ A)]: ((((finite_finite2 @ A) @ B7) & ((((ord_less_eq @ (set @ A)) @ B7) @ ((((image @ B) @ A) @ F2) @ A5)) & (P @ B7))))) = (? [B7:(set @ B)]: ((((finite_finite2 @ B) @ B7) & ((((ord_less_eq @ (set @ B)) @ B7) @ A5) & (P @ ((((image @ B) @ A) @ F2) @ B7)))))))))).
% 2.63/1.52 thf(fact_237_all__finite__subset__image, axiom, (! [A:$tType,B:$tType,F2:(B > A),A5:(set @ B),P:((set @ A) > $o)]: (((! [B7:(set @ A)]: (((((finite_finite2 @ A) @ B7) & (((ord_less_eq @ (set @ A)) @ B7) @ ((((image @ B) @ A) @ F2) @ A5))) => (P @ B7)))) = (! [B7:(set @ B)]: (((((finite_finite2 @ B) @ B7) & (((ord_less_eq @ (set @ B)) @ B7) @ A5)) => (P @ ((((image @ B) @ A) @ F2) @ B7))))))))).
% 2.63/1.52 thf(fact_238_finite__imageD, axiom, (! [A:$tType,B:$tType,F2:(B > A),A5:(set @ B)]: ((((finite_finite2 @ A) @ ((((image @ B) @ A) @ F2) @ A5)) => (((((inj_on @ B) @ A) @ F2) @ A5) => ((finite_finite2 @ B) @ A5)))))).
% 2.63/1.52 thf(fact_239_finite__image__iff, axiom, (! [B:$tType,A:$tType,F2:(A > B),A5:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ A5) => (((finite_finite2 @ B) @ ((((image @ A) @ B) @ F2) @ A5)) = ((finite_finite2 @ A) @ A5)))))).
% 2.63/1.52 thf(fact_240_inj__on__image__eq__iff, axiom, (! [B:$tType,A:$tType,F2:(A > B),C4:(set @ A),A5:(set @ A),B5:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ C4) => ((((ord_less_eq @ (set @ A)) @ A5) @ C4) => ((((ord_less_eq @ (set @ A)) @ B5) @ C4) => ((((((image @ A) @ B) @ F2) @ A5) = ((((image @ A) @ B) @ F2) @ B5)) = (A5 = B5)))))))).
% 2.63/1.52 thf(fact_241_inj__on__image__mem__iff, axiom, (! [B:$tType,A:$tType,F2:(A > B),B5:(set @ A),A3:A,A5:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ B5) => ((((member @ A) @ A3) @ B5) => ((((ord_less_eq @ (set @ A)) @ A5) @ B5) => ((((member @ B) @ (F2 @ A3)) @ ((((image @ A) @ B) @ F2) @ A5)) = (((member @ A) @ A3) @ A5)))))))).
% 2.63/1.52 thf(fact_242_inj__img__insertE, axiom, (! [B:$tType,A:$tType,F2:(A > B),A5:(set @ A),X2:B,B5:(set @ B)]: ((((((inj_on @ A) @ B) @ F2) @ A5) => ((~ ((((member @ B) @ X2) @ B5))) => (((((insert @ B) @ X2) @ B5) = ((((image @ A) @ B) @ F2) @ A5)) => (~ ((! [X6:A,A8:(set @ A)]: (((~ ((((member @ A) @ X6) @ A8))) => ((A5 = (((insert @ A) @ X6) @ A8)) => ((X2 = (F2 @ X6)) => (B5 != ((((image @ A) @ B) @ F2) @ A8))))))))))))))).
% 2.63/1.52 thf(fact_243_card__image, axiom, (! [B:$tType,A:$tType,F2:(A > B),A5:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ A5) => (((finite_card @ B) @ ((((image @ A) @ B) @ F2) @ A5)) = ((finite_card @ A) @ A5)))))).
% 2.63/1.52 thf(fact_244_Diff__insert__absorb, axiom, (! [A:$tType,X2:A,A5:(set @ A)]: (((~ ((((member @ A) @ X2) @ A5))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X2) @ A5)) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) = A5))))).
% 2.63/1.52 thf(fact_245_Diff__insert2, axiom, (! [A:$tType,A5:(set @ A),A3:A,B5:(set @ A)]: (((((minus_minus @ (set @ A)) @ A5) @ (((insert @ A) @ A3) @ B5)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A5) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) @ B5))))).
% 2.63/1.52 thf(fact_246_insert__Diff, axiom, (! [A:$tType,A3:A,A5:(set @ A)]: (((((member @ A) @ A3) @ A5) => ((((insert @ A) @ A3) @ (((minus_minus @ (set @ A)) @ A5) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) = A5))))).
% 2.63/1.52 thf(fact_247_Diff__insert, axiom, (! [A:$tType,A5:(set @ A),A3:A,B5:(set @ A)]: (((((minus_minus @ (set @ A)) @ A5) @ (((insert @ A) @ A3) @ B5)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A5) @ B5)) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))))))).
% 2.63/1.52 thf(fact_248_subset__Diff__insert, axiom, (! [A:$tType,A5:(set @ A),B5:(set @ A),X2:A,C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ (((minus_minus @ (set @ A)) @ B5) @ (((insert @ A) @ X2) @ C4))) = ((((ord_less_eq @ (set @ A)) @ A5) @ (((minus_minus @ (set @ A)) @ B5) @ C4)) & (~ ((((member @ A) @ X2) @ A5)))))))).
% 2.63/1.52 thf(fact_249_in__image__insert__iff, axiom, (! [A:$tType,B5:(set @ (set @ A)),X2:A,A5:(set @ A)]: (((! [C5:(set @ A)]: (((((member @ (set @ A)) @ C5) @ B5) => (~ ((((member @ A) @ X2) @ C5)))))) => ((((member @ (set @ A)) @ A5) @ ((((image @ (set @ A)) @ (set @ A)) @ ((insert @ A) @ X2)) @ B5)) = ((((member @ A) @ X2) @ A5) & (((member @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A5) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))) @ B5))))))).
% 2.63/1.52 thf(fact_250_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B4:A,C:A,D2:A]: ((((((minus_minus @ A) @ A3) @ B4) = (((minus_minus @ A) @ C) @ D2)) => ((A3 = B4) = (C = D2))))))))).
% 2.63/1.52 thf(fact_251_diff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A3:A,C:A,B4:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A3) @ C)) @ B4) = (((minus_minus @ A) @ (((minus_minus @ A) @ A3) @ B4)) @ C)))))))).
% 2.63/1.52 thf(fact_252_DiffE, axiom, (! [A:$tType,C:A,A5:(set @ A),B5:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A5) @ B5)) => (~ (((((member @ A) @ C) @ A5) => (((member @ A) @ C) @ B5)))))))).
% 2.63/1.52 thf(fact_253_DiffD1, axiom, (! [A:$tType,C:A,A5:(set @ A),B5:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A5) @ B5)) => (((member @ A) @ C) @ A5))))).
% 2.63/1.52 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A9:$tType,A10:$tType]: (((order_bot @ A10) => (order_bot @ (A9 > A10)))))).
% 2.63/1.52 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A9:$tType,A10:$tType]: (((preorder @ A10) => (preorder @ (A9 > A10)))))).
% 2.63/1.52 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ (A9 > A10)))))).
% 2.63/1.52 thf(tcon_fun___Orderings_Oorder, axiom, (! [A9:$tType,A10:$tType]: (((order @ A10) => (order @ (A9 > A10)))))).
% 2.63/1.52 thf(tcon_fun___Orderings_Oord, axiom, (! [A9:$tType,A10:$tType]: (((ord @ A10) => (ord @ (A9 > A10)))))).
% 2.63/1.52 thf(tcon_fun___Orderings_Obot, axiom, (! [A9:$tType,A10:$tType]: (((bot @ A10) => (bot @ (A9 > A10)))))).
% 2.63/1.52 thf(tcon_Nat_Onat___Parity_Ounique__euclidean__semiring__with__bit__shifts, axiom, (unique788075200shifts @ nat)).
% 2.63/1.52 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.63/1.52 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ nat)).
% 2.63/1.52 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add, axiom, (cancel146912293up_add @ nat)).
% 2.63/1.52 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ nat)).
% 2.63/1.52 thf(tcon_Nat_Onat___Parity_Osemiring__bit__shifts, axiom, (semiring_bit_shifts @ nat)).
% 2.63/1.52 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff, axiom, (comm_monoid_diff @ nat)).
% 2.63/1.52 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1, axiom, (order_bot @ nat)).
% 2.63/1.52 thf(tcon_Nat_Onat___Orderings_Opreorder_2, axiom, (preorder @ nat)).
% 2.63/1.52 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 2.63/1.52 thf(tcon_Nat_Onat___Rings_Osemiring__1, axiom, (semiring_1 @ nat)).
% 2.63/1.52 thf(tcon_Nat_Onat___Orderings_Oorder_3, axiom, (order @ nat)).
% 2.63/1.52 thf(tcon_Nat_Onat___Orderings_Oord_4, axiom, (ord @ nat)).
% 2.63/1.52 thf(tcon_Nat_Onat___Orderings_Obot_5, axiom, (bot @ nat)).
% 2.63/1.52 thf(tcon_Nat_Onat___Groups_Ozero, axiom, (zero @ nat)).
% 2.63/1.52 thf(tcon_Set_Oset___Orderings_Oorder__bot_6, axiom, (! [A9:$tType]: ((order_bot @ (set @ A9))))).
% 2.63/1.52 thf(tcon_Set_Oset___Orderings_Opreorder_7, axiom, (! [A9:$tType]: ((preorder @ (set @ A9))))).
% 2.63/1.52 thf(tcon_Set_Oset___Finite__Set_Ofinite_8, axiom, (! [A9:$tType]: (((finite_finite @ A9) => (finite_finite @ (set @ A9)))))).
% 2.63/1.52 thf(tcon_Set_Oset___Orderings_Oorder_9, axiom, (! [A9:$tType]: ((order @ (set @ A9))))).
% 2.63/1.52 thf(tcon_Set_Oset___Orderings_Oord_10, axiom, (! [A9:$tType]: ((ord @ (set @ A9))))).
% 2.63/1.52 thf(tcon_Set_Oset___Orderings_Obot_11, axiom, (! [A9:$tType]: ((bot @ (set @ A9))))).
% 2.63/1.52 thf(tcon_HOL_Obool___Orderings_Oorder__bot_12, axiom, (order_bot @ $o)).
% 2.63/1.52 thf(tcon_HOL_Obool___Orderings_Opreorder_13, axiom, (preorder @ $o)).
% 2.63/1.52 thf(tcon_HOL_Obool___Orderings_Olinorder_14, axiom, (linorder @ $o)).
% 2.63/1.52 thf(tcon_HOL_Obool___Finite__Set_Ofinite_15, axiom, (finite_finite @ $o)).
% 2.63/1.52 thf(tcon_HOL_Obool___Orderings_Oorder_16, axiom, (order @ $o)).
% 2.63/1.52 thf(tcon_HOL_Obool___Orderings_Oord_17, axiom, (ord @ $o)).
% 2.63/1.52 thf(tcon_HOL_Obool___Orderings_Obot_18, axiom, (bot @ $o)).
% 2.63/1.52 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ozero_19, axiom, (zero @ hF_Mirabelle_hf)).
% 2.63/1.52 thf(conj_0, conjecture, ((z = (zero_zero @ hF_Mirabelle_hf)) = (! [X:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ X) @ z))))))).
% 2.63/1.52 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.2mXs7qeb79/DTF2DTF32116.p
% 2.63/1.52 ------------------------
% 2.63/1.56 ---- Cleaned THF ---
% 2.63/1.56 thf(ty_t_HF__Mirabelle__fsbjehakzm_Ohf,type,
% 2.63/1.56 hF_Mirabelle_hf: $tType ).
% 2.63/1.56
% 2.63/1.56 thf(ty_t_Set_Oset,type,
% 2.63/1.56 set: $tType > $tType ).
% 2.63/1.56
% 2.63/1.56 thf(ty_t_Nat_Onat,type,
% 2.63/1.56 nat: $tType ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_HOL_Otype,type,
% 2.63/1.56 type:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Groups_Ozero,type,
% 2.63/1.56 zero:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Orderings_Obot,type,
% 2.63/1.56 bot:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Orderings_Oord,type,
% 2.63/1.56 ord:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Num_Oneg__numeral,type,
% 2.63/1.56 neg_numeral:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Orderings_Oorder,type,
% 2.63/1.56 order:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Groups_Ogroup__add,type,
% 2.63/1.56 group_add:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Rings_Osemiring__1,type,
% 2.63/1.56 semiring_1:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.63/1.56 finite_finite:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Orderings_Olinorder,type,
% 2.63/1.56 linorder:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Orderings_Opreorder,type,
% 2.63/1.56 preorder:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.63/1.56 order_bot:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 2.63/1.56 comm_monoid_diff:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Parity_Osemiring__bit__shifts,type,
% 2.63/1.56 semiring_bit_shifts:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.63/1.56 ordered_ab_group_add:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.63/1.56 cancel1352612707id_add:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.63/1.56 cancel146912293up_add:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.63/1.56 linord1659791738miring:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.63/1.56 canoni770627133id_add:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_cl_Parity_Ounique__euclidean__semiring__with__bit__shifts,type,
% 2.63/1.56 unique788075200shifts:
% 2.63/1.56 !>[A: $tType] : $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Finite__Set_Ocard,type,
% 2.63/1.56 finite_card:
% 2.63/1.56 !>[B: $tType] : ( ( set @ B ) > nat ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Finite__Set_Ofinite,type,
% 2.63/1.56 finite_finite2:
% 2.63/1.56 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Fun_Oinj__on,type,
% 2.63/1.56 inj_on:
% 2.63/1.56 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.63/1.56 minus_minus:
% 2.63/1.56 !>[A: $tType] : ( A > A > A ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.63/1.56 zero_zero:
% 2.63/1.56 !>[A: $tType] : A ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHF,type,
% 2.63/1.56 hF_Mirabelle_HF: ( set @ hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohf_OAbs__hf,type,
% 2.63/1.56 hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfset,type,
% 2.63/1.56 hF_Mirabelle_hfset: hF_Mirabelle_hf > ( set @ hF_Mirabelle_hf ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohinsert,type,
% 2.63/1.56 hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohmem,type,
% 2.63/1.56 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
% 2.63/1.56 lattic1704895705min_on:
% 2.63/1.56 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Nat_Osemiring__1__class_ONats,type,
% 2.63/1.56 semiring_1_Nats:
% 2.63/1.56 !>[A: $tType] : ( set @ A ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
% 2.63/1.56 semiri532925092at_aux:
% 2.63/1.56 !>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
% 2.63/1.56 neg_numeral_dbl:
% 2.63/1.56 !>[A: $tType] : ( A > A ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.63/1.56 bot_bot:
% 2.63/1.56 !>[A: $tType] : A ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.63/1.56 ord_less_eq:
% 2.63/1.56 !>[A: $tType] : ( A > A > $o ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Parity_Osemiring__bit__shifts__class_Odrop__bit,type,
% 2.63/1.56 semiri2097166173op_bit:
% 2.63/1.56 !>[A: $tType] : ( nat > A > A ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Parity_Osemiring__bit__shifts__class_Opush__bit,type,
% 2.63/1.56 semiri1924326578sh_bit:
% 2.63/1.56 !>[A: $tType] : ( nat > A > A ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Relation_Oinv__imagep,type,
% 2.63/1.56 inv_imagep:
% 2.63/1.56 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Set_OCollect,type,
% 2.63/1.56 collect:
% 2.63/1.56 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Set_Oimage,type,
% 2.63/1.56 image:
% 2.63/1.56 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Set_Oinsert,type,
% 2.63/1.56 insert:
% 2.63/1.56 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Set_Ois__empty,type,
% 2.63/1.56 is_empty:
% 2.63/1.56 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Set_Ois__singleton,type,
% 2.63/1.56 is_singleton:
% 2.63/1.56 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_Set_Othe__elem,type,
% 2.63/1.56 the_elem:
% 2.63/1.56 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_c_member,type,
% 2.63/1.56 member:
% 2.63/1.56 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.63/1.56
% 2.63/1.56 thf(sy_v_z,type,
% 2.63/1.56 z: hF_Mirabelle_hf ).
% 2.63/1.56
% 2.63/1.56 thf(fact_0_hf__ext,axiom,
% 2.63/1.56 ( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z ) )
% 2.63/1.56 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 2.63/1.56 ! [X: hF_Mirabelle_hf] :
% 2.63/1.56 ( ( hF_Mirabelle_hmem @ X @ A2 )
% 2.63/1.56 = ( hF_Mirabelle_hmem @ X @ B2 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_1_hemptyE,axiom,
% 2.63/1.56 ! [A3: hF_Mirabelle_hf] :
% 2.63/1.56 ~ ( hF_Mirabelle_hmem @ A3 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_2_hmem__hempty,axiom,
% 2.63/1.56 ! [A3: hF_Mirabelle_hf] :
% 2.63/1.56 ~ ( hF_Mirabelle_hmem @ A3 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_3_hf__cases,axiom,
% 2.63/1.56 ! [Y2: hF_Mirabelle_hf] :
% 2.63/1.56 ( ( Y2
% 2.63/1.56 != ( zero_zero @ hF_Mirabelle_hf ) )
% 2.63/1.56 => ~ ! [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 2.63/1.56 ( ( Y2
% 2.63/1.56 = ( hF_Mirabelle_hinsert @ A4 @ B3 ) )
% 2.63/1.56 => ( hF_Mirabelle_hmem @ A4 @ B3 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_4_hmem__hinsert,axiom,
% 2.63/1.56 ! [A3: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.63/1.56 ( ( hF_Mirabelle_hmem @ A3 @ ( hF_Mirabelle_hinsert @ B4 @ C ) )
% 2.63/1.56 = ( ( A3 = B4 )
% 2.63/1.56 | ( hF_Mirabelle_hmem @ A3 @ C ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_5_hmem__def,axiom,
% 2.63/1.56 ( hF_Mirabelle_hmem
% 2.63/1.56 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( member @ hF_Mirabelle_hf @ A2 @ ( hF_Mirabelle_hfset @ B2 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_6_zero__reorient,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( zero @ A )
% 2.63/1.56 => ! [X2: A] :
% 2.63/1.56 ( ( ( zero_zero @ A )
% 2.63/1.56 = X2 )
% 2.63/1.56 = ( X2
% 2.63/1.56 = ( zero_zero @ A ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_7_Abs__hf__0,axiom,
% 2.63/1.56 ( ( hF_Mirabelle_Abs_hf @ ( zero_zero @ nat ) )
% 2.63/1.56 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_8_dbl__simps_I2_J,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( neg_numeral @ A )
% 2.63/1.56 => ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
% 2.63/1.56 = ( zero_zero @ A ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_9_Zero__hf__def,axiom,
% 2.63/1.56 ( ( zero_zero @ hF_Mirabelle_hf )
% 2.63/1.56 = ( hF_Mirabelle_HF @ ( bot_bot @ ( set @ hF_Mirabelle_hf ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_10_push__bit__of__0,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( semiring_bit_shifts @ A )
% 2.63/1.56 => ! [N: nat] :
% 2.63/1.56 ( ( semiri1924326578sh_bit @ A @ N @ ( zero_zero @ A ) )
% 2.63/1.56 = ( zero_zero @ A ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_11_push__bit__eq__0__iff,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( unique788075200shifts @ A )
% 2.63/1.56 => ! [N: nat,A3: A] :
% 2.63/1.56 ( ( ( semiri1924326578sh_bit @ A @ N @ A3 )
% 2.63/1.56 = ( zero_zero @ A ) )
% 2.63/1.56 = ( A3
% 2.63/1.56 = ( zero_zero @ A ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_12_drop__bit__of__0,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( semiring_bit_shifts @ A )
% 2.63/1.56 => ! [N: nat] :
% 2.63/1.56 ( ( semiri2097166173op_bit @ A @ N @ ( zero_zero @ A ) )
% 2.63/1.56 = ( zero_zero @ A ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_13_Nats__0,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( semiring_1 @ A )
% 2.63/1.56 => ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_14_HF__hfset,axiom,
% 2.63/1.56 ! [A3: hF_Mirabelle_hf] :
% 2.63/1.56 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A3 ) )
% 2.63/1.56 = A3 ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_15_empty__iff,axiom,
% 2.63/1.56 ! [A: $tType,C: A] :
% 2.63/1.56 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_16_all__not__in__conv,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A] :
% 2.63/1.56 ( ( ! [X: A] :
% 2.63/1.56 ~ ( member @ A @ X @ A5 ) )
% 2.63/1.56 = ( A5
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_17_Collect__empty__eq,axiom,
% 2.63/1.56 ! [A: $tType,P: A > $o] :
% 2.63/1.56 ( ( ( collect @ A @ P )
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 = ( ! [X: A] :
% 2.63/1.56 ~ ( P @ X ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_18_empty__Collect__eq,axiom,
% 2.63/1.56 ! [A: $tType,P: A > $o] :
% 2.63/1.56 ( ( ( bot_bot @ ( set @ A ) )
% 2.63/1.56 = ( collect @ A @ P ) )
% 2.63/1.56 = ( ! [X: A] :
% 2.63/1.56 ~ ( P @ X ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_19_bot__apply,axiom,
% 2.63/1.56 ! [C2: $tType,D: $tType] :
% 2.63/1.56 ( ( bot @ C2 )
% 2.63/1.56 => ( ( bot_bot @ ( D > C2 ) )
% 2.63/1.56 = ( ^ [X: D] : ( bot_bot @ C2 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_20_hinsert__def,axiom,
% 2.63/1.56 ( hF_Mirabelle_hinsert
% 2.63/1.56 = ( ^ [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( hF_Mirabelle_HF @ ( insert @ hF_Mirabelle_hf @ A2 @ ( hF_Mirabelle_hfset @ B2 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_21_hfset__HF,axiom,
% 2.63/1.56 ! [A5: set @ hF_Mirabelle_hf] :
% 2.63/1.56 ( ( finite_finite2 @ hF_Mirabelle_hf @ A5 )
% 2.63/1.56 => ( ( hF_Mirabelle_hfset @ ( hF_Mirabelle_HF @ A5 ) )
% 2.63/1.56 = A5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_22_zero__natural_Orsp,axiom,
% 2.63/1.56 ( ( zero_zero @ nat )
% 2.63/1.56 = ( zero_zero @ nat ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_23_emptyE,axiom,
% 2.63/1.56 ! [A: $tType,A3: A] :
% 2.63/1.56 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_24_equals0D,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,A3: A] :
% 2.63/1.56 ( ( A5
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 => ~ ( member @ A @ A3 @ A5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_25_insert__absorb2,axiom,
% 2.63/1.56 ! [A: $tType,X2: A,A5: set @ A] :
% 2.63/1.56 ( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A5 ) )
% 2.63/1.56 = ( insert @ A @ X2 @ A5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_26_insert__iff,axiom,
% 2.63/1.56 ! [A: $tType,A3: A,B4: A,A5: set @ A] :
% 2.63/1.56 ( ( member @ A @ A3 @ ( insert @ A @ B4 @ A5 ) )
% 2.63/1.56 = ( ( A3 = B4 )
% 2.63/1.56 | ( member @ A @ A3 @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_27_insertCI,axiom,
% 2.63/1.56 ! [A: $tType,A3: A,B5: set @ A,B4: A] :
% 2.63/1.56 ( ( ~ ( member @ A @ A3 @ B5 )
% 2.63/1.56 => ( A3 = B4 ) )
% 2.63/1.56 => ( member @ A @ A3 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_28_singletonI,axiom,
% 2.63/1.56 ! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_29_mk__disjoint__insert,axiom,
% 2.63/1.56 ! [A: $tType,A3: A,A5: set @ A] :
% 2.63/1.56 ( ( member @ A @ A3 @ A5 )
% 2.63/1.56 => ? [B6: set @ A] :
% 2.63/1.56 ( ( A5
% 2.63/1.56 = ( insert @ A @ A3 @ B6 ) )
% 2.63/1.56 & ~ ( member @ A @ A3 @ B6 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_30_insert__commute,axiom,
% 2.63/1.56 ! [A: $tType,X2: A,Y2: A,A5: set @ A] :
% 2.63/1.56 ( ( insert @ A @ X2 @ ( insert @ A @ Y2 @ A5 ) )
% 2.63/1.56 = ( insert @ A @ Y2 @ ( insert @ A @ X2 @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_31_insert__eq__iff,axiom,
% 2.63/1.56 ! [A: $tType,A3: A,A5: set @ A,B4: A,B5: set @ A] :
% 2.63/1.56 ( ~ ( member @ A @ A3 @ A5 )
% 2.63/1.56 => ( ~ ( member @ A @ B4 @ B5 )
% 2.63/1.56 => ( ( ( insert @ A @ A3 @ A5 )
% 2.63/1.56 = ( insert @ A @ B4 @ B5 ) )
% 2.63/1.56 = ( ( ( A3 = B4 )
% 2.63/1.56 => ( A5 = B5 ) )
% 2.63/1.56 & ( ( A3 != B4 )
% 2.63/1.56 => ? [C3: set @ A] :
% 2.63/1.56 ( ( A5
% 2.63/1.56 = ( insert @ A @ B4 @ C3 ) )
% 2.63/1.56 & ~ ( member @ A @ B4 @ C3 )
% 2.63/1.56 & ( B5
% 2.63/1.56 = ( insert @ A @ A3 @ C3 ) )
% 2.63/1.56 & ~ ( member @ A @ A3 @ C3 ) ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_32_insert__absorb,axiom,
% 2.63/1.56 ! [A: $tType,A3: A,A5: set @ A] :
% 2.63/1.56 ( ( member @ A @ A3 @ A5 )
% 2.63/1.56 => ( ( insert @ A @ A3 @ A5 )
% 2.63/1.56 = A5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_33_insert__ident,axiom,
% 2.63/1.56 ! [A: $tType,X2: A,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ~ ( member @ A @ X2 @ A5 )
% 2.63/1.56 => ( ~ ( member @ A @ X2 @ B5 )
% 2.63/1.56 => ( ( ( insert @ A @ X2 @ A5 )
% 2.63/1.56 = ( insert @ A @ X2 @ B5 ) )
% 2.63/1.56 = ( A5 = B5 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_34_Set_Oset__insert,axiom,
% 2.63/1.56 ! [A: $tType,X2: A,A5: set @ A] :
% 2.63/1.56 ( ( member @ A @ X2 @ A5 )
% 2.63/1.56 => ~ ! [B6: set @ A] :
% 2.63/1.56 ( ( A5
% 2.63/1.56 = ( insert @ A @ X2 @ B6 ) )
% 2.63/1.56 => ( member @ A @ X2 @ B6 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_35_insertI2,axiom,
% 2.63/1.56 ! [A: $tType,A3: A,B5: set @ A,B4: A] :
% 2.63/1.56 ( ( member @ A @ A3 @ B5 )
% 2.63/1.56 => ( member @ A @ A3 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_36_insertI1,axiom,
% 2.63/1.56 ! [A: $tType,A3: A,B5: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_37_insertE,axiom,
% 2.63/1.56 ! [A: $tType,A3: A,B4: A,A5: set @ A] :
% 2.63/1.56 ( ( member @ A @ A3 @ ( insert @ A @ B4 @ A5 ) )
% 2.63/1.56 => ( ( A3 != B4 )
% 2.63/1.56 => ( member @ A @ A3 @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_38_finite__cases,axiom,
% 2.63/1.56 ! [A: $tType,F: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ F )
% 2.63/1.56 => ( ( F
% 2.63/1.56 != ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 => ~ ! [A6: set @ A,X3: A] :
% 2.63/1.56 ( ( F
% 2.63/1.56 = ( insert @ A @ X3 @ A6 ) )
% 2.63/1.56 => ( ~ ( member @ A @ X3 @ A6 )
% 2.63/1.56 => ~ ( finite_finite2 @ A @ A6 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_39_bot__set__def,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( bot_bot @ ( set @ A ) )
% 2.63/1.56 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_40_singleton__inject,axiom,
% 2.63/1.56 ! [A: $tType,A3: A,B4: A] :
% 2.63/1.56 ( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 = ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.63/1.56 => ( A3 = B4 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_41_insert__not__empty,axiom,
% 2.63/1.56 ! [A: $tType,A3: A,A5: set @ A] :
% 2.63/1.56 ( ( insert @ A @ A3 @ A5 )
% 2.63/1.56 != ( bot_bot @ ( set @ A ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_42_doubleton__eq__iff,axiom,
% 2.63/1.56 ! [A: $tType,A3: A,B4: A,C: A,D2: A] :
% 2.63/1.56 ( ( ( insert @ A @ A3 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.63/1.56 = ( insert @ A @ C @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.63/1.56 = ( ( ( A3 = C )
% 2.63/1.56 & ( B4 = D2 ) )
% 2.63/1.56 | ( ( A3 = D2 )
% 2.63/1.56 & ( B4 = C ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_43_mem__Collect__eq,axiom,
% 2.63/1.56 ! [A: $tType,A3: A,P: A > $o] :
% 2.63/1.56 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 2.63/1.56 = ( P @ A3 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_44_Collect__mem__eq,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A] :
% 2.63/1.56 ( ( collect @ A
% 2.63/1.56 @ ^ [X: A] : ( member @ A @ X @ A5 ) )
% 2.63/1.56 = A5 ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_45_Collect__cong,axiom,
% 2.63/1.56 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.63/1.56 ( ! [X3: A] :
% 2.63/1.56 ( ( P @ X3 )
% 2.63/1.56 = ( Q @ X3 ) )
% 2.63/1.56 => ( ( collect @ A @ P )
% 2.63/1.56 = ( collect @ A @ Q ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_46_ext,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,F2: A > B,G: A > B] :
% 2.63/1.56 ( ! [X3: A] :
% 2.63/1.56 ( ( F2 @ X3 )
% 2.63/1.56 = ( G @ X3 ) )
% 2.63/1.56 => ( F2 = G ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_47_singleton__iff,axiom,
% 2.63/1.56 ! [A: $tType,B4: A,A3: A] :
% 2.63/1.56 ( ( member @ A @ B4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.63/1.56 = ( B4 = A3 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_48_singletonD,axiom,
% 2.63/1.56 ! [A: $tType,B4: A,A3: A] :
% 2.63/1.56 ( ( member @ A @ B4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.63/1.56 => ( B4 = A3 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_49_bot__nat__def,axiom,
% 2.63/1.56 ( ( bot_bot @ nat )
% 2.63/1.56 = ( zero_zero @ nat ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_50_finite__hfset,axiom,
% 2.63/1.56 ! [A3: hF_Mirabelle_hf] : ( finite_finite2 @ hF_Mirabelle_hf @ ( hF_Mirabelle_hfset @ A3 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_51_bot__fun__def,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType] :
% 2.63/1.56 ( ( bot @ B )
% 2.63/1.56 => ( ( bot_bot @ ( A > B ) )
% 2.63/1.56 = ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_52_ex__in__conv,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A] :
% 2.63/1.56 ( ( ? [X: A] : ( member @ A @ X @ A5 ) )
% 2.63/1.56 = ( A5
% 2.63/1.56 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_53_equals0I,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A] :
% 2.63/1.56 ( ! [Y3: A] :
% 2.63/1.56 ~ ( member @ A @ Y3 @ A5 )
% 2.63/1.56 => ( A5
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_54_finite__insert,axiom,
% 2.63/1.56 ! [A: $tType,A3: A,A5: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ ( insert @ A @ A3 @ A5 ) )
% 2.63/1.56 = ( finite_finite2 @ A @ A5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_55_finite_Ocases,axiom,
% 2.63/1.56 ! [A: $tType,A3: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ A3 )
% 2.63/1.56 => ( ( A3
% 2.63/1.56 != ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 => ~ ! [A6: set @ A] :
% 2.63/1.56 ( ? [A4: A] :
% 2.63/1.56 ( A3
% 2.63/1.56 = ( insert @ A @ A4 @ A6 ) )
% 2.63/1.56 => ~ ( finite_finite2 @ A @ A6 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_56_finite_Osimps,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( finite_finite2 @ A )
% 2.63/1.56 = ( ^ [A2: set @ A] :
% 2.63/1.56 ( ( A2
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 | ? [A7: set @ A,B2: A] :
% 2.63/1.56 ( ( A2
% 2.63/1.56 = ( insert @ A @ B2 @ A7 ) )
% 2.63/1.56 & ( finite_finite2 @ A @ A7 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_57_finite__induct,axiom,
% 2.63/1.56 ! [A: $tType,F: set @ A,P: ( set @ A ) > $o] :
% 2.63/1.56 ( ( finite_finite2 @ A @ F )
% 2.63/1.56 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 => ( ! [X3: A,F3: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ F3 )
% 2.63/1.56 => ( ~ ( member @ A @ X3 @ F3 )
% 2.63/1.56 => ( ( P @ F3 )
% 2.63/1.56 => ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) )
% 2.63/1.56 => ( P @ F ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_58_finite_Oinducts,axiom,
% 2.63/1.56 ! [A: $tType,X2: set @ A,P: ( set @ A ) > $o] :
% 2.63/1.56 ( ( finite_finite2 @ A @ X2 )
% 2.63/1.56 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 => ( ! [A6: set @ A,A4: A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ A6 )
% 2.63/1.56 => ( ( P @ A6 )
% 2.63/1.56 => ( P @ ( insert @ A @ A4 @ A6 ) ) ) )
% 2.63/1.56 => ( P @ X2 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_59_finite__ne__induct,axiom,
% 2.63/1.56 ! [A: $tType,F: set @ A,P: ( set @ A ) > $o] :
% 2.63/1.56 ( ( finite_finite2 @ A @ F )
% 2.63/1.56 => ( ( F
% 2.63/1.56 != ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 => ( ! [X3: A] : ( P @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.63/1.56 => ( ! [X3: A,F3: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ F3 )
% 2.63/1.56 => ( ( F3
% 2.63/1.56 != ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 => ( ~ ( member @ A @ X3 @ F3 )
% 2.63/1.56 => ( ( P @ F3 )
% 2.63/1.56 => ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) ) )
% 2.63/1.56 => ( P @ F ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_60_infinite__finite__induct,axiom,
% 2.63/1.56 ! [A: $tType,P: ( set @ A ) > $o,A5: set @ A] :
% 2.63/1.56 ( ! [A6: set @ A] :
% 2.63/1.56 ( ~ ( finite_finite2 @ A @ A6 )
% 2.63/1.56 => ( P @ A6 ) )
% 2.63/1.56 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 => ( ! [X3: A,F3: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ F3 )
% 2.63/1.56 => ( ~ ( member @ A @ X3 @ F3 )
% 2.63/1.56 => ( ( P @ F3 )
% 2.63/1.56 => ( P @ ( insert @ A @ X3 @ F3 ) ) ) ) )
% 2.63/1.56 => ( P @ A5 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_61_the__elem__eq,axiom,
% 2.63/1.56 ! [A: $tType,X2: A] :
% 2.63/1.56 ( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.63/1.56 = X2 ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_62_finite__code,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( finite_finite @ A )
% 2.63/1.56 => ( ( finite_finite2 @ A )
% 2.63/1.56 = ( ^ [A7: set @ A] : $true ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_63_finite__set__choice,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,A5: set @ A,P: A > B > $o] :
% 2.63/1.56 ( ( finite_finite2 @ A @ A5 )
% 2.63/1.56 => ( ! [X3: A] :
% 2.63/1.56 ( ( member @ A @ X3 @ A5 )
% 2.63/1.56 => ? [X_1: B] : ( P @ X3 @ X_1 ) )
% 2.63/1.56 => ? [F4: A > B] :
% 2.63/1.56 ! [X4: A] :
% 2.63/1.56 ( ( member @ A @ X4 @ A5 )
% 2.63/1.56 => ( P @ X4 @ ( F4 @ X4 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_64_finite,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( finite_finite @ A )
% 2.63/1.56 => ! [A5: set @ A] : ( finite_finite2 @ A @ A5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_65_finite_OemptyI,axiom,
% 2.63/1.56 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_66_infinite__imp__nonempty,axiom,
% 2.63/1.56 ! [A: $tType,S: set @ A] :
% 2.63/1.56 ( ~ ( finite_finite2 @ A @ S )
% 2.63/1.56 => ( S
% 2.63/1.56 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_67_finite_OinsertI,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,A3: A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ A5 )
% 2.63/1.56 => ( finite_finite2 @ A @ ( insert @ A @ A3 @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_68_bot__empty__eq,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( bot_bot @ ( A > $o ) )
% 2.63/1.56 = ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_69_Collect__empty__eq__bot,axiom,
% 2.63/1.56 ! [A: $tType,P: A > $o] :
% 2.63/1.56 ( ( ( collect @ A @ P )
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 = ( P
% 2.63/1.56 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_70_is__singleton__the__elem,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( is_singleton @ A )
% 2.63/1.56 = ( ^ [A7: set @ A] :
% 2.63/1.56 ( A7
% 2.63/1.56 = ( insert @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_71_is__singletonI,axiom,
% 2.63/1.56 ! [A: $tType,X2: A] : ( is_singleton @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_72_Set_Ois__empty__def,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( is_empty @ A )
% 2.63/1.56 = ( ^ [A7: set @ A] :
% 2.63/1.56 ( A7
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_73_is__singleton__def,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( is_singleton @ A )
% 2.63/1.56 = ( ^ [A7: set @ A] :
% 2.63/1.56 ? [X: A] :
% 2.63/1.56 ( A7
% 2.63/1.56 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_74_is__singletonE,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A] :
% 2.63/1.56 ( ( is_singleton @ A @ A5 )
% 2.63/1.56 => ~ ! [X3: A] :
% 2.63/1.56 ( A5
% 2.63/1.56 != ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_75_is__singletonI_H,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A] :
% 2.63/1.56 ( ( A5
% 2.63/1.56 != ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 => ( ! [X3: A,Y3: A] :
% 2.63/1.56 ( ( member @ A @ X3 @ A5 )
% 2.63/1.56 => ( ( member @ A @ Y3 @ A5 )
% 2.63/1.56 => ( X3 = Y3 ) ) )
% 2.63/1.56 => ( is_singleton @ A @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_76_card__0__eq,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ A5 )
% 2.63/1.56 => ( ( ( finite_card @ A @ A5 )
% 2.63/1.56 = ( zero_zero @ nat ) )
% 2.63/1.56 = ( A5
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_77_finite__subset__induct,axiom,
% 2.63/1.56 ! [A: $tType,F: set @ A,A5: set @ A,P: ( set @ A ) > $o] :
% 2.63/1.56 ( ( finite_finite2 @ A @ F )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ F @ A5 )
% 2.63/1.56 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 => ( ! [A4: A,F3: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ F3 )
% 2.63/1.56 => ( ( member @ A @ A4 @ A5 )
% 2.63/1.56 => ( ~ ( member @ A @ A4 @ F3 )
% 2.63/1.56 => ( ( P @ F3 )
% 2.63/1.56 => ( P @ ( insert @ A @ A4 @ F3 ) ) ) ) ) )
% 2.63/1.56 => ( P @ F ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_78_finite__subset__induct_H,axiom,
% 2.63/1.56 ! [A: $tType,F: set @ A,A5: set @ A,P: ( set @ A ) > $o] :
% 2.63/1.56 ( ( finite_finite2 @ A @ F )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ F @ A5 )
% 2.63/1.56 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 => ( ! [A4: A,F3: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ F3 )
% 2.63/1.56 => ( ( member @ A @ A4 @ A5 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ F3 @ A5 )
% 2.63/1.56 => ( ~ ( member @ A @ A4 @ F3 )
% 2.63/1.56 => ( ( P @ F3 )
% 2.63/1.56 => ( P @ ( insert @ A @ A4 @ F3 ) ) ) ) ) ) )
% 2.63/1.56 => ( P @ F ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_79_in__inv__imagep,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType] :
% 2.63/1.56 ( ( inv_imagep @ A @ B )
% 2.63/1.56 = ( ^ [R: A > A > $o,F5: B > A,X: B,Y4: B] : ( R @ ( F5 @ X ) @ ( F5 @ Y4 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_80_inj__on__HF,axiom,
% 2.63/1.56 inj_on @ ( set @ hF_Mirabelle_hf ) @ hF_Mirabelle_hf @ hF_Mirabelle_HF @ ( collect @ ( set @ hF_Mirabelle_hf ) @ ( finite_finite2 @ hF_Mirabelle_hf ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_81_of__nat__aux_Osimps_I1_J,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( semiring_1 @ A )
% 2.63/1.56 => ! [Inc: A > A,I: A] :
% 2.63/1.56 ( ( semiri532925092at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
% 2.63/1.56 = I ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_82_order__refl,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( preorder @ A )
% 2.63/1.56 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_83_subset__antisym,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
% 2.63/1.56 => ( A5 = B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_84_subsetI,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ! [X3: A] :
% 2.63/1.56 ( ( member @ A @ X3 @ A5 )
% 2.63/1.56 => ( member @ A @ X3 @ B5 ) )
% 2.63/1.56 => ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_85_le__zero__eq,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( canoni770627133id_add @ A )
% 2.63/1.56 => ! [N: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.63/1.56 = ( N
% 2.63/1.56 = ( zero_zero @ A ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_86_subset__empty,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 = ( A5
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_87_empty__subsetI,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_88_insert__subset,axiom,
% 2.63/1.56 ! [A: $tType,X2: A,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ A5 ) @ B5 )
% 2.63/1.56 = ( ( member @ A @ X2 @ B5 )
% 2.63/1.56 & ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_89_singleton__insert__inj__eq_H,axiom,
% 2.63/1.56 ! [A: $tType,A3: A,A5: set @ A,B4: A] :
% 2.63/1.56 ( ( ( insert @ A @ A3 @ A5 )
% 2.63/1.56 = ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.63/1.56 = ( ( A3 = B4 )
% 2.63/1.56 & ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_90_singleton__insert__inj__eq,axiom,
% 2.63/1.56 ! [A: $tType,B4: A,A3: A,A5: set @ A] :
% 2.63/1.56 ( ( ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 = ( insert @ A @ A3 @ A5 ) )
% 2.63/1.56 = ( ( A3 = B4 )
% 2.63/1.56 & ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_91_card_Oempty,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 = ( zero_zero @ nat ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_92_card_Oinfinite,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A] :
% 2.63/1.56 ( ~ ( finite_finite2 @ A @ A5 )
% 2.63/1.56 => ( ( finite_card @ A @ A5 )
% 2.63/1.56 = ( zero_zero @ nat ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_93_dual__order_Oantisym,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order @ A )
% 2.63/1.56 => ! [B4: A,A3: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ B4 @ A3 )
% 2.63/1.56 => ( ( ord_less_eq @ A @ A3 @ B4 )
% 2.63/1.56 => ( A3 = B4 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_94_dual__order_Oeq__iff,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order @ A )
% 2.63/1.56 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 2.63/1.56 = ( ^ [A2: A,B2: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.63/1.56 & ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_95_dual__order_Otrans,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order @ A )
% 2.63/1.56 => ! [B4: A,A3: A,C: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ B4 @ A3 )
% 2.63/1.56 => ( ( ord_less_eq @ A @ C @ B4 )
% 2.63/1.56 => ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_96_linorder__wlog,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( linorder @ A )
% 2.63/1.56 => ! [P: A > A > $o,A3: A,B4: A] :
% 2.63/1.56 ( ! [A4: A,B3: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.63/1.56 => ( P @ A4 @ B3 ) )
% 2.63/1.56 => ( ! [A4: A,B3: A] :
% 2.63/1.56 ( ( P @ B3 @ A4 )
% 2.63/1.56 => ( P @ A4 @ B3 ) )
% 2.63/1.56 => ( P @ A3 @ B4 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_97_dual__order_Orefl,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order @ A )
% 2.63/1.56 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_98_order__trans,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( preorder @ A )
% 2.63/1.56 => ! [X2: A,Y2: A,Z2: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.63/1.56 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.63/1.56 => ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_99_order__class_Oorder_Oantisym,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order @ A )
% 2.63/1.56 => ! [A3: A,B4: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ A3 @ B4 )
% 2.63/1.56 => ( ( ord_less_eq @ A @ B4 @ A3 )
% 2.63/1.56 => ( A3 = B4 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_100_ord__le__eq__trans,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( ord @ A )
% 2.63/1.56 => ! [A3: A,B4: A,C: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ A3 @ B4 )
% 2.63/1.56 => ( ( B4 = C )
% 2.63/1.56 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_101_ord__eq__le__trans,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( ord @ A )
% 2.63/1.56 => ! [A3: A,B4: A,C: A] :
% 2.63/1.56 ( ( A3 = B4 )
% 2.63/1.56 => ( ( ord_less_eq @ A @ B4 @ C )
% 2.63/1.56 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_102_order__class_Oorder_Oeq__iff,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order @ A )
% 2.63/1.56 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 2.63/1.56 = ( ^ [A2: A,B2: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.63/1.56 & ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_103_antisym__conv,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order @ A )
% 2.63/1.56 => ! [Y2: A,X2: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ Y2 @ X2 )
% 2.63/1.56 => ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.63/1.56 = ( X2 = Y2 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_104_le__cases3,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( linorder @ A )
% 2.63/1.56 => ! [X2: A,Y2: A,Z2: A] :
% 2.63/1.56 ( ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.63/1.56 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 2.63/1.56 => ( ( ( ord_less_eq @ A @ Y2 @ X2 )
% 2.63/1.56 => ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
% 2.63/1.56 => ( ( ( ord_less_eq @ A @ X2 @ Z2 )
% 2.63/1.56 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 2.63/1.56 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 2.63/1.56 => ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
% 2.63/1.56 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.63/1.56 => ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
% 2.63/1.56 => ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
% 2.63/1.56 => ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_105_order_Otrans,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order @ A )
% 2.63/1.56 => ! [A3: A,B4: A,C: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ A3 @ B4 )
% 2.63/1.56 => ( ( ord_less_eq @ A @ B4 @ C )
% 2.63/1.56 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_106_le__cases,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( linorder @ A )
% 2.63/1.56 => ! [X2: A,Y2: A] :
% 2.63/1.56 ( ~ ( ord_less_eq @ A @ X2 @ Y2 )
% 2.63/1.56 => ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_107_eq__refl,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( preorder @ A )
% 2.63/1.56 => ! [X2: A,Y2: A] :
% 2.63/1.56 ( ( X2 = Y2 )
% 2.63/1.56 => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_108_linear,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( linorder @ A )
% 2.63/1.56 => ! [X2: A,Y2: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.63/1.56 | ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_109_antisym,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order @ A )
% 2.63/1.56 => ! [X2: A,Y2: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.63/1.56 => ( ( ord_less_eq @ A @ Y2 @ X2 )
% 2.63/1.56 => ( X2 = Y2 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_110_eq__iff,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order @ A )
% 2.63/1.56 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 2.63/1.56 = ( ^ [X: A,Y4: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.63/1.56 & ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_111_ord__le__eq__subst,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType] :
% 2.63/1.56 ( ( ( ord @ B )
% 2.63/1.56 & ( ord @ A ) )
% 2.63/1.56 => ! [A3: A,B4: A,F2: A > B,C: B] :
% 2.63/1.56 ( ( ord_less_eq @ A @ A3 @ B4 )
% 2.63/1.56 => ( ( ( F2 @ B4 )
% 2.63/1.56 = C )
% 2.63/1.56 => ( ! [X3: A,Y3: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.63/1.56 => ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 2.63/1.56 => ( ord_less_eq @ B @ ( F2 @ A3 ) @ C ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_112_ord__eq__le__subst,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType] :
% 2.63/1.56 ( ( ( ord @ B )
% 2.63/1.56 & ( ord @ A ) )
% 2.63/1.56 => ! [A3: A,F2: B > A,B4: B,C: B] :
% 2.63/1.56 ( ( A3
% 2.63/1.56 = ( F2 @ B4 ) )
% 2.63/1.56 => ( ( ord_less_eq @ B @ B4 @ C )
% 2.63/1.56 => ( ! [X3: B,Y3: B] :
% 2.63/1.56 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 2.63/1.56 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 2.63/1.56 => ( ord_less_eq @ A @ A3 @ ( F2 @ C ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_113_order__subst2,axiom,
% 2.63/1.56 ! [A: $tType,C2: $tType] :
% 2.63/1.56 ( ( ( order @ C2 )
% 2.63/1.56 & ( order @ A ) )
% 2.63/1.56 => ! [A3: A,B4: A,F2: A > C2,C: C2] :
% 2.63/1.56 ( ( ord_less_eq @ A @ A3 @ B4 )
% 2.63/1.56 => ( ( ord_less_eq @ C2 @ ( F2 @ B4 ) @ C )
% 2.63/1.56 => ( ! [X3: A,Y3: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.63/1.56 => ( ord_less_eq @ C2 @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 2.63/1.56 => ( ord_less_eq @ C2 @ ( F2 @ A3 ) @ C ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_114_order__subst1,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType] :
% 2.63/1.56 ( ( ( order @ B )
% 2.63/1.56 & ( order @ A ) )
% 2.63/1.56 => ! [A3: A,F2: B > A,B4: B,C: B] :
% 2.63/1.56 ( ( ord_less_eq @ A @ A3 @ ( F2 @ B4 ) )
% 2.63/1.56 => ( ( ord_less_eq @ B @ B4 @ C )
% 2.63/1.56 => ( ! [X3: B,Y3: B] :
% 2.63/1.56 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 2.63/1.56 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
% 2.63/1.56 => ( ord_less_eq @ A @ A3 @ ( F2 @ C ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_115_Collect__mono__iff,axiom,
% 2.63/1.56 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.63/1.56 = ( ! [X: A] :
% 2.63/1.56 ( ( P @ X )
% 2.63/1.56 => ( Q @ X ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_116_le__fun__def,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType] :
% 2.63/1.56 ( ( ord @ B )
% 2.63/1.56 => ( ( ord_less_eq @ ( A > B ) )
% 2.63/1.56 = ( ^ [F5: A > B,G2: A > B] :
% 2.63/1.56 ! [X: A] : ( ord_less_eq @ B @ ( F5 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_117_set__eq__subset,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( ^ [Y: set @ A,Z: set @ A] : ( Y = Z ) )
% 2.63/1.56 = ( ^ [A7: set @ A,B7: set @ A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.63/1.56 & ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_118_le__funI,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType] :
% 2.63/1.56 ( ( ord @ B )
% 2.63/1.56 => ! [F2: A > B,G: A > B] :
% 2.63/1.56 ( ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
% 2.63/1.56 => ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_119_le__funE,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType] :
% 2.63/1.56 ( ( ord @ B )
% 2.63/1.56 => ! [F2: A > B,G: A > B,X2: A] :
% 2.63/1.56 ( ( ord_less_eq @ ( A > B ) @ F2 @ G )
% 2.63/1.56 => ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_120_le__funD,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType] :
% 2.63/1.56 ( ( ord @ B )
% 2.63/1.56 => ! [F2: A > B,G: A > B,X2: A] :
% 2.63/1.56 ( ( ord_less_eq @ ( A > B ) @ F2 @ G )
% 2.63/1.56 => ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_121_subset__trans,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 2.63/1.56 => ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_122_Collect__mono,axiom,
% 2.63/1.56 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.63/1.56 ( ! [X3: A] :
% 2.63/1.56 ( ( P @ X3 )
% 2.63/1.56 => ( Q @ X3 ) )
% 2.63/1.56 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_123_subset__refl,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_124_subset__iff,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) )
% 2.63/1.56 = ( ^ [A7: set @ A,B7: set @ A] :
% 2.63/1.56 ! [T: A] :
% 2.63/1.56 ( ( member @ A @ T @ A7 )
% 2.63/1.56 => ( member @ A @ T @ B7 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_125_equalityD2,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( A5 = B5 )
% 2.63/1.56 => ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_126_equalityD1,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( A5 = B5 )
% 2.63/1.56 => ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_127_subset__eq,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) )
% 2.63/1.56 = ( ^ [A7: set @ A,B7: set @ A] :
% 2.63/1.56 ! [X: A] :
% 2.63/1.56 ( ( member @ A @ X @ A7 )
% 2.63/1.56 => ( member @ A @ X @ B7 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_128_equalityE,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( A5 = B5 )
% 2.63/1.56 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.63/1.56 => ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_129_subsetD,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,B5: set @ A,C: A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.63/1.56 => ( ( member @ A @ C @ A5 )
% 2.63/1.56 => ( member @ A @ C @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_130_in__mono,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,B5: set @ A,X2: A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.63/1.56 => ( ( member @ A @ X2 @ A5 )
% 2.63/1.56 => ( member @ A @ X2 @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_131_card__subset__eq,axiom,
% 2.63/1.56 ! [A: $tType,B5: set @ A,A5: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ B5 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.63/1.56 => ( ( ( finite_card @ A @ A5 )
% 2.63/1.56 = ( finite_card @ A @ B5 ) )
% 2.63/1.56 => ( A5 = B5 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_132_infinite__arbitrarily__large,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,N: nat] :
% 2.63/1.56 ( ~ ( finite_finite2 @ A @ A5 )
% 2.63/1.56 => ? [B6: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ B6 )
% 2.63/1.56 & ( ( finite_card @ A @ B6 )
% 2.63/1.56 = N )
% 2.63/1.56 & ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_133_zero__le,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( canoni770627133id_add @ A )
% 2.63/1.56 => ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_134_le__numeral__extra_I3_J,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( linord1659791738miring @ A )
% 2.63/1.56 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_135_bot_Oextremum,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order_bot @ A )
% 2.63/1.56 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_136_bot_Oextremum__unique,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order_bot @ A )
% 2.63/1.56 => ! [A3: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 2.63/1.56 = ( A3
% 2.63/1.56 = ( bot_bot @ A ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_137_bot_Oextremum__uniqueI,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order_bot @ A )
% 2.63/1.56 => ! [A3: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 2.63/1.56 => ( A3
% 2.63/1.56 = ( bot_bot @ A ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_138_finite__has__maximal2,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order @ A )
% 2.63/1.56 => ! [A5: set @ A,A3: A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ A5 )
% 2.63/1.56 => ( ( member @ A @ A3 @ A5 )
% 2.63/1.56 => ? [X3: A] :
% 2.63/1.56 ( ( member @ A @ X3 @ A5 )
% 2.63/1.56 & ( ord_less_eq @ A @ A3 @ X3 )
% 2.63/1.56 & ! [Xa: A] :
% 2.63/1.56 ( ( member @ A @ Xa @ A5 )
% 2.63/1.56 => ( ( ord_less_eq @ A @ X3 @ Xa )
% 2.63/1.56 => ( X3 = Xa ) ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_139_finite__has__minimal2,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order @ A )
% 2.63/1.56 => ! [A5: set @ A,A3: A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ A5 )
% 2.63/1.56 => ( ( member @ A @ A3 @ A5 )
% 2.63/1.56 => ? [X3: A] :
% 2.63/1.56 ( ( member @ A @ X3 @ A5 )
% 2.63/1.56 & ( ord_less_eq @ A @ X3 @ A3 )
% 2.63/1.56 & ! [Xa: A] :
% 2.63/1.56 ( ( member @ A @ Xa @ A5 )
% 2.63/1.56 => ( ( ord_less_eq @ A @ Xa @ X3 )
% 2.63/1.56 => ( X3 = Xa ) ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_140_finite__subset,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.63/1.56 => ( ( finite_finite2 @ A @ B5 )
% 2.63/1.56 => ( finite_finite2 @ A @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_141_infinite__super,axiom,
% 2.63/1.56 ! [A: $tType,S: set @ A,T2: set @ A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
% 2.63/1.56 => ( ~ ( finite_finite2 @ A @ S )
% 2.63/1.56 => ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_142_rev__finite__subset,axiom,
% 2.63/1.56 ! [A: $tType,B5: set @ A,A5: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ B5 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.63/1.56 => ( finite_finite2 @ A @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_143_insert__mono,axiom,
% 2.63/1.56 ! [A: $tType,C4: set @ A,D3: set @ A,A3: A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
% 2.63/1.56 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C4 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_144_subset__insert,axiom,
% 2.63/1.56 ! [A: $tType,X2: A,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ~ ( member @ A @ X2 @ A5 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ B5 ) )
% 2.63/1.56 = ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_145_subset__insertI,axiom,
% 2.63/1.56 ! [A: $tType,B5: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A3 @ B5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_146_subset__insertI2,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,B5: set @ A,B4: A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.63/1.56 => ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B4 @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_147_finite__has__minimal,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order @ A )
% 2.63/1.56 => ! [A5: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ A5 )
% 2.63/1.56 => ( ( A5
% 2.63/1.56 != ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 => ? [X3: A] :
% 2.63/1.56 ( ( member @ A @ X3 @ A5 )
% 2.63/1.56 & ! [Xa: A] :
% 2.63/1.56 ( ( member @ A @ Xa @ A5 )
% 2.63/1.56 => ( ( ord_less_eq @ A @ Xa @ X3 )
% 2.63/1.56 => ( X3 = Xa ) ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_148_finite__has__maximal,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( order @ A )
% 2.63/1.56 => ! [A5: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ A5 )
% 2.63/1.56 => ( ( A5
% 2.63/1.56 != ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 => ? [X3: A] :
% 2.63/1.56 ( ( member @ A @ X3 @ A5 )
% 2.63/1.56 & ! [Xa: A] :
% 2.63/1.56 ( ( member @ A @ Xa @ A5 )
% 2.63/1.56 => ( ( ord_less_eq @ A @ X3 @ Xa )
% 2.63/1.56 => ( X3 = Xa ) ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_149_subset__singleton__iff,axiom,
% 2.63/1.56 ! [A: $tType,X5: set @ A,A3: A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.63/1.56 = ( ( X5
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 | ( X5
% 2.63/1.56 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_150_subset__singletonD,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,X2: A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.63/1.56 => ( ( A5
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 | ( A5
% 2.63/1.56 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_151_card__eq__0__iff,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A] :
% 2.63/1.56 ( ( ( finite_card @ A @ A5 )
% 2.63/1.56 = ( zero_zero @ nat ) )
% 2.63/1.56 = ( ( A5
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 | ~ ( finite_finite2 @ A @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_152_inj__on__empty,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,F2: A > B] : ( inj_on @ A @ B @ F2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_153_finite__ranking__induct,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType] :
% 2.63/1.56 ( ( linorder @ A )
% 2.63/1.56 => ! [S: set @ B,P: ( set @ B ) > $o,F2: B > A] :
% 2.63/1.56 ( ( finite_finite2 @ B @ S )
% 2.63/1.56 => ( ( P @ ( bot_bot @ ( set @ B ) ) )
% 2.63/1.56 => ( ! [X3: B,S2: set @ B] :
% 2.63/1.56 ( ( finite_finite2 @ B @ S2 )
% 2.63/1.56 => ( ! [Y5: B] :
% 2.63/1.56 ( ( member @ B @ Y5 @ S2 )
% 2.63/1.56 => ( ord_less_eq @ A @ ( F2 @ Y5 ) @ ( F2 @ X3 ) ) )
% 2.63/1.56 => ( ( P @ S2 )
% 2.63/1.56 => ( P @ ( insert @ B @ X3 @ S2 ) ) ) ) )
% 2.63/1.56 => ( P @ S ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_154_inj__on__subset,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,F2: A > B,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( inj_on @ A @ B @ F2 @ A5 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
% 2.63/1.56 => ( inj_on @ A @ B @ F2 @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_155_subset__inj__on,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,F2: A > B,B5: set @ A,A5: set @ A] :
% 2.63/1.56 ( ( inj_on @ A @ B @ F2 @ B5 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.63/1.56 => ( inj_on @ A @ B @ F2 @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_156_insert__subsetI,axiom,
% 2.63/1.56 ! [A: $tType,X2: A,A5: set @ A,X5: set @ A] :
% 2.63/1.56 ( ( member @ A @ X2 @ A5 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ X5 @ A5 )
% 2.63/1.56 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ X5 ) @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_157_subset__emptyI,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A] :
% 2.63/1.56 ( ! [X3: A] :
% 2.63/1.56 ~ ( member @ A @ X3 @ A5 )
% 2.63/1.56 => ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_158_bot__nat__0_Oextremum,axiom,
% 2.63/1.56 ! [A3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A3 ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_159_le0,axiom,
% 2.63/1.56 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_160_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.63/1.56 ! [A3: nat] :
% 2.63/1.56 ( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
% 2.63/1.56 => ( A3
% 2.63/1.56 = ( zero_zero @ nat ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_161_bot__nat__0_Oextremum__unique,axiom,
% 2.63/1.56 ! [A3: nat] :
% 2.63/1.56 ( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
% 2.63/1.56 = ( A3
% 2.63/1.56 = ( zero_zero @ nat ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_162_le__0__eq,axiom,
% 2.63/1.56 ! [N: nat] :
% 2.63/1.56 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 2.63/1.56 = ( N
% 2.63/1.56 = ( zero_zero @ nat ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_163_less__eq__nat_Osimps_I1_J,axiom,
% 2.63/1.56 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_164_finite__if__finite__subsets__card__bdd,axiom,
% 2.63/1.56 ! [A: $tType,F: set @ A,C4: nat] :
% 2.63/1.56 ( ! [G3: set @ A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ G3 @ F )
% 2.63/1.56 => ( ( finite_finite2 @ A @ G3 )
% 2.63/1.56 => ( ord_less_eq @ nat @ ( finite_card @ A @ G3 ) @ C4 ) ) )
% 2.63/1.56 => ( ( finite_finite2 @ A @ F )
% 2.63/1.56 & ( ord_less_eq @ nat @ ( finite_card @ A @ F ) @ C4 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_165_card__seteq,axiom,
% 2.63/1.56 ! [A: $tType,B5: set @ A,A5: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ B5 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.63/1.56 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ B5 ) @ ( finite_card @ A @ A5 ) )
% 2.63/1.56 => ( A5 = B5 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_166_card__mono,axiom,
% 2.63/1.56 ! [A: $tType,B5: set @ A,A5: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ B5 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.63/1.56 => ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_167_card__insert__le,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,X2: A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ A5 )
% 2.63/1.56 => ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ ( insert @ A @ X2 @ A5 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_168_inj__on__inverseI,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,A5: set @ A,G: B > A,F2: A > B] :
% 2.63/1.56 ( ! [X3: A] :
% 2.63/1.56 ( ( member @ A @ X3 @ A5 )
% 2.63/1.56 => ( ( G @ ( F2 @ X3 ) )
% 2.63/1.56 = X3 ) )
% 2.63/1.56 => ( inj_on @ A @ B @ F2 @ A5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_169_inj__on__contraD,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,F2: A > B,A5: set @ A,X2: A,Y2: A] :
% 2.63/1.56 ( ( inj_on @ A @ B @ F2 @ A5 )
% 2.63/1.56 => ( ( X2 != Y2 )
% 2.63/1.56 => ( ( member @ A @ X2 @ A5 )
% 2.63/1.56 => ( ( member @ A @ Y2 @ A5 )
% 2.63/1.56 => ( ( F2 @ X2 )
% 2.63/1.56 != ( F2 @ Y2 ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_170_inj__on__eq__iff,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,F2: A > B,A5: set @ A,X2: A,Y2: A] :
% 2.63/1.56 ( ( inj_on @ A @ B @ F2 @ A5 )
% 2.63/1.56 => ( ( member @ A @ X2 @ A5 )
% 2.63/1.56 => ( ( member @ A @ Y2 @ A5 )
% 2.63/1.56 => ( ( ( F2 @ X2 )
% 2.63/1.56 = ( F2 @ Y2 ) )
% 2.63/1.56 = ( X2 = Y2 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_171_inj__on__cong,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,A5: set @ A,F2: A > B,G: A > B] :
% 2.63/1.56 ( ! [A4: A] :
% 2.63/1.56 ( ( member @ A @ A4 @ A5 )
% 2.63/1.56 => ( ( F2 @ A4 )
% 2.63/1.56 = ( G @ A4 ) ) )
% 2.63/1.56 => ( ( inj_on @ A @ B @ F2 @ A5 )
% 2.63/1.56 = ( inj_on @ A @ B @ G @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_172_inj__on__def,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType] :
% 2.63/1.56 ( ( inj_on @ A @ B )
% 2.63/1.56 = ( ^ [F5: A > B,A7: set @ A] :
% 2.63/1.56 ! [X: A] :
% 2.63/1.56 ( ( member @ A @ X @ A7 )
% 2.63/1.56 => ! [Y4: A] :
% 2.63/1.56 ( ( member @ A @ Y4 @ A7 )
% 2.63/1.56 => ( ( ( F5 @ X )
% 2.63/1.56 = ( F5 @ Y4 ) )
% 2.63/1.56 => ( X = Y4 ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_173_inj__onI,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,A5: set @ A,F2: A > B] :
% 2.63/1.56 ( ! [X3: A,Y3: A] :
% 2.63/1.56 ( ( member @ A @ X3 @ A5 )
% 2.63/1.56 => ( ( member @ A @ Y3 @ A5 )
% 2.63/1.56 => ( ( ( F2 @ X3 )
% 2.63/1.56 = ( F2 @ Y3 ) )
% 2.63/1.56 => ( X3 = Y3 ) ) ) )
% 2.63/1.56 => ( inj_on @ A @ B @ F2 @ A5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_174_inj__onD,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,F2: A > B,A5: set @ A,X2: A,Y2: A] :
% 2.63/1.56 ( ( inj_on @ A @ B @ F2 @ A5 )
% 2.63/1.56 => ( ( ( F2 @ X2 )
% 2.63/1.56 = ( F2 @ Y2 ) )
% 2.63/1.56 => ( ( member @ A @ X2 @ A5 )
% 2.63/1.56 => ( ( member @ A @ Y2 @ A5 )
% 2.63/1.56 => ( X2 = Y2 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_175_obtain__subset__with__card__n,axiom,
% 2.63/1.56 ! [A: $tType,N: nat,S: set @ A] :
% 2.63/1.56 ( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S ) )
% 2.63/1.56 => ~ ! [T3: set @ A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ T3 @ S )
% 2.63/1.56 => ( ( ( finite_card @ A @ T3 )
% 2.63/1.56 = N )
% 2.63/1.56 => ~ ( finite_finite2 @ A @ T3 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_176_card__le__if__inj__on__rel,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,B5: set @ A,A5: set @ B,R2: B > A > $o] :
% 2.63/1.56 ( ( finite_finite2 @ A @ B5 )
% 2.63/1.56 => ( ! [A4: B] :
% 2.63/1.56 ( ( member @ B @ A4 @ A5 )
% 2.63/1.56 => ? [B8: A] :
% 2.63/1.56 ( ( member @ A @ B8 @ B5 )
% 2.63/1.56 & ( R2 @ A4 @ B8 ) ) )
% 2.63/1.56 => ( ! [A1: B,A22: B,B3: A] :
% 2.63/1.56 ( ( member @ B @ A1 @ A5 )
% 2.63/1.56 => ( ( member @ B @ A22 @ A5 )
% 2.63/1.56 => ( ( member @ A @ B3 @ B5 )
% 2.63/1.56 => ( ( R2 @ A1 @ B3 )
% 2.63/1.56 => ( ( R2 @ A22 @ B3 )
% 2.63/1.56 => ( A1 = A22 ) ) ) ) ) )
% 2.63/1.56 => ( ord_less_eq @ nat @ ( finite_card @ B @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_177_arg__min__least,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType] :
% 2.63/1.56 ( ( linorder @ B )
% 2.63/1.56 => ! [S: set @ A,Y2: A,F2: A > B] :
% 2.63/1.56 ( ( finite_finite2 @ A @ S )
% 2.63/1.56 => ( ( S
% 2.63/1.56 != ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 => ( ( member @ A @ Y2 @ S )
% 2.63/1.56 => ( ord_less_eq @ B @ ( F2 @ ( lattic1704895705min_on @ A @ B @ F2 @ S ) ) @ ( F2 @ Y2 ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_178_bounded__Max__nat,axiom,
% 2.63/1.56 ! [P: nat > $o,X2: nat,M: nat] :
% 2.63/1.56 ( ( P @ X2 )
% 2.63/1.56 => ( ! [X3: nat] :
% 2.63/1.56 ( ( P @ X3 )
% 2.63/1.56 => ( ord_less_eq @ nat @ X3 @ M ) )
% 2.63/1.56 => ~ ! [M2: nat] :
% 2.63/1.56 ( ( P @ M2 )
% 2.63/1.56 => ~ ! [X4: nat] :
% 2.63/1.56 ( ( P @ X4 )
% 2.63/1.56 => ( ord_less_eq @ nat @ X4 @ M2 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_179_ex__has__least__nat,axiom,
% 2.63/1.56 ! [A: $tType,P: A > $o,K: A,M3: A > nat] :
% 2.63/1.56 ( ( P @ K )
% 2.63/1.56 => ? [X3: A] :
% 2.63/1.56 ( ( P @ X3 )
% 2.63/1.56 & ! [Y5: A] :
% 2.63/1.56 ( ( P @ Y5 )
% 2.63/1.56 => ( ord_less_eq @ nat @ ( M3 @ X3 ) @ ( M3 @ Y5 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_180_Nat_Oex__has__greatest__nat,axiom,
% 2.63/1.56 ! [P: nat > $o,K: nat,B4: nat] :
% 2.63/1.56 ( ( P @ K )
% 2.63/1.56 => ( ! [Y3: nat] :
% 2.63/1.56 ( ( P @ Y3 )
% 2.63/1.56 => ( ord_less_eq @ nat @ Y3 @ B4 ) )
% 2.63/1.56 => ? [X3: nat] :
% 2.63/1.56 ( ( P @ X3 )
% 2.63/1.56 & ! [Y5: nat] :
% 2.63/1.56 ( ( P @ Y5 )
% 2.63/1.56 => ( ord_less_eq @ nat @ Y5 @ X3 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_181_nat__le__linear,axiom,
% 2.63/1.56 ! [M3: nat,N: nat] :
% 2.63/1.56 ( ( ord_less_eq @ nat @ M3 @ N )
% 2.63/1.56 | ( ord_less_eq @ nat @ N @ M3 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_182_le__antisym,axiom,
% 2.63/1.56 ! [M3: nat,N: nat] :
% 2.63/1.56 ( ( ord_less_eq @ nat @ M3 @ N )
% 2.63/1.56 => ( ( ord_less_eq @ nat @ N @ M3 )
% 2.63/1.56 => ( M3 = N ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_183_eq__imp__le,axiom,
% 2.63/1.56 ! [M3: nat,N: nat] :
% 2.63/1.56 ( ( M3 = N )
% 2.63/1.56 => ( ord_less_eq @ nat @ M3 @ N ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_184_le__trans,axiom,
% 2.63/1.56 ! [I: nat,J: nat,K: nat] :
% 2.63/1.56 ( ( ord_less_eq @ nat @ I @ J )
% 2.63/1.56 => ( ( ord_less_eq @ nat @ J @ K )
% 2.63/1.56 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_185_le__refl,axiom,
% 2.63/1.56 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_186_finite__nat__set__iff__bounded__le,axiom,
% 2.63/1.56 ( ( finite_finite2 @ nat )
% 2.63/1.56 = ( ^ [N2: set @ nat] :
% 2.63/1.56 ? [M4: nat] :
% 2.63/1.56 ! [X: nat] :
% 2.63/1.56 ( ( member @ nat @ X @ N2 )
% 2.63/1.56 => ( ord_less_eq @ nat @ X @ M4 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_187_arg__min__if__finite_I1_J,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType] :
% 2.63/1.56 ( ( order @ B )
% 2.63/1.56 => ! [S: set @ A,F2: A > B] :
% 2.63/1.56 ( ( finite_finite2 @ A @ S )
% 2.63/1.56 => ( ( S
% 2.63/1.56 != ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 => ( member @ A @ ( lattic1704895705min_on @ A @ B @ F2 @ S ) @ S ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_188_card__Diff1__le,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,X2: A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ A5 )
% 2.63/1.56 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_189_card__le__inj,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,A5: set @ A,B5: set @ B] :
% 2.63/1.56 ( ( finite_finite2 @ A @ A5 )
% 2.63/1.56 => ( ( finite_finite2 @ B @ B5 )
% 2.63/1.56 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B5 ) )
% 2.63/1.56 => ? [F4: A > B] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A5 ) @ B5 )
% 2.63/1.56 & ( inj_on @ A @ B @ F4 @ A5 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_190_card__inj__on__le,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType,F2: A > B,A5: set @ A,B5: set @ B] :
% 2.63/1.56 ( ( inj_on @ A @ B @ F2 @ A5 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A5 ) @ B5 )
% 2.63/1.56 => ( ( finite_finite2 @ B @ B5 )
% 2.63/1.56 => ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B5 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_191_image__eqI,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType,B4: A,F2: B > A,X2: B,A5: set @ B] :
% 2.63/1.56 ( ( B4
% 2.63/1.56 = ( F2 @ X2 ) )
% 2.63/1.56 => ( ( member @ B @ X2 @ A5 )
% 2.63/1.56 => ( member @ A @ B4 @ ( image @ B @ A @ F2 @ A5 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_192_DiffI,axiom,
% 2.63/1.56 ! [A: $tType,C: A,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( member @ A @ C @ A5 )
% 2.63/1.56 => ( ~ ( member @ A @ C @ B5 )
% 2.63/1.56 => ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_193_Diff__iff,axiom,
% 2.63/1.56 ! [A: $tType,C: A,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
% 2.63/1.56 = ( ( member @ A @ C @ A5 )
% 2.63/1.56 & ~ ( member @ A @ C @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_194_Diff__idemp,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ B5 )
% 2.63/1.56 = ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_195_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( cancel1352612707id_add @ A )
% 2.63/1.56 => ! [A3: A] :
% 2.63/1.56 ( ( minus_minus @ A @ A3 @ A3 )
% 2.63/1.56 = ( zero_zero @ A ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_196_diff__zero,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( cancel1352612707id_add @ A )
% 2.63/1.56 => ! [A3: A] :
% 2.63/1.56 ( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
% 2.63/1.56 = A3 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_197_zero__diff,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( comm_monoid_diff @ A )
% 2.63/1.56 => ! [A3: A] :
% 2.63/1.56 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
% 2.63/1.56 = ( zero_zero @ A ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_198_diff__0__right,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( group_add @ A )
% 2.63/1.56 => ! [A3: A] :
% 2.63/1.56 ( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
% 2.63/1.56 = A3 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_199_diff__self,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( group_add @ A )
% 2.63/1.56 => ! [A3: A] :
% 2.63/1.56 ( ( minus_minus @ A @ A3 @ A3 )
% 2.63/1.56 = ( zero_zero @ A ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_200_image__empty,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,F2: B > A] :
% 2.63/1.56 ( ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) )
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_201_empty__is__image,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType,F2: B > A,A5: set @ B] :
% 2.63/1.56 ( ( ( bot_bot @ ( set @ A ) )
% 2.63/1.56 = ( image @ B @ A @ F2 @ A5 ) )
% 2.63/1.56 = ( A5
% 2.63/1.56 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_202_image__is__empty,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType,F2: B > A,A5: set @ B] :
% 2.63/1.56 ( ( ( image @ B @ A @ F2 @ A5 )
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 = ( A5
% 2.63/1.56 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_203_finite__imageI,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,F: set @ A,H: A > B] :
% 2.63/1.56 ( ( finite_finite2 @ A @ F )
% 2.63/1.56 => ( finite_finite2 @ B @ ( image @ A @ B @ H @ F ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_204_image__insert,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType,F2: B > A,A3: B,B5: set @ B] :
% 2.63/1.56 ( ( image @ B @ A @ F2 @ ( insert @ B @ A3 @ B5 ) )
% 2.63/1.56 = ( insert @ A @ ( F2 @ A3 ) @ ( image @ B @ A @ F2 @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_205_insert__image,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,X2: A,A5: set @ A,F2: A > B] :
% 2.63/1.56 ( ( member @ A @ X2 @ A5 )
% 2.63/1.56 => ( ( insert @ B @ ( F2 @ X2 ) @ ( image @ A @ B @ F2 @ A5 ) )
% 2.63/1.56 = ( image @ A @ B @ F2 @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_206_Diff__empty,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A] :
% 2.63/1.56 ( ( minus_minus @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 = A5 ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_207_empty__Diff,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A] :
% 2.63/1.56 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 )
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_208_Diff__cancel,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A] :
% 2.63/1.56 ( ( minus_minus @ ( set @ A ) @ A5 @ A5 )
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_209_finite__Diff,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ A5 )
% 2.63/1.56 => ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_210_finite__Diff2,axiom,
% 2.63/1.56 ! [A: $tType,B5: set @ A,A5: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ B5 )
% 2.63/1.56 => ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
% 2.63/1.56 = ( finite_finite2 @ A @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_211_Diff__insert0,axiom,
% 2.63/1.56 ! [A: $tType,X2: A,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ~ ( member @ A @ X2 @ A5 )
% 2.63/1.56 => ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ B5 ) )
% 2.63/1.56 = ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_212_insert__Diff1,axiom,
% 2.63/1.56 ! [A: $tType,X2: A,B5: set @ A,A5: set @ A] :
% 2.63/1.56 ( ( member @ A @ X2 @ B5 )
% 2.63/1.56 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A5 ) @ B5 )
% 2.63/1.56 = ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_213_diff__ge__0__iff__ge,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( ordered_ab_group_add @ A )
% 2.63/1.56 => ! [A3: A,B4: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B4 ) )
% 2.63/1.56 = ( ord_less_eq @ A @ B4 @ A3 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_214_Diff__eq__empty__iff,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( ( minus_minus @ ( set @ A ) @ A5 @ B5 )
% 2.63/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.63/1.56 = ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_215_insert__Diff__single,axiom,
% 2.63/1.56 ! [A: $tType,A3: A,A5: set @ A] :
% 2.63/1.56 ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.63/1.56 = ( insert @ A @ A3 @ A5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_216_finite__Diff__insert,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,A3: A,B5: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ B5 ) ) )
% 2.63/1.56 = ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_217_inj__on__insert,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,F2: A > B,A3: A,A5: set @ A] :
% 2.63/1.56 ( ( inj_on @ A @ B @ F2 @ ( insert @ A @ A3 @ A5 ) )
% 2.63/1.56 = ( ( inj_on @ A @ B @ F2 @ A5 )
% 2.63/1.56 & ~ ( member @ B @ ( F2 @ A3 ) @ ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_218_image__diff__subset,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType,F2: B > A,A5: set @ B,B5: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A5 ) @ ( image @ B @ A @ F2 @ B5 ) ) @ ( image @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A5 @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_219_inj__on__image__set__diff,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,F2: A > B,C4: set @ A,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( inj_on @ A @ B @ F2 @ C4 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ C4 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 2.63/1.56 => ( ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
% 2.63/1.56 = ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A5 ) @ ( image @ A @ B @ F2 @ B5 ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_220_diff__mono,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( ordered_ab_group_add @ A )
% 2.63/1.56 => ! [A3: A,B4: A,D2: A,C: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ A3 @ B4 )
% 2.63/1.56 => ( ( ord_less_eq @ A @ D2 @ C )
% 2.63/1.56 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B4 @ D2 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_221_diff__left__mono,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( ordered_ab_group_add @ A )
% 2.63/1.56 => ! [B4: A,A3: A,C: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ B4 @ A3 )
% 2.63/1.56 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B4 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_222_diff__right__mono,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( ordered_ab_group_add @ A )
% 2.63/1.56 => ! [A3: A,B4: A,C: A] :
% 2.63/1.56 ( ( ord_less_eq @ A @ A3 @ B4 )
% 2.63/1.56 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B4 @ C ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_223_diff__eq__diff__less__eq,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( ordered_ab_group_add @ A )
% 2.63/1.56 => ! [A3: A,B4: A,C: A,D2: A] :
% 2.63/1.56 ( ( ( minus_minus @ A @ A3 @ B4 )
% 2.63/1.56 = ( minus_minus @ A @ C @ D2 ) )
% 2.63/1.56 => ( ( ord_less_eq @ A @ A3 @ B4 )
% 2.63/1.56 = ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_224_all__subset__image,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType,F2: B > A,A5: set @ B,P: ( set @ A ) > $o] :
% 2.63/1.56 ( ( ! [B7: set @ A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F2 @ A5 ) )
% 2.63/1.56 => ( P @ B7 ) ) )
% 2.63/1.56 = ( ! [B7: set @ B] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ B ) @ B7 @ A5 )
% 2.63/1.56 => ( P @ ( image @ B @ A @ F2 @ B7 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_225_image__mono,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,A5: set @ A,B5: set @ A,F2: A > B] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.63/1.56 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A5 ) @ ( image @ A @ B @ F2 @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_226_image__subsetI,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType,A5: set @ A,F2: A > B,B5: set @ B] :
% 2.63/1.56 ( ! [X3: A] :
% 2.63/1.56 ( ( member @ A @ X3 @ A5 )
% 2.63/1.56 => ( member @ B @ ( F2 @ X3 ) @ B5 ) )
% 2.63/1.56 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A5 ) @ B5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_227_subset__imageE,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType,B5: set @ A,F2: B > A,A5: set @ B] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F2 @ A5 ) )
% 2.63/1.56 => ~ ! [C5: set @ B] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A5 )
% 2.63/1.56 => ( B5
% 2.63/1.56 != ( image @ B @ A @ F2 @ C5 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_228_image__subset__iff,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType,F2: B > A,A5: set @ B,B5: set @ A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A5 ) @ B5 )
% 2.63/1.56 = ( ! [X: B] :
% 2.63/1.56 ( ( member @ B @ X @ A5 )
% 2.63/1.56 => ( member @ A @ ( F2 @ X ) @ B5 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_229_subset__image__iff,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType,B5: set @ A,F2: B > A,A5: set @ B] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F2 @ A5 ) )
% 2.63/1.56 = ( ? [AA: set @ B] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ B ) @ AA @ A5 )
% 2.63/1.56 & ( B5
% 2.63/1.56 = ( image @ B @ A @ F2 @ AA ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_230_Diff__mono,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,C4: set @ A,D3: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ D3 @ B5 )
% 2.63/1.56 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_231_Diff__subset,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ A5 ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_232_double__diff,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 2.63/1.56 => ( ( minus_minus @ ( set @ A ) @ B5 @ ( minus_minus @ ( set @ A ) @ C4 @ A5 ) )
% 2.63/1.56 = A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_233_le__iff__diff__le__0,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( ordered_ab_group_add @ A )
% 2.63/1.56 => ( ( ord_less_eq @ A )
% 2.63/1.56 = ( ^ [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_234_finite__surj,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType,A5: set @ A,B5: set @ B,F2: A > B] :
% 2.63/1.56 ( ( finite_finite2 @ A @ A5 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( image @ A @ B @ F2 @ A5 ) )
% 2.63/1.56 => ( finite_finite2 @ B @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_235_finite__subset__image,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType,B5: set @ A,F2: B > A,A5: set @ B] :
% 2.63/1.56 ( ( finite_finite2 @ A @ B5 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F2 @ A5 ) )
% 2.63/1.56 => ? [C5: set @ B] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A5 )
% 2.63/1.56 & ( finite_finite2 @ B @ C5 )
% 2.63/1.56 & ( B5
% 2.63/1.56 = ( image @ B @ A @ F2 @ C5 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_236_ex__finite__subset__image,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType,F2: B > A,A5: set @ B,P: ( set @ A ) > $o] :
% 2.63/1.56 ( ( ? [B7: set @ A] :
% 2.63/1.56 ( ( finite_finite2 @ A @ B7 )
% 2.63/1.56 & ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F2 @ A5 ) )
% 2.63/1.56 & ( P @ B7 ) ) )
% 2.63/1.56 = ( ? [B7: set @ B] :
% 2.63/1.56 ( ( finite_finite2 @ B @ B7 )
% 2.63/1.56 & ( ord_less_eq @ ( set @ B ) @ B7 @ A5 )
% 2.63/1.56 & ( P @ ( image @ B @ A @ F2 @ B7 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_237_all__finite__subset__image,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType,F2: B > A,A5: set @ B,P: ( set @ A ) > $o] :
% 2.63/1.56 ( ( ! [B7: set @ A] :
% 2.63/1.56 ( ( ( finite_finite2 @ A @ B7 )
% 2.63/1.56 & ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F2 @ A5 ) ) )
% 2.63/1.56 => ( P @ B7 ) ) )
% 2.63/1.56 = ( ! [B7: set @ B] :
% 2.63/1.56 ( ( ( finite_finite2 @ B @ B7 )
% 2.63/1.56 & ( ord_less_eq @ ( set @ B ) @ B7 @ A5 ) )
% 2.63/1.56 => ( P @ ( image @ B @ A @ F2 @ B7 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_238_finite__imageD,axiom,
% 2.63/1.56 ! [A: $tType,B: $tType,F2: B > A,A5: set @ B] :
% 2.63/1.56 ( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ A5 ) )
% 2.63/1.56 => ( ( inj_on @ B @ A @ F2 @ A5 )
% 2.63/1.56 => ( finite_finite2 @ B @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_239_finite__image__iff,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,F2: A > B,A5: set @ A] :
% 2.63/1.56 ( ( inj_on @ A @ B @ F2 @ A5 )
% 2.63/1.56 => ( ( finite_finite2 @ B @ ( image @ A @ B @ F2 @ A5 ) )
% 2.63/1.56 = ( finite_finite2 @ A @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_240_inj__on__image__eq__iff,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,F2: A > B,C4: set @ A,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( inj_on @ A @ B @ F2 @ C4 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
% 2.63/1.56 => ( ( ( image @ A @ B @ F2 @ A5 )
% 2.63/1.56 = ( image @ A @ B @ F2 @ B5 ) )
% 2.63/1.56 = ( A5 = B5 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_241_inj__on__image__mem__iff,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,F2: A > B,B5: set @ A,A3: A,A5: set @ A] :
% 2.63/1.56 ( ( inj_on @ A @ B @ F2 @ B5 )
% 2.63/1.56 => ( ( member @ A @ A3 @ B5 )
% 2.63/1.56 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.63/1.56 => ( ( member @ B @ ( F2 @ A3 ) @ ( image @ A @ B @ F2 @ A5 ) )
% 2.63/1.56 = ( member @ A @ A3 @ A5 ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_242_inj__img__insertE,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,F2: A > B,A5: set @ A,X2: B,B5: set @ B] :
% 2.63/1.56 ( ( inj_on @ A @ B @ F2 @ A5 )
% 2.63/1.56 => ( ~ ( member @ B @ X2 @ B5 )
% 2.63/1.56 => ( ( ( insert @ B @ X2 @ B5 )
% 2.63/1.56 = ( image @ A @ B @ F2 @ A5 ) )
% 2.63/1.56 => ~ ! [X6: A,A8: set @ A] :
% 2.63/1.56 ( ~ ( member @ A @ X6 @ A8 )
% 2.63/1.56 => ( ( A5
% 2.63/1.56 = ( insert @ A @ X6 @ A8 ) )
% 2.63/1.56 => ( ( X2
% 2.63/1.56 = ( F2 @ X6 ) )
% 2.63/1.56 => ( B5
% 2.63/1.56 != ( image @ A @ B @ F2 @ A8 ) ) ) ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_243_card__image,axiom,
% 2.63/1.56 ! [B: $tType,A: $tType,F2: A > B,A5: set @ A] :
% 2.63/1.56 ( ( inj_on @ A @ B @ F2 @ A5 )
% 2.63/1.56 => ( ( finite_card @ B @ ( image @ A @ B @ F2 @ A5 ) )
% 2.63/1.56 = ( finite_card @ A @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_244_Diff__insert__absorb,axiom,
% 2.63/1.56 ! [A: $tType,X2: A,A5: set @ A] :
% 2.63/1.56 ( ~ ( member @ A @ X2 @ A5 )
% 2.63/1.56 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A5 ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.63/1.56 = A5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_245_Diff__insert2,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,A3: A,B5: set @ A] :
% 2.63/1.56 ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ B5 ) )
% 2.63/1.56 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_246_insert__Diff,axiom,
% 2.63/1.56 ! [A: $tType,A3: A,A5: set @ A] :
% 2.63/1.56 ( ( member @ A @ A3 @ A5 )
% 2.63/1.56 => ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.63/1.56 = A5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_247_Diff__insert,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,A3: A,B5: set @ A] :
% 2.63/1.56 ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ B5 ) )
% 2.63/1.56 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_248_subset__Diff__insert,axiom,
% 2.63/1.56 ! [A: $tType,A5: set @ A,B5: set @ A,X2: A,C4: set @ A] :
% 2.63/1.56 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B5 @ ( insert @ A @ X2 @ C4 ) ) )
% 2.63/1.56 = ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B5 @ C4 ) )
% 2.63/1.56 & ~ ( member @ A @ X2 @ A5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_249_in__image__insert__iff,axiom,
% 2.63/1.56 ! [A: $tType,B5: set @ ( set @ A ),X2: A,A5: set @ A] :
% 2.63/1.56 ( ! [C5: set @ A] :
% 2.63/1.56 ( ( member @ ( set @ A ) @ C5 @ B5 )
% 2.63/1.56 => ~ ( member @ A @ X2 @ C5 ) )
% 2.63/1.56 => ( ( member @ ( set @ A ) @ A5 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X2 ) @ B5 ) )
% 2.63/1.56 = ( ( member @ A @ X2 @ A5 )
% 2.63/1.56 & ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_250_diff__eq__diff__eq,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( group_add @ A )
% 2.63/1.56 => ! [A3: A,B4: A,C: A,D2: A] :
% 2.63/1.56 ( ( ( minus_minus @ A @ A3 @ B4 )
% 2.63/1.56 = ( minus_minus @ A @ C @ D2 ) )
% 2.63/1.56 => ( ( A3 = B4 )
% 2.63/1.56 = ( C = D2 ) ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_251_diff__right__commute,axiom,
% 2.63/1.56 ! [A: $tType] :
% 2.63/1.56 ( ( cancel146912293up_add @ A )
% 2.63/1.56 => ! [A3: A,C: A,B4: A] :
% 2.63/1.56 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C ) @ B4 )
% 2.63/1.56 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B4 ) @ C ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_252_DiffE,axiom,
% 2.63/1.56 ! [A: $tType,C: A,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
% 2.63/1.56 => ~ ( ( member @ A @ C @ A5 )
% 2.63/1.56 => ( member @ A @ C @ B5 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(fact_253_DiffD1,axiom,
% 2.63/1.56 ! [A: $tType,C: A,A5: set @ A,B5: set @ A] :
% 2.63/1.56 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) )
% 2.63/1.56 => ( member @ A @ C @ A5 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.63/1.56 ! [A9: $tType,A10: $tType] :
% 2.63/1.56 ( ( order_bot @ A10 )
% 2.63/1.56 => ( order_bot @ ( A9 > A10 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.63/1.56 ! [A9: $tType,A10: $tType] :
% 2.63/1.56 ( ( preorder @ A10 )
% 2.63/1.56 => ( preorder @ ( A9 > A10 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.63/1.56 ! [A9: $tType,A10: $tType] :
% 2.63/1.56 ( ( ( finite_finite @ A9 )
% 2.63/1.56 & ( finite_finite @ A10 ) )
% 2.63/1.56 => ( finite_finite @ ( A9 > A10 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.63/1.56 ! [A9: $tType,A10: $tType] :
% 2.63/1.56 ( ( order @ A10 )
% 2.63/1.56 => ( order @ ( A9 > A10 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_fun___Orderings_Oord,axiom,
% 2.63/1.56 ! [A9: $tType,A10: $tType] :
% 2.63/1.56 ( ( ord @ A10 )
% 2.63/1.56 => ( ord @ ( A9 > A10 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_fun___Orderings_Obot,axiom,
% 2.63/1.56 ! [A9: $tType,A10: $tType] :
% 2.63/1.56 ( ( bot @ A10 )
% 2.63/1.56 => ( bot @ ( A9 > A10 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Nat_Onat___Parity_Ounique__euclidean__semiring__with__bit__shifts,axiom,
% 2.63/1.56 unique788075200shifts @ nat ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.63/1.56 canoni770627133id_add @ nat ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 2.63/1.56 linord1659791738miring @ nat ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.63/1.56 cancel146912293up_add @ nat ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 2.63/1.56 cancel1352612707id_add @ nat ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Nat_Onat___Parity_Osemiring__bit__shifts,axiom,
% 2.63/1.56 semiring_bit_shifts @ nat ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 2.63/1.56 comm_monoid_diff @ nat ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
% 2.63/1.56 order_bot @ nat ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
% 2.63/1.56 preorder @ nat ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.63/1.56 linorder @ nat ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Nat_Onat___Rings_Osemiring__1,axiom,
% 2.63/1.56 semiring_1 @ nat ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
% 2.63/1.56 order @ nat ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
% 2.63/1.56 ord @ nat ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
% 2.63/1.56 bot @ nat ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 2.63/1.56 zero @ nat ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
% 2.63/1.56 ! [A9: $tType] : ( order_bot @ ( set @ A9 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 2.63/1.56 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Set_Oset___Finite__Set_Ofinite_8,axiom,
% 2.63/1.56 ! [A9: $tType] :
% 2.63/1.56 ( ( finite_finite @ A9 )
% 2.63/1.56 => ( finite_finite @ ( set @ A9 ) ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
% 2.63/1.56 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
% 2.63/1.56 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
% 2.63/1.56 ! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_HOL_Obool___Orderings_Oorder__bot_12,axiom,
% 2.63/1.56 order_bot @ $o ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_HOL_Obool___Orderings_Opreorder_13,axiom,
% 2.63/1.56 preorder @ $o ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_HOL_Obool___Orderings_Olinorder_14,axiom,
% 2.63/1.56 linorder @ $o ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_HOL_Obool___Finite__Set_Ofinite_15,axiom,
% 2.63/1.56 finite_finite @ $o ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
% 2.63/1.56 order @ $o ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_HOL_Obool___Orderings_Oord_17,axiom,
% 2.63/1.56 ord @ $o ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_HOL_Obool___Orderings_Obot_18,axiom,
% 2.63/1.56 bot @ $o ).
% 2.63/1.56
% 2.63/1.56 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ozero_19,axiom,
% 2.63/1.56 zero @ hF_Mirabelle_hf ).
% 2.63/1.56
% 2.63/1.56 thf(conj_0,conjecture,
% 2.63/1.56 ( ( z
% 2.63/1.56 = ( zero_zero @ hF_Mirabelle_hf ) )
% 2.63/1.56 = ( ! [X: hF_Mirabelle_hf] :
% 2.63/1.56 ~ ( hF_Mirabelle_hmem @ X @ z ) ) ) ).
% 2.63/1.56 ------------------------
% 2.63/1.58 % (32297)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/183Mi)
% 2.63/1.58 % (32304)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/18Mi)
% 2.63/1.59 % (32301)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/2Mi)
% 2.63/1.59 % (32300)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_32116 for (2999ds/27Mi)
% 2.63/1.59 % (32303)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_32116 for (2999ds/275Mi)
% 2.63/1.59 % (32302)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_32116 for (2999ds/2Mi)
% 2.63/1.59 % (32301)Instruction limit reached!
% 2.63/1.59 % (32301)------------------------------
% 2.63/1.59 % (32301)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.59 % (32301)Termination reason: Unknown
% 2.63/1.59 % (32301)Termination phase: shuffling
% 2.63/1.59 % (32302)Instruction limit reached!
% 2.63/1.59 % (32302)------------------------------
% 2.63/1.59 % (32302)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.59 % (32302)Termination reason: Unknown
% 2.63/1.59 % (32302)Termination phase: shuffling
% 2.63/1.59
% 2.63/1.59 % (32302)Memory used [KB]: 1279
% 2.63/1.59 % (32302)Time elapsed: 0.003 s
% 2.63/1.59 % (32302)Instructions burned: 3 (million)
% 2.63/1.59 % (32302)------------------------------
% 2.63/1.59 % (32302)------------------------------
% 2.63/1.59
% 2.63/1.59 % (32301)Memory used [KB]: 1279
% 2.63/1.59 % (32301)Time elapsed: 0.003 s
% 2.63/1.59 % (32301)Instructions burned: 3 (million)
% 2.63/1.59 % (32301)------------------------------
% 2.63/1.59 % (32301)------------------------------
% 2.63/1.59 % (32304)Instruction limit reached!
% 2.63/1.59 % (32304)------------------------------
% 2.63/1.59 % (32304)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.59 % (32304)Termination reason: Unknown
% 2.63/1.59 % (32304)Termination phase: shuffling
% 2.63/1.59
% 2.63/1.59 % (32304)Memory used [KB]: 1663
% 2.63/1.59 % (32304)Time elapsed: 0.008 s
% 2.63/1.59 % (32304)Instructions burned: 18 (million)
% 2.63/1.59 % (32304)------------------------------
% 2.63/1.59 % (32304)------------------------------
% 2.63/1.59 % (32299)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_32116 for (2999ds/4Mi)
% 2.63/1.59 % (32299)Instruction limit reached!
% 2.63/1.59 % (32299)------------------------------
% 2.63/1.59 % (32299)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.59 % (32299)Termination reason: Unknown
% 2.63/1.59 % (32299)Termination phase: shuffling
% 2.63/1.59
% 2.63/1.59 % (32299)Memory used [KB]: 1407
% 2.63/1.59 % (32299)Time elapsed: 0.004 s
% 2.63/1.59 % (32299)Instructions burned: 5 (million)
% 2.63/1.59 % (32299)------------------------------
% 2.63/1.59 % (32299)------------------------------
% 2.71/1.60 % (32300)Instruction limit reached!
% 2.71/1.60 % (32300)------------------------------
% 2.71/1.60 % (32300)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.60 % (32300)Termination reason: Unknown
% 2.71/1.60 % (32300)Termination phase: Property scanning
% 2.71/1.60
% 2.71/1.60 % (32300)Memory used [KB]: 1663
% 2.71/1.60 % (32300)Time elapsed: 0.016 s
% 2.71/1.60 % (32300)Instructions burned: 29 (million)
% 2.71/1.60 % (32300)------------------------------
% 2.71/1.60 % (32300)------------------------------
% 2.71/1.60 % (32308)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/15Mi)
% 2.71/1.61 % (32307)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_32116 for (2999ds/37Mi)
% 2.71/1.61 % (32306)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/3Mi)
% 2.71/1.61 % (32306)Instruction limit reached!
% 2.71/1.61 % (32306)------------------------------
% 2.71/1.61 % (32306)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.61 % (32306)Termination reason: Unknown
% 2.71/1.61 % (32306)Termination phase: shuffling
% 2.71/1.61
% 2.71/1.61 % (32306)Memory used [KB]: 1279
% 2.71/1.61 % (32306)Time elapsed: 0.003 s
% 2.71/1.61 % (32306)Instructions burned: 3 (million)
% 2.71/1.61 % (32306)------------------------------
% 2.71/1.61 % (32306)------------------------------
% 2.71/1.61 % (32303)Refutation not found, incomplete strategy
% 2.71/1.61 % (32303)------------------------------
% 2.71/1.61 % (32303)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.61 % (32303)Termination reason: Refutation not found, incomplete strategy
% 2.71/1.61
% 2.71/1.61
% 2.71/1.61 % (32303)Memory used [KB]: 6268
% 2.71/1.61 % (32303)Time elapsed: 0.022 s
% 2.71/1.61 % (32303)Instructions burned: 39 (million)
% 2.71/1.61 % (32303)------------------------------
% 2.71/1.61 % (32303)------------------------------
% 2.71/1.61 % (32309)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/3Mi)
% 2.71/1.61 % (32309)Instruction limit reached!
% 2.71/1.61 % (32309)------------------------------
% 2.71/1.61 % (32309)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.61 % (32309)Termination reason: Unknown
% 2.71/1.61 % (32309)Termination phase: shuffling
% 2.71/1.61
% 2.71/1.61 % (32309)Memory used [KB]: 1407
% 2.71/1.61 % (32309)Time elapsed: 0.004 s
% 2.71/1.61 % (32309)Instructions burned: 4 (million)
% 2.71/1.61 % (32309)------------------------------
% 2.71/1.61 % (32309)------------------------------
% 2.71/1.61 % (32308)Instruction limit reached!
% 2.71/1.61 % (32308)------------------------------
% 2.71/1.61 % (32308)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.61 % (32308)Termination reason: Unknown
% 2.71/1.61 % (32308)Termination phase: shuffling
% 2.71/1.61
% 2.71/1.61 % (32308)Memory used [KB]: 1663
% 2.71/1.61 % (32308)Time elapsed: 0.010 s
% 2.71/1.61 % (32308)Instructions burned: 16 (million)
% 2.71/1.61 % (32308)------------------------------
% 2.71/1.61 % (32308)------------------------------
% 2.71/1.62 % (32311)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/1041Mi)
% 2.71/1.62 % (32312)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/7Mi)
% 2.71/1.62 % (32307)Instruction limit reached!
% 2.71/1.62 % (32307)------------------------------
% 2.71/1.62 % (32307)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.62 % (32307)Termination reason: Unknown
% 2.71/1.62 % (32307)Termination phase: shuffling
% 2.71/1.62
% 2.71/1.62 % (32307)Memory used [KB]: 2046
% 2.71/1.62 % (32307)Time elapsed: 0.019 s
% 2.71/1.62 % (32307)Instructions burned: 37 (million)
% 2.71/1.62 % (32307)------------------------------
% 2.71/1.62 % (32307)------------------------------
% 2.71/1.62 % (32313)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/16Mi)
% 2.71/1.63 % (32312)Instruction limit reached!
% 2.71/1.63 % (32312)------------------------------
% 2.71/1.63 % (32312)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.63 % (32312)Termination reason: Unknown
% 2.71/1.63 % (32312)Termination phase: shuffling
% 2.71/1.63
% 2.71/1.63 % (32312)Memory used [KB]: 1407
% 2.71/1.63 % (32312)Time elapsed: 0.005 s
% 2.71/1.63 % (32312)Instructions burned: 7 (million)
% 2.71/1.63 % (32312)------------------------------
% 2.71/1.63 % (32312)------------------------------
% 2.71/1.63 % (32314)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/3Mi)
% 2.71/1.63 % (32315)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/3Mi)
% 2.71/1.63 % (32314)Instruction limit reached!
% 2.71/1.63 % (32314)------------------------------
% 2.71/1.63 % (32314)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.63 % (32314)Termination reason: Unknown
% 2.71/1.63 % (32314)Termination phase: shuffling
% 2.71/1.63
% 2.71/1.63 % (32314)Memory used [KB]: 1407
% 2.71/1.63 % (32314)Time elapsed: 0.004 s
% 2.71/1.63 % (32315)Instruction limit reached!
% 2.71/1.63 % (32315)------------------------------
% 2.71/1.63 % (32315)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.63 % (32315)Termination reason: Unknown
% 2.71/1.63 % (32315)Termination phase: shuffling
% 2.71/1.63
% 2.71/1.63 % (32315)Memory used [KB]: 1407
% 2.71/1.63 % (32315)Time elapsed: 0.003 s
% 2.71/1.63 % (32315)Instructions burned: 3 (million)
% 2.71/1.63 % (32315)------------------------------
% 2.71/1.63 % (32315)------------------------------
% 2.71/1.63 % (32314)Instructions burned: 4 (million)
% 2.71/1.63 % (32314)------------------------------
% 2.71/1.63 % (32314)------------------------------
% 2.71/1.63 % (32313)Instruction limit reached!
% 2.71/1.63 % (32313)------------------------------
% 2.71/1.63 % (32313)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.63 % (32313)Termination reason: Unknown
% 2.71/1.63 % (32313)Termination phase: shuffling
% 2.71/1.63
% 2.71/1.63 % (32313)Memory used [KB]: 1663
% 2.71/1.63 % (32313)Time elapsed: 0.010 s
% 2.71/1.63 % (32313)Instructions burned: 17 (million)
% 2.71/1.63 % (32313)------------------------------
% 2.71/1.63 % (32313)------------------------------
% 2.71/1.64 % (32316)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/7Mi)
% 2.71/1.64 % (32297)Instruction limit reached!
% 2.71/1.64 % (32297)------------------------------
% 2.71/1.64 % (32297)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.64 % (32297)Termination reason: Unknown
% 2.71/1.64 % (32297)Termination phase: Saturation
% 2.71/1.64
% 2.71/1.64 % (32297)Memory used [KB]: 7164
% 2.71/1.64 % (32297)Time elapsed: 0.064 s
% 2.71/1.64 % (32297)Instructions burned: 183 (million)
% 2.71/1.64 % (32297)------------------------------
% 2.71/1.64 % (32297)------------------------------
% 2.71/1.64 % (32317)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/3Mi)
% 2.71/1.64 % (32316)Instruction limit reached!
% 2.71/1.64 % (32316)------------------------------
% 2.71/1.64 % (32316)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.64 % (32316)Termination reason: Unknown
% 2.71/1.64 % (32316)Termination phase: shuffling
% 2.71/1.64
% 2.71/1.64 % (32316)Memory used [KB]: 1407
% 2.71/1.64 % (32316)Time elapsed: 0.006 s
% 2.71/1.64 % (32316)Instructions burned: 8 (million)
% 2.71/1.64 % (32316)------------------------------
% 2.71/1.64 % (32316)------------------------------
% 2.71/1.65 % (32317)Instruction limit reached!
% 2.71/1.65 % (32317)------------------------------
% 2.71/1.65 % (32317)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.65 % (32317)Termination reason: Unknown
% 2.71/1.65 % (32317)Termination phase: shuffling
% 2.71/1.65
% 2.71/1.65 % (32317)Memory used [KB]: 1407
% 2.71/1.65 % (32317)Time elapsed: 0.003 s
% 2.71/1.65 % (32317)Instructions burned: 3 (million)
% 2.71/1.65 % (32317)------------------------------
% 2.71/1.65 % (32317)------------------------------
% 2.71/1.65 % (32318)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/4Mi)
% 2.71/1.65 % (32319)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/18Mi)
% 2.71/1.65 % (32318)Instruction limit reached!
% 2.71/1.65 % (32318)------------------------------
% 2.71/1.65 % (32318)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.65 % (32318)Termination reason: Unknown
% 2.71/1.65 % (32318)Termination phase: shuffling
% 2.71/1.65
% 2.71/1.65 % (32318)Memory used [KB]: 1407
% 2.71/1.65 % (32318)Time elapsed: 0.004 s
% 2.71/1.65 % (32318)Instructions burned: 5 (million)
% 2.71/1.65 % (32318)------------------------------
% 2.71/1.65 % (32318)------------------------------
% 2.71/1.65 % (32320)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/710Mi)
% 2.71/1.66 % (32321)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/6Mi)
% 2.71/1.66 % (32319)Instruction limit reached!
% 2.71/1.66 % (32319)------------------------------
% 2.71/1.66 % (32319)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.66 % (32319)Termination reason: Unknown
% 2.71/1.66 % (32319)Termination phase: shuffling
% 2.71/1.66
% 2.71/1.66 % (32319)Memory used [KB]: 1663
% 2.71/1.66 % (32319)Time elapsed: 0.012 s
% 2.71/1.66 % (32319)Instructions burned: 18 (million)
% 2.71/1.66 % (32319)------------------------------
% 2.71/1.66 % (32319)------------------------------
% 2.71/1.66 % (32321)Instruction limit reached!
% 2.71/1.66 % (32321)------------------------------
% 2.71/1.66 % (32321)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.66 % (32321)Termination reason: Unknown
% 2.71/1.66 % (32321)Termination phase: shuffling
% 2.71/1.66
% 2.71/1.66 % (32321)Memory used [KB]: 1407
% 2.71/1.66 % (32321)Time elapsed: 0.004 s
% 2.71/1.66 % (32321)Instructions burned: 6 (million)
% 2.71/1.66 % (32321)------------------------------
% 2.71/1.66 % (32321)------------------------------
% 2.71/1.66 % (32322)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/902Mi)
% 2.71/1.66 % (32323)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/21Mi)
% 2.71/1.67 % (32324)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_32116 for (2999ds/5Mi)
% 2.71/1.67 % (32320)First to succeed.
% 2.71/1.67 % (32324)Instruction limit reached!
% 2.71/1.67 % (32324)------------------------------
% 2.71/1.67 % (32324)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.67 % (32324)Termination reason: Unknown
% 2.71/1.67 % (32324)Termination phase: shuffling
% 2.71/1.67
% 2.71/1.67 % (32324)Memory used [KB]: 1407
% 2.71/1.67 % (32324)Time elapsed: 0.003 s
% 2.71/1.67 % (32324)Instructions burned: 6 (million)
% 2.71/1.67 % (32324)------------------------------
% 2.71/1.67 % (32324)------------------------------
% 2.71/1.67 % (32323)Instruction limit reached!
% 2.71/1.67 % (32323)------------------------------
% 2.71/1.67 % (32323)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.67 % (32323)Termination reason: Unknown
% 2.71/1.67 % (32323)Termination phase: shuffling
% 2.71/1.67
% 2.71/1.67 % (32323)Memory used [KB]: 1791
% 2.71/1.67 % (32323)Time elapsed: 0.013 s
% 2.71/1.67 % (32323)Instructions burned: 22 (million)
% 2.71/1.67 % (32323)------------------------------
% 2.71/1.67 % (32323)------------------------------
% 2.71/1.67 % (32320)Refutation found. Thanks to Tanya!
% 2.71/1.67 % SZS status Theorem for DTF2THF_32116
% 2.71/1.67 % SZS output start Proof for DTF2THF_32116
% 2.71/1.67 thf(type_def_5, type, set: ($tType) > $tType).
% 2.71/1.67 thf(type_def_6, type, nat: $tType).
% 2.71/1.67 thf(type_def_8, type, hF_Mirabelle_hf: $tType).
% 2.71/1.67 thf(func_def_0, type, hF_Mirabelle_hf: $tType).
% 2.71/1.67 thf(func_def_1, type, set: ($tType) > $tType).
% 2.71/1.67 thf(func_def_2, type, nat: $tType).
% 2.71/1.67 thf(func_def_3, type, type: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_4, type, zero: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_5, type, bot: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_6, type, ord: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_7, type, neg_numeral: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_8, type, order: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_9, type, group_add: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_10, type, semiring_1: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_11, type, finite_finite: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_12, type, linorder: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_13, type, preorder: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_14, type, order_bot: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_15, type, comm_monoid_diff: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_16, type, semiring_bit_shifts: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_17, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_18, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_19, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_20, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_21, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_22, type, unique788075200shifts: !>[X0: $tType]:($o)).
% 2.71/1.67 thf(func_def_23, type, finite_card: !>[X0: $tType]:(set @ X0 > nat)).
% 2.71/1.67 thf(func_def_24, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 2.71/1.67 thf(func_def_25, type, inj_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 2.71/1.67 thf(func_def_26, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.71/1.67 thf(func_def_27, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.71/1.67 thf(func_def_28, type, hF_Mirabelle_HF: set @ hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.71/1.67 thf(func_def_29, type, hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf).
% 2.71/1.67 thf(func_def_30, type, hF_Mirabelle_hfset: hF_Mirabelle_hf > set @ hF_Mirabelle_hf).
% 2.71/1.67 thf(func_def_31, type, hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.71/1.67 thf(func_def_32, type, hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o).
% 2.71/1.67 thf(func_def_33, type, lattic1704895705min_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X0)).
% 2.71/1.67 thf(func_def_34, type, semiring_1_Nats: !>[X0: $tType]:(set @ X0)).
% 2.71/1.67 thf(func_def_35, type, semiri532925092at_aux: !>[X0: $tType]:((X0 > X0) > nat > X0 > X0)).
% 2.71/1.67 thf(func_def_36, type, neg_numeral_dbl: !>[X0: $tType]:(X0 > X0)).
% 2.71/1.67 thf(func_def_37, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.71/1.67 thf(func_def_38, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.71/1.67 thf(func_def_39, type, semiri2097166173op_bit: !>[X0: $tType]:(nat > X0 > X0)).
% 2.71/1.67 thf(func_def_40, type, semiri1924326578sh_bit: !>[X0: $tType]:(nat > X0 > X0)).
% 2.71/1.67 thf(func_def_41, type, inv_imagep: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > X1 > X1 > $o)).
% 2.71/1.67 thf(func_def_42, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.71/1.67 thf(func_def_43, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 2.71/1.67 thf(func_def_44, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 2.71/1.67 thf(func_def_45, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 2.71/1.67 thf(func_def_46, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 2.71/1.67 thf(func_def_47, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 2.71/1.67 thf(func_def_48, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.71/1.67 thf(func_def_49, type, z: hF_Mirabelle_hf).
% 2.71/1.67 thf(func_def_64, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.71/1.67 thf(func_def_65, type, sK1: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.71/1.67 thf(func_def_66, type, sK2: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.71/1.67 thf(func_def_67, type, sK3: hF_Mirabelle_hf).
% 2.71/1.67 thf(f911,plain,(
% 2.71/1.67 $false),
% 2.71/1.67 inference(avatar_sat_refutation,[status(thm)],[f891,f896,f903,f909,f910])).
% 2.71/1.67 thf(f910,plain,(
% 2.71/1.67 spl4_2 | ~spl4_1),
% 2.71/1.67 inference(avatar_split_clause,[status(thm)],[f907,f885,f889])).
% 2.71/1.67 thf(f889,definition,(
% 2.71/1.67 spl4_2 <=> ! [X1 : hF_Mirabelle_hf] : ((member @ hF_Mirabelle_hf @ X1 @ (hF_Mirabelle_hfset @ z)) != $true)),
% 2.71/1.67 introduced(definition,[new_symbols(naming,[spl4_2])],[avatar_definition])).
% 2.71/1.67 thf(f885,definition,(
% 2.71/1.67 spl4_1 <=> (zero_zero @ hF_Mirabelle_hf = z)),
% 2.71/1.67 introduced(definition,[new_symbols(naming,[spl4_1])],[avatar_definition])).
% 2.71/1.67 thf(f907,plain,(
% 2.71/1.67 ( ! [X0 : hF_Mirabelle_hf] : (((member @ hF_Mirabelle_hf @ X0 @ (hF_Mirabelle_hfset @ z)) != $true)) ) | ~spl4_1),
% 2.71/1.67 inference(backward_demodulation,[status(thm)],[f882,f887])).
% 2.71/1.67 thf(f887,plain,(
% 2.71/1.67 (zero_zero @ hF_Mirabelle_hf = z) | ~spl4_1),
% 2.71/1.67 inference(avatar_component_clause,[status(thm)],[f885])).
% 2.71/1.67 thf(f882,plain,(
% 2.71/1.67 ( ! [X0 : hF_Mirabelle_hf] : (($true != (member @ hF_Mirabelle_hf @ X0 @ (hF_Mirabelle_hfset @ zero_zero @ hF_Mirabelle_hf)))) )),
% 2.71/1.67 inference(beta_eta_normalization,[status(thm)],[f864])).
% 2.71/1.67 thf(f864,plain,(
% 2.71/1.67 ( ! [X0 : hF_Mirabelle_hf] : (($true != ((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member @ hF_Mirabelle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))) @ X0 @ zero_zero @ hF_Mirabelle_hf))) )),
% 2.71/1.67 inference(definition_unfolding,[status(thm)],[f846,f862])).
% 2.71/1.67 thf(f862,plain,(
% 2.71/1.67 (hF_Mirabelle_hmem = (^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member @ hF_Mirabelle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))))),
% 2.71/1.67 inference(cnf_transformation,[status(thm)],[f424])).
% 2.71/1.67 thf(f424,plain,(
% 2.71/1.67 (hF_Mirabelle_hmem = (^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member @ hF_Mirabelle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))))),
% 2.71/1.67 inference(fool_elimination,[status(thm)],[f423])).
% 2.71/1.67 thf(f423,plain,(
% 2.71/1.67 (hF_Mirabelle_hmem = (^[X0 : hF_Mirabelle_hf, X1 : hF_Mirabelle_hf] : (member @ hF_Mirabelle_hf @ X0 @ (hF_Mirabelle_hfset @ X1))))),
% 2.71/1.67 inference(rectify,[status(thm)],[f6])).
% 2.71/1.67 thf(f6,axiom,(
% 2.71/1.67 (hF_Mirabelle_hmem = (^[X4 : hF_Mirabelle_hf, X5 : hF_Mirabelle_hf] : (member @ hF_Mirabelle_hf @ X4 @ (hF_Mirabelle_hfset @ X5))))),
% 2.71/1.67 file('/export/starexec/sandbox2/tmp/tmp.2mXs7qeb79/DTF2THF_32116.p',fact_5_hmem__def)).
% 2.71/1.67 thf(f846,plain,(
% 2.71/1.67 ( ! [X0 : hF_Mirabelle_hf] : (((hF_Mirabelle_hmem @ X0 @ zero_zero @ hF_Mirabelle_hf) != $true)) )),
% 2.71/1.67 inference(cnf_transformation,[status(thm)],[f823])).
% 2.71/1.67 thf(f823,plain,(
% 2.71/1.67 ! [X0 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X0 @ zero_zero @ hF_Mirabelle_hf) != $true)),
% 2.71/1.67 inference(flattening,[status(thm)],[f392])).
% 2.71/1.67 thf(f392,plain,(
% 2.71/1.67 ! [X0 : hF_Mirabelle_hf] : ~((hF_Mirabelle_hmem @ X0 @ zero_zero @ hF_Mirabelle_hf) = $true)),
% 2.71/1.67 inference(fool_elimination,[status(thm)],[f391])).
% 2.71/1.67 thf(f391,plain,(
% 2.71/1.67 ! [X0 : hF_Mirabelle_hf] : ~(hF_Mirabelle_hmem @ X0 @ zero_zero @ hF_Mirabelle_hf)),
% 2.71/1.67 inference(rectify,[status(thm)],[f3])).
% 2.71/1.67 thf(f3,axiom,(
% 2.71/1.67 ! [X7 : hF_Mirabelle_hf] : ~(hF_Mirabelle_hmem @ X7 @ zero_zero @ hF_Mirabelle_hf)),
% 2.71/1.67 file('/export/starexec/sandbox2/tmp/tmp.2mXs7qeb79/DTF2THF_32116.p',fact_2_hmem__hempty)).
% 2.71/1.67 thf(f909,plain,(
% 2.71/1.67 ~spl4_2 | ~spl4_3),
% 2.71/1.67 inference(avatar_contradiction_clause,[status(thm)],[f908])).
% 2.71/1.67 thf(f908,plain,(
% 2.71/1.67 $false | (~spl4_2 | ~spl4_3)),
% 2.71/1.67 inference(subsumption_resolution,[status(thm)],[f895,f890])).
% 2.71/1.67 thf(f890,plain,(
% 2.71/1.67 ( ! [X1 : hF_Mirabelle_hf] : (((member @ hF_Mirabelle_hf @ X1 @ (hF_Mirabelle_hfset @ z)) != $true)) ) | ~spl4_2),
% 2.71/1.67 inference(avatar_component_clause,[status(thm)],[f889])).
% 2.71/1.67 thf(f895,plain,(
% 2.71/1.67 ((member @ hF_Mirabelle_hf @ sK3 @ (hF_Mirabelle_hfset @ z)) = $true) | ~spl4_3),
% 2.71/1.67 inference(avatar_component_clause,[status(thm)],[f893])).
% 2.71/1.67 thf(f893,definition,(
% 2.71/1.67 spl4_3 <=> ((member @ hF_Mirabelle_hf @ sK3 @ (hF_Mirabelle_hfset @ z)) = $true)),
% 2.71/1.67 introduced(definition,[new_symbols(naming,[spl4_3])],[avatar_definition])).
% 2.71/1.67 thf(f903,plain,(
% 2.71/1.67 spl4_1 | ~spl4_2),
% 2.71/1.67 inference(avatar_contradiction_clause,[status(thm)],[f902])).
% 2.71/1.67 thf(f902,plain,(
% 2.71/1.67 $false | (spl4_1 | ~spl4_2)),
% 2.71/1.67 inference(subsumption_resolution,[status(thm)],[f901,f886])).
% 2.71/1.67 thf(f886,plain,(
% 2.71/1.67 (zero_zero @ hF_Mirabelle_hf != z) | spl4_1),
% 2.71/1.67 inference(avatar_component_clause,[status(thm)],[f885])).
% 2.71/1.67 thf(f901,plain,(
% 2.71/1.67 (zero_zero @ hF_Mirabelle_hf = z) | ~spl4_2),
% 2.71/1.67 inference(trivial_inequality_removal,[status(thm)],[f899])).
% 2.71/1.67 thf(f899,plain,(
% 2.71/1.67 ($true != $true) | (zero_zero @ hF_Mirabelle_hf = z) | ~spl4_2),
% 2.71/1.67 inference(superposition,[status(thm)],[f890,f898])).
% 2.71/1.67 thf(f898,plain,(
% 2.71/1.67 ( ! [X0 : hF_Mirabelle_hf] : (((member @ hF_Mirabelle_hf @ (sK1 @ X0) @ (hF_Mirabelle_hfset @ X0)) = $true) | (zero_zero @ hF_Mirabelle_hf = X0)) )),
% 2.71/1.67 inference(superposition,[status(thm)],[f880,f851])).
% 2.71/1.67 thf(f851,plain,(
% 2.71/1.67 ( ! [X0 : hF_Mirabelle_hf] : (((hF_Mirabelle_hinsert @ (sK1 @ X0) @ (sK2 @ X0)) = X0) | (zero_zero @ hF_Mirabelle_hf = X0)) )),
% 2.71/1.67 inference(cnf_transformation,[status(thm)],[f833])).
% 2.71/1.67 thf(f833,plain,(
% 2.71/1.67 ! [X0 : hF_Mirabelle_hf] : ((((hF_Mirabelle_hinsert @ (sK1 @ X0) @ (sK2 @ X0)) = X0) & ((hF_Mirabelle_hmem @ (sK1 @ X0) @ (sK2 @ X0)) != $true)) | (zero_zero @ hF_Mirabelle_hf = X0))),
% 2.71/1.67 inference(skolemisation,[status(esa),new_symbols(skolem,[sK1,sK2])],[f826,f832])).
% 2.71/1.67 thf(f832,plain,(
% 2.71/1.67 ! [X0 : hF_Mirabelle_hf] : (? [X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (((hF_Mirabelle_hinsert @ X1 @ X2) = X0) & ((hF_Mirabelle_hmem @ X1 @ X2) != $true)) => (((hF_Mirabelle_hinsert @ (sK1 @ X0) @ (sK2 @ X0)) = X0) & ((hF_Mirabelle_hmem @ (sK1 @ X0) @ (sK2 @ X0)) != $true)))),
% 2.71/1.67 introduced(definition,[],[choice_axiom])).
% 2.71/1.67 thf(f826,plain,(
% 2.71/1.67 ! [X0 : hF_Mirabelle_hf] : (? [X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (((hF_Mirabelle_hinsert @ X1 @ X2) = X0) & ((hF_Mirabelle_hmem @ X1 @ X2) != $true)) | (zero_zero @ hF_Mirabelle_hf = X0))),
% 2.71/1.67 inference(ennf_transformation,[status(thm)],[f386])).
% 2.71/1.67 thf(f386,plain,(
% 2.71/1.67 ! [X0 : hF_Mirabelle_hf] : ((zero_zero @ hF_Mirabelle_hf != X0) => ~! [X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (((hF_Mirabelle_hinsert @ X1 @ X2) = X0) => ((hF_Mirabelle_hmem @ X1 @ X2) = $true)))),
% 2.71/1.67 inference(fool_elimination,[status(thm)],[f385])).
% 2.71/1.67 thf(f385,plain,(
% 2.71/1.67 ! [X0 : hF_Mirabelle_hf] : ((zero_zero @ hF_Mirabelle_hf != X0) => ~! [X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (((hF_Mirabelle_hinsert @ X1 @ X2) = X0) => (hF_Mirabelle_hmem @ X1 @ X2)))),
% 2.71/1.67 inference(rectify,[status(thm)],[f4])).
% 2.71/1.67 thf(f4,axiom,(
% 2.71/1.67 ! [X8 : hF_Mirabelle_hf] : ((zero_zero @ hF_Mirabelle_hf != X8) => ~! [X9 : hF_Mirabelle_hf,X10 : hF_Mirabelle_hf] : (((hF_Mirabelle_hinsert @ X9 @ X10) = X8) => (hF_Mirabelle_hmem @ X9 @ X10)))),
% 2.71/1.67 file('/export/starexec/sandbox2/tmp/tmp.2mXs7qeb79/DTF2THF_32116.p',fact_3_hf__cases)).
% 2.71/1.67 thf(f880,plain,(
% 2.71/1.67 ( ! [X2 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (($true = (member @ hF_Mirabelle_hf @ X1 @ (hF_Mirabelle_hfset @ (hF_Mirabelle_hinsert @ X1 @ X2))))) )),
% 2.71/1.67 inference(beta_eta_normalization,[status(thm)],[f873])).
% 2.71/1.67 thf(f873,plain,(
% 2.71/1.67 ( ! [X2 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : ((((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member @ hF_Mirabelle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))) @ X1 @ (hF_Mirabelle_hinsert @ X1 @ X2)) = $true)) )),
% 2.71/1.67 inference(equality_resolution,[status(thm)],[f868])).
% 2.71/1.67 thf(f868,plain,(
% 2.71/1.67 ( ! [X2 : hF_Mirabelle_hf,X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : ((((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member @ hF_Mirabelle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))) @ X0 @ (hF_Mirabelle_hinsert @ X1 @ X2)) = $true) | (X0 != X1)) )),
% 2.71/1.67 inference(definition_unfolding,[status(thm)],[f852,f862])).
% 2.71/1.67 thf(f852,plain,(
% 2.71/1.67 ( ! [X2 : hF_Mirabelle_hf,X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (($true = (hF_Mirabelle_hmem @ X0 @ (hF_Mirabelle_hinsert @ X1 @ X2))) | (X0 != X1)) )),
% 2.71/1.67 inference(cnf_transformation,[status(thm)],[f835])).
% 2.71/1.67 thf(f835,plain,(
% 2.71/1.67 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : ((((hF_Mirabelle_hmem @ X0 @ X2) = $true) | (X0 = X1) | ($true != (hF_Mirabelle_hmem @ X0 @ (hF_Mirabelle_hinsert @ X1 @ X2)))) & (($true = (hF_Mirabelle_hmem @ X0 @ (hF_Mirabelle_hinsert @ X1 @ X2))) | (((hF_Mirabelle_hmem @ X0 @ X2) != $true) & (X0 != X1))))),
% 2.71/1.67 inference(flattening,[status(thm)],[f834])).
% 2.71/1.67 thf(f834,plain,(
% 2.71/1.67 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (((((hF_Mirabelle_hmem @ X0 @ X2) = $true) | (X0 = X1)) | ($true != (hF_Mirabelle_hmem @ X0 @ (hF_Mirabelle_hinsert @ X1 @ X2)))) & (($true = (hF_Mirabelle_hmem @ X0 @ (hF_Mirabelle_hinsert @ X1 @ X2))) | (((hF_Mirabelle_hmem @ X0 @ X2) != $true) & (X0 != X1))))),
% 2.71/1.67 inference(nnf_transformation,[status(thm)],[f422])).
% 2.71/1.67 thf(f422,plain,(
% 2.71/1.67 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : ((((hF_Mirabelle_hmem @ X0 @ X2) = $true) | (X0 = X1)) <=> ($true = (hF_Mirabelle_hmem @ X0 @ (hF_Mirabelle_hinsert @ X1 @ X2))))),
% 2.71/1.67 inference(fool_elimination,[status(thm)],[f421])).
% 2.71/1.67 thf(f421,plain,(
% 2.71/1.67 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X0 @ (hF_Mirabelle_hinsert @ X1 @ X2)) = (X0 = X1) | (hF_Mirabelle_hmem @ X0 @ X2))),
% 2.71/1.67 inference(rectify,[status(thm)],[f5])).
% 2.71/1.67 thf(f5,axiom,(
% 2.71/1.67 ! [X7 : hF_Mirabelle_hf,X11 : hF_Mirabelle_hf,X12 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X7 @ (hF_Mirabelle_hinsert @ X11 @ X12)) = (X7 = X11) | (hF_Mirabelle_hmem @ X7 @ X12))),
% 2.71/1.67 file('/export/starexec/sandbox2/tmp/tmp.2mXs7qeb79/DTF2THF_32116.p',fact_4_hmem__hinsert)).
% 2.71/1.67 thf(f896,plain,(
% 2.71/1.67 spl4_3 | ~spl4_1),
% 2.71/1.67 inference(avatar_split_clause,[status(thm)],[f879,f885,f893])).
% 2.71/1.67 thf(f879,plain,(
% 2.71/1.67 ((member @ hF_Mirabelle_hf @ sK3 @ (hF_Mirabelle_hfset @ z)) = $true) | (zero_zero @ hF_Mirabelle_hf != z)),
% 2.71/1.67 inference(beta_eta_normalization,[status(thm)],[f869])).
% 2.71/1.67 thf(f869,plain,(
% 2.71/1.67 (((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member @ hF_Mirabelle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))) @ sK3 @ z) = $true) | (zero_zero @ hF_Mirabelle_hf != z)),
% 2.71/1.67 inference(definition_unfolding,[status(thm)],[f857,f862])).
% 2.71/1.67 thf(f857,plain,(
% 2.71/1.67 ($true = (hF_Mirabelle_hmem @ sK3 @ z)) | (zero_zero @ hF_Mirabelle_hf != z)),
% 2.71/1.67 inference(cnf_transformation,[status(thm)],[f839])).
% 2.71/1.67 thf(f839,plain,(
% 2.71/1.67 (($true = (hF_Mirabelle_hmem @ sK3 @ z)) | (zero_zero @ hF_Mirabelle_hf != z)) & (! [X1 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X1 @ z) != $true) | (zero_zero @ hF_Mirabelle_hf = z))),
% 2.71/1.67 inference(skolemisation,[status(esa),new_symbols(skolem,[sK3])],[f837,f838])).
% 2.71/1.67 thf(f838,plain,(
% 2.71/1.67 ? [X0 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X0 @ z) = $true) => ($true = (hF_Mirabelle_hmem @ sK3 @ z))),
% 2.71/1.67 introduced(definition,[],[choice_axiom])).
% 2.71/1.67 thf(f837,plain,(
% 2.71/1.67 (? [X0 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X0 @ z) = $true) | (zero_zero @ hF_Mirabelle_hf != z)) & (! [X1 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X1 @ z) != $true) | (zero_zero @ hF_Mirabelle_hf = z))),
% 2.71/1.67 inference(rectify,[status(thm)],[f836])).
% 2.71/1.67 thf(f836,plain,(
% 2.71/1.67 (? [X0 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X0 @ z) = $true) | (zero_zero @ hF_Mirabelle_hf != z)) & (! [X0 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X0 @ z) != $true) | (zero_zero @ hF_Mirabelle_hf = z))),
% 2.71/1.67 inference(nnf_transformation,[status(thm)],[f827])).
% 2.71/1.67 thf(f827,plain,(
% 2.71/1.67 (zero_zero @ hF_Mirabelle_hf = z) <~> ! [X0 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X0 @ z) != $true)),
% 2.71/1.67 inference(ennf_transformation,[status(thm)],[f821])).
% 2.71/1.67 thf(f821,plain,(
% 2.71/1.67 ~(! [X0 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X0 @ z) != $true) <=> (zero_zero @ hF_Mirabelle_hf = z))),
% 2.71/1.67 inference(flattening,[status(thm)],[f344])).
% 2.71/1.67 thf(f344,plain,(
% 2.71/1.67 ~(! [X0 : hF_Mirabelle_hf] : ~((hF_Mirabelle_hmem @ X0 @ z) = $true) <=> (zero_zero @ hF_Mirabelle_hf = z))),
% 2.71/1.67 inference(fool_elimination,[status(thm)],[f343])).
% 2.71/1.67 thf(f343,plain,(
% 2.71/1.67 ~(! [X0 : hF_Mirabelle_hf] : ~(hF_Mirabelle_hmem @ X0 @ z) = (zero_zero @ hF_Mirabelle_hf = z))),
% 2.71/1.67 inference(rectify,[status(thm)],[f291])).
% 2.71/1.67 thf(f291,negated_conjecture,(
% 2.71/1.67 ~(! [X6 : hF_Mirabelle_hf] : ~(hF_Mirabelle_hmem @ X6 @ z) = (zero_zero @ hF_Mirabelle_hf = z))),
% 2.71/1.67 inference(negated_conjecture,[status(cth)],[f290])).
% 2.71/1.67 thf(f290,conjecture,(
% 2.71/1.67 (! [X6 : hF_Mirabelle_hf] : ~(hF_Mirabelle_hmem @ X6 @ z) = (zero_zero @ hF_Mirabelle_hf = z))),
% 2.71/1.67 file('/export/starexec/sandbox2/tmp/tmp.2mXs7qeb79/DTF2THF_32116.p',conj_0)).
% 2.71/1.67 thf(f891,plain,(
% 2.71/1.67 spl4_1 | spl4_2),
% 2.71/1.67 inference(avatar_split_clause,[status(thm)],[f881,f889,f885])).
% 2.71/1.67 thf(f881,plain,(
% 2.71/1.67 ( ! [X1 : hF_Mirabelle_hf] : (((member @ hF_Mirabelle_hf @ X1 @ (hF_Mirabelle_hfset @ z)) != $true) | (zero_zero @ hF_Mirabelle_hf = z)) )),
% 2.71/1.67 inference(beta_eta_normalization,[status(thm)],[f870])).
% 2.71/1.67 thf(f870,plain,(
% 2.71/1.67 ( ! [X1 : hF_Mirabelle_hf] : ((zero_zero @ hF_Mirabelle_hf = z) | ($true != ((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf]: (member @ hF_Mirabelle_hf @ Y0 @ (hF_Mirabelle_hfset @ Y1))))) @ X1 @ z))) )),
% 2.71/1.68 inference(definition_unfolding,[status(thm)],[f856,f862])).
% 2.71/1.68 thf(f856,plain,(
% 2.71/1.68 ( ! [X1 : hF_Mirabelle_hf] : (((hF_Mirabelle_hmem @ X1 @ z) != $true) | (zero_zero @ hF_Mirabelle_hf = z)) )),
% 2.71/1.68 inference(cnf_transformation,[status(thm)],[f839])).
% 2.71/1.68 % SZS output end Proof for DTF2THF_32116
% 2.71/1.68 % (32320)------------------------------
% 2.71/1.68 % (32320)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.68 % (32320)Termination reason: Refutation
% 2.71/1.68
% 2.71/1.68 % (32320)Memory used [KB]: 6396
% 2.71/1.68 % (32320)Time elapsed: 0.024 s
% 2.71/1.68 % (32320)Instructions burned: 40 (million)
% 2.71/1.68 % (32320)------------------------------
% 2.71/1.68 % (32320)------------------------------
% 2.71/1.68 % (32295)Success in time 0.102 s
%------------------------------------------------------------------------------