%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM183^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n015.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 08:41:52 AM UTC 2026
% Result : Theorem 2.97s 1.60s
% Output : Refutation 2.97s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : COM183^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n015.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 04:16:46 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.45 ---- Original DTF file ---
% 0.30/0.45 thf(spec,logic,$$dhol).
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 % File : COM183^1 : TPTP v9.2.1. Released v7.0.0.
% 0.30/0.45 % Domain : Computing Theory
% 0.30/0.45 % Problem : Grammars and languages 157
% 0.30/0.45 % Version : [Bla16] axioms : Especial.
% 0.30/0.45 % English :
% 0.30/0.45
% 0.30/0.45 % Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% 0.30/0.45 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.30/0.45 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.30/0.45 % Source : [Bla16]
% 0.30/0.45 % Names : gram_lang__157.p [Bla16]
% 0.30/0.45
% 0.30/0.45 % Status : Theorem
% 0.30/0.45 % Rating : 0.00 v7.2.0, 0.25 v7.1.0
% 0.30/0.45 % Syntax : Number of formulae : 354 ( 92 unt; 57 typ; 0 def)
% 0.30/0.45 % Number of atoms : 773 ( 196 equ; 0 cnn)
% 0.30/0.45 % Maximal formula atoms : 13 ( 2 avg)
% 0.30/0.45 % Number of connectives : 3486 ( 45 ~; 4 |; 25 &;3068 @)
% 0.30/0.45 % ( 0 <=>; 344 =>; 0 <=; 0 <~>)
% 0.30/0.45 % Maximal formula depth : 18 ( 7 avg)
% 0.30/0.45 % Number of types : 4 ( 3 usr)
% 0.30/0.45 % Number of type conns : 159 ( 159 >; 0 *; 0 +; 0 <<)
% 0.30/0.45 % Number of symbols : 55 ( 54 usr; 4 con; 0-3 aty)
% 0.30/0.45 % Number of variables : 935 ( 61 ^; 826 !; 4 ?; 935 :)
% 0.30/0.45 % ( 44 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.45 % SPC : TH1_THM_EQU_NAR
% 0.30/0.45
% 0.30/0.45 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.30/0.45 % 2016-07-13 14:40:45.275
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 %----Could-be-implicit typings (5)
% 0.30/0.45 thf(ty_t_DTree_Odtree,type,
% 0.30/0.45 dtree: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_Set_Oset,type,
% 0.30/0.45 set: $tType > $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_DTree_OT,type,
% 0.30/0.45 t: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_DTree_ON,type,
% 0.30/0.45 n: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_itself,type,
% 0.30/0.45 itself: $tType > $tType ).
% 0.30/0.45
% 0.30/0.45 %----Explicit typings (52)
% 0.30/0.45 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.30/0.45 bounded_lattice:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_HOL_Otype,type,
% 0.30/0.45 type:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Groups_Ominus,type,
% 0.30/0.45 minus:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Groups_Ouminus,type,
% 0.30/0.45 uminus:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Orderings_Obot,type,
% 0.30/0.45 bot:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Orderings_Oord,type,
% 0.30/0.45 ord:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Num_Oneg__numeral,type,
% 0.30/0.45 neg_numeral:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Orderings_Oorder,type,
% 0.30/0.45 order:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Groups_Ogroup__add,type,
% 0.30/0.45 group_add:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Lattices_Olattice,type,
% 0.30/0.45 lattice:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Orderings_Olinorder,type,
% 0.30/0.45 linorder:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Orderings_Opreorder,type,
% 0.30/0.45 preorder:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.30/0.45 order_bot:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.30/0.45 linordered_idom:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.30/0.45 comm_monoid_diff:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 0.30/0.45 boolean_algebra:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.30/0.45 semilattice_sup:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.30/0.45 linordered_semidom:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.30/0.45 ordered_ab_group_add:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Lattices_Obounded__lattice__bot,type,
% 0.30/0.45 bounded_lattice_bot:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.30/0.45 cancel1352612707id_add:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.30/0.45 cancel146912293up_add:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.30/0.45 linord219039673up_add:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.30/0.45 linord1659791738miring:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 0.30/0.45 bounde1808546759up_bot:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.30/0.45 canoni770627133id_add:
% 0.30/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_DTree_Odtree_Oroot,type,
% 0.30/0.45 root: dtree > n ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OFr,type,
% 0.30/0.45 gram_L861583724lle_Fr: ( set @ n ) > dtree > ( set @ t ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr,type,
% 0.30/0.45 gram_L1333338417e_inFr: ( set @ n ) > dtree > t > $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr2,type,
% 0.30/0.45 gram_L805317441_inFr2: ( set @ n ) > dtree > t > $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinItr,type,
% 0.30/0.45 gram_L830233218_inItr: ( set @ n ) > dtree > n > $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Osubtr,type,
% 0.30/0.45 gram_L716654942_subtr: ( set @ n ) > dtree > dtree > $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.30/0.45 minus_minus:
% 0.30/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.30/0.45 one_one:
% 0.30/0.45 !>[A: $tType] : A ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.30/0.45 uminus_uminus:
% 0.30/0.45 !>[A: $tType] : ( A > A ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.30/0.45 zero_zero:
% 0.30/0.45 !>[A: $tType] : A ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Lattices_Osemilattice__neutr,type,
% 0.30/0.45 semilattice_neutr:
% 0.30/0.45 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.30/0.45 sup_sup:
% 0.30/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.30/0.45 bot_bot:
% 0.30/0.45 !>[A: $tType] : A ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.45 ord_less_eq:
% 0.30/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 0.30/0.45 order_antimono:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 0.30/0.45 order_strict_mono:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Pure_Otype,type,
% 0.30/0.45 type2:
% 0.30/0.45 !>[A: $tType] : ( itself @ A ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Set_OCollect,type,
% 0.30/0.45 collect:
% 0.30/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Set_Oinsert,type,
% 0.30/0.45 insert:
% 0.30/0.45 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Set_Ois__empty,type,
% 0.30/0.45 is_empty:
% 0.30/0.45 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Set_Opairwise,type,
% 0.30/0.45 pairwise:
% 0.30/0.45 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_member,type,
% 0.30/0.45 member:
% 0.30/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v_ns,type,
% 0.30/0.45 ns: set @ n ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v_tr1,type,
% 0.30/0.45 tr1: dtree ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v_tr2,type,
% 0.30/0.45 tr2: dtree ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v_tr3,type,
% 0.30/0.45 tr3: dtree ).
% 0.30/0.45
% 0.30/0.45 %----Relevant facts (256)
% 0.30/0.45 thf(fact_0__092_060open_062subtr_A_Ins_A_092_060union_062_Ans_J_Atr1_Atr3_092_060close_062,axiom,
% 0.30/0.45 gram_L716654942_subtr @ ( sup_sup @ ( set @ n ) @ ns @ ns ) @ tr1 @ tr3 ).
% 0.30/0.45
% 0.30/0.45 % \<open>subtr (ns \<union> ns) tr1 tr3\<close>
% 0.30/0.45 thf(fact_1_assms_I2_J,axiom,
% 0.30/0.45 gram_L716654942_subtr @ ns @ tr2 @ tr3 ).
% 0.30/0.45
% 0.30/0.45 % assms(2)
% 0.30/0.45 thf(fact_2_assms_I1_J,axiom,
% 0.30/0.45 gram_L716654942_subtr @ ns @ tr1 @ tr2 ).
% 0.30/0.45
% 0.30/0.45 % assms(1)
% 0.30/0.45 thf(fact_3_subtr__trans__Un,axiom,
% 0.30/0.45 ! [Ns12: set @ n,Tr1: dtree,Tr2: dtree,Ns23: set @ n,Tr3: dtree] :
% 0.30/0.45 ( ( gram_L716654942_subtr @ Ns12 @ Tr1 @ Tr2 )
% 0.30/0.45 => ( ( gram_L716654942_subtr @ Ns23 @ Tr2 @ Tr3 )
% 0.30/0.45 => ( gram_L716654942_subtr @ ( sup_sup @ ( set @ n ) @ Ns12 @ Ns23 ) @ Tr1 @ Tr3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % subtr_trans_Un
% 0.30/0.45 thf(fact_4_subtr__mono,axiom,
% 0.30/0.45 ! [Ns: set @ n,Tr1: dtree,Tr2: dtree,Ns2: set @ n] :
% 0.30/0.45 ( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr2 )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns2 )
% 0.30/0.45 => ( gram_L716654942_subtr @ Ns2 @ Tr1 @ Tr2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % subtr_mono
% 0.30/0.45 thf(fact_5_Refl,axiom,
% 0.30/0.45 ! [Tr: dtree,Ns: set @ n] :
% 0.30/0.45 ( ( member @ n @ ( root @ Tr ) @ Ns )
% 0.30/0.45 => ( gram_L716654942_subtr @ Ns @ Tr @ Tr ) ) ).
% 0.30/0.45
% 0.30/0.45 % Refl
% 0.30/0.45 thf(fact_6_subtr__rootL__in,axiom,
% 0.30/0.45 ! [Ns: set @ n,Tr1: dtree,Tr2: dtree] :
% 0.30/0.45 ( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr2 )
% 0.30/0.45 => ( member @ n @ ( root @ Tr1 ) @ Ns ) ) ).
% 0.30/0.45
% 0.30/0.45 % subtr_rootL_in
% 0.30/0.45 thf(fact_7_subtr__rootR__in,axiom,
% 0.30/0.45 ! [Ns: set @ n,Tr1: dtree,Tr2: dtree] :
% 0.30/0.45 ( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr2 )
% 0.30/0.45 => ( member @ n @ ( root @ Tr2 ) @ Ns ) ) ).
% 0.30/0.45
% 0.30/0.45 % subtr_rootR_in
% 0.30/0.45 thf(fact_8_Un__subset__iff,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,C: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C )
% 0.30/0.45 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ C )
% 0.30/0.45 & ( ord_less_eq @ ( set @ A ) @ B2 @ C ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Un_subset_iff
% 0.30/0.45 thf(fact_9_le__sup__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 0.30/0.45 = ( ( ord_less_eq @ A @ X @ Z )
% 0.30/0.45 & ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_sup_iff
% 0.30/0.45 thf(fact_10_sup_Obounded__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [B3: A,C2: A,A3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
% 0.30/0.45 = ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.45 & ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.bounded_iff
% 0.30/0.45 thf(fact_11_UnCI,axiom,
% 0.30/0.45 ! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
% 0.30/0.45 ( ( ~ ( member @ A @ C2 @ B2 )
% 0.30/0.45 => ( member @ A @ C2 @ A2 ) )
% 0.30/0.45 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % UnCI
% 0.30/0.45 thf(fact_12_Un__iff,axiom,
% 0.30/0.45 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 0.30/0.45 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.30/0.45 = ( ( member @ A @ C2 @ A2 )
% 0.30/0.45 | ( member @ A @ C2 @ B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Un_iff
% 0.30/0.45 thf(fact_13_sup__apply,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 0.30/0.45 => ( ( sup_sup @ ( A > B ) )
% 0.30/0.45 = ( ^ [F: A > B,G: A > B,X2: A] : ( sup_sup @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup_apply
% 0.30/0.45 thf(fact_14_sup_Oidem,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [A3: A] :
% 0.30/0.45 ( ( sup_sup @ A @ A3 @ A3 )
% 0.30/0.45 = A3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.idem
% 0.30/0.45 thf(fact_15_sup__idem,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A] :
% 0.30/0.45 ( ( sup_sup @ A @ X @ X )
% 0.30/0.45 = X ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup_idem
% 0.30/0.45 thf(fact_16_sup_Oleft__idem,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [A3: A,B3: A] :
% 0.30/0.45 ( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) )
% 0.30/0.45 = ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.left_idem
% 0.30/0.45 thf(fact_17_sup__left__idem,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.30/0.45 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup_left_idem
% 0.30/0.45 thf(fact_18_sup_Oright__idem,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [A3: A,B3: A] :
% 0.30/0.45 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ B3 )
% 0.30/0.45 = ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.right_idem
% 0.30/0.45 thf(fact_19_subset__antisym,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.30/0.45 => ( A2 = B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % subset_antisym
% 0.30/0.45 thf(fact_20_subsetI,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.45 ( ! [X3: A] :
% 0.30/0.45 ( ( member @ A @ X3 @ A2 )
% 0.30/0.45 => ( member @ A @ X3 @ B2 ) )
% 0.30/0.45 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % subsetI
% 0.30/0.45 thf(fact_21_Collect__mono__iff,axiom,
% 0.30/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.30/0.45 = ( ! [X2: A] :
% 0.30/0.45 ( ( P @ X2 )
% 0.30/0.45 => ( Q @ X2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Collect_mono_iff
% 0.30/0.45 thf(fact_22_contra__subsetD,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.45 => ( ~ ( member @ A @ C2 @ B2 )
% 0.30/0.45 => ~ ( member @ A @ C2 @ A2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % contra_subsetD
% 0.30/0.45 thf(fact_23_set__eq__subset,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ^ [Y2: set @ A,Z2: set @ A] : ( Y2 = Z2 ) )
% 0.30/0.45 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.30/0.45 & ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % set_eq_subset
% 0.30/0.45 thf(fact_24_subset__trans,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,C: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C )
% 0.30/0.45 => ( ord_less_eq @ ( set @ A ) @ A2 @ C ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % subset_trans
% 0.30/0.45 thf(fact_25_Collect__mono,axiom,
% 0.30/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.45 ( ! [X3: A] :
% 0.30/0.45 ( ( P @ X3 )
% 0.30/0.45 => ( Q @ X3 ) )
% 0.30/0.45 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Collect_mono
% 0.30/0.45 thf(fact_26_subset__refl,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.30/0.45
% 0.30/0.45 % subset_refl
% 0.30/0.45 thf(fact_27_rev__subsetD,axiom,
% 0.30/0.45 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 0.30/0.45 ( ( member @ A @ C2 @ A2 )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.45 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % rev_subsetD
% 0.30/0.45 thf(fact_28_subset__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.45 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.30/0.45 ! [T: A] :
% 0.30/0.45 ( ( member @ A @ T @ A4 )
% 0.30/0.45 => ( member @ A @ T @ B4 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % subset_iff
% 0.30/0.45 thf(fact_29_set__rev__mp,axiom,
% 0.30/0.45 ! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
% 0.30/0.45 ( ( member @ A @ X @ A2 )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.45 => ( member @ A @ X @ B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % set_rev_mp
% 0.30/0.45 thf(fact_30_equalityD2,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.45 ( ( A2 = B2 )
% 0.30/0.45 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % equalityD2
% 0.30/0.45 thf(fact_31_equalityD1,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.45 ( ( A2 = B2 )
% 0.30/0.45 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % equalityD1
% 0.30/0.45 thf(fact_32_subset__eq,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.45 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.30/0.45 ! [X2: A] :
% 0.30/0.45 ( ( member @ A @ X2 @ A4 )
% 0.30/0.45 => ( member @ A @ X2 @ B4 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % subset_eq
% 0.30/0.45 thf(fact_33_equalityE,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.45 ( ( A2 = B2 )
% 0.30/0.45 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.45 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % equalityE
% 0.30/0.45 thf(fact_34_subsetCE,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.45 => ( ( member @ A @ C2 @ A2 )
% 0.30/0.45 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % subsetCE
% 0.30/0.45 thf(fact_35_subsetD,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.45 => ( ( member @ A @ C2 @ A2 )
% 0.30/0.45 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % subsetD
% 0.30/0.45 thf(fact_36_in__mono,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.45 => ( ( member @ A @ X @ A2 )
% 0.30/0.45 => ( member @ A @ X @ B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % in_mono
% 0.30/0.45 thf(fact_37_set__mp,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.45 => ( ( member @ A @ X @ A2 )
% 0.30/0.45 => ( member @ A @ X @ B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % set_mp
% 0.30/0.45 thf(fact_38_sup__left__commute,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 0.30/0.45 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup_left_commute
% 0.30/0.45 thf(fact_39_sup_Oleft__commute,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [B3: A,A3: A,C2: A] :
% 0.30/0.45 ( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A3 @ C2 ) )
% 0.30/0.45 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.left_commute
% 0.30/0.45 thf(fact_40_sup__commute,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ( ( sup_sup @ A )
% 0.30/0.45 = ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup_commute
% 0.30/0.45 thf(fact_41_sup_Ocommute,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ( ( sup_sup @ A )
% 0.30/0.45 = ( ^ [A5: A,B5: A] : ( sup_sup @ A @ B5 @ A5 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.commute
% 0.30/0.45 thf(fact_42_sup__assoc,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 0.30/0.45 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup_assoc
% 0.30/0.45 thf(fact_43_sup_Oassoc,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [A3: A,B3: A,C2: A] :
% 0.30/0.45 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ C2 )
% 0.30/0.45 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.assoc
% 0.30/0.45 thf(fact_44_sup__fun__def,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 0.30/0.45 => ( ( sup_sup @ ( A > B ) )
% 0.30/0.45 = ( ^ [F: A > B,G: A > B,X2: A] : ( sup_sup @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup_fun_def
% 0.30/0.45 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.45 ! [A: $tType,A3: A,P: A > $o] :
% 0.30/0.45 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.30/0.45 = ( P @ A3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % mem_Collect_eq
% 0.30/0.45 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A] :
% 0.30/0.45 ( ( collect @ A
% 0.30/0.45 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 0.30/0.45 = A2 ) ).
% 0.30/0.45
% 0.30/0.45 % Collect_mem_eq
% 0.30/0.45 thf(fact_47_Collect__cong,axiom,
% 0.30/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.45 ( ! [X3: A] :
% 0.30/0.45 ( ( P @ X3 )
% 0.30/0.45 = ( Q @ X3 ) )
% 0.30/0.45 => ( ( collect @ A @ P )
% 0.30/0.45 = ( collect @ A @ Q ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Collect_cong
% 0.30/0.45 thf(fact_48_ext,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
% 0.30/0.45 ( ! [X3: A] :
% 0.30/0.45 ( ( F2 @ X3 )
% 0.30/0.45 = ( G2 @ X3 ) )
% 0.30/0.45 => ( F2 = G2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % ext
% 0.30/0.45 thf(fact_49_inf__sup__aci_I5_J,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.30/0.45 => ( ( sup_sup @ A )
% 0.30/0.45 = ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % inf_sup_aci(5)
% 0.30/0.45 thf(fact_50_inf__sup__aci_I6_J,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 0.30/0.45 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % inf_sup_aci(6)
% 0.30/0.45 thf(fact_51_inf__sup__aci_I7_J,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 0.30/0.45 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % inf_sup_aci(7)
% 0.30/0.45 thf(fact_52_inf__sup__aci_I8_J,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.30/0.45 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % inf_sup_aci(8)
% 0.30/0.45 thf(fact_53_Un__left__commute,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,C: set @ A] :
% 0.30/0.45 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C ) )
% 0.30/0.45 = ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ C ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Un_left_commute
% 0.30/0.45 thf(fact_54_Un__left__absorb,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.45 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.30/0.45 = ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % Un_left_absorb
% 0.30/0.45 thf(fact_55_Un__commute,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( sup_sup @ ( set @ A ) )
% 0.30/0.45 = ( ^ [A4: set @ A,B4: set @ A] : ( sup_sup @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Un_commute
% 0.30/0.45 thf(fact_56_Un__absorb,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A] :
% 0.30/0.45 ( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
% 0.30/0.45 = A2 ) ).
% 0.30/0.45
% 0.30/0.45 % Un_absorb
% 0.30/0.45 thf(fact_57_Un__assoc,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,C: set @ A] :
% 0.30/0.45 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C )
% 0.30/0.45 = ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Un_assoc
% 0.30/0.45 thf(fact_58_ball__Un,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
% 0.30/0.45 ( ( ! [X2: A] :
% 0.30/0.45 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.30/0.45 => ( P @ X2 ) ) )
% 0.30/0.45 = ( ! [X2: A] :
% 0.30/0.45 ( ( member @ A @ X2 @ A2 )
% 0.30/0.45 => ( P @ X2 ) )
% 0.30/0.45 & ! [X2: A] :
% 0.30/0.45 ( ( member @ A @ X2 @ B2 )
% 0.30/0.45 => ( P @ X2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ball_Un
% 0.30/0.45 thf(fact_59_bex__Un,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
% 0.30/0.45 ( ( ? [X2: A] :
% 0.30/0.45 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.30/0.45 & ( P @ X2 ) ) )
% 0.30/0.45 = ( ? [X2: A] :
% 0.30/0.45 ( ( member @ A @ X2 @ A2 )
% 0.30/0.45 & ( P @ X2 ) )
% 0.30/0.45 | ? [X2: A] :
% 0.30/0.45 ( ( member @ A @ X2 @ B2 )
% 0.30/0.45 & ( P @ X2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % bex_Un
% 0.30/0.45 thf(fact_60_UnI2,axiom,
% 0.30/0.45 ! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
% 0.30/0.45 ( ( member @ A @ C2 @ B2 )
% 0.30/0.45 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % UnI2
% 0.30/0.45 thf(fact_61_UnI1,axiom,
% 0.30/0.45 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 0.30/0.45 ( ( member @ A @ C2 @ A2 )
% 0.30/0.45 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % UnI1
% 0.30/0.45 thf(fact_62_UnE,axiom,
% 0.30/0.45 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 0.30/0.45 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.30/0.45 => ( ~ ( member @ A @ C2 @ A2 )
% 0.30/0.45 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % UnE
% 0.30/0.45 thf(fact_63_sup_OcoboundedI2,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [C2: A,B3: A,A3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.30/0.45 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.coboundedI2
% 0.30/0.45 thf(fact_64_sup_OcoboundedI1,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [C2: A,A3: A,B3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.45 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.coboundedI1
% 0.30/0.45 thf(fact_65_sup_Oabsorb__iff2,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ( ( ord_less_eq @ A )
% 0.30/0.45 = ( ^ [A5: A,B5: A] :
% 0.30/0.45 ( ( sup_sup @ A @ A5 @ B5 )
% 0.30/0.45 = B5 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.absorb_iff2
% 0.30/0.45 thf(fact_66_sup_Oabsorb__iff1,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ( ( ord_less_eq @ A )
% 0.30/0.45 = ( ^ [B5: A,A5: A] :
% 0.30/0.45 ( ( sup_sup @ A @ A5 @ B5 )
% 0.30/0.45 = A5 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.absorb_iff1
% 0.30/0.45 thf(fact_67_sup_Ocobounded2,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [B3: A,A3: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.cobounded2
% 0.30/0.45 thf(fact_68_sup_Ocobounded1,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [A3: A,B3: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.cobounded1
% 0.30/0.45 thf(fact_69_sup_Oorder__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ( ( ord_less_eq @ A )
% 0.30/0.45 = ( ^ [B5: A,A5: A] :
% 0.30/0.45 ( A5
% 0.30/0.45 = ( sup_sup @ A @ A5 @ B5 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.order_iff
% 0.30/0.45 thf(fact_70_sup_OboundedI,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [B3: A,A3: A,C2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.45 => ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.boundedI
% 0.30/0.45 thf(fact_71_sup_OboundedE,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [B3: A,C2: A,A3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
% 0.30/0.45 => ~ ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.boundedE
% 0.30/0.45 thf(fact_72_sup__absorb2,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 => ( ( sup_sup @ A @ X @ Y )
% 0.30/0.45 = Y ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup_absorb2
% 0.30/0.45 thf(fact_73_sup__absorb1,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [Y: A,X: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.45 => ( ( sup_sup @ A @ X @ Y )
% 0.30/0.45 = X ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup_absorb1
% 0.30/0.45 thf(fact_74_sup_Oabsorb2,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [A3: A,B3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.45 => ( ( sup_sup @ A @ A3 @ B3 )
% 0.30/0.45 = B3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.absorb2
% 0.30/0.45 thf(fact_75_sup_Oabsorb1,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [B3: A,A3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.45 => ( ( sup_sup @ A @ A3 @ B3 )
% 0.30/0.45 = A3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.absorb1
% 0.30/0.45 thf(fact_76_sup__unique,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [F2: A > A > A,X: A,Y: A] :
% 0.30/0.45 ( ! [X3: A,Y4: A] : ( ord_less_eq @ A @ X3 @ ( F2 @ X3 @ Y4 ) )
% 0.30/0.45 => ( ! [X3: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ ( F2 @ X3 @ Y4 ) )
% 0.30/0.45 => ( ! [X3: A,Y4: A,Z3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ Y4 @ X3 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ Z3 @ X3 )
% 0.30/0.45 => ( ord_less_eq @ A @ ( F2 @ Y4 @ Z3 ) @ X3 ) ) )
% 0.30/0.45 => ( ( sup_sup @ A @ X @ Y )
% 0.30/0.45 = ( F2 @ X @ Y ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup_unique
% 0.30/0.45 thf(fact_77_sup_OorderI,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [A3: A,B3: A] :
% 0.30/0.45 ( ( A3
% 0.30/0.45 = ( sup_sup @ A @ A3 @ B3 ) )
% 0.30/0.45 => ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.orderI
% 0.30/0.45 thf(fact_78_sup_OorderE,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [B3: A,A3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.45 => ( A3
% 0.30/0.45 = ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.orderE
% 0.30/0.45 thf(fact_79_le__iff__sup,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ( ( ord_less_eq @ A )
% 0.30/0.45 = ( ^ [X2: A,Y3: A] :
% 0.30/0.45 ( ( sup_sup @ A @ X2 @ Y3 )
% 0.30/0.45 = Y3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_iff_sup
% 0.30/0.45 thf(fact_80_sup__least,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [Y: A,X: A,Z: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.45 => ( ( ord_less_eq @ A @ Z @ X )
% 0.30/0.45 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z ) @ X ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup_least
% 0.30/0.45 thf(fact_81_sup__mono,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [A3: A,C2: A,B3: A,D: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A3 @ C2 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ B3 @ D )
% 0.30/0.45 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ ( sup_sup @ A @ C2 @ D ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup_mono
% 0.30/0.45 thf(fact_82_sup_Omono,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [C2: A,A3: A,D: A,B3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ D @ B3 )
% 0.30/0.45 => ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D ) @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup.mono
% 0.30/0.45 thf(fact_83_le__supI2,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,B3: A,A3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X @ B3 )
% 0.30/0.45 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_supI2
% 0.30/0.45 thf(fact_84_le__supI1,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,A3: A,B3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X @ A3 )
% 0.30/0.45 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_supI1
% 0.30/0.45 thf(fact_85_sup__ge2,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup_ge2
% 0.30/0.45 thf(fact_86_sup__ge1,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % sup_ge1
% 0.30/0.45 thf(fact_87_le__supI,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [A3: A,X: A,B3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A3 @ X )
% 0.30/0.45 => ( ( ord_less_eq @ A @ B3 @ X )
% 0.30/0.45 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_supI
% 0.30/0.45 thf(fact_88_le__supE,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [A3: A,B3: A,X: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X )
% 0.30/0.45 => ~ ( ( ord_less_eq @ A @ A3 @ X )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ B3 @ X ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_supE
% 0.30/0.45 thf(fact_89_inf__sup__ord_I3_J,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % inf_sup_ord(3)
% 0.30/0.45 thf(fact_90_inf__sup__ord_I4_J,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( lattice @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % inf_sup_ord(4)
% 0.30/0.45 thf(fact_91_subset__Un__eq,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.45 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.30/0.45 ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
% 0.30/0.45 = B4 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % subset_Un_eq
% 0.30/0.45 thf(fact_92_Un__absorb2,axiom,
% 0.30/0.45 ! [A: $tType,B2: set @ A,A2: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.30/0.45 => ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.45 = A2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % Un_absorb2
% 0.30/0.45 thf(fact_93_Un__absorb1,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.45 => ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.45 = B2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % Un_absorb1
% 0.30/0.45 thf(fact_94_Un__upper2,axiom,
% 0.30/0.45 ! [A: $tType,B2: set @ A,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % Un_upper2
% 0.30/0.45 thf(fact_95_Un__upper1,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % Un_upper1
% 0.30/0.45 thf(fact_96_Un__least,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,C: set @ A,B2: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C )
% 0.30/0.45 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Un_least
% 0.30/0.45 thf(fact_97_Un__mono,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,C: set @ A,B2: set @ A,D2: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ D2 )
% 0.30/0.45 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C @ D2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Un_mono
% 0.30/0.45 thf(fact_98_order__refl,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_refl
% 0.30/0.45 thf(fact_99_dual__order_Oantisym,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [B3: A,A3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.45 => ( A3 = B3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % dual_order.antisym
% 0.30/0.45 thf(fact_100_dual__order_Otrans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [B3: A,A3: A,C2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.30/0.45 => ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % dual_order.trans
% 0.30/0.45 thf(fact_101_linorder__wlog,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [P: A > A > $o,A3: A,B3: A] :
% 0.30/0.45 ( ! [A6: A,B6: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A6 @ B6 )
% 0.30/0.45 => ( P @ A6 @ B6 ) )
% 0.30/0.45 => ( ! [A6: A,B6: A] :
% 0.30/0.45 ( ( P @ B6 @ A6 )
% 0.30/0.45 => ( P @ A6 @ B6 ) )
% 0.30/0.45 => ( P @ A3 @ B3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % linorder_wlog
% 0.30/0.45 thf(fact_102_dual__order_Orefl,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % dual_order.refl
% 0.30/0.45 thf(fact_103_order__trans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.30/0.45 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_trans
% 0.30/0.45 thf(fact_104_order__class_Oorder_Oantisym,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [A3: A,B3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.45 => ( A3 = B3 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_class.order.antisym
% 0.30/0.45 thf(fact_105_ord__le__eq__trans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [A3: A,B3: A,C2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.45 => ( ( B3 = C2 )
% 0.30/0.45 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ord_le_eq_trans
% 0.30/0.45 thf(fact_106_ord__eq__le__trans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [A3: A,B3: A,C2: A] :
% 0.30/0.45 ( ( A3 = B3 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.30/0.45 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ord_eq_le_trans
% 0.30/0.45 thf(fact_107_antisym__conv,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [Y: A,X: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.45 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 = ( X = Y ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % antisym_conv
% 0.30/0.45 thf(fact_108_le__funD,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.45 => ! [F2: A > B,G2: A > B,X: A] :
% 0.30/0.45 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_funD
% 0.30/0.45 thf(fact_109_le__funE,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.45 => ! [F2: A > B,G2: A > B,X: A] :
% 0.30/0.45 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_funE
% 0.30/0.45 thf(fact_110_le__funI,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.45 => ! [F2: A > B,G2: A > B] :
% 0.30/0.45 ( ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.30/0.45 => ( ord_less_eq @ ( A > B ) @ F2 @ G2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_funI
% 0.30/0.45 thf(fact_111_le__fun__def,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.45 => ( ( ord_less_eq @ ( A > B ) )
% 0.30/0.45 = ( ^ [F: A > B,G: A > B] :
% 0.30/0.45 ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_fun_def
% 0.30/0.45 thf(fact_112_order__subst1,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType] :
% 0.30/0.45 ( ( ( order @ B @ ( type2 @ B ) )
% 0.30/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.45 => ! [A3: A,F2: B > A,B3: B,C2: B] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A3 @ ( F2 @ B3 ) )
% 0.30/0.45 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.30/0.45 => ( ! [X3: B,Y4: B] :
% 0.30/0.45 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 0.30/0.45 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
% 0.30/0.45 => ( ord_less_eq @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_subst1
% 0.30/0.45 thf(fact_113_order__subst2,axiom,
% 0.30/0.45 ! [A: $tType,C3: $tType] :
% 0.30/0.45 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.30/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.45 => ! [A3: A,B3: A,F2: A > C3,C2: C3] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.45 => ( ( ord_less_eq @ C3 @ ( F2 @ B3 ) @ C2 )
% 0.30/0.45 => ( ! [X3: A,Y4: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.30/0.45 => ( ord_less_eq @ C3 @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
% 0.30/0.45 => ( ord_less_eq @ C3 @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order_subst2
% 0.30/0.45 thf(fact_114_ord__eq__le__subst,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType] :
% 0.30/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.45 => ! [A3: A,F2: B > A,B3: B,C2: B] :
% 0.30/0.45 ( ( A3
% 0.30/0.45 = ( F2 @ B3 ) )
% 0.30/0.45 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.30/0.45 => ( ! [X3: B,Y4: B] :
% 0.30/0.45 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 0.30/0.45 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
% 0.30/0.45 => ( ord_less_eq @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ord_eq_le_subst
% 0.30/0.45 thf(fact_115_ord__le__eq__subst,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType] :
% 0.30/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.45 => ! [A3: A,B3: A,F2: A > B,C2: B] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.45 => ( ( ( F2 @ B3 )
% 0.30/0.45 = C2 )
% 0.30/0.45 => ( ! [X3: A,Y4: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ord_le_eq_subst
% 0.30/0.45 thf(fact_116_eq__iff,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.45 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 0.30/0.45 = ( ^ [X2: A,Y3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.45 & ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % eq_iff
% 0.30/0.45 thf(fact_117_antisym,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 => ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.45 => ( X = Y ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % antisym
% 0.30/0.45 thf(fact_118_linear,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % linear
% 0.30/0.45 thf(fact_119_eq__refl,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ( X = Y )
% 0.30/0.45 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % eq_refl
% 0.30/0.45 thf(fact_120_le__cases,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,Y: A] :
% 0.30/0.45 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_cases
% 0.30/0.45 thf(fact_121_order_Otrans,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [A3: A,B3: A,C2: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.30/0.45 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % order.trans
% 0.30/0.45 thf(fact_122_le__cases3,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.45 => ! [X: A,Y: A,Z: A] :
% 0.30/0.45 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.30/0.45 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 0.30/0.45 => ( ( ( ord_less_eq @ A @ X @ Z )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.30/0.45 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.30/0.45 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 0.30/0.45 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 0.30/0.45 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % le_cases3
% 0.30/0.45 thf(fact_123_inItr__mono,axiom,
% 0.30/0.45 ! [Ns: set @ n,Tr: dtree,N: n,Ns2: set @ n] :
% 0.30/0.45 ( ( gram_L830233218_inItr @ Ns @ Tr @ N )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns2 )
% 0.30/0.45 => ( gram_L830233218_inItr @ Ns2 @ Tr @ N ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % inItr_mono
% 0.30/0.45 thf(fact_124_inFr2__mono,axiom,
% 0.30/0.45 ! [Ns: set @ n,Tr: dtree,T2: t,Ns2: set @ n] :
% 0.30/0.45 ( ( gram_L805317441_inFr2 @ Ns @ Tr @ T2 )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns2 )
% 0.30/0.45 => ( gram_L805317441_inFr2 @ Ns2 @ Tr @ T2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % inFr2_mono
% 0.30/0.45 thf(fact_125_inFr__mono,axiom,
% 0.30/0.45 ! [Ns: set @ n,Tr: dtree,T2: t,Ns2: set @ n] :
% 0.30/0.45 ( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns2 )
% 0.30/0.45 => ( gram_L1333338417e_inFr @ Ns2 @ Tr @ T2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % inFr_mono
% 0.30/0.45 thf(fact_126_antimonoD,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.45 => ! [F2: A > B,X: A,Y: A] :
% 0.30/0.45 ( ( order_antimono @ A @ B @ F2 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % antimonoD
% 0.30/0.45 thf(fact_127_antimonoE,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.45 => ! [F2: A > B,X: A,Y: A] :
% 0.30/0.45 ( ( order_antimono @ A @ B @ F2 )
% 0.30/0.45 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % antimonoE
% 0.30/0.45 thf(fact_128_inFr__inFr2,axiom,
% 0.30/0.45 gram_L1333338417e_inFr = gram_L805317441_inFr2 ).
% 0.30/0.45
% 0.30/0.45 % inFr_inFr2
% 0.30/0.45 thf(fact_129_not__root__inFr,axiom,
% 0.30/0.45 ! [Tr: dtree,Ns: set @ n,T2: t] :
% 0.30/0.45 ( ~ ( member @ n @ ( root @ Tr ) @ Ns )
% 0.30/0.45 => ~ ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % not_root_inFr
% 0.30/0.45 thf(fact_130_inFr__root__in,axiom,
% 0.30/0.45 ! [Ns: set @ n,Tr: dtree,T2: t] :
% 0.30/0.45 ( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
% 0.30/0.45 => ( member @ n @ ( root @ Tr ) @ Ns ) ) ).
% 0.30/0.45
% 0.30/0.45 % inFr_root_in
% 0.30/0.45 thf(fact_131_inFr2__root__in,axiom,
% 0.30/0.45 ! [Ns: set @ n,Tr: dtree,T2: t] :
% 0.30/0.45 ( ( gram_L805317441_inFr2 @ Ns @ Tr @ T2 )
% 0.30/0.45 => ( member @ n @ ( root @ Tr ) @ Ns ) ) ).
% 0.30/0.45
% 0.30/0.45 % inFr2_root_in
% 0.30/0.45 thf(fact_132_inItr__root__in,axiom,
% 0.30/0.45 ! [Ns: set @ n,Tr: dtree,N: n] :
% 0.30/0.45 ( ( gram_L830233218_inItr @ Ns @ Tr @ N )
% 0.30/0.45 => ( member @ n @ ( root @ Tr ) @ Ns ) ) ).
% 0.30/0.45
% 0.30/0.45 % inItr_root_in
% 0.30/0.45 thf(fact_133_inItr_OBase,axiom,
% 0.30/0.45 ! [Tr: dtree,Ns: set @ n] :
% 0.30/0.45 ( ( member @ n @ ( root @ Tr ) @ Ns )
% 0.30/0.45 => ( gram_L830233218_inItr @ Ns @ Tr @ ( root @ Tr ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % inItr.Base
% 0.30/0.45 thf(fact_134_antimono__def,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.45 => ( ( order_antimono @ A @ B )
% 0.30/0.45 = ( ^ [F: A > B] :
% 0.30/0.45 ! [X2: A,Y3: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % antimono_def
% 0.30/0.45 thf(fact_135_antimonoI,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.45 => ! [F2: A > B] :
% 0.30/0.45 ( ! [X3: A,Y4: A] :
% 0.30/0.45 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.30/0.45 => ( ord_less_eq @ B @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) )
% 0.30/0.45 => ( order_antimono @ A @ B @ F2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % antimonoI
% 0.30/0.45 thf(fact_136_strict__mono__less__eq,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.45 => ! [F2: A > B,X: A,Y: A] :
% 0.30/0.45 ( ( order_strict_mono @ A @ B @ F2 )
% 0.30/0.45 => ( ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y ) )
% 0.30/0.45 = ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % strict_mono_less_eq
% 0.30/0.45 thf(fact_137_pairwise__subset,axiom,
% 0.30/0.45 ! [A: $tType,P: A > A > $o,S: set @ A,T3: set @ A] :
% 0.30/0.45 ( ( pairwise @ A @ P @ S )
% 0.30/0.45 => ( ( ord_less_eq @ ( set @ A ) @ T3 @ S )
% 0.30/0.45 => ( pairwise @ A @ P @ T3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % pairwise_subset
% 0.30/0.45 thf(fact_138_Diff__subset__conv,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,C: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ C )
% 0.30/0.45 = ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Diff_subset_conv
% 0.30/0.45 thf(fact_139_Diff__partition,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.45 => ( ( sup_sup @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) )
% 0.30/0.45 = B2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % Diff_partition
% 0.30/0.45 thf(fact_140_minus__apply,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( minus @ B @ ( type2 @ B ) )
% 0.30/0.46 => ( ( minus_minus @ ( A > B ) )
% 0.30/0.46 = ( ^ [A4: A > B,B4: A > B,X2: A] : ( minus_minus @ B @ ( A4 @ X2 ) @ ( B4 @ X2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % minus_apply
% 0.30/0.46 thf(fact_141_Diff__idemp,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ B2 )
% 0.30/0.46 = ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_idemp
% 0.30/0.46 thf(fact_142_Diff__iff,axiom,
% 0.30/0.46 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 0.30/0.46 = ( ( member @ A @ C2 @ A2 )
% 0.30/0.46 & ~ ( member @ A @ C2 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_iff
% 0.30/0.46 thf(fact_143_DiffI,axiom,
% 0.30/0.46 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( member @ A @ C2 @ A2 )
% 0.30/0.46 => ( ~ ( member @ A @ C2 @ B2 )
% 0.30/0.46 => ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % DiffI
% 0.30/0.46 thf(fact_144_Un__Diff__cancel2,axiom,
% 0.30/0.46 ! [A: $tType,B2: set @ A,A2: set @ A] :
% 0.30/0.46 ( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) @ A2 )
% 0.30/0.46 = ( sup_sup @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Un_Diff_cancel2
% 0.30/0.46 thf(fact_145_Un__Diff__cancel,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( sup_sup @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) )
% 0.30/0.46 = ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Un_Diff_cancel
% 0.30/0.46 thf(fact_146_Diff__mono,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,C: set @ A,D2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ D2 @ B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( set @ A ) @ C @ D2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_mono
% 0.30/0.46 thf(fact_147_Diff__subset,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ A2 ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_subset
% 0.30/0.46 thf(fact_148_double__diff,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C )
% 0.30/0.46 => ( ( minus_minus @ ( set @ A ) @ B2 @ ( minus_minus @ ( set @ A ) @ C @ A2 ) )
% 0.30/0.46 = A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % double_diff
% 0.30/0.46 thf(fact_149_fun__diff__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( minus @ B @ ( type2 @ B ) )
% 0.30/0.46 => ( ( minus_minus @ ( A > B ) )
% 0.30/0.46 = ( ^ [A4: A > B,B4: A > B,X2: A] : ( minus_minus @ B @ ( A4 @ X2 ) @ ( B4 @ X2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_diff_def
% 0.30/0.46 thf(fact_150_pairwise__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( pairwise @ A )
% 0.30/0.46 = ( ^ [R: A > A > $o,S2: set @ A] :
% 0.30/0.46 ! [X2: A] :
% 0.30/0.46 ( ( member @ A @ X2 @ S2 )
% 0.30/0.46 => ! [Y3: A] :
% 0.30/0.46 ( ( member @ A @ Y3 @ S2 )
% 0.30/0.46 => ( ( X2 != Y3 )
% 0.30/0.46 => ( R @ X2 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pairwise_def
% 0.30/0.46 thf(fact_151_DiffD2,axiom,
% 0.30/0.46 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 0.30/0.46 => ~ ( member @ A @ C2 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % DiffD2
% 0.30/0.46 thf(fact_152_DiffD1,axiom,
% 0.30/0.46 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 0.30/0.46 => ( member @ A @ C2 @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % DiffD1
% 0.30/0.46 thf(fact_153_DiffE,axiom,
% 0.30/0.46 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 0.30/0.46 => ~ ( ( member @ A @ C2 @ A2 )
% 0.30/0.46 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % DiffE
% 0.30/0.46 thf(fact_154_Un__Diff,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C: set @ A] :
% 0.30/0.46 ( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C )
% 0.30/0.46 = ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ C ) @ ( minus_minus @ ( set @ A ) @ B2 @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Un_Diff
% 0.30/0.46 thf(fact_155_strict__mono__eq,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.46 => ! [F2: A > B,X: A,Y: A] :
% 0.30/0.46 ( ( order_strict_mono @ A @ B @ F2 )
% 0.30/0.46 => ( ( ( F2 @ X )
% 0.30/0.46 = ( F2 @ Y ) )
% 0.30/0.46 = ( X = Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % strict_mono_eq
% 0.30/0.46 thf(fact_156_diff__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A,B3: A,D: A,C2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ D @ C2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_mono
% 0.30/0.46 thf(fact_157_diff__left__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [B3: A,A3: A,C2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_left_mono
% 0.30/0.46 thf(fact_158_diff__right__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A,B3: A,C2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B3 @ C2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_right_mono
% 0.30/0.46 thf(fact_159_diff__eq__diff__less__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A,B3: A,C2: A,D: A] :
% 0.30/0.46 ( ( ( minus_minus @ A @ A3 @ B3 )
% 0.30/0.46 = ( minus_minus @ A @ C2 @ D ) )
% 0.30/0.46 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.46 = ( ord_less_eq @ A @ C2 @ D ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_eq_diff_less_eq
% 0.30/0.46 thf(fact_160_diff__right__commute,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( cancel146912293up_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A,C2: A,B3: A] :
% 0.30/0.46 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B3 )
% 0.30/0.46 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_right_commute
% 0.30/0.46 thf(fact_161_diff__eq__diff__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A,B3: A,C2: A,D: A] :
% 0.30/0.46 ( ( ( minus_minus @ A @ A3 @ B3 )
% 0.30/0.46 = ( minus_minus @ A @ C2 @ D ) )
% 0.30/0.46 => ( ( A3 = B3 )
% 0.30/0.46 = ( C2 = D ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_eq_diff_eq
% 0.30/0.46 thf(fact_162_Diff__eq__empty__iff,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ( minus_minus @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_eq_empty_iff
% 0.30/0.46 thf(fact_163_diff__ge__0__iff__ge,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A,B3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B3 ) )
% 0.30/0.46 = ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_ge_0_iff_ge
% 0.30/0.46 thf(fact_164_Compl__Diff__eq,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 0.30/0.46 = ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Compl_Diff_eq
% 0.30/0.46 thf(fact_165_bot__apply,axiom,
% 0.30/0.46 ! [C3: $tType,D3: $tType] :
% 0.30/0.46 ( ( bot @ C3 @ ( type2 @ C3 ) )
% 0.30/0.46 => ( ( bot_bot @ ( D3 > C3 ) )
% 0.30/0.46 = ( ^ [X2: D3] : ( bot_bot @ C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot_apply
% 0.30/0.46 thf(fact_166_empty__Collect__eq,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o] :
% 0.30/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/0.46 = ( collect @ A @ P ) )
% 0.30/0.46 = ( ! [X2: A] :
% 0.30/0.46 ~ ( P @ X2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % empty_Collect_eq
% 0.30/0.46 thf(fact_167_Collect__empty__eq,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o] :
% 0.30/0.46 ( ( ( collect @ A @ P )
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = ( ! [X2: A] :
% 0.30/0.46 ~ ( P @ X2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_empty_eq
% 0.30/0.46 thf(fact_168_all__not__in__conv,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ( ! [X2: A] :
% 0.30/0.46 ~ ( member @ A @ X2 @ A2 ) )
% 0.30/0.46 = ( A2
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % all_not_in_conv
% 0.30/0.46 thf(fact_169_empty__iff,axiom,
% 0.30/0.46 ! [A: $tType,C2: A] :
% 0.30/0.46 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % empty_iff
% 0.30/0.46 thf(fact_170_compl__eq__compl__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ( uminus_uminus @ A @ X )
% 0.30/0.46 = ( uminus_uminus @ A @ Y ) )
% 0.30/0.46 = ( X = Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % compl_eq_compl_iff
% 0.30/0.46 thf(fact_171_double__compl,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A] :
% 0.30/0.46 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
% 0.30/0.46 = X ) ) ).
% 0.30/0.46
% 0.30/0.46 % double_compl
% 0.30/0.46 thf(fact_172_uminus__apply,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( uminus @ B @ ( type2 @ B ) )
% 0.30/0.46 => ( ( uminus_uminus @ ( A > B ) )
% 0.30/0.46 = ( ^ [A4: A > B,X2: A] : ( uminus_uminus @ B @ ( A4 @ X2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % uminus_apply
% 0.30/0.46 thf(fact_173_Compl__eq__Compl__iff,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ( uminus_uminus @ ( set @ A ) @ A2 )
% 0.30/0.46 = ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 0.30/0.46 = ( A2 = B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Compl_eq_Compl_iff
% 0.30/0.46 thf(fact_174_Compl__iff,axiom,
% 0.30/0.46 ! [A: $tType,C2: A,A2: set @ A] :
% 0.30/0.46 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.30/0.46 = ( ~ ( member @ A @ C2 @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Compl_iff
% 0.30/0.46 thf(fact_175_ComplI,axiom,
% 0.30/0.46 ! [A: $tType,C2: A,A2: set @ A] :
% 0.30/0.46 ( ~ ( member @ A @ C2 @ A2 )
% 0.30/0.46 => ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ComplI
% 0.30/0.46 thf(fact_176_le__zero__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [N: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.30/0.46 = ( N
% 0.30/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_zero_eq
% 0.30/0.46 thf(fact_177_diff__self,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( minus_minus @ A @ A3 @ A3 )
% 0.30/0.46 = ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_self
% 0.30/0.46 thf(fact_178_diff__0__right,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.46 = A3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_0_right
% 0.30/0.46 thf(fact_179_zero__diff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comm_monoid_diff @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.46 = ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_diff
% 0.30/0.46 thf(fact_180_diff__zero,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
% 0.30/0.46 = A3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_zero
% 0.30/0.46 thf(fact_181_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( minus_minus @ A @ A3 @ A3 )
% 0.30/0.46 = ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.46 thf(fact_182_neg__le__iff__le,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [B3: A,A3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) )
% 0.30/0.46 = ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % neg_le_iff_le
% 0.30/0.46 thf(fact_183_compl__le__compl__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
% 0.30/0.46 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % compl_le_compl_iff
% 0.30/0.46 thf(fact_184_sup__bot_Oright__neutral,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
% 0.30/0.46 = A3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % sup_bot.right_neutral
% 0.30/0.46 thf(fact_185_sup__bot_Oleft__neutral,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
% 0.30/0.46 = A3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % sup_bot.left_neutral
% 0.30/0.46 thf(fact_186_sup__eq__bot__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ( sup_sup @ A @ X @ Y )
% 0.30/0.46 = ( bot_bot @ A ) )
% 0.30/0.46 = ( ( X
% 0.30/0.46 = ( bot_bot @ A ) )
% 0.30/0.46 & ( Y
% 0.30/0.46 = ( bot_bot @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sup_eq_bot_iff
% 0.30/0.46 thf(fact_187_bot__eq__sup__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ( bot_bot @ A )
% 0.30/0.46 = ( sup_sup @ A @ X @ Y ) )
% 0.30/0.46 = ( ( X
% 0.30/0.46 = ( bot_bot @ A ) )
% 0.30/0.46 & ( Y
% 0.30/0.46 = ( bot_bot @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot_eq_sup_iff
% 0.30/0.46 thf(fact_188_minus__diff__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A,B3: A] :
% 0.30/0.46 ( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B3 ) )
% 0.30/0.46 = ( minus_minus @ A @ B3 @ A3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % minus_diff_eq
% 0.30/0.46 thf(fact_189_empty__subsetI,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% 0.30/0.46
% 0.30/0.46 % empty_subsetI
% 0.30/0.46 thf(fact_190_subset__empty,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = ( A2
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_empty
% 0.30/0.46 thf(fact_191_Un__empty,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = ( ( A2
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 & ( B2
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Un_empty
% 0.30/0.46 thf(fact_192_Diff__cancel,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ( minus_minus @ ( set @ A ) @ A2 @ A2 )
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_cancel
% 0.30/0.46 thf(fact_193_empty__Diff,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 )
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % empty_Diff
% 0.30/0.46 thf(fact_194_Diff__empty,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ( minus_minus @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = A2 ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_empty
% 0.30/0.46 thf(fact_195_Compl__subset__Compl__iff,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 0.30/0.46 = ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Compl_subset_Compl_iff
% 0.30/0.46 thf(fact_196_Compl__anti__mono,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Compl_anti_mono
% 0.30/0.46 thf(fact_197_neg__0__le__iff__le,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
% 0.30/0.46 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % neg_0_le_iff_le
% 0.30/0.46 thf(fact_198_neg__le__0__iff__le,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
% 0.30/0.46 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % neg_le_0_iff_le
% 0.30/0.46 thf(fact_199_less__eq__neg__nonpos,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
% 0.30/0.46 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_eq_neg_nonpos
% 0.30/0.46 thf(fact_200_neg__less__eq__nonneg,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 0.30/0.46 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % neg_less_eq_nonneg
% 0.30/0.46 thf(fact_201_diff__0,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
% 0.30/0.46 = ( uminus_uminus @ A @ A3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_0
% 0.30/0.46 thf(fact_202_eq__iff__diff__eq__0,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 0.30/0.46 = ( ^ [A5: A,B5: A] :
% 0.30/0.46 ( ( minus_minus @ A @ A5 @ B5 )
% 0.30/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_iff_diff_eq_0
% 0.30/0.46 thf(fact_203_zero__le,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_le
% 0.30/0.46 thf(fact_204_le__imp__neg__le,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A,B3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_imp_neg_le
% 0.30/0.46 thf(fact_205_minus__le__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A,B3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
% 0.30/0.46 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ A3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % minus_le_iff
% 0.30/0.46 thf(fact_206_le__minus__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A,B3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
% 0.30/0.46 = ( ord_less_eq @ A @ B3 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_minus_iff
% 0.30/0.46 thf(fact_207_pairwise__empty,axiom,
% 0.30/0.46 ! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pairwise_empty
% 0.30/0.46 thf(fact_208_subset__Compl__self__eq,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.30/0.46 = ( A2
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_Compl_self_eq
% 0.30/0.46 thf(fact_209_bot_Oextremum,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot.extremum
% 0.30/0.46 thf(fact_210_bot_Oextremum__unique,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.30/0.46 = ( A3
% 0.30/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot.extremum_unique
% 0.30/0.46 thf(fact_211_bot_Oextremum__uniqueI,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.30/0.46 => ( A3
% 0.30/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot.extremum_uniqueI
% 0.30/0.46 thf(fact_212_compl__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.46 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % compl_mono
% 0.30/0.46 thf(fact_213_compl__le__swap1,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [Y: A,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
% 0.30/0.46 => ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % compl_le_swap1
% 0.30/0.46 thf(fact_214_compl__le__swap2,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [Y: A,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
% 0.30/0.46 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % compl_le_swap2
% 0.30/0.46 thf(fact_215_Un__empty__right,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ( sup_sup @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = A2 ) ).
% 0.30/0.46
% 0.30/0.46 % Un_empty_right
% 0.30/0.46 thf(fact_216_Un__empty__left,axiom,
% 0.30/0.46 ! [A: $tType,B2: set @ A] :
% 0.30/0.46 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
% 0.30/0.46 = B2 ) ).
% 0.30/0.46
% 0.30/0.46 % Un_empty_left
% 0.30/0.46 thf(fact_217_fun__Compl__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( uminus @ B @ ( type2 @ B ) )
% 0.30/0.46 => ( ( uminus_uminus @ ( A > B ) )
% 0.30/0.46 = ( ^ [A4: A > B,X2: A] : ( uminus_uminus @ B @ ( A4 @ X2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_Compl_def
% 0.30/0.46 thf(fact_218_double__complement,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.30/0.46 = A2 ) ).
% 0.30/0.46
% 0.30/0.46 % double_complement
% 0.30/0.46 thf(fact_219_ex__in__conv,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ( ? [X2: A] : ( member @ A @ X2 @ A2 ) )
% 0.30/0.46 = ( A2
% 0.30/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ex_in_conv
% 0.30/0.46 thf(fact_220_equals0I,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ! [Y4: A] :
% 0.30/0.46 ~ ( member @ A @ Y4 @ A2 )
% 0.30/0.46 => ( A2
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % equals0I
% 0.30/0.46 thf(fact_221_equals0D,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,A3: A] :
% 0.30/0.46 ( ( A2
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ~ ( member @ A @ A3 @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % equals0D
% 0.30/0.46 thf(fact_222_emptyE,axiom,
% 0.30/0.46 ! [A: $tType,A3: A] :
% 0.30/0.46 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % emptyE
% 0.30/0.46 thf(fact_223_ComplD,axiom,
% 0.30/0.46 ! [A: $tType,C2: A,A2: set @ A] :
% 0.30/0.46 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.30/0.46 => ~ ( member @ A @ C2 @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % ComplD
% 0.30/0.46 thf(fact_224_sup__bot__right,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A] :
% 0.30/0.46 ( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
% 0.30/0.46 = X ) ) ).
% 0.30/0.46
% 0.30/0.46 % sup_bot_right
% 0.30/0.46 thf(fact_225_sup__bot__left,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A] :
% 0.30/0.46 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
% 0.30/0.46 = X ) ) ).
% 0.30/0.46
% 0.30/0.46 % sup_bot_left
% 0.30/0.46 thf(fact_226_bot__fun__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( bot @ B @ ( type2 @ B ) )
% 0.30/0.46 => ( ( bot_bot @ ( A > B ) )
% 0.30/0.46 = ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot_fun_def
% 0.30/0.46 thf(fact_227_le__iff__diff__le__0,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ( ord_less_eq @ A )
% 0.30/0.46 = ( ^ [A5: A,B5: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_iff_diff_le_0
% 0.30/0.46 thf(fact_228_subset__emptyI,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ~ ( member @ A @ X3 @ A2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_emptyI
% 0.30/0.46 thf(fact_229_le__numeral__extra_I3_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_numeral_extra(3)
% 0.30/0.46 thf(fact_230_not__root__Fr,axiom,
% 0.30/0.46 ! [Tr: dtree,Ns: set @ n] :
% 0.30/0.46 ( ~ ( member @ n @ ( root @ Tr ) @ Ns )
% 0.30/0.46 => ( ( gram_L861583724lle_Fr @ Ns @ Tr )
% 0.30/0.46 = ( bot_bot @ ( set @ t ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % not_root_Fr
% 0.30/0.46 thf(fact_231_bot__set__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( bot_bot @ ( set @ A ) )
% 0.30/0.46 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot_set_def
% 0.30/0.46 thf(fact_232_Set_Ois__empty__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( is_empty @ A )
% 0.30/0.46 = ( ^ [A4: set @ A] :
% 0.30/0.46 ( A4
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Set.is_empty_def
% 0.30/0.46 thf(fact_233_diff__numeral__special_I12_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( neg_numeral @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.30/0.46 = ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_numeral_special(12)
% 0.30/0.46 thf(fact_234_diff__numeral__special_I9_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( neg_numeral @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 0.30/0.46 = ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_numeral_special(9)
% 0.30/0.46 thf(fact_235_le__numeral__extra_I1_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_numeral_extra(1)
% 0.30/0.46 thf(fact_236_le__numeral__extra_I2_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_numeral_extra(2)
% 0.30/0.46 thf(fact_237_le__minus__one__simps_I2_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_minus_one_simps(2)
% 0.30/0.46 thf(fact_238_le__minus__one__simps_I4_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_minus_one_simps(4)
% 0.30/0.46 thf(fact_239_le__numeral__extra_I4_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_numeral_extra(4)
% 0.30/0.46 thf(fact_240_le__minus__one__simps_I1_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_minus_one_simps(1)
% 0.30/0.46 thf(fact_241_le__minus__one__simps_I3_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_minus_one_simps(3)
% 0.30/0.46 thf(fact_242_zero__le__one,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linord1659791738miring @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_le_one
% 0.30/0.46 thf(fact_243_not__one__le__zero,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linord1659791738miring @ A @ ( type2 @ A ) )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % not_one_le_zero
% 0.30/0.46 thf(fact_244_subset__Compl__singleton,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,B3: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.46 = ( ~ ( member @ A @ B3 @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_Compl_singleton
% 0.30/0.46 thf(fact_245_sup__bot_Osemilattice__neutr__axioms,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( semilattice_neutr @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sup_bot.semilattice_neutr_axioms
% 0.30/0.46 thf(fact_246_insert__absorb2,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A2: set @ A] :
% 0.30/0.46 ( ( insert @ A @ X @ ( insert @ A @ X @ A2 ) )
% 0.30/0.46 = ( insert @ A @ X @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_absorb2
% 0.30/0.46 thf(fact_247_insert__iff,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,B3: A,A2: set @ A] :
% 0.30/0.46 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
% 0.30/0.46 = ( ( A3 = B3 )
% 0.30/0.46 | ( member @ A @ A3 @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_iff
% 0.30/0.46 thf(fact_248_insertCI,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 0.30/0.46 ( ( ~ ( member @ A @ A3 @ B2 )
% 0.30/0.46 => ( A3 = B3 ) )
% 0.30/0.46 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insertCI
% 0.30/0.46 thf(fact_249_singletonI,axiom,
% 0.30/0.46 ! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % singletonI
% 0.30/0.46 thf(fact_250_insert__subset,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
% 0.30/0.46 = ( ( member @ A @ X @ B2 )
% 0.30/0.46 & ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_subset
% 0.30/0.46 thf(fact_251_Un__insert__right,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
% 0.30/0.46 ( ( sup_sup @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
% 0.30/0.46 = ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Un_insert_right
% 0.30/0.46 thf(fact_252_Un__insert__left,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,B2: set @ A,C: set @ A] :
% 0.30/0.46 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C )
% 0.30/0.46 = ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Un_insert_left
% 0.30/0.46 thf(fact_253_insert__Diff1,axiom,
% 0.30/0.46 ! [A: $tType,X: A,B2: set @ A,A2: set @ A] :
% 0.30/0.46 ( ( member @ A @ X @ B2 )
% 0.30/0.46 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
% 0.30/0.46 = ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_Diff1
% 0.30/0.46 thf(fact_254_Diff__insert0,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ~ ( member @ A @ X @ A2 )
% 0.30/0.46 => ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) )
% 0.30/0.46 = ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_insert0
% 0.30/0.46 thf(fact_255_singleton__insert__inj__eq_H,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,A2: set @ A,B3: A] :
% 0.30/0.46 ( ( ( insert @ A @ A3 @ A2 )
% 0.30/0.46 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 = ( ( A3 = B3 )
% 0.30/0.46 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % singleton_insert_inj_eq'
% 0.30/0.46
% 0.30/0.46 %----Type constructors (40)
% 0.30/0.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 0.30/0.46 bounded_lattice @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 0.30/0.46 ! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( bounded_lattice @ A8 @ ( type2 @ A8 ) )
% 0.30/0.46 => ( bounded_lattice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( bounded_lattice @ A8 @ ( type2 @ A8 ) )
% 0.30/0.46 => ( bounde1808546759up_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( bounded_lattice @ A8 @ ( type2 @ A8 ) )
% 0.30/0.46 => ( bounded_lattice_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( semilattice_sup @ A8 @ ( type2 @ A8 ) )
% 0.30/0.46 => ( semilattice_sup @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( boolean_algebra @ A8 @ ( type2 @ A8 ) )
% 0.30/0.46 => ( boolean_algebra @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( order_bot @ A8 @ ( type2 @ A8 ) )
% 0.30/0.46 => ( order_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 0.30/0.46 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( lattice @ A8 @ ( type2 @ A8 ) )
% 0.30/0.46 => ( lattice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( order @ A8 @ ( type2 @ A8 ) )
% 0.30/0.46 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 0.30/0.46 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Obot,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( bot @ A8 @ ( type2 @ A8 ) )
% 0.30/0.46 => ( bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Groups_Ouminus,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( uminus @ A8 @ ( type2 @ A8 ) )
% 0.30/0.46 => ( uminus @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Groups_Ominus,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( minus @ A8 @ ( type2 @ A8 ) )
% 0.30/0.46 => ( minus @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_3,axiom,
% 0.30/0.46 ! [A7: $tType] : ( bounde1808546759up_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_4,axiom,
% 0.30/0.46 ! [A7: $tType] : ( bounded_lattice_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5,axiom,
% 0.30/0.46 ! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_6,axiom,
% 0.30/0.46 ! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Oorder__bot_7,axiom,
% 0.30/0.46 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_8,axiom,
% 0.30/0.46 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Lattices_Olattice_9,axiom,
% 0.30/0.46 ! [A7: $tType] : ( lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Oorder_10,axiom,
% 0.30/0.46 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Oord_11,axiom,
% 0.30/0.46 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Obot_12,axiom,
% 0.30/0.46 ! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Groups_Ouminus_13,axiom,
% 0.30/0.46 ! [A7: $tType] : ( uminus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Groups_Ominus_14,axiom,
% 0.30/0.46 ! [A7: $tType] : ( minus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_15,axiom,
% 0.30/0.46 bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_16,axiom,
% 0.30/0.46 bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_17,axiom,
% 0.30/0.46 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_18,axiom,
% 0.30/0.46 boolean_algebra @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__bot_19,axiom,
% 0.30/0.46 order_bot @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_20,axiom,
% 0.30/0.46 preorder @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.30/0.46 linorder @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Lattices_Olattice_21,axiom,
% 0.30/0.46 lattice @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_22,axiom,
% 0.30/0.46 order @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Oord_23,axiom,
% 0.30/0.46 ord @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Obot_24,axiom,
% 0.30/0.46 bot @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Groups_Ouminus_25,axiom,
% 0.30/0.46 uminus @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Groups_Ominus_26,axiom,
% 0.30/0.46 minus @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 %----Conjectures (1)
% 0.30/0.46 thf(conj_0,conjecture,
% 0.30/0.46 gram_L716654942_subtr @ ns @ tr1 @ tr3 ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 2.90/1.51 ---- Embedded in THF ---
% 2.90/1.51 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.90/1.51 %%% Generated on Tue Feb 24 04:16:47 EST 2026
% 2.90/1.51 %%% using '$$dhol' embedding, version 1.3.0.
% 2.90/1.51 %%% Logic specification used:
% 2.90/1.51 %%% thf(spec, logic, $$dhol).
% 2.90/1.51
% 2.90/1.51 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.xTiNdgqN7t/DTF2DTF9630.p
% 2.90/1.51 thf(ty_t_DTree_Odtree, type, dtree: $tType).
% 2.90/1.51 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.90/1.51 thf(ty_t_DTree_OT, type, t: $tType).
% 2.90/1.51 thf(ty_t_DTree_ON, type, n: $tType).
% 2.90/1.51 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.90/1.51 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Groups_Ouminus, type, uminus: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Num_Oneg__numeral, type, neg_numeral: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Groups_Ocomm__monoid__diff, type, comm_monoid_diff: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Lattices_Oboolean__algebra, type, boolean_algebra: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Lattices_Obounded__lattice__bot, type, bounded_lattice_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot, type, bounde1808546759up_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.90/1.51 thf(sy_c_DTree_Odtree_Oroot, type, root: (dtree > n)).
% 2.90/1.51 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OFr, type, gram_L861583724lle_Fr: ((set @ n) > (dtree > (set @ t)))).
% 2.90/1.51 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr, type, gram_L1333338417e_inFr: ((set @ n) > (dtree > (t > $o)))).
% 2.90/1.51 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr2, type, gram_L805317441_inFr2: ((set @ n) > (dtree > (t > $o)))).
% 2.90/1.51 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinItr, type, gram_L830233218_inItr: ((set @ n) > (dtree > (n > $o)))).
% 2.90/1.51 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Osubtr, type, gram_L716654942_subtr: ((set @ n) > (dtree > (dtree > $o)))).
% 2.90/1.51 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.90/1.51 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 2.90/1.51 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.90/1.51 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.90/1.51 thf(sy_c_Lattices_Osemilattice__neutr, type, semilattice_neutr: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 2.90/1.51 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 2.90/1.51 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.90/1.51 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.90/1.51 thf(sy_c_Orderings_Oorder__class_Oantimono, type, order_antimono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.90/1.51 thf(sy_c_Orderings_Oorder__class_Ostrict__mono, type, order_strict_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.90/1.51 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.90/1.51 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.90/1.51 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.90/1.51 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.90/1.51 thf(sy_c_Set_Opairwise, type, pairwise: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > $o))))).
% 2.90/1.51 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.90/1.51 thf(sy_v_ns, type, ns: (set @ n)).
% 2.90/1.51 thf(sy_v_tr1, type, tr1: dtree).
% 2.90/1.51 thf(sy_v_tr2, type, tr2: dtree).
% 2.90/1.51 thf(sy_v_tr3, type, tr3: dtree).
% 2.90/1.51 thf(fact_0__092_060open_062subtr_A_Ins_A_092_060union_062_Ans_J_Atr1_Atr3_092_060close_062, axiom, (((gram_L716654942_subtr @ (((sup_sup @ (set @ n)) @ ns) @ ns)) @ tr1) @ tr3)).
% 2.90/1.51 thf(fact_1_assms_I2_J, axiom, (((gram_L716654942_subtr @ ns) @ tr2) @ tr3)).
% 2.90/1.51 thf(fact_2_assms_I1_J, axiom, (((gram_L716654942_subtr @ ns) @ tr1) @ tr2)).
% 2.90/1.51 thf(fact_3_subtr__trans__Un, axiom, (! [Ns12:(set @ n),Tr1:dtree,Tr2:dtree,Ns23:(set @ n),Tr3:dtree]: (((((gram_L716654942_subtr @ Ns12) @ Tr1) @ Tr2) => ((((gram_L716654942_subtr @ Ns23) @ Tr2) @ Tr3) => (((gram_L716654942_subtr @ (((sup_sup @ (set @ n)) @ Ns12) @ Ns23)) @ Tr1) @ Tr3)))))).
% 2.90/1.51 thf(fact_4_subtr__mono, axiom, (! [Ns:(set @ n),Tr1:dtree,Tr2:dtree,Ns2:(set @ n)]: (((((gram_L716654942_subtr @ Ns) @ Tr1) @ Tr2) => ((((ord_less_eq @ (set @ n)) @ Ns) @ Ns2) => (((gram_L716654942_subtr @ Ns2) @ Tr1) @ Tr2)))))).
% 2.90/1.51 thf(fact_5_Refl, axiom, (! [Tr:dtree,Ns:(set @ n)]: (((((member @ n) @ (root @ Tr)) @ Ns) => (((gram_L716654942_subtr @ Ns) @ Tr) @ Tr))))).
% 2.90/1.51 thf(fact_6_subtr__rootL__in, axiom, (! [Ns:(set @ n),Tr1:dtree,Tr2:dtree]: (((((gram_L716654942_subtr @ Ns) @ Tr1) @ Tr2) => (((member @ n) @ (root @ Tr1)) @ Ns))))).
% 2.90/1.51 thf(fact_7_subtr__rootR__in, axiom, (! [Ns:(set @ n),Tr1:dtree,Tr2:dtree]: (((((gram_L716654942_subtr @ Ns) @ Tr1) @ Tr2) => (((member @ n) @ (root @ Tr2)) @ Ns))))).
% 2.90/1.51 thf(fact_8_Un__subset__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ C) = ((((ord_less_eq @ (set @ A)) @ A2) @ C) & (((ord_less_eq @ (set @ A)) @ B2) @ C)))))).
% 2.90/1.51 thf(fact_9_le__sup__iff, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z) = ((((ord_less_eq @ A) @ X) @ Z) & (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 2.90/1.51 thf(fact_10_sup_Obounded__iff, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,C2:A,A3:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B3) @ C2)) @ A3) = ((((ord_less_eq @ A) @ B3) @ A3) & (((ord_less_eq @ A) @ C2) @ A3))))))))).
% 2.90/1.51 thf(fact_11_UnCI, axiom, (! [A:$tType,C2:A,B2:(set @ A),A2:(set @ A)]: ((((~ ((((member @ A) @ C2) @ B2))) => (((member @ A) @ C2) @ A2)) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)))))).
% 2.90/1.51 thf(fact_12_Un__iff, axiom, (! [A:$tType,C2:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) = ((((member @ A) @ C2) @ A2) | (((member @ A) @ C2) @ B2)))))).
% 2.90/1.51 thf(fact_13_sup__apply, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ B) @ (type2 @ B)) => ((sup_sup @ (A > B)) = (^ [F:(A > B),G:(A > B),X2:A]: ((((sup_sup @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.90/1.51 thf(fact_14_sup_Oidem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A]: (((((sup_sup @ A) @ A3) @ A3) = A3))))))).
% 2.90/1.51 thf(fact_15_sup__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ X) @ X) = X))))))).
% 2.90/1.51 thf(fact_16_sup_Oleft__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ A3) @ B3)) = (((sup_sup @ A) @ A3) @ B3)))))))).
% 2.90/1.51 thf(fact_17_sup__left__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = (((sup_sup @ A) @ X) @ Y)))))))).
% 2.90/1.51 thf(fact_18_sup_Oright__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A3) @ B3)) @ B3) = (((sup_sup @ A) @ A3) @ B3)))))))).
% 2.90/1.51 thf(fact_19_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => (A2 = B2)))))).
% 2.90/1.51 thf(fact_20_subsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A2) => (((member @ A) @ X3) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.90/1.51 thf(fact_21_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X2:A]: (((P @ X2) => (Q @ X2)))))))).
% 2.90/1.51 thf(fact_22_contra__subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((~ ((((member @ A) @ C2) @ B2))) => (~ ((((member @ A) @ C2) @ A2)))))))).
% 2.90/1.51 thf(fact_23_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y2:(set @ A),Z2:(set @ A)]: ((Y2 = Z2))) = (^ [A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) & (((ord_less_eq @ (set @ A)) @ B4) @ A4)))))))).
% 2.90/1.51 thf(fact_24_subset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C) => (((ord_less_eq @ (set @ A)) @ A2) @ C)))))).
% 2.90/1.51 thf(fact_25_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.90/1.51 thf(fact_26_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 2.90/1.51 thf(fact_27_rev__subsetD, axiom, (! [A:$tType,C2:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C2) @ A2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((member @ A) @ C2) @ B2)))))).
% 2.90/1.51 thf(fact_28_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A4:(set @ A),B4:(set @ A)]: ((! [T:A]: (((((member @ A) @ T) @ A4) => (((member @ A) @ T) @ B4)))))))))).
% 2.90/1.51 thf(fact_29_set__rev__mp, axiom, (! [A:$tType,X:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ X) @ A2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((member @ A) @ X) @ B2)))))).
% 2.90/1.51 thf(fact_30_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 2.90/1.51 thf(fact_31_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.90/1.51 thf(fact_32_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A4:(set @ A),B4:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ A4) => (((member @ A) @ X2) @ B4)))))))))).
% 2.90/1.51 thf(fact_33_equalityE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A2)))))))))).
% 2.90/1.51 thf(fact_34_subsetCE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ B2)))))).
% 2.90/1.51 thf(fact_35_subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ B2)))))).
% 2.90/1.51 thf(fact_36_in__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X) @ A2) => (((member @ A) @ X) @ B2)))))).
% 2.90/1.51 thf(fact_37_set__mp, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X) @ A2) => (((member @ A) @ X) @ B2)))))).
% 2.90/1.51 thf(fact_38_sup__left__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 2.90/1.51 thf(fact_39_sup_Oleft__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,A3:A,C2:A]: (((((sup_sup @ A) @ B3) @ (((sup_sup @ A) @ A3) @ C2)) = (((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ B3) @ C2))))))))).
% 2.90/1.51 thf(fact_40_sup__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [X2:A,Y3:A]: ((((sup_sup @ A) @ Y3) @ X2)))))))).
% 2.90/1.51 thf(fact_41_sup_Ocommute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [A5:A,B5:A]: ((((sup_sup @ A) @ B5) @ A5)))))))).
% 2.90/1.51 thf(fact_42_sup__assoc, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z))))))))).
% 2.90/1.51 thf(fact_43_sup_Oassoc, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A3) @ B3)) @ C2) = (((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ B3) @ C2))))))))).
% 2.90/1.51 thf(fact_44_sup__fun__def, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ B) @ (type2 @ B)) => ((sup_sup @ (A > B)) = (^ [F:(A > B),G:(A > B),X2:A]: ((((sup_sup @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.90/1.51 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 2.90/1.51 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A2)))) = A2)))).
% 2.90/1.51 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.90/1.51 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F2:(A > B),G2:(A > B)]: (((! [X3:A]: (((F2 @ X3) = (G2 @ X3)))) => (F2 = G2))))).
% 2.90/1.51 thf(fact_49_inf__sup__aci_I5_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [X2:A,Y3:A]: ((((sup_sup @ A) @ Y3) @ X2)))))))).
% 2.90/1.51 thf(fact_50_inf__sup__aci_I6_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z))))))))).
% 2.90/1.51 thf(fact_51_inf__sup__aci_I7_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 2.90/1.51 thf(fact_52_inf__sup__aci_I8_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = (((sup_sup @ A) @ X) @ Y)))))))).
% 2.90/1.51 thf(fact_53_Un__left__commute, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ B2) @ C)) = (((sup_sup @ (set @ A)) @ B2) @ (((sup_sup @ (set @ A)) @ A2) @ C)))))).
% 2.90/1.51 thf(fact_54_Un__left__absorb, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) = (((sup_sup @ (set @ A)) @ A2) @ B2))))).
% 2.90/1.51 thf(fact_55_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A4:(set @ A),B4:(set @ A)]: ((((sup_sup @ (set @ A)) @ B4) @ A4))))))).
% 2.90/1.51 thf(fact_56_Un__absorb, axiom, (! [A:$tType,A2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ A2) = A2)))).
% 2.90/1.51 thf(fact_57_Un__assoc, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ C) = (((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ B2) @ C)))))).
% 2.90/1.51 thf(fact_58_ball__Un, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),P:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) => (P @ X2)))) = ((! [X2:A]: (((((member @ A) @ X2) @ A2) => (P @ X2)))) & (! [X2:A]: (((((member @ A) @ X2) @ B2) => (P @ X2))))))))).
% 2.90/1.51 thf(fact_59_bex__Un, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),P:(A > $o)]: (((? [X2:A]: (((((member @ A) @ X2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) & (P @ X2)))) = ((? [X2:A]: (((((member @ A) @ X2) @ A2) & (P @ X2)))) | (? [X2:A]: (((((member @ A) @ X2) @ B2) & (P @ X2))))))))).
% 2.90/1.51 thf(fact_60_UnI2, axiom, (! [A:$tType,C2:A,B2:(set @ A),A2:(set @ A)]: (((((member @ A) @ C2) @ B2) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)))))).
% 2.90/1.51 thf(fact_61_UnI1, axiom, (! [A:$tType,C2:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)))))).
% 2.90/1.51 thf(fact_62_UnE, axiom, (! [A:$tType,C2:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) => ((~ ((((member @ A) @ C2) @ A2))) => (((member @ A) @ C2) @ B2)))))).
% 2.90/1.51 thf(fact_63_sup_OcoboundedI2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [C2:A,B3:A,A3:A]: (((((ord_less_eq @ A) @ C2) @ B3) => (((ord_less_eq @ A) @ C2) @ (((sup_sup @ A) @ A3) @ B3))))))))).
% 2.90/1.51 thf(fact_64_sup_OcoboundedI1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [C2:A,A3:A,B3:A]: (((((ord_less_eq @ A) @ C2) @ A3) => (((ord_less_eq @ A) @ C2) @ (((sup_sup @ A) @ A3) @ B3))))))))).
% 2.90/1.51 thf(fact_65_sup_Oabsorb__iff2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A5:A,B5:A]: (((((sup_sup @ A) @ A5) @ B5) = B5)))))))).
% 2.90/1.51 thf(fact_66_sup_Oabsorb__iff1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B5:A,A5:A]: (((((sup_sup @ A) @ A5) @ B5) = A5)))))))).
% 2.90/1.51 thf(fact_67_sup_Ocobounded2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,A3:A]: ((((ord_less_eq @ A) @ B3) @ (((sup_sup @ A) @ A3) @ B3)))))))).
% 2.90/1.51 thf(fact_68_sup_Ocobounded1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: ((((ord_less_eq @ A) @ A3) @ (((sup_sup @ A) @ A3) @ B3)))))))).
% 2.90/1.51 thf(fact_69_sup_Oorder__iff, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B5:A,A5:A]: ((A5 = (((sup_sup @ A) @ A5) @ B5))))))))).
% 2.90/1.51 thf(fact_70_sup_OboundedI, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C2) @ A3) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ B3) @ C2)) @ A3))))))))).
% 2.90/1.51 thf(fact_71_sup_OboundedE, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,C2:A,A3:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B3) @ C2)) @ A3) => (~ (((((ord_less_eq @ A) @ B3) @ A3) => (~ ((((ord_less_eq @ A) @ C2) @ A3))))))))))))).
% 2.90/1.51 thf(fact_72_sup__absorb2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((sup_sup @ A) @ X) @ Y) = Y)))))))).
% 2.90/1.51 thf(fact_73_sup__absorb1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((sup_sup @ A) @ X) @ Y) = X)))))))).
% 2.90/1.51 thf(fact_74_sup_Oabsorb2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((sup_sup @ A) @ A3) @ B3) = B3)))))))).
% 2.90/1.51 thf(fact_75_sup_Oabsorb1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((sup_sup @ A) @ A3) @ B3) = A3)))))))).
% 2.90/1.51 thf(fact_76_sup__unique, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [F2:(A > (A > A)),X:A,Y:A]: (((! [X3:A,Y4:A]: ((((ord_less_eq @ A) @ X3) @ ((F2 @ X3) @ Y4)))) => ((! [X3:A,Y4:A]: ((((ord_less_eq @ A) @ Y4) @ ((F2 @ X3) @ Y4)))) => ((! [X3:A,Y4:A,Z3:A]: (((((ord_less_eq @ A) @ Y4) @ X3) => ((((ord_less_eq @ A) @ Z3) @ X3) => (((ord_less_eq @ A) @ ((F2 @ Y4) @ Z3)) @ X3))))) => ((((sup_sup @ A) @ X) @ Y) = ((F2 @ X) @ Y))))))))))).
% 2.90/1.51 thf(fact_77_sup_OorderI, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((A3 = (((sup_sup @ A) @ A3) @ B3)) => (((ord_less_eq @ A) @ B3) @ A3)))))))).
% 2.90/1.51 thf(fact_78_sup_OorderE, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => (A3 = (((sup_sup @ A) @ A3) @ B3))))))))).
% 2.90/1.51 thf(fact_79_le__iff__sup, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X2:A,Y3:A]: (((((sup_sup @ A) @ X2) @ Y3) = Y3)))))))).
% 2.90/1.51 thf(fact_80_sup__least, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [Y:A,X:A,Z:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ Z) @ X) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ Y) @ Z)) @ X))))))))).
% 2.90/1.51 thf(fact_81_sup__mono, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,C2:A,B3:A,D:A]: (((((ord_less_eq @ A) @ A3) @ C2) => ((((ord_less_eq @ A) @ B3) @ D) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A3) @ B3)) @ (((sup_sup @ A) @ C2) @ D)))))))))).
% 2.90/1.51 thf(fact_82_sup_Omono, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [C2:A,A3:A,D:A,B3:A]: (((((ord_less_eq @ A) @ C2) @ A3) => ((((ord_less_eq @ A) @ D) @ B3) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ C2) @ D)) @ (((sup_sup @ A) @ A3) @ B3)))))))))).
% 2.90/1.51 thf(fact_83_le__supI2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,B3:A,A3:A]: (((((ord_less_eq @ A) @ X) @ B3) => (((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ A3) @ B3))))))))).
% 2.90/1.51 thf(fact_84_le__supI1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,A3:A,B3:A]: (((((ord_less_eq @ A) @ X) @ A3) => (((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ A3) @ B3))))))))).
% 2.90/1.51 thf(fact_85_sup__ge2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [Y:A,X:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 2.90/1.51 thf(fact_86_sup__ge1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 2.90/1.51 thf(fact_87_le__supI, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,X:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ X) => ((((ord_less_eq @ A) @ B3) @ X) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A3) @ B3)) @ X))))))))).
% 2.90/1.51 thf(fact_88_le__supE, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,B3:A,X:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ A3) @ B3)) @ X) => (~ (((((ord_less_eq @ A) @ A3) @ X) => (~ ((((ord_less_eq @ A) @ B3) @ X))))))))))))).
% 2.90/1.51 thf(fact_89_inf__sup__ord_I3_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 2.90/1.51 thf(fact_90_inf__sup__ord_I4_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [Y:A,X:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 2.90/1.51 thf(fact_91_subset__Un__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A4:(set @ A),B4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ B4) = B4))))))).
% 2.90/1.51 thf(fact_92_Un__absorb2, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A2) => ((((sup_sup @ (set @ A)) @ A2) @ B2) = A2))))).
% 2.90/1.51 thf(fact_93_Un__absorb1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((sup_sup @ (set @ A)) @ A2) @ B2) = B2))))).
% 2.90/1.51 thf(fact_94_Un__upper2, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B2) @ (((sup_sup @ (set @ A)) @ A2) @ B2))))).
% 2.90/1.51 thf(fact_95_Un__upper1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ A2) @ B2))))).
% 2.90/1.51 thf(fact_96_Un__least, axiom, (! [A:$tType,A2:(set @ A),C:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ C) => ((((ord_less_eq @ (set @ A)) @ B2) @ C) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ C)))))).
% 2.90/1.51 thf(fact_97_Un__mono, axiom, (! [A:$tType,A2:(set @ A),C:(set @ A),B2:(set @ A),D2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ C) => ((((ord_less_eq @ (set @ A)) @ B2) @ D2) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ (((sup_sup @ (set @ A)) @ C) @ D2))))))).
% 2.90/1.51 thf(fact_98_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.90/1.51 thf(fact_99_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ A3) @ B3) => (A3 = B3))))))))).
% 2.90/1.51 thf(fact_100_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C2) @ B3) => (((ord_less_eq @ A) @ C2) @ A3))))))))).
% 2.90/1.51 thf(fact_101_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A3:A,B3:A]: (((! [A6:A,B6:A]: (((((ord_less_eq @ A) @ A6) @ B6) => ((P @ A6) @ B6)))) => ((! [A6:A,B6:A]: ((((P @ B6) @ A6) => ((P @ A6) @ B6)))) => ((P @ A3) @ B3))))))))).
% 2.90/1.51 thf(fact_102_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 2.90/1.51 thf(fact_103_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X) @ Z))))))))).
% 2.90/1.51 thf(fact_104_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ A3) => (A3 = B3))))))))).
% 2.90/1.51 thf(fact_105_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((B3 = C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 2.90/1.51 thf(fact_106_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C2:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 2.90/1.51 thf(fact_107_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.90/1.51 thf(fact_108_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F2:(A > B),G2:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F2) @ G2) => (((ord_less_eq @ B) @ (F2 @ X)) @ (G2 @ X))))))))).
% 2.90/1.51 thf(fact_109_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F2:(A > B),G2:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F2) @ G2) => (((ord_less_eq @ B) @ (F2 @ X)) @ (G2 @ X))))))))).
% 2.90/1.51 thf(fact_110_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F2:(A > B),G2:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F2 @ X3)) @ (G2 @ X3)))) => (((ord_less_eq @ (A > B)) @ F2) @ G2)))))))).
% 2.90/1.51 thf(fact_111_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F:(A > B),G:(A > B)]: ((! [X2:A]: ((((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))))).
% 2.90/1.51 thf(fact_112_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,F2:(B > A),B3:B,C2:B]: (((((ord_less_eq @ A) @ A3) @ (F2 @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X3:B,Y4:B]: (((((ord_less_eq @ B) @ X3) @ Y4) => (((ord_less_eq @ A) @ (F2 @ X3)) @ (F2 @ Y4))))) => (((ord_less_eq @ A) @ A3) @ (F2 @ C2))))))))))).
% 2.90/1.51 thf(fact_113_order__subst2, axiom, (! [A:$tType,C3:$tType]: (((((order @ C3) @ (type2 @ C3)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,B3:A,F2:(A > C3),C2:C3]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ C3) @ (F2 @ B3)) @ C2) => ((! [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) => (((ord_less_eq @ C3) @ (F2 @ X3)) @ (F2 @ Y4))))) => (((ord_less_eq @ C3) @ (F2 @ A3)) @ C2)))))))))).
% 2.90/1.51 thf(fact_114_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A3:A,F2:(B > A),B3:B,C2:B]: (((A3 = (F2 @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X3:B,Y4:B]: (((((ord_less_eq @ B) @ X3) @ Y4) => (((ord_less_eq @ A) @ (F2 @ X3)) @ (F2 @ Y4))))) => (((ord_less_eq @ A) @ A3) @ (F2 @ C2))))))))))).
% 2.90/1.51 thf(fact_115_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A3:A,B3:A,F2:(A > B),C2:B]: (((((ord_less_eq @ A) @ A3) @ B3) => (((F2 @ B3) = C2) => ((! [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) => (((ord_less_eq @ B) @ (F2 @ X3)) @ (F2 @ Y4))))) => (((ord_less_eq @ B) @ (F2 @ A3)) @ C2)))))))))).
% 2.90/1.51 thf(fact_116_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y2:A,Z2:A]: ((Y2 = Z2))) = (^ [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X2))))))))).
% 2.90/1.51 thf(fact_117_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.90/1.51 thf(fact_118_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.90/1.51 thf(fact_119_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.90/1.51 thf(fact_120_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.90/1.51 thf(fact_121_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 2.90/1.51 thf(fact_122_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z)))) => (((((ord_less_eq @ A) @ X) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X)))) => (~ (((((ord_less_eq @ A) @ Z) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 2.90/1.51 thf(fact_123_inItr__mono, axiom, (! [Ns:(set @ n),Tr:dtree,N:n,Ns2:(set @ n)]: (((((gram_L830233218_inItr @ Ns) @ Tr) @ N) => ((((ord_less_eq @ (set @ n)) @ Ns) @ Ns2) => (((gram_L830233218_inItr @ Ns2) @ Tr) @ N)))))).
% 2.90/1.51 thf(fact_124_inFr2__mono, axiom, (! [Ns:(set @ n),Tr:dtree,T2:t,Ns2:(set @ n)]: (((((gram_L805317441_inFr2 @ Ns) @ Tr) @ T2) => ((((ord_less_eq @ (set @ n)) @ Ns) @ Ns2) => (((gram_L805317441_inFr2 @ Ns2) @ Tr) @ T2)))))).
% 2.90/1.51 thf(fact_125_inFr__mono, axiom, (! [Ns:(set @ n),Tr:dtree,T2:t,Ns2:(set @ n)]: (((((gram_L1333338417e_inFr @ Ns) @ Tr) @ T2) => ((((ord_less_eq @ (set @ n)) @ Ns) @ Ns2) => (((gram_L1333338417e_inFr @ Ns2) @ Tr) @ T2)))))).
% 2.90/1.51 thf(fact_126_antimonoD, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F2:(A > B),X:A,Y:A]: (((((order_antimono @ A) @ B) @ F2) => ((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ B) @ (F2 @ Y)) @ (F2 @ X)))))))))).
% 2.90/1.51 thf(fact_127_antimonoE, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F2:(A > B),X:A,Y:A]: (((((order_antimono @ A) @ B) @ F2) => ((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ B) @ (F2 @ Y)) @ (F2 @ X)))))))))).
% 2.90/1.51 thf(fact_128_inFr__inFr2, axiom, (gram_L1333338417e_inFr = gram_L805317441_inFr2)).
% 2.90/1.51 thf(fact_129_not__root__inFr, axiom, (! [Tr:dtree,Ns:(set @ n),T2:t]: (((~ ((((member @ n) @ (root @ Tr)) @ Ns))) => (~ ((((gram_L1333338417e_inFr @ Ns) @ Tr) @ T2))))))).
% 2.90/1.51 thf(fact_130_inFr__root__in, axiom, (! [Ns:(set @ n),Tr:dtree,T2:t]: (((((gram_L1333338417e_inFr @ Ns) @ Tr) @ T2) => (((member @ n) @ (root @ Tr)) @ Ns))))).
% 2.90/1.51 thf(fact_131_inFr2__root__in, axiom, (! [Ns:(set @ n),Tr:dtree,T2:t]: (((((gram_L805317441_inFr2 @ Ns) @ Tr) @ T2) => (((member @ n) @ (root @ Tr)) @ Ns))))).
% 2.90/1.51 thf(fact_132_inItr__root__in, axiom, (! [Ns:(set @ n),Tr:dtree,N:n]: (((((gram_L830233218_inItr @ Ns) @ Tr) @ N) => (((member @ n) @ (root @ Tr)) @ Ns))))).
% 2.90/1.51 thf(fact_133_inItr_OBase, axiom, (! [Tr:dtree,Ns:(set @ n)]: (((((member @ n) @ (root @ Tr)) @ Ns) => (((gram_L830233218_inItr @ Ns) @ Tr) @ (root @ Tr)))))).
% 2.90/1.51 thf(fact_134_antimono__def, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (((order_antimono @ A) @ B) = (^ [F:(A > B)]: ((! [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) => (((ord_less_eq @ B) @ (F @ Y3)) @ (F @ X2)))))))))))).
% 2.90/1.51 thf(fact_135_antimonoI, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F2:(A > B)]: (((! [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) => (((ord_less_eq @ B) @ (F2 @ Y4)) @ (F2 @ X3))))) => (((order_antimono @ A) @ B) @ F2)))))))).
% 2.90/1.51 thf(fact_136_strict__mono__less__eq, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F2:(A > B),X:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F2) => ((((ord_less_eq @ B) @ (F2 @ X)) @ (F2 @ Y)) = (((ord_less_eq @ A) @ X) @ Y))))))))).
% 2.90/1.51 thf(fact_137_pairwise__subset, axiom, (! [A:$tType,P:(A > (A > $o)),S:(set @ A),T3:(set @ A)]: (((((pairwise @ A) @ P) @ S) => ((((ord_less_eq @ (set @ A)) @ T3) @ S) => (((pairwise @ A) @ P) @ T3)))))).
% 2.90/1.51 thf(fact_138_Diff__subset__conv, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) @ C) = (((ord_less_eq @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ B2) @ C)))))).
% 2.90/1.51 thf(fact_139_Diff__partition, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((sup_sup @ (set @ A)) @ A2) @ (((minus_minus @ (set @ A)) @ B2) @ A2)) = B2))))).
% 2.90/1.51 thf(fact_140_minus__apply, axiom, (! [B:$tType,A:$tType]: ((((minus @ B) @ (type2 @ B)) => ((minus_minus @ (A > B)) = (^ [A4:(A > B),B4:(A > B),X2:A]: ((((minus_minus @ B) @ (A4 @ X2)) @ (B4 @ X2))))))))).
% 2.90/1.51 thf(fact_141_Diff__idemp, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) @ B2) = (((minus_minus @ (set @ A)) @ A2) @ B2))))).
% 2.90/1.51 thf(fact_142_Diff__iff, axiom, (! [A:$tType,C2:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) = ((((member @ A) @ C2) @ A2) & (~ ((((member @ A) @ C2) @ B2)))))))).
% 2.90/1.51 thf(fact_143_DiffI, axiom, (! [A:$tType,C2:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C2) @ A2) => ((~ ((((member @ A) @ C2) @ B2))) => (((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A2) @ B2))))))).
% 2.90/1.51 thf(fact_144_Un__Diff__cancel2, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((minus_minus @ (set @ A)) @ B2) @ A2)) @ A2) = (((sup_sup @ (set @ A)) @ B2) @ A2))))).
% 2.90/1.51 thf(fact_145_Un__Diff__cancel, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((minus_minus @ (set @ A)) @ B2) @ A2)) = (((sup_sup @ (set @ A)) @ A2) @ B2))))).
% 2.90/1.51 thf(fact_146_Diff__mono, axiom, (! [A:$tType,A2:(set @ A),C:(set @ A),D2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ C) => ((((ord_less_eq @ (set @ A)) @ D2) @ B2) => (((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) @ (((minus_minus @ (set @ A)) @ C) @ D2))))))).
% 2.90/1.51 thf(fact_147_Diff__subset, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) @ A2)))).
% 2.90/1.51 thf(fact_148_double__diff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C) => ((((minus_minus @ (set @ A)) @ B2) @ (((minus_minus @ (set @ A)) @ C) @ A2)) = A2)))))).
% 2.90/1.51 thf(fact_149_fun__diff__def, axiom, (! [B:$tType,A:$tType]: ((((minus @ B) @ (type2 @ B)) => ((minus_minus @ (A > B)) = (^ [A4:(A > B),B4:(A > B),X2:A]: ((((minus_minus @ B) @ (A4 @ X2)) @ (B4 @ X2))))))))).
% 2.90/1.51 thf(fact_150_pairwise__def, axiom, (! [A:$tType]: (((pairwise @ A) = (^ [R:(A > (A > $o)),S2:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ S2) => (! [Y3:A]: (((((member @ A) @ Y3) @ S2) => ((X2 != Y3) => ((R @ X2) @ Y3)))))))))))))).
% 2.90/1.51 thf(fact_151_DiffD2, axiom, (! [A:$tType,C2:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) => (~ ((((member @ A) @ C2) @ B2))))))).
% 2.90/1.51 thf(fact_152_DiffD1, axiom, (! [A:$tType,C2:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) => (((member @ A) @ C2) @ A2))))).
% 2.90/1.51 thf(fact_153_DiffE, axiom, (! [A:$tType,C2:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) => (~ (((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ B2)))))))).
% 2.90/1.51 thf(fact_154_Un__Diff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ C) = (((sup_sup @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ C)) @ (((minus_minus @ (set @ A)) @ B2) @ C)))))).
% 2.90/1.51 thf(fact_155_strict__mono__eq, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F2:(A > B),X:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F2) => (((F2 @ X) = (F2 @ Y)) = (X = Y))))))))).
% 2.90/1.51 thf(fact_156_diff__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A,D:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ D) @ C2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ (((minus_minus @ A) @ B3) @ D)))))))))).
% 2.90/1.51 thf(fact_157_diff__left__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [B3:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B3) @ A3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C2) @ A3)) @ (((minus_minus @ A) @ C2) @ B3))))))))).
% 2.90/1.51 thf(fact_158_diff__right__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ (((minus_minus @ A) @ B3) @ C2))))))))).
% 2.90/1.51 thf(fact_159_diff__eq__diff__less__eq, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C2:A,D:A]: ((((((minus_minus @ A) @ A3) @ B3) = (((minus_minus @ A) @ C2) @ D)) => ((((ord_less_eq @ A) @ A3) @ B3) = (((ord_less_eq @ A) @ C2) @ D))))))))).
% 2.90/1.51 thf(fact_160_diff__right__commute, axiom, (! [A:$tType]: ((((cancel146912293up_add @ A) @ (type2 @ A)) => (! [A3:A,C2:A,B3:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ B3) = (((minus_minus @ A) @ (((minus_minus @ A) @ A3) @ B3)) @ C2)))))))).
% 2.90/1.51 thf(fact_161_diff__eq__diff__eq, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C2:A,D:A]: ((((((minus_minus @ A) @ A3) @ B3) = (((minus_minus @ A) @ C2) @ D)) => ((A3 = B3) = (C2 = D))))))))).
% 2.90/1.51 thf(fact_162_Diff__eq__empty__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: ((((((minus_minus @ (set @ A)) @ A2) @ B2) = (bot_bot @ (set @ A))) = (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.90/1.51 thf(fact_163_diff__ge__0__iff__ge, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A3) @ B3)) = (((ord_less_eq @ A) @ B3) @ A3)))))))).
% 2.90/1.51 thf(fact_164_Compl__Diff__eq, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ B2)) = (((sup_sup @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A2)) @ B2))))).
% 2.90/1.51 thf(fact_165_bot__apply, axiom, (! [C3:$tType,D3:$tType]: ((((bot @ C3) @ (type2 @ C3)) => ((bot_bot @ (D3 > C3)) = (^ [X2:D3]: ((bot_bot @ C3)))))))).
% 2.90/1.51 thf(fact_166_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X2:A]: ((~ ((P @ X2))))))))).
% 2.90/1.51 thf(fact_167_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X2:A]: ((~ ((P @ X2))))))))).
% 2.90/1.51 thf(fact_168_all__not__in__conv, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X2:A]: ((~ ((((member @ A) @ X2) @ A2))))) = (A2 = (bot_bot @ (set @ A))))))).
% 2.90/1.51 thf(fact_169_empty__iff, axiom, (! [A:$tType,C2:A]: ((~ ((((member @ A) @ C2) @ (bot_bot @ (set @ A)))))))).
% 2.90/1.51 thf(fact_170_compl__eq__compl__iff, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((uminus_uminus @ A) @ X) = ((uminus_uminus @ A) @ Y)) = (X = Y)))))))).
% 2.90/1.51 thf(fact_171_double__compl, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ X)) = X))))))).
% 2.90/1.51 thf(fact_172_uminus__apply, axiom, (! [B:$tType,A:$tType]: ((((uminus @ B) @ (type2 @ B)) => ((uminus_uminus @ (A > B)) = (^ [A4:(A > B),X2:A]: (((uminus_uminus @ B) @ (A4 @ X2))))))))).
% 2.90/1.51 thf(fact_173_Compl__eq__Compl__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((uminus_uminus @ (set @ A)) @ A2) = ((uminus_uminus @ (set @ A)) @ B2)) = (A2 = B2))))).
% 2.90/1.51 thf(fact_174_Compl__iff, axiom, (! [A:$tType,C2:A,A2:(set @ A)]: (((((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A2)) = (~ ((((member @ A) @ C2) @ A2))))))).
% 2.90/1.51 thf(fact_175_ComplI, axiom, (! [A:$tType,C2:A,A2:(set @ A)]: (((~ ((((member @ A) @ C2) @ A2))) => (((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A2)))))).
% 2.90/1.51 thf(fact_176_le__zero__eq, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.90/1.51 thf(fact_177_diff__self, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A3:A]: (((((minus_minus @ A) @ A3) @ A3) = (zero_zero @ A)))))))).
% 2.90/1.51 thf(fact_178_diff__0__right, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A3:A]: (((((minus_minus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 2.90/1.51 thf(fact_179_zero__diff, axiom, (! [A:$tType]: ((((comm_monoid_diff @ A) @ (type2 @ A)) => (! [A3:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A3) = (zero_zero @ A)))))))).
% 2.90/1.51 thf(fact_180_diff__zero, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type2 @ A)) => (! [A3:A]: (((((minus_minus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 2.90/1.51 thf(fact_181_cancel__comm__monoid__add__class_Odiff__cancel, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type2 @ A)) => (! [A3:A]: (((((minus_minus @ A) @ A3) @ A3) = (zero_zero @ A)))))))).
% 2.90/1.51 thf(fact_182_neg__le__iff__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B3)) @ ((uminus_uminus @ A) @ A3)) = (((ord_less_eq @ A) @ A3) @ B3)))))))).
% 2.90/1.51 thf(fact_183_compl__le__compl__iff, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X)) @ ((uminus_uminus @ A) @ Y)) = (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.90/1.51 thf(fact_184_sup__bot_Oright__neutral, axiom, (! [A:$tType]: ((((bounde1808546759up_bot @ A) @ (type2 @ A)) => (! [A3:A]: (((((sup_sup @ A) @ A3) @ (bot_bot @ A)) = A3))))))).
% 2.90/1.51 thf(fact_185_sup__bot_Oleft__neutral, axiom, (! [A:$tType]: ((((bounde1808546759up_bot @ A) @ (type2 @ A)) => (! [A3:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ A3) = A3))))))).
% 2.90/1.51 thf(fact_186_sup__eq__bot__iff, axiom, (! [A:$tType]: ((((bounded_lattice_bot @ A) @ (type2 @ A)) => (! [X:A,Y:A]: ((((((sup_sup @ A) @ X) @ Y) = (bot_bot @ A)) = ((X = (bot_bot @ A)) & (Y = (bot_bot @ A)))))))))).
% 2.90/1.51 thf(fact_187_bot__eq__sup__iff, axiom, (! [A:$tType]: ((((bounded_lattice_bot @ A) @ (type2 @ A)) => (! [X:A,Y:A]: ((((bot_bot @ A) = (((sup_sup @ A) @ X) @ Y)) = ((X = (bot_bot @ A)) & (Y = (bot_bot @ A)))))))))).
% 2.90/1.51 thf(fact_188_minus__diff__eq, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: ((((uminus_uminus @ A) @ (((minus_minus @ A) @ A3) @ B3)) = (((minus_minus @ A) @ B3) @ A3)))))))).
% 2.90/1.51 thf(fact_189_empty__subsetI, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A2)))).
% 2.90/1.51 thf(fact_190_subset__empty, axiom, (! [A:$tType,A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))) = (A2 = (bot_bot @ (set @ A))))))).
% 2.90/1.51 thf(fact_191_Un__empty, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: ((((((sup_sup @ (set @ A)) @ A2) @ B2) = (bot_bot @ (set @ A))) = ((A2 = (bot_bot @ (set @ A))) & (B2 = (bot_bot @ (set @ A)))))))).
% 2.90/1.51 thf(fact_192_Diff__cancel, axiom, (! [A:$tType,A2:(set @ A)]: (((((minus_minus @ (set @ A)) @ A2) @ A2) = (bot_bot @ (set @ A)))))).
% 2.90/1.51 thf(fact_193_empty__Diff, axiom, (! [A:$tType,A2:(set @ A)]: (((((minus_minus @ (set @ A)) @ (bot_bot @ (set @ A))) @ A2) = (bot_bot @ (set @ A)))))).
% 2.90/1.51 thf(fact_194_Diff__empty, axiom, (! [A:$tType,A2:(set @ A)]: (((((minus_minus @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))) = A2)))).
% 2.90/1.51 thf(fact_195_Compl__subset__Compl__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A2)) @ ((uminus_uminus @ (set @ A)) @ B2)) = (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 2.90/1.51 thf(fact_196_Compl__anti__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ B2)) @ ((uminus_uminus @ (set @ A)) @ A2)))))).
% 2.90/1.51 thf(fact_197_neg__0__le__iff__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A3)) = (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))).
% 2.90/1.51 thf(fact_198_neg__le__0__iff__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))).
% 2.90/1.51 thf(fact_199_less__eq__neg__nonpos, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ ((uminus_uminus @ A) @ A3)) = (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))).
% 2.90/1.51 thf(fact_200_neg__less__eq__nonneg, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ A3) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))).
% 2.90/1.51 thf(fact_201_diff__0, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A3:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A3) = ((uminus_uminus @ A) @ A3)))))))).
% 2.90/1.51 thf(fact_202_eq__iff__diff__eq__0, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => ((^ [Y2:A,Z2:A]: ((Y2 = Z2))) = (^ [A5:A,B5:A]: (((((minus_minus @ A) @ A5) @ B5) = (zero_zero @ A))))))))).
% 2.90/1.51 thf(fact_203_zero__le, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.90/1.51 thf(fact_204_le__imp__neg__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B3)) @ ((uminus_uminus @ A) @ A3))))))))).
% 2.90/1.51 thf(fact_205_minus__le__iff, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ B3) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B3)) @ A3)))))))).
% 2.90/1.51 thf(fact_206_le__minus__iff, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ ((uminus_uminus @ A) @ B3)) = (((ord_less_eq @ A) @ B3) @ ((uminus_uminus @ A) @ A3))))))))).
% 2.90/1.51 thf(fact_207_pairwise__empty, axiom, (! [A:$tType,P:(A > (A > $o))]: ((((pairwise @ A) @ P) @ (bot_bot @ (set @ A)))))).
% 2.90/1.51 thf(fact_208_subset__Compl__self__eq, axiom, (! [A:$tType,A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ ((uminus_uminus @ (set @ A)) @ A2)) = (A2 = (bot_bot @ (set @ A))))))).
% 2.90/1.51 thf(fact_209_bot_Oextremum, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A3:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A3))))))).
% 2.90/1.51 thf(fact_210_bot_Oextremum__unique, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) = (A3 = (bot_bot @ A))))))))).
% 2.90/1.51 thf(fact_211_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) => (A3 = (bot_bot @ A))))))))).
% 2.90/1.51 thf(fact_212_compl__mono, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y)) @ ((uminus_uminus @ A) @ X))))))))).
% 2.90/1.51 thf(fact_213_compl__le__swap1, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ ((uminus_uminus @ A) @ X)) => (((ord_less_eq @ A) @ X) @ ((uminus_uminus @ A) @ Y))))))))).
% 2.90/1.51 thf(fact_214_compl__le__swap2, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y)) @ X) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X)) @ Y)))))))).
% 2.90/1.51 thf(fact_215_Un__empty__right, axiom, (! [A:$tType,A2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))) = A2)))).
% 2.90/1.51 thf(fact_216_Un__empty__left, axiom, (! [A:$tType,B2:(set @ A)]: (((((sup_sup @ (set @ A)) @ (bot_bot @ (set @ A))) @ B2) = B2)))).
% 2.90/1.51 thf(fact_217_fun__Compl__def, axiom, (! [B:$tType,A:$tType]: ((((uminus @ B) @ (type2 @ B)) => ((uminus_uminus @ (A > B)) = (^ [A4:(A > B),X2:A]: (((uminus_uminus @ B) @ (A4 @ X2))))))))).
% 2.90/1.51 thf(fact_218_double__complement, axiom, (! [A:$tType,A2:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A2)) = A2)))).
% 2.90/1.51 thf(fact_219_ex__in__conv, axiom, (! [A:$tType,A2:(set @ A)]: (((? [X2:A]: ((((member @ A) @ X2) @ A2))) = (A2 != (bot_bot @ (set @ A))))))).
% 2.90/1.51 thf(fact_220_equals0I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [Y4:A]: ((~ ((((member @ A) @ Y4) @ A2))))) => (A2 = (bot_bot @ (set @ A))))))).
% 2.90/1.51 thf(fact_221_equals0D, axiom, (! [A:$tType,A2:(set @ A),A3:A]: (((A2 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A3) @ A2))))))).
% 2.90/1.51 thf(fact_222_emptyE, axiom, (! [A:$tType,A3:A]: ((~ ((((member @ A) @ A3) @ (bot_bot @ (set @ A)))))))).
% 2.90/1.51 thf(fact_223_ComplD, axiom, (! [A:$tType,C2:A,A2:(set @ A)]: (((((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A2)) => (~ ((((member @ A) @ C2) @ A2))))))).
% 2.90/1.51 thf(fact_224_sup__bot__right, axiom, (! [A:$tType]: ((((bounded_lattice_bot @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ X) @ (bot_bot @ A)) = X))))))).
% 2.90/1.51 thf(fact_225_sup__bot__left, axiom, (! [A:$tType]: ((((bounded_lattice_bot @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ X) = X))))))).
% 2.90/1.51 thf(fact_226_bot__fun__def, axiom, (! [B:$tType,A:$tType]: ((((bot @ B) @ (type2 @ B)) => ((bot_bot @ (A > B)) = (^ [X2:A]: ((bot_bot @ B)))))))).
% 2.90/1.51 thf(fact_227_le__iff__diff__le__0, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A5:A,B5:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ A5) @ B5)) @ (zero_zero @ A))))))))).
% 2.90/1.51 thf(fact_228_subset__emptyI, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X3:A]: ((~ ((((member @ A) @ X3) @ A2))))) => (((ord_less_eq @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.90/1.51 thf(fact_229_le__numeral__extra_I3_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.90/1.51 thf(fact_230_not__root__Fr, axiom, (! [Tr:dtree,Ns:(set @ n)]: (((~ ((((member @ n) @ (root @ Tr)) @ Ns))) => (((gram_L861583724lle_Fr @ Ns) @ Tr) = (bot_bot @ (set @ t))))))).
% 2.90/1.51 thf(fact_231_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.90/1.51 thf(fact_232_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A4:(set @ A)]: ((A4 = (bot_bot @ (set @ A))))))))).
% 2.90/1.51 thf(fact_233_diff__numeral__special_I12_J, axiom, (! [A:$tType]: ((((neg_numeral @ A) @ (type2 @ A)) => ((((minus_minus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ A) @ (one_one @ A))) = (zero_zero @ A)))))).
% 2.90/1.51 thf(fact_234_diff__numeral__special_I9_J, axiom, (! [A:$tType]: ((((neg_numeral @ A) @ (type2 @ A)) => ((((minus_minus @ A) @ (one_one @ A)) @ (one_one @ A)) = (zero_zero @ A)))))).
% 2.90/1.51 thf(fact_235_le__numeral__extra_I1_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 2.90/1.51 thf(fact_236_le__numeral__extra_I2_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 2.90/1.51 thf(fact_237_le__minus__one__simps_I2_J, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (one_one @ A)))))).
% 2.90/1.51 thf(fact_238_le__minus__one__simps_I4_J, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 2.90/1.51 thf(fact_239_le__numeral__extra_I4_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ (one_one @ A)))))).
% 2.90/1.51 thf(fact_240_le__minus__one__simps_I1_J, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (zero_zero @ A)))))).
% 2.90/1.51 thf(fact_241_le__minus__one__simps_I3_J, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 2.90/1.51 thf(fact_242_zero__le__one, axiom, (! [A:$tType]: ((((linord1659791738miring @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 2.90/1.51 thf(fact_243_not__one__le__zero, axiom, (! [A:$tType]: ((((linord1659791738miring @ A) @ (type2 @ A)) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 2.90/1.51 thf(fact_244_subset__Compl__singleton, axiom, (! [A:$tType,A2:(set @ A),B3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ ((uminus_uminus @ (set @ A)) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A))))) = (~ ((((member @ A) @ B3) @ A2))))))).
% 2.90/1.51 thf(fact_245_sup__bot_Osemilattice__neutr__axioms, axiom, (! [A:$tType]: ((((bounde1808546759up_bot @ A) @ (type2 @ A)) => (((semilattice_neutr @ A) @ (sup_sup @ A)) @ (bot_bot @ A)))))).
% 2.90/1.51 thf(fact_246_insert__absorb2, axiom, (! [A:$tType,X:A,A2:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ X) @ A2)) = (((insert @ A) @ X) @ A2))))).
% 2.90/1.51 thf(fact_247_insert__iff, axiom, (! [A:$tType,A3:A,B3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B3) @ A2)) = ((A3 = B3) | (((member @ A) @ A3) @ A2)))))).
% 2.90/1.51 thf(fact_248_insertCI, axiom, (! [A:$tType,A3:A,B2:(set @ A),B3:A]: ((((~ ((((member @ A) @ A3) @ B2))) => (A3 = B3)) => (((member @ A) @ A3) @ (((insert @ A) @ B3) @ B2)))))).
% 2.90/1.51 thf(fact_249_singletonI, axiom, (! [A:$tType,A3:A]: ((((member @ A) @ A3) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))).
% 2.90/1.51 thf(fact_250_insert__subset, axiom, (! [A:$tType,X:A,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X) @ A2)) @ B2) = ((((member @ A) @ X) @ B2) & (((ord_less_eq @ (set @ A)) @ A2) @ B2)))))).
% 2.90/1.51 thf(fact_251_Un__insert__right, axiom, (! [A:$tType,A2:(set @ A),A3:A,B2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ B2)) = (((insert @ A) @ A3) @ (((sup_sup @ (set @ A)) @ A2) @ B2)))))).
% 2.90/1.51 thf(fact_252_Un__insert__left, axiom, (! [A:$tType,A3:A,B2:(set @ A),C:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((insert @ A) @ A3) @ B2)) @ C) = (((insert @ A) @ A3) @ (((sup_sup @ (set @ A)) @ B2) @ C)))))).
% 2.90/1.51 thf(fact_253_insert__Diff1, axiom, (! [A:$tType,X:A,B2:(set @ A),A2:(set @ A)]: (((((member @ A) @ X) @ B2) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X) @ A2)) @ B2) = (((minus_minus @ (set @ A)) @ A2) @ B2)))))).
% 2.90/1.51 thf(fact_254_Diff__insert0, axiom, (! [A:$tType,X:A,A2:(set @ A),B2:(set @ A)]: (((~ ((((member @ A) @ X) @ A2))) => ((((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ X) @ B2)) = (((minus_minus @ (set @ A)) @ A2) @ B2)))))).
% 2.90/1.51 thf(fact_255_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A3:A,A2:(set @ A),B3:A]: ((((((insert @ A) @ A3) @ A2) = (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) = ((A3 = B3) & (((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A))))))))).
% 2.90/1.51 thf(tcon_HOL_Obool___Lattices_Obounded__lattice, axiom, ((bounded_lattice @ $o) @ (type2 @ $o))).
% 2.90/1.51 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1, axiom, (! [A7:$tType]: (((bounded_lattice @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.51 thf(tcon_fun___Lattices_Obounded__lattice_2, axiom, (! [A7:$tType,A8:$tType]: ((((bounded_lattice @ A8) @ (type2 @ A8)) => ((bounded_lattice @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.51 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot, axiom, (! [A7:$tType,A8:$tType]: ((((bounded_lattice @ A8) @ (type2 @ A8)) => ((bounde1808546759up_bot @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.51 thf(tcon_fun___Lattices_Obounded__lattice__bot, axiom, (! [A7:$tType,A8:$tType]: ((((bounded_lattice @ A8) @ (type2 @ A8)) => ((bounded_lattice_bot @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.51 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A7:$tType,A8:$tType]: ((((semilattice_sup @ A8) @ (type2 @ A8)) => ((semilattice_sup @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.51 thf(tcon_fun___Lattices_Oboolean__algebra, axiom, (! [A7:$tType,A8:$tType]: ((((boolean_algebra @ A8) @ (type2 @ A8)) => ((boolean_algebra @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.51 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A7:$tType,A8:$tType]: ((((order_bot @ A8) @ (type2 @ A8)) => ((order_bot @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.51 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: ((((preorder @ A8) @ (type2 @ A8)) => ((preorder @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.51 thf(tcon_fun___Lattices_Olattice, axiom, (! [A7:$tType,A8:$tType]: ((((lattice @ A8) @ (type2 @ A8)) => ((lattice @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.51 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: ((((order @ A8) @ (type2 @ A8)) => ((order @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.51 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: ((((ord @ A8) @ (type2 @ A8)) => ((ord @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.51 thf(tcon_fun___Orderings_Obot, axiom, (! [A7:$tType,A8:$tType]: ((((bot @ A8) @ (type2 @ A8)) => ((bot @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.51 thf(tcon_fun___Groups_Ouminus, axiom, (! [A7:$tType,A8:$tType]: ((((uminus @ A8) @ (type2 @ A8)) => ((uminus @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.51 thf(tcon_fun___Groups_Ominus, axiom, (! [A7:$tType,A8:$tType]: ((((minus @ A8) @ (type2 @ A8)) => ((minus @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.90/1.51 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_3, axiom, (! [A7:$tType]: (((bounde1808546759up_bot @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.51 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_4, axiom, (! [A7:$tType]: (((bounded_lattice_bot @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.51 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5, axiom, (! [A7:$tType]: (((semilattice_sup @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.51 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_6, axiom, (! [A7:$tType]: (((boolean_algebra @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.51 thf(tcon_Set_Oset___Orderings_Oorder__bot_7, axiom, (! [A7:$tType]: (((order_bot @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.51 thf(tcon_Set_Oset___Orderings_Opreorder_8, axiom, (! [A7:$tType]: (((preorder @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.51 thf(tcon_Set_Oset___Lattices_Olattice_9, axiom, (! [A7:$tType]: (((lattice @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.51 thf(tcon_Set_Oset___Orderings_Oorder_10, axiom, (! [A7:$tType]: (((order @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.51 thf(tcon_Set_Oset___Orderings_Oord_11, axiom, (! [A7:$tType]: (((ord @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.51 thf(tcon_Set_Oset___Orderings_Obot_12, axiom, (! [A7:$tType]: (((bot @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.51 thf(tcon_Set_Oset___Groups_Ouminus_13, axiom, (! [A7:$tType]: (((uminus @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.51 thf(tcon_Set_Oset___Groups_Ominus_14, axiom, (! [A7:$tType]: (((minus @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.90/1.51 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_15, axiom, ((bounde1808546759up_bot @ $o) @ (type2 @ $o))).
% 2.90/1.51 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_16, axiom, ((bounded_lattice_bot @ $o) @ (type2 @ $o))).
% 2.90/1.51 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_17, axiom, ((semilattice_sup @ $o) @ (type2 @ $o))).
% 2.90/1.51 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_18, axiom, ((boolean_algebra @ $o) @ (type2 @ $o))).
% 2.90/1.51 thf(tcon_HOL_Obool___Orderings_Oorder__bot_19, axiom, ((order_bot @ $o) @ (type2 @ $o))).
% 2.90/1.51 thf(tcon_HOL_Obool___Orderings_Opreorder_20, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.90/1.51 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.90/1.51 thf(tcon_HOL_Obool___Lattices_Olattice_21, axiom, ((lattice @ $o) @ (type2 @ $o))).
% 2.90/1.51 thf(tcon_HOL_Obool___Orderings_Oorder_22, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.90/1.51 thf(tcon_HOL_Obool___Orderings_Oord_23, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.90/1.51 thf(tcon_HOL_Obool___Orderings_Obot_24, axiom, ((bot @ $o) @ (type2 @ $o))).
% 2.90/1.51 thf(tcon_HOL_Obool___Groups_Ouminus_25, axiom, ((uminus @ $o) @ (type2 @ $o))).
% 2.90/1.51 thf(tcon_HOL_Obool___Groups_Ominus_26, axiom, ((minus @ $o) @ (type2 @ $o))).
% 2.90/1.51 thf(conj_0, conjecture, (((gram_L716654942_subtr @ ns) @ tr1) @ tr3)).
% 2.90/1.51 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.xTiNdgqN7t/DTF2DTF9630.p
% 2.90/1.51 ------------------------
% 2.90/1.56 ---- Cleaned THF ---
% 2.90/1.56 thf(ty_t_DTree_Odtree,type,
% 2.90/1.56 dtree: $tType ).
% 2.90/1.56
% 2.90/1.56 thf(ty_t_Set_Oset,type,
% 2.90/1.56 set: $tType > $tType ).
% 2.90/1.56
% 2.90/1.56 thf(ty_t_DTree_OT,type,
% 2.90/1.56 t: $tType ).
% 2.90/1.56
% 2.90/1.56 thf(ty_t_DTree_ON,type,
% 2.90/1.56 n: $tType ).
% 2.90/1.56
% 2.90/1.56 thf(ty_t_itself,type,
% 2.90/1.56 itself: $tType > $tType ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Lattices_Obounded__lattice,type,
% 2.90/1.56 bounded_lattice:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_HOL_Otype,type,
% 2.90/1.56 type:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Groups_Ominus,type,
% 2.90/1.56 minus:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Groups_Ouminus,type,
% 2.90/1.56 uminus:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Orderings_Obot,type,
% 2.90/1.56 bot:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Orderings_Oord,type,
% 2.90/1.56 ord:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Num_Oneg__numeral,type,
% 2.90/1.56 neg_numeral:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Orderings_Oorder,type,
% 2.90/1.56 order:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Groups_Ogroup__add,type,
% 2.90/1.56 group_add:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Lattices_Olattice,type,
% 2.90/1.56 lattice:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Orderings_Olinorder,type,
% 2.90/1.56 linorder:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Orderings_Opreorder,type,
% 2.90/1.56 preorder:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.90/1.56 order_bot:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.90/1.56 linordered_idom:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 2.90/1.56 comm_monoid_diff:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 2.90/1.56 boolean_algebra:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 2.90/1.56 semilattice_sup:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.90/1.56 linordered_semidom:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.90/1.56 ordered_ab_group_add:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Lattices_Obounded__lattice__bot,type,
% 2.90/1.56 bounded_lattice_bot:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.90/1.56 cancel1352612707id_add:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.90/1.56 cancel146912293up_add:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 2.90/1.56 linord219039673up_add:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.90/1.56 linord1659791738miring:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 2.90/1.56 bounde1808546759up_bot:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.90/1.56 canoni770627133id_add:
% 2.90/1.56 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_DTree_Odtree_Oroot,type,
% 2.90/1.56 root: dtree > n ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OFr,type,
% 2.90/1.56 gram_L861583724lle_Fr: ( set @ n ) > dtree > ( set @ t ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr,type,
% 2.90/1.56 gram_L1333338417e_inFr: ( set @ n ) > dtree > t > $o ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr2,type,
% 2.90/1.56 gram_L805317441_inFr2: ( set @ n ) > dtree > t > $o ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinItr,type,
% 2.90/1.56 gram_L830233218_inItr: ( set @ n ) > dtree > n > $o ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Osubtr,type,
% 2.90/1.56 gram_L716654942_subtr: ( set @ n ) > dtree > dtree > $o ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.90/1.56 minus_minus:
% 2.90/1.56 !>[A: $tType] : ( A > A > A ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Groups_Oone__class_Oone,type,
% 2.90/1.56 one_one:
% 2.90/1.56 !>[A: $tType] : A ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 2.90/1.56 uminus_uminus:
% 2.90/1.56 !>[A: $tType] : ( A > A ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.90/1.56 zero_zero:
% 2.90/1.56 !>[A: $tType] : A ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Lattices_Osemilattice__neutr,type,
% 2.90/1.56 semilattice_neutr:
% 2.90/1.56 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Lattices_Osup__class_Osup,type,
% 2.90/1.56 sup_sup:
% 2.90/1.56 !>[A: $tType] : ( A > A > A ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.90/1.56 bot_bot:
% 2.90/1.56 !>[A: $tType] : A ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.90/1.56 ord_less_eq:
% 2.90/1.56 !>[A: $tType] : ( A > A > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 2.90/1.56 order_antimono:
% 2.90/1.56 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 2.90/1.56 order_strict_mono:
% 2.90/1.56 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Pure_Otype,type,
% 2.90/1.56 type2:
% 2.90/1.56 !>[A: $tType] : ( itself @ A ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Set_OCollect,type,
% 2.90/1.56 collect:
% 2.90/1.56 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Set_Oinsert,type,
% 2.90/1.56 insert:
% 2.90/1.56 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Set_Ois__empty,type,
% 2.90/1.56 is_empty:
% 2.90/1.56 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_Set_Opairwise,type,
% 2.90/1.56 pairwise:
% 2.90/1.56 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_c_member,type,
% 2.90/1.56 member:
% 2.90/1.56 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(sy_v_ns,type,
% 2.90/1.56 ns: set @ n ).
% 2.90/1.56
% 2.90/1.56 thf(sy_v_tr1,type,
% 2.90/1.56 tr1: dtree ).
% 2.90/1.56
% 2.90/1.56 thf(sy_v_tr2,type,
% 2.90/1.56 tr2: dtree ).
% 2.90/1.56
% 2.90/1.56 thf(sy_v_tr3,type,
% 2.90/1.56 tr3: dtree ).
% 2.90/1.56
% 2.90/1.56 thf(fact_0__092_060open_062subtr_A_Ins_A_092_060union_062_Ans_J_Atr1_Atr3_092_060close_062,axiom,
% 2.90/1.56 gram_L716654942_subtr @ ( sup_sup @ ( set @ n ) @ ns @ ns ) @ tr1 @ tr3 ).
% 2.90/1.56
% 2.90/1.56 thf(fact_1_assms_I2_J,axiom,
% 2.90/1.56 gram_L716654942_subtr @ ns @ tr2 @ tr3 ).
% 2.90/1.56
% 2.90/1.56 thf(fact_2_assms_I1_J,axiom,
% 2.90/1.56 gram_L716654942_subtr @ ns @ tr1 @ tr2 ).
% 2.90/1.56
% 2.90/1.56 thf(fact_3_subtr__trans__Un,axiom,
% 2.90/1.56 ! [Ns12: set @ n,Tr1: dtree,Tr2: dtree,Ns23: set @ n,Tr3: dtree] :
% 2.90/1.56 ( ( gram_L716654942_subtr @ Ns12 @ Tr1 @ Tr2 )
% 2.90/1.56 => ( ( gram_L716654942_subtr @ Ns23 @ Tr2 @ Tr3 )
% 2.90/1.56 => ( gram_L716654942_subtr @ ( sup_sup @ ( set @ n ) @ Ns12 @ Ns23 ) @ Tr1 @ Tr3 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_4_subtr__mono,axiom,
% 2.90/1.56 ! [Ns: set @ n,Tr1: dtree,Tr2: dtree,Ns2: set @ n] :
% 2.90/1.56 ( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr2 )
% 2.90/1.56 => ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns2 )
% 2.90/1.56 => ( gram_L716654942_subtr @ Ns2 @ Tr1 @ Tr2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_5_Refl,axiom,
% 2.90/1.56 ! [Tr: dtree,Ns: set @ n] :
% 2.90/1.56 ( ( member @ n @ ( root @ Tr ) @ Ns )
% 2.90/1.56 => ( gram_L716654942_subtr @ Ns @ Tr @ Tr ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_6_subtr__rootL__in,axiom,
% 2.90/1.56 ! [Ns: set @ n,Tr1: dtree,Tr2: dtree] :
% 2.90/1.56 ( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr2 )
% 2.90/1.56 => ( member @ n @ ( root @ Tr1 ) @ Ns ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_7_subtr__rootR__in,axiom,
% 2.90/1.56 ! [Ns: set @ n,Tr1: dtree,Tr2: dtree] :
% 2.90/1.56 ( ( gram_L716654942_subtr @ Ns @ Tr1 @ Tr2 )
% 2.90/1.56 => ( member @ n @ ( root @ Tr2 ) @ Ns ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_8_Un__subset__iff,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A,C: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C )
% 2.90/1.56 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ C )
% 2.90/1.56 & ( ord_less_eq @ ( set @ A ) @ B2 @ C ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_9_le__sup__iff,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A,Z: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 2.90/1.56 = ( ( ord_less_eq @ A @ X @ Z )
% 2.90/1.56 & ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_10_sup_Obounded__iff,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [B3: A,C2: A,A3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
% 2.90/1.56 = ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.90/1.56 & ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_11_UnCI,axiom,
% 2.90/1.56 ! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
% 2.90/1.56 ( ( ~ ( member @ A @ C2 @ B2 )
% 2.90/1.56 => ( member @ A @ C2 @ A2 ) )
% 2.90/1.56 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_12_Un__iff,axiom,
% 2.90/1.56 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.90/1.56 = ( ( member @ A @ C2 @ A2 )
% 2.90/1.56 | ( member @ A @ C2 @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_13_sup__apply,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 2.90/1.56 => ( ( sup_sup @ ( A > B ) )
% 2.90/1.56 = ( ^ [F: A > B,G: A > B,X2: A] : ( sup_sup @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_14_sup_Oidem,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] :
% 2.90/1.56 ( ( sup_sup @ A @ A3 @ A3 )
% 2.90/1.56 = A3 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_15_sup__idem,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A] :
% 2.90/1.56 ( ( sup_sup @ A @ X @ X )
% 2.90/1.56 = X ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_16_sup_Oleft__idem,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A] :
% 2.90/1.56 ( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) )
% 2.90/1.56 = ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_17_sup__left__idem,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A] :
% 2.90/1.56 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 2.90/1.56 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_18_sup_Oright__idem,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A] :
% 2.90/1.56 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ B3 )
% 2.90/1.56 = ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_19_subset__antisym,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.90/1.56 => ( A2 = B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_20_subsetI,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ! [X3: A] :
% 2.90/1.56 ( ( member @ A @ X3 @ A2 )
% 2.90/1.56 => ( member @ A @ X3 @ B2 ) )
% 2.90/1.56 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_21_Collect__mono__iff,axiom,
% 2.90/1.56 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.90/1.56 = ( ! [X2: A] :
% 2.90/1.56 ( ( P @ X2 )
% 2.90/1.56 => ( Q @ X2 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_22_contra__subsetD,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 => ( ~ ( member @ A @ C2 @ B2 )
% 2.90/1.56 => ~ ( member @ A @ C2 @ A2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_23_set__eq__subset,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ^ [Y2: set @ A,Z2: set @ A] : ( Y2 = Z2 ) )
% 2.90/1.56 = ( ^ [A4: set @ A,B4: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.90/1.56 & ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_24_subset__trans,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A,C: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C )
% 2.90/1.56 => ( ord_less_eq @ ( set @ A ) @ A2 @ C ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_25_Collect__mono,axiom,
% 2.90/1.56 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.90/1.56 ( ! [X3: A] :
% 2.90/1.56 ( ( P @ X3 )
% 2.90/1.56 => ( Q @ X3 ) )
% 2.90/1.56 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_26_subset__refl,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_27_rev__subsetD,axiom,
% 2.90/1.56 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( member @ A @ C2 @ A2 )
% 2.90/1.56 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_28_subset__iff,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) )
% 2.90/1.56 = ( ^ [A4: set @ A,B4: set @ A] :
% 2.90/1.56 ! [T: A] :
% 2.90/1.56 ( ( member @ A @ T @ A4 )
% 2.90/1.56 => ( member @ A @ T @ B4 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_29_set__rev__mp,axiom,
% 2.90/1.56 ! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( member @ A @ X @ A2 )
% 2.90/1.56 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 => ( member @ A @ X @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_30_equalityD2,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( A2 = B2 )
% 2.90/1.56 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_31_equalityD1,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( A2 = B2 )
% 2.90/1.56 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_32_subset__eq,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) )
% 2.90/1.56 = ( ^ [A4: set @ A,B4: set @ A] :
% 2.90/1.56 ! [X2: A] :
% 2.90/1.56 ( ( member @ A @ X2 @ A4 )
% 2.90/1.56 => ( member @ A @ X2 @ B4 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_33_equalityE,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( A2 = B2 )
% 2.90/1.56 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_34_subsetCE,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 => ( ( member @ A @ C2 @ A2 )
% 2.90/1.56 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_35_subsetD,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 => ( ( member @ A @ C2 @ A2 )
% 2.90/1.56 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_36_in__mono,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 => ( ( member @ A @ X @ A2 )
% 2.90/1.56 => ( member @ A @ X @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_37_set__mp,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 => ( ( member @ A @ X @ A2 )
% 2.90/1.56 => ( member @ A @ X @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_38_sup__left__commute,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A,Z: A] :
% 2.90/1.56 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 2.90/1.56 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_39_sup_Oleft__commute,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [B3: A,A3: A,C2: A] :
% 2.90/1.56 ( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A3 @ C2 ) )
% 2.90/1.56 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_40_sup__commute,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ( sup_sup @ A )
% 2.90/1.56 = ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_41_sup_Ocommute,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ( sup_sup @ A )
% 2.90/1.56 = ( ^ [A5: A,B5: A] : ( sup_sup @ A @ B5 @ A5 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_42_sup__assoc,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A,Z: A] :
% 2.90/1.56 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 2.90/1.56 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_43_sup_Oassoc,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A,C2: A] :
% 2.90/1.56 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ C2 )
% 2.90/1.56 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_44_sup__fun__def,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 2.90/1.56 => ( ( sup_sup @ ( A > B ) )
% 2.90/1.56 = ( ^ [F: A > B,G: A > B,X2: A] : ( sup_sup @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_45_mem__Collect__eq,axiom,
% 2.90/1.56 ! [A: $tType,A3: A,P: A > $o] :
% 2.90/1.56 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 2.90/1.56 = ( P @ A3 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_46_Collect__mem__eq,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A] :
% 2.90/1.56 ( ( collect @ A
% 2.90/1.56 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 2.90/1.56 = A2 ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_47_Collect__cong,axiom,
% 2.90/1.56 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.90/1.56 ( ! [X3: A] :
% 2.90/1.56 ( ( P @ X3 )
% 2.90/1.56 = ( Q @ X3 ) )
% 2.90/1.56 => ( ( collect @ A @ P )
% 2.90/1.56 = ( collect @ A @ Q ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_48_ext,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
% 2.90/1.56 ( ! [X3: A] :
% 2.90/1.56 ( ( F2 @ X3 )
% 2.90/1.56 = ( G2 @ X3 ) )
% 2.90/1.56 => ( F2 = G2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_49_inf__sup__aci_I5_J,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( lattice @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ( sup_sup @ A )
% 2.90/1.56 = ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_50_inf__sup__aci_I6_J,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( lattice @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A,Z: A] :
% 2.90/1.56 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
% 2.90/1.56 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_51_inf__sup__aci_I7_J,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( lattice @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A,Z: A] :
% 2.90/1.56 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
% 2.90/1.56 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_52_inf__sup__aci_I8_J,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( lattice @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A] :
% 2.90/1.56 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 2.90/1.56 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_53_Un__left__commute,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A,C: set @ A] :
% 2.90/1.56 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C ) )
% 2.90/1.56 = ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ C ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_54_Un__left__absorb,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.90/1.56 = ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_55_Un__commute,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( sup_sup @ ( set @ A ) )
% 2.90/1.56 = ( ^ [A4: set @ A,B4: set @ A] : ( sup_sup @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_56_Un__absorb,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A] :
% 2.90/1.56 ( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
% 2.90/1.56 = A2 ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_57_Un__assoc,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A,C: set @ A] :
% 2.90/1.56 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C )
% 2.90/1.56 = ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_58_ball__Un,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
% 2.90/1.56 ( ( ! [X2: A] :
% 2.90/1.56 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.90/1.56 => ( P @ X2 ) ) )
% 2.90/1.56 = ( ! [X2: A] :
% 2.90/1.56 ( ( member @ A @ X2 @ A2 )
% 2.90/1.56 => ( P @ X2 ) )
% 2.90/1.56 & ! [X2: A] :
% 2.90/1.56 ( ( member @ A @ X2 @ B2 )
% 2.90/1.56 => ( P @ X2 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_59_bex__Un,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
% 2.90/1.56 ( ( ? [X2: A] :
% 2.90/1.56 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.90/1.56 & ( P @ X2 ) ) )
% 2.90/1.56 = ( ? [X2: A] :
% 2.90/1.56 ( ( member @ A @ X2 @ A2 )
% 2.90/1.56 & ( P @ X2 ) )
% 2.90/1.56 | ? [X2: A] :
% 2.90/1.56 ( ( member @ A @ X2 @ B2 )
% 2.90/1.56 & ( P @ X2 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_60_UnI2,axiom,
% 2.90/1.56 ! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
% 2.90/1.56 ( ( member @ A @ C2 @ B2 )
% 2.90/1.56 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_61_UnI1,axiom,
% 2.90/1.56 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( member @ A @ C2 @ A2 )
% 2.90/1.56 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_62_UnE,axiom,
% 2.90/1.56 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.90/1.56 => ( ~ ( member @ A @ C2 @ A2 )
% 2.90/1.56 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_63_sup_OcoboundedI2,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [C2: A,B3: A,A3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ C2 @ B3 )
% 2.90/1.56 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_64_sup_OcoboundedI1,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [C2: A,A3: A,B3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ C2 @ A3 )
% 2.90/1.56 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_65_sup_Oabsorb__iff2,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ( ord_less_eq @ A )
% 2.90/1.56 = ( ^ [A5: A,B5: A] :
% 2.90/1.56 ( ( sup_sup @ A @ A5 @ B5 )
% 2.90/1.56 = B5 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_66_sup_Oabsorb__iff1,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ( ord_less_eq @ A )
% 2.90/1.56 = ( ^ [B5: A,A5: A] :
% 2.90/1.56 ( ( sup_sup @ A @ A5 @ B5 )
% 2.90/1.56 = A5 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_67_sup_Ocobounded2,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [B3: A,A3: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_68_sup_Ocobounded1,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_69_sup_Oorder__iff,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ( ord_less_eq @ A )
% 2.90/1.56 = ( ^ [B5: A,A5: A] :
% 2.90/1.56 ( A5
% 2.90/1.56 = ( sup_sup @ A @ A5 @ B5 ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_70_sup_OboundedI,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [B3: A,A3: A,C2: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.90/1.56 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 2.90/1.56 => ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_71_sup_OboundedE,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [B3: A,C2: A,A3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
% 2.90/1.56 => ~ ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.90/1.56 => ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_72_sup__absorb2,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.56 => ( ( sup_sup @ A @ X @ Y )
% 2.90/1.56 = Y ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_73_sup__absorb1,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [Y: A,X: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.56 => ( ( sup_sup @ A @ X @ Y )
% 2.90/1.56 = X ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_74_sup_Oabsorb2,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.56 => ( ( sup_sup @ A @ A3 @ B3 )
% 2.90/1.56 = B3 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_75_sup_Oabsorb1,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [B3: A,A3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.90/1.56 => ( ( sup_sup @ A @ A3 @ B3 )
% 2.90/1.56 = A3 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_76_sup__unique,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [F2: A > A > A,X: A,Y: A] :
% 2.90/1.56 ( ! [X3: A,Y4: A] : ( ord_less_eq @ A @ X3 @ ( F2 @ X3 @ Y4 ) )
% 2.90/1.56 => ( ! [X3: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ ( F2 @ X3 @ Y4 ) )
% 2.90/1.56 => ( ! [X3: A,Y4: A,Z3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ Y4 @ X3 )
% 2.90/1.56 => ( ( ord_less_eq @ A @ Z3 @ X3 )
% 2.90/1.56 => ( ord_less_eq @ A @ ( F2 @ Y4 @ Z3 ) @ X3 ) ) )
% 2.90/1.56 => ( ( sup_sup @ A @ X @ Y )
% 2.90/1.56 = ( F2 @ X @ Y ) ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_77_sup_OorderI,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A] :
% 2.90/1.56 ( ( A3
% 2.90/1.56 = ( sup_sup @ A @ A3 @ B3 ) )
% 2.90/1.56 => ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_78_sup_OorderE,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [B3: A,A3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.90/1.56 => ( A3
% 2.90/1.56 = ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_79_le__iff__sup,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ( ord_less_eq @ A )
% 2.90/1.56 = ( ^ [X2: A,Y3: A] :
% 2.90/1.56 ( ( sup_sup @ A @ X2 @ Y3 )
% 2.90/1.56 = Y3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_80_sup__least,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [Y: A,X: A,Z: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.56 => ( ( ord_less_eq @ A @ Z @ X )
% 2.90/1.56 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z ) @ X ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_81_sup__mono,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,C2: A,B3: A,D: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A3 @ C2 )
% 2.90/1.56 => ( ( ord_less_eq @ A @ B3 @ D )
% 2.90/1.56 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ ( sup_sup @ A @ C2 @ D ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_82_sup_Omono,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [C2: A,A3: A,D: A,B3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ C2 @ A3 )
% 2.90/1.56 => ( ( ord_less_eq @ A @ D @ B3 )
% 2.90/1.56 => ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D ) @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_83_le__supI2,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,B3: A,A3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ X @ B3 )
% 2.90/1.56 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_84_le__supI1,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,A3: A,B3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ X @ A3 )
% 2.90/1.56 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_85_sup__ge2,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_86_sup__ge1,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_87_le__supI,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,X: A,B3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A3 @ X )
% 2.90/1.56 => ( ( ord_less_eq @ A @ B3 @ X )
% 2.90/1.56 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_88_le__supE,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A,X: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X )
% 2.90/1.56 => ~ ( ( ord_less_eq @ A @ A3 @ X )
% 2.90/1.56 => ~ ( ord_less_eq @ A @ B3 @ X ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_89_inf__sup__ord_I3_J,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( lattice @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_90_inf__sup__ord_I4_J,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( lattice @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_91_subset__Un__eq,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) )
% 2.90/1.56 = ( ^ [A4: set @ A,B4: set @ A] :
% 2.90/1.56 ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
% 2.90/1.56 = B4 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_92_Un__absorb2,axiom,
% 2.90/1.56 ! [A: $tType,B2: set @ A,A2: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.90/1.56 => ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 = A2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_93_Un__absorb1,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 => ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 = B2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_94_Un__upper2,axiom,
% 2.90/1.56 ! [A: $tType,B2: set @ A,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_95_Un__upper1,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_96_Un__least,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,C: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C )
% 2.90/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C )
% 2.90/1.56 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_97_Un__mono,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,C: set @ A,B2: set @ A,D2: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C )
% 2.90/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ D2 )
% 2.90/1.56 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C @ D2 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_98_order__refl,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( preorder @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_99_dual__order_Oantisym,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [B3: A,A3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.90/1.56 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.56 => ( A3 = B3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_100_dual__order_Otrans,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [B3: A,A3: A,C2: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.90/1.56 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 2.90/1.56 => ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_101_linorder__wlog,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( linorder @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [P: A > A > $o,A3: A,B3: A] :
% 2.90/1.56 ( ! [A6: A,B6: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A6 @ B6 )
% 2.90/1.56 => ( P @ A6 @ B6 ) )
% 2.90/1.56 => ( ! [A6: A,B6: A] :
% 2.90/1.56 ( ( P @ B6 @ A6 )
% 2.90/1.56 => ( P @ A6 @ B6 ) )
% 2.90/1.56 => ( P @ A3 @ B3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_102_dual__order_Orefl,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_103_order__trans,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( preorder @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A,Z: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.56 => ( ( ord_less_eq @ A @ Y @ Z )
% 2.90/1.56 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_104_order__class_Oorder_Oantisym,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.56 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.90/1.56 => ( A3 = B3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_105_ord__le__eq__trans,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ord @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A,C2: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.56 => ( ( B3 = C2 )
% 2.90/1.56 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_106_ord__eq__le__trans,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ord @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A,C2: A] :
% 2.90/1.56 ( ( A3 = B3 )
% 2.90/1.56 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 2.90/1.56 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_107_antisym__conv,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [Y: A,X: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.56 => ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.56 = ( X = Y ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_108_le__funD,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.56 => ! [F2: A > B,G2: A > B,X: A] :
% 2.90/1.56 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 2.90/1.56 => ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_109_le__funE,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.56 => ! [F2: A > B,G2: A > B,X: A] :
% 2.90/1.56 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 2.90/1.56 => ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_110_le__funI,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.56 => ! [F2: A > B,G2: A > B] :
% 2.90/1.56 ( ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 2.90/1.56 => ( ord_less_eq @ ( A > B ) @ F2 @ G2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_111_le__fun__def,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.56 => ( ( ord_less_eq @ ( A > B ) )
% 2.90/1.56 = ( ^ [F: A > B,G: A > B] :
% 2.90/1.56 ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_112_order__subst1,axiom,
% 2.90/1.56 ! [A: $tType,B: $tType] :
% 2.90/1.56 ( ( ( order @ B @ ( type2 @ B ) )
% 2.90/1.56 & ( order @ A @ ( type2 @ A ) ) )
% 2.90/1.56 => ! [A3: A,F2: B > A,B3: B,C2: B] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A3 @ ( F2 @ B3 ) )
% 2.90/1.56 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 2.90/1.56 => ( ! [X3: B,Y4: B] :
% 2.90/1.56 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 2.90/1.56 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
% 2.90/1.56 => ( ord_less_eq @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_113_order__subst2,axiom,
% 2.90/1.56 ! [A: $tType,C3: $tType] :
% 2.90/1.56 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 2.90/1.56 & ( order @ A @ ( type2 @ A ) ) )
% 2.90/1.56 => ! [A3: A,B3: A,F2: A > C3,C2: C3] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.56 => ( ( ord_less_eq @ C3 @ ( F2 @ B3 ) @ C2 )
% 2.90/1.56 => ( ! [X3: A,Y4: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 2.90/1.56 => ( ord_less_eq @ C3 @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
% 2.90/1.56 => ( ord_less_eq @ C3 @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_114_ord__eq__le__subst,axiom,
% 2.90/1.56 ! [A: $tType,B: $tType] :
% 2.90/1.56 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.56 & ( ord @ A @ ( type2 @ A ) ) )
% 2.90/1.56 => ! [A3: A,F2: B > A,B3: B,C2: B] :
% 2.90/1.56 ( ( A3
% 2.90/1.56 = ( F2 @ B3 ) )
% 2.90/1.56 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 2.90/1.56 => ( ! [X3: B,Y4: B] :
% 2.90/1.56 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 2.90/1.56 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
% 2.90/1.56 => ( ord_less_eq @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_115_ord__le__eq__subst,axiom,
% 2.90/1.56 ! [A: $tType,B: $tType] :
% 2.90/1.56 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.56 & ( ord @ A @ ( type2 @ A ) ) )
% 2.90/1.56 => ! [A3: A,B3: A,F2: A > B,C2: B] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.56 => ( ( ( F2 @ B3 )
% 2.90/1.56 = C2 )
% 2.90/1.56 => ( ! [X3: A,Y4: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 2.90/1.56 => ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
% 2.90/1.56 => ( ord_less_eq @ B @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_116_eq__iff,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 2.90/1.56 = ( ^ [X2: A,Y3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.90/1.56 & ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_117_antisym,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.56 => ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.56 => ( X = Y ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_118_linear,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( linorder @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.56 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_119_eq__refl,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( preorder @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A] :
% 2.90/1.56 ( ( X = Y )
% 2.90/1.56 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_120_le__cases,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( linorder @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A] :
% 2.90/1.56 ( ~ ( ord_less_eq @ A @ X @ Y )
% 2.90/1.56 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_121_order_Otrans,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A,C2: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.56 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 2.90/1.56 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_122_le__cases3,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( linorder @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A,Z: A] :
% 2.90/1.56 ( ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.56 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 2.90/1.56 => ( ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.56 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 2.90/1.56 => ( ( ( ord_less_eq @ A @ X @ Z )
% 2.90/1.56 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 2.90/1.56 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 2.90/1.56 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 2.90/1.56 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 2.90/1.56 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 2.90/1.56 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 2.90/1.56 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_123_inItr__mono,axiom,
% 2.90/1.56 ! [Ns: set @ n,Tr: dtree,N: n,Ns2: set @ n] :
% 2.90/1.56 ( ( gram_L830233218_inItr @ Ns @ Tr @ N )
% 2.90/1.56 => ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns2 )
% 2.90/1.56 => ( gram_L830233218_inItr @ Ns2 @ Tr @ N ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_124_inFr2__mono,axiom,
% 2.90/1.56 ! [Ns: set @ n,Tr: dtree,T2: t,Ns2: set @ n] :
% 2.90/1.56 ( ( gram_L805317441_inFr2 @ Ns @ Tr @ T2 )
% 2.90/1.56 => ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns2 )
% 2.90/1.56 => ( gram_L805317441_inFr2 @ Ns2 @ Tr @ T2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_125_inFr__mono,axiom,
% 2.90/1.56 ! [Ns: set @ n,Tr: dtree,T2: t,Ns2: set @ n] :
% 2.90/1.56 ( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
% 2.90/1.56 => ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns2 )
% 2.90/1.56 => ( gram_L1333338417e_inFr @ Ns2 @ Tr @ T2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_126_antimonoD,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.56 & ( order @ B @ ( type2 @ B ) ) )
% 2.90/1.56 => ! [F2: A > B,X: A,Y: A] :
% 2.90/1.56 ( ( order_antimono @ A @ B @ F2 )
% 2.90/1.56 => ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.56 => ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_127_antimonoE,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.56 & ( order @ B @ ( type2 @ B ) ) )
% 2.90/1.56 => ! [F2: A > B,X: A,Y: A] :
% 2.90/1.56 ( ( order_antimono @ A @ B @ F2 )
% 2.90/1.56 => ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.56 => ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_128_inFr__inFr2,axiom,
% 2.90/1.56 gram_L1333338417e_inFr = gram_L805317441_inFr2 ).
% 2.90/1.56
% 2.90/1.56 thf(fact_129_not__root__inFr,axiom,
% 2.90/1.56 ! [Tr: dtree,Ns: set @ n,T2: t] :
% 2.90/1.56 ( ~ ( member @ n @ ( root @ Tr ) @ Ns )
% 2.90/1.56 => ~ ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_130_inFr__root__in,axiom,
% 2.90/1.56 ! [Ns: set @ n,Tr: dtree,T2: t] :
% 2.90/1.56 ( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
% 2.90/1.56 => ( member @ n @ ( root @ Tr ) @ Ns ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_131_inFr2__root__in,axiom,
% 2.90/1.56 ! [Ns: set @ n,Tr: dtree,T2: t] :
% 2.90/1.56 ( ( gram_L805317441_inFr2 @ Ns @ Tr @ T2 )
% 2.90/1.56 => ( member @ n @ ( root @ Tr ) @ Ns ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_132_inItr__root__in,axiom,
% 2.90/1.56 ! [Ns: set @ n,Tr: dtree,N: n] :
% 2.90/1.56 ( ( gram_L830233218_inItr @ Ns @ Tr @ N )
% 2.90/1.56 => ( member @ n @ ( root @ Tr ) @ Ns ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_133_inItr_OBase,axiom,
% 2.90/1.56 ! [Tr: dtree,Ns: set @ n] :
% 2.90/1.56 ( ( member @ n @ ( root @ Tr ) @ Ns )
% 2.90/1.56 => ( gram_L830233218_inItr @ Ns @ Tr @ ( root @ Tr ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_134_antimono__def,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.56 & ( order @ B @ ( type2 @ B ) ) )
% 2.90/1.56 => ( ( order_antimono @ A @ B )
% 2.90/1.56 = ( ^ [F: A > B] :
% 2.90/1.56 ! [X2: A,Y3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.90/1.56 => ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_135_antimonoI,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.56 & ( order @ B @ ( type2 @ B ) ) )
% 2.90/1.56 => ! [F2: A > B] :
% 2.90/1.56 ( ! [X3: A,Y4: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 2.90/1.56 => ( ord_less_eq @ B @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) )
% 2.90/1.56 => ( order_antimono @ A @ B @ F2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_136_strict__mono__less__eq,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( ( linorder @ A @ ( type2 @ A ) )
% 2.90/1.56 & ( order @ B @ ( type2 @ B ) ) )
% 2.90/1.56 => ! [F2: A > B,X: A,Y: A] :
% 2.90/1.56 ( ( order_strict_mono @ A @ B @ F2 )
% 2.90/1.56 => ( ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y ) )
% 2.90/1.56 = ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_137_pairwise__subset,axiom,
% 2.90/1.56 ! [A: $tType,P: A > A > $o,S: set @ A,T3: set @ A] :
% 2.90/1.56 ( ( pairwise @ A @ P @ S )
% 2.90/1.56 => ( ( ord_less_eq @ ( set @ A ) @ T3 @ S )
% 2.90/1.56 => ( pairwise @ A @ P @ T3 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_138_Diff__subset__conv,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A,C: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ C )
% 2.90/1.56 = ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_139_Diff__partition,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 => ( ( sup_sup @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) )
% 2.90/1.56 = B2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_140_minus__apply,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( minus @ B @ ( type2 @ B ) )
% 2.90/1.56 => ( ( minus_minus @ ( A > B ) )
% 2.90/1.56 = ( ^ [A4: A > B,B4: A > B,X2: A] : ( minus_minus @ B @ ( A4 @ X2 ) @ ( B4 @ X2 ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_141_Diff__idemp,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ B2 )
% 2.90/1.56 = ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_142_Diff__iff,axiom,
% 2.90/1.56 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 2.90/1.56 = ( ( member @ A @ C2 @ A2 )
% 2.90/1.56 & ~ ( member @ A @ C2 @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_143_DiffI,axiom,
% 2.90/1.56 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( member @ A @ C2 @ A2 )
% 2.90/1.56 => ( ~ ( member @ A @ C2 @ B2 )
% 2.90/1.56 => ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_144_Un__Diff__cancel2,axiom,
% 2.90/1.56 ! [A: $tType,B2: set @ A,A2: set @ A] :
% 2.90/1.56 ( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) @ A2 )
% 2.90/1.56 = ( sup_sup @ ( set @ A ) @ B2 @ A2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_145_Un__Diff__cancel,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( sup_sup @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B2 @ A2 ) )
% 2.90/1.56 = ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_146_Diff__mono,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,C: set @ A,D2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C )
% 2.90/1.56 => ( ( ord_less_eq @ ( set @ A ) @ D2 @ B2 )
% 2.90/1.56 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( set @ A ) @ C @ D2 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_147_Diff__subset,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) @ A2 ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_148_double__diff,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A,C: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C )
% 2.90/1.56 => ( ( minus_minus @ ( set @ A ) @ B2 @ ( minus_minus @ ( set @ A ) @ C @ A2 ) )
% 2.90/1.56 = A2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_149_fun__diff__def,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( minus @ B @ ( type2 @ B ) )
% 2.90/1.56 => ( ( minus_minus @ ( A > B ) )
% 2.90/1.56 = ( ^ [A4: A > B,B4: A > B,X2: A] : ( minus_minus @ B @ ( A4 @ X2 ) @ ( B4 @ X2 ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_150_pairwise__def,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( pairwise @ A )
% 2.90/1.56 = ( ^ [R: A > A > $o,S2: set @ A] :
% 2.90/1.56 ! [X2: A] :
% 2.90/1.56 ( ( member @ A @ X2 @ S2 )
% 2.90/1.56 => ! [Y3: A] :
% 2.90/1.56 ( ( member @ A @ Y3 @ S2 )
% 2.90/1.56 => ( ( X2 != Y3 )
% 2.90/1.56 => ( R @ X2 @ Y3 ) ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_151_DiffD2,axiom,
% 2.90/1.56 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 2.90/1.56 => ~ ( member @ A @ C2 @ B2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_152_DiffD1,axiom,
% 2.90/1.56 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 2.90/1.56 => ( member @ A @ C2 @ A2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_153_DiffE,axiom,
% 2.90/1.56 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 2.90/1.56 => ~ ( ( member @ A @ C2 @ A2 )
% 2.90/1.56 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_154_Un__Diff,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A,C: set @ A] :
% 2.90/1.56 ( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C )
% 2.90/1.56 = ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ C ) @ ( minus_minus @ ( set @ A ) @ B2 @ C ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_155_strict__mono__eq,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( ( linorder @ A @ ( type2 @ A ) )
% 2.90/1.56 & ( order @ B @ ( type2 @ B ) ) )
% 2.90/1.56 => ! [F2: A > B,X: A,Y: A] :
% 2.90/1.56 ( ( order_strict_mono @ A @ B @ F2 )
% 2.90/1.56 => ( ( ( F2 @ X )
% 2.90/1.56 = ( F2 @ Y ) )
% 2.90/1.56 = ( X = Y ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_156_diff__mono,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A,D: A,C2: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.56 => ( ( ord_less_eq @ A @ D @ C2 )
% 2.90/1.56 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_157_diff__left__mono,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [B3: A,A3: A,C2: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.90/1.56 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_158_diff__right__mono,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A,C2: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.56 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B3 @ C2 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_159_diff__eq__diff__less__eq,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A,C2: A,D: A] :
% 2.90/1.56 ( ( ( minus_minus @ A @ A3 @ B3 )
% 2.90/1.56 = ( minus_minus @ A @ C2 @ D ) )
% 2.90/1.56 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.56 = ( ord_less_eq @ A @ C2 @ D ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_160_diff__right__commute,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( cancel146912293up_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,C2: A,B3: A] :
% 2.90/1.56 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B3 )
% 2.90/1.56 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_161_diff__eq__diff__eq,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A,C2: A,D: A] :
% 2.90/1.56 ( ( ( minus_minus @ A @ A3 @ B3 )
% 2.90/1.56 = ( minus_minus @ A @ C2 @ D ) )
% 2.90/1.56 => ( ( A3 = B3 )
% 2.90/1.56 = ( C2 = D ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_162_Diff__eq__empty__iff,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( ( minus_minus @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.90/1.56 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_163_diff__ge__0__iff__ge,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B3 ) )
% 2.90/1.56 = ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_164_Compl__Diff__eq,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B2 ) )
% 2.90/1.56 = ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ B2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_165_bot__apply,axiom,
% 2.90/1.56 ! [C3: $tType,D3: $tType] :
% 2.90/1.56 ( ( bot @ C3 @ ( type2 @ C3 ) )
% 2.90/1.56 => ( ( bot_bot @ ( D3 > C3 ) )
% 2.90/1.56 = ( ^ [X2: D3] : ( bot_bot @ C3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_166_empty__Collect__eq,axiom,
% 2.90/1.56 ! [A: $tType,P: A > $o] :
% 2.90/1.56 ( ( ( bot_bot @ ( set @ A ) )
% 2.90/1.56 = ( collect @ A @ P ) )
% 2.90/1.56 = ( ! [X2: A] :
% 2.90/1.56 ~ ( P @ X2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_167_Collect__empty__eq,axiom,
% 2.90/1.56 ! [A: $tType,P: A > $o] :
% 2.90/1.56 ( ( ( collect @ A @ P )
% 2.90/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.90/1.56 = ( ! [X2: A] :
% 2.90/1.56 ~ ( P @ X2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_168_all__not__in__conv,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A] :
% 2.90/1.56 ( ( ! [X2: A] :
% 2.90/1.56 ~ ( member @ A @ X2 @ A2 ) )
% 2.90/1.56 = ( A2
% 2.90/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_169_empty__iff,axiom,
% 2.90/1.56 ! [A: $tType,C2: A] :
% 2.90/1.56 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_170_compl__eq__compl__iff,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A] :
% 2.90/1.56 ( ( ( uminus_uminus @ A @ X )
% 2.90/1.56 = ( uminus_uminus @ A @ Y ) )
% 2.90/1.56 = ( X = Y ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_171_double__compl,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A] :
% 2.90/1.56 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
% 2.90/1.56 = X ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_172_uminus__apply,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( uminus @ B @ ( type2 @ B ) )
% 2.90/1.56 => ( ( uminus_uminus @ ( A > B ) )
% 2.90/1.56 = ( ^ [A4: A > B,X2: A] : ( uminus_uminus @ B @ ( A4 @ X2 ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_173_Compl__eq__Compl__iff,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( ( uminus_uminus @ ( set @ A ) @ A2 )
% 2.90/1.56 = ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 2.90/1.56 = ( A2 = B2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_174_Compl__iff,axiom,
% 2.90/1.56 ! [A: $tType,C2: A,A2: set @ A] :
% 2.90/1.56 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.90/1.56 = ( ~ ( member @ A @ C2 @ A2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_175_ComplI,axiom,
% 2.90/1.56 ! [A: $tType,C2: A,A2: set @ A] :
% 2.90/1.56 ( ~ ( member @ A @ C2 @ A2 )
% 2.90/1.56 => ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_176_le__zero__eq,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [N: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.90/1.56 = ( N
% 2.90/1.56 = ( zero_zero @ A ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_177_diff__self,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] :
% 2.90/1.56 ( ( minus_minus @ A @ A3 @ A3 )
% 2.90/1.56 = ( zero_zero @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_178_diff__0__right,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] :
% 2.90/1.56 ( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
% 2.90/1.56 = A3 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_179_zero__diff,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( comm_monoid_diff @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] :
% 2.90/1.56 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
% 2.90/1.56 = ( zero_zero @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_180_diff__zero,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] :
% 2.90/1.56 ( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
% 2.90/1.56 = A3 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_181_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] :
% 2.90/1.56 ( ( minus_minus @ A @ A3 @ A3 )
% 2.90/1.56 = ( zero_zero @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_182_neg__le__iff__le,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [B3: A,A3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) )
% 2.90/1.56 = ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_183_compl__le__compl__iff,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
% 2.90/1.56 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_184_sup__bot_Oright__neutral,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] :
% 2.90/1.56 ( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
% 2.90/1.56 = A3 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_185_sup__bot_Oleft__neutral,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] :
% 2.90/1.56 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
% 2.90/1.56 = A3 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_186_sup__eq__bot__iff,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A] :
% 2.90/1.56 ( ( ( sup_sup @ A @ X @ Y )
% 2.90/1.56 = ( bot_bot @ A ) )
% 2.90/1.56 = ( ( X
% 2.90/1.56 = ( bot_bot @ A ) )
% 2.90/1.56 & ( Y
% 2.90/1.56 = ( bot_bot @ A ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_187_bot__eq__sup__iff,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A] :
% 2.90/1.56 ( ( ( bot_bot @ A )
% 2.90/1.56 = ( sup_sup @ A @ X @ Y ) )
% 2.90/1.56 = ( ( X
% 2.90/1.56 = ( bot_bot @ A ) )
% 2.90/1.56 & ( Y
% 2.90/1.56 = ( bot_bot @ A ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_188_minus__diff__eq,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A] :
% 2.90/1.56 ( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B3 ) )
% 2.90/1.56 = ( minus_minus @ A @ B3 @ A3 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_189_empty__subsetI,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_190_subset__empty,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.90/1.56 = ( A2
% 2.90/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_191_Un__empty,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.90/1.56 = ( ( A2
% 2.90/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.90/1.56 & ( B2
% 2.90/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_192_Diff__cancel,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A] :
% 2.90/1.56 ( ( minus_minus @ ( set @ A ) @ A2 @ A2 )
% 2.90/1.56 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_193_empty__Diff,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A] :
% 2.90/1.56 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 )
% 2.90/1.56 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_194_Diff__empty,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A] :
% 2.90/1.56 ( ( minus_minus @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.90/1.56 = A2 ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_195_Compl__subset__Compl__iff,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 2.90/1.56 = ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_196_Compl__anti__mono,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.90/1.56 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_197_neg__0__le__iff__le,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
% 2.90/1.56 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_198_neg__le__0__iff__le,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
% 2.90/1.56 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_199_less__eq__neg__nonpos,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
% 2.90/1.56 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_200_neg__less__eq__nonneg,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 2.90/1.56 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_201_diff__0,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] :
% 2.90/1.56 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
% 2.90/1.56 = ( uminus_uminus @ A @ A3 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_202_eq__iff__diff__eq__0,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 2.90/1.56 = ( ^ [A5: A,B5: A] :
% 2.90/1.56 ( ( minus_minus @ A @ A5 @ B5 )
% 2.90/1.56 = ( zero_zero @ A ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_203_zero__le,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_204_le__imp__neg__le,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.56 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_205_minus__le__iff,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
% 2.90/1.56 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ A3 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_206_le__minus__iff,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A,B3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
% 2.90/1.56 = ( ord_less_eq @ A @ B3 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_207_pairwise__empty,axiom,
% 2.90/1.56 ! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_208_subset__Compl__self__eq,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.90/1.56 = ( A2
% 2.90/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_209_bot_Oextremum,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_210_bot_Oextremum__unique,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 2.90/1.56 = ( A3
% 2.90/1.56 = ( bot_bot @ A ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_211_bot_Oextremum__uniqueI,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [A3: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 2.90/1.56 => ( A3
% 2.90/1.56 = ( bot_bot @ A ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_212_compl__mono,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A,Y: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.56 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_213_compl__le__swap1,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [Y: A,X: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
% 2.90/1.56 => ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_214_compl__le__swap2,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [Y: A,X: A] :
% 2.90/1.56 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
% 2.90/1.56 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_215_Un__empty__right,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A] :
% 2.90/1.56 ( ( sup_sup @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.90/1.56 = A2 ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_216_Un__empty__left,axiom,
% 2.90/1.56 ! [A: $tType,B2: set @ A] :
% 2.90/1.56 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
% 2.90/1.56 = B2 ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_217_fun__Compl__def,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( uminus @ B @ ( type2 @ B ) )
% 2.90/1.56 => ( ( uminus_uminus @ ( A > B ) )
% 2.90/1.56 = ( ^ [A4: A > B,X2: A] : ( uminus_uminus @ B @ ( A4 @ X2 ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_218_double__complement,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A] :
% 2.90/1.56 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.90/1.56 = A2 ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_219_ex__in__conv,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A] :
% 2.90/1.56 ( ( ? [X2: A] : ( member @ A @ X2 @ A2 ) )
% 2.90/1.56 = ( A2
% 2.90/1.56 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_220_equals0I,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A] :
% 2.90/1.56 ( ! [Y4: A] :
% 2.90/1.56 ~ ( member @ A @ Y4 @ A2 )
% 2.90/1.56 => ( A2
% 2.90/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_221_equals0D,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,A3: A] :
% 2.90/1.56 ( ( A2
% 2.90/1.56 = ( bot_bot @ ( set @ A ) ) )
% 2.90/1.56 => ~ ( member @ A @ A3 @ A2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_222_emptyE,axiom,
% 2.90/1.56 ! [A: $tType,A3: A] :
% 2.90/1.56 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_223_ComplD,axiom,
% 2.90/1.56 ! [A: $tType,C2: A,A2: set @ A] :
% 2.90/1.56 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.90/1.56 => ~ ( member @ A @ C2 @ A2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_224_sup__bot__right,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A] :
% 2.90/1.56 ( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
% 2.90/1.56 = X ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_225_sup__bot__left,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 2.90/1.56 => ! [X: A] :
% 2.90/1.56 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
% 2.90/1.56 = X ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_226_bot__fun__def,axiom,
% 2.90/1.56 ! [B: $tType,A: $tType] :
% 2.90/1.56 ( ( bot @ B @ ( type2 @ B ) )
% 2.90/1.56 => ( ( bot_bot @ ( A > B ) )
% 2.90/1.56 = ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_227_le__iff__diff__le__0,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ( ord_less_eq @ A )
% 2.90/1.56 = ( ^ [A5: A,B5: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_228_subset__emptyI,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A] :
% 2.90/1.56 ( ! [X3: A] :
% 2.90/1.56 ~ ( member @ A @ X3 @ A2 )
% 2.90/1.56 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_229_le__numeral__extra_I3_J,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_230_not__root__Fr,axiom,
% 2.90/1.56 ! [Tr: dtree,Ns: set @ n] :
% 2.90/1.56 ( ~ ( member @ n @ ( root @ Tr ) @ Ns )
% 2.90/1.56 => ( ( gram_L861583724lle_Fr @ Ns @ Tr )
% 2.90/1.56 = ( bot_bot @ ( set @ t ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_231_bot__set__def,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( bot_bot @ ( set @ A ) )
% 2.90/1.56 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_232_Set_Ois__empty__def,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( is_empty @ A )
% 2.90/1.56 = ( ^ [A4: set @ A] :
% 2.90/1.56 ( A4
% 2.90/1.56 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_233_diff__numeral__special_I12_J,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( neg_numeral @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 2.90/1.56 = ( zero_zero @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_234_diff__numeral__special_I9_J,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( neg_numeral @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 2.90/1.56 = ( zero_zero @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_235_le__numeral__extra_I1_J,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_236_le__numeral__extra_I2_J,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.90/1.56 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_237_le__minus__one__simps_I2_J,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_238_le__minus__one__simps_I4_J,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.90/1.56 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_239_le__numeral__extra_I4_J,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_240_le__minus__one__simps_I1_J,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_241_le__minus__one__simps_I3_J,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.90/1.56 => ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_242_zero__le__one,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( linord1659791738miring @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_243_not__one__le__zero,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( linord1659791738miring @ A @ ( type2 @ A ) )
% 2.90/1.56 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_244_subset__Compl__singleton,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,B3: A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.90/1.56 = ( ~ ( member @ A @ B3 @ A2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_245_sup__bot_Osemilattice__neutr__axioms,axiom,
% 2.90/1.56 ! [A: $tType] :
% 2.90/1.56 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 2.90/1.56 => ( semilattice_neutr @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_246_insert__absorb2,axiom,
% 2.90/1.56 ! [A: $tType,X: A,A2: set @ A] :
% 2.90/1.56 ( ( insert @ A @ X @ ( insert @ A @ X @ A2 ) )
% 2.90/1.56 = ( insert @ A @ X @ A2 ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_247_insert__iff,axiom,
% 2.90/1.56 ! [A: $tType,A3: A,B3: A,A2: set @ A] :
% 2.90/1.56 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
% 2.90/1.56 = ( ( A3 = B3 )
% 2.90/1.56 | ( member @ A @ A3 @ A2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_248_insertCI,axiom,
% 2.90/1.56 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 2.90/1.56 ( ( ~ ( member @ A @ A3 @ B2 )
% 2.90/1.56 => ( A3 = B3 ) )
% 2.90/1.56 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_249_singletonI,axiom,
% 2.90/1.56 ! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_250_insert__subset,axiom,
% 2.90/1.56 ! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
% 2.90/1.56 = ( ( member @ A @ X @ B2 )
% 2.90/1.56 & ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_251_Un__insert__right,axiom,
% 2.90/1.56 ! [A: $tType,A2: set @ A,A3: A,B2: set @ A] :
% 2.90/1.56 ( ( sup_sup @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B2 ) )
% 2.90/1.56 = ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_252_Un__insert__left,axiom,
% 2.90/1.56 ! [A: $tType,A3: A,B2: set @ A,C: set @ A] :
% 2.90/1.56 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ B2 ) @ C )
% 2.90/1.56 = ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_253_insert__Diff1,axiom,
% 2.90/1.56 ! [A: $tType,X: A,B2: set @ A,A2: set @ A] :
% 2.90/1.56 ( ( member @ A @ X @ B2 )
% 2.90/1.56 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B2 )
% 2.90/1.56 = ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_254_Diff__insert0,axiom,
% 2.90/1.56 ! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
% 2.90/1.56 ( ~ ( member @ A @ X @ A2 )
% 2.90/1.56 => ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B2 ) )
% 2.90/1.56 = ( minus_minus @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(fact_255_singleton__insert__inj__eq_H,axiom,
% 2.90/1.56 ! [A: $tType,A3: A,A2: set @ A,B3: A] :
% 2.90/1.56 ( ( ( insert @ A @ A3 @ A2 )
% 2.90/1.56 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.90/1.56 = ( ( A3 = B3 )
% 2.90/1.56 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 2.90/1.56 bounded_lattice @ $o @ ( type2 @ $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 2.90/1.56 ! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 2.90/1.56 ! [A7: $tType,A8: $tType] :
% 2.90/1.56 ( ( bounded_lattice @ A8 @ ( type2 @ A8 ) )
% 2.90/1.56 => ( bounded_lattice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 2.90/1.56 ! [A7: $tType,A8: $tType] :
% 2.90/1.56 ( ( bounded_lattice @ A8 @ ( type2 @ A8 ) )
% 2.90/1.56 => ( bounde1808546759up_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
% 2.90/1.56 ! [A7: $tType,A8: $tType] :
% 2.90/1.56 ( ( bounded_lattice @ A8 @ ( type2 @ A8 ) )
% 2.90/1.56 => ( bounded_lattice_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 2.90/1.56 ! [A7: $tType,A8: $tType] :
% 2.90/1.56 ( ( semilattice_sup @ A8 @ ( type2 @ A8 ) )
% 2.90/1.56 => ( semilattice_sup @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 2.90/1.56 ! [A7: $tType,A8: $tType] :
% 2.90/1.56 ( ( boolean_algebra @ A8 @ ( type2 @ A8 ) )
% 2.90/1.56 => ( boolean_algebra @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.90/1.56 ! [A7: $tType,A8: $tType] :
% 2.90/1.56 ( ( order_bot @ A8 @ ( type2 @ A8 ) )
% 2.90/1.56 => ( order_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.90/1.56 ! [A7: $tType,A8: $tType] :
% 2.90/1.56 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 2.90/1.56 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_fun___Lattices_Olattice,axiom,
% 2.90/1.56 ! [A7: $tType,A8: $tType] :
% 2.90/1.56 ( ( lattice @ A8 @ ( type2 @ A8 ) )
% 2.90/1.56 => ( lattice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.90/1.56 ! [A7: $tType,A8: $tType] :
% 2.90/1.56 ( ( order @ A8 @ ( type2 @ A8 ) )
% 2.90/1.56 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_fun___Orderings_Oord,axiom,
% 2.90/1.56 ! [A7: $tType,A8: $tType] :
% 2.90/1.56 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 2.90/1.56 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_fun___Orderings_Obot,axiom,
% 2.90/1.56 ! [A7: $tType,A8: $tType] :
% 2.90/1.56 ( ( bot @ A8 @ ( type2 @ A8 ) )
% 2.90/1.56 => ( bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_fun___Groups_Ouminus,axiom,
% 2.90/1.56 ! [A7: $tType,A8: $tType] :
% 2.90/1.56 ( ( uminus @ A8 @ ( type2 @ A8 ) )
% 2.90/1.56 => ( uminus @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_fun___Groups_Ominus,axiom,
% 2.90/1.56 ! [A7: $tType,A8: $tType] :
% 2.90/1.56 ( ( minus @ A8 @ ( type2 @ A8 ) )
% 2.90/1.56 => ( minus @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_3,axiom,
% 2.90/1.56 ! [A7: $tType] : ( bounde1808546759up_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_4,axiom,
% 2.90/1.56 ! [A7: $tType] : ( bounded_lattice_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5,axiom,
% 2.90/1.56 ! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_6,axiom,
% 2.90/1.56 ! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_Set_Oset___Orderings_Oorder__bot_7,axiom,
% 2.90/1.56 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_Set_Oset___Orderings_Opreorder_8,axiom,
% 2.90/1.56 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_Set_Oset___Lattices_Olattice_9,axiom,
% 2.90/1.56 ! [A7: $tType] : ( lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_Set_Oset___Orderings_Oorder_10,axiom,
% 2.90/1.56 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_Set_Oset___Orderings_Oord_11,axiom,
% 2.90/1.56 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_Set_Oset___Orderings_Obot_12,axiom,
% 2.90/1.56 ! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_Set_Oset___Groups_Ouminus_13,axiom,
% 2.90/1.56 ! [A7: $tType] : ( uminus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_Set_Oset___Groups_Ominus_14,axiom,
% 2.90/1.56 ! [A7: $tType] : ( minus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_15,axiom,
% 2.90/1.56 bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_16,axiom,
% 2.90/1.56 bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_17,axiom,
% 2.90/1.56 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_18,axiom,
% 2.90/1.56 boolean_algebra @ $o @ ( type2 @ $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_HOL_Obool___Orderings_Oorder__bot_19,axiom,
% 2.90/1.56 order_bot @ $o @ ( type2 @ $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_HOL_Obool___Orderings_Opreorder_20,axiom,
% 2.90/1.56 preorder @ $o @ ( type2 @ $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.90/1.56 linorder @ $o @ ( type2 @ $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_HOL_Obool___Lattices_Olattice_21,axiom,
% 2.90/1.56 lattice @ $o @ ( type2 @ $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_HOL_Obool___Orderings_Oorder_22,axiom,
% 2.90/1.56 order @ $o @ ( type2 @ $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_HOL_Obool___Orderings_Oord_23,axiom,
% 2.90/1.56 ord @ $o @ ( type2 @ $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_HOL_Obool___Orderings_Obot_24,axiom,
% 2.90/1.56 bot @ $o @ ( type2 @ $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_HOL_Obool___Groups_Ouminus_25,axiom,
% 2.90/1.56 uminus @ $o @ ( type2 @ $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(tcon_HOL_Obool___Groups_Ominus_26,axiom,
% 2.90/1.56 minus @ $o @ ( type2 @ $o ) ).
% 2.90/1.56
% 2.90/1.56 thf(conj_0,conjecture,
% 2.90/1.56 gram_L716654942_subtr @ ns @ tr1 @ tr3 ).
% 2.90/1.56 ------------------------
% 2.90/1.58 % (9748)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_9630 for (2999ds/27Mi)
% 2.90/1.58 % (9752)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_9630 for (2999ds/18Mi)
% 2.90/1.58 % (9751)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_9630 for (2999ds/275Mi)
% 2.90/1.58 % (9747)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_9630 for (2999ds/4Mi)
% 2.90/1.58 % (9747)Instruction limit reached!
% 2.90/1.58 % (9747)------------------------------
% 2.90/1.58 % (9747)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.58 % (9747)Termination reason: Unknown
% 2.90/1.58 % (9747)Termination phase: shuffling
% 2.90/1.58
% 2.90/1.58 % (9747)Memory used [KB]: 1407
% 2.90/1.58 % (9747)Time elapsed: 0.003 s
% 2.90/1.58 % (9747)Instructions burned: 6 (million)
% 2.90/1.58 % (9747)------------------------------
% 2.90/1.58 % (9747)------------------------------
% 2.90/1.58 % (9752)Instruction limit reached!
% 2.90/1.58 % (9752)------------------------------
% 2.90/1.58 % (9752)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.58 % (9752)Termination reason: Unknown
% 2.90/1.58 % (9752)Termination phase: shuffling
% 2.90/1.58
% 2.90/1.58 % (9752)Memory used [KB]: 1663
% 2.90/1.58 % (9752)Time elapsed: 0.007 s
% 2.90/1.58 % (9752)Instructions burned: 18 (million)
% 2.90/1.58 % (9752)------------------------------
% 2.90/1.58 % (9752)------------------------------
% 2.90/1.59 % (9746)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_9630 for (2999ds/183Mi)
% 2.90/1.59 % (9750)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_9630 for (2999ds/2Mi)
% 2.90/1.59 % (9749)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_9630 for (2999ds/2Mi)
% 2.90/1.59 % (9748)Instruction limit reached!
% 2.90/1.59 % (9748)------------------------------
% 2.90/1.59 % (9748)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.59 % (9748)Termination reason: Unknown
% 2.90/1.59 % (9748)Termination phase: Property scanning
% 2.90/1.59
% 2.90/1.59 % (9748)Memory used [KB]: 1663
% 2.90/1.59 % (9748)Time elapsed: 0.010 s
% 2.90/1.59 % (9748)Instructions burned: 28 (million)
% 2.90/1.59 % (9748)------------------------------
% 2.90/1.59 % (9748)------------------------------
% 2.90/1.59 % (9750)Instruction limit reached!
% 2.90/1.59 % (9750)------------------------------
% 2.90/1.59 % (9750)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.59 % (9750)Termination reason: Unknown
% 2.90/1.59 % (9750)Termination phase: shuffling
% 2.90/1.59
% 2.90/1.59 % (9750)Memory used [KB]: 1279
% 2.90/1.59 % (9750)Time elapsed: 0.003 s
% 2.90/1.59 % (9750)Instructions burned: 2 (million)
% 2.90/1.59 % (9750)------------------------------
% 2.90/1.59 % (9750)------------------------------
% 2.90/1.59 % (9749)Instruction limit reached!
% 2.90/1.59 % (9749)------------------------------
% 2.90/1.59 % (9749)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.59 % (9749)Termination reason: Unknown
% 2.90/1.59 % (9749)Termination phase: shuffling
% 2.90/1.59
% 2.90/1.59 % (9749)Memory used [KB]: 1279
% 2.90/1.59 % (9749)Time elapsed: 0.004 s
% 2.90/1.59 % (9749)Instructions burned: 3 (million)
% 2.90/1.59 % (9749)------------------------------
% 2.90/1.59 % (9749)------------------------------
% 2.90/1.59 % (9753)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9630 for (2999ds/3Mi)
% 2.90/1.59 % (9753)Instruction limit reached!
% 2.90/1.59 % (9753)------------------------------
% 2.90/1.59 % (9753)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.59 % (9753)Termination reason: Unknown
% 2.90/1.59 % (9753)Termination phase: shuffling
% 2.90/1.59
% 2.90/1.59 % (9753)Memory used [KB]: 1407
% 2.90/1.59 % (9753)Time elapsed: 0.003 s
% 2.90/1.59 % (9753)Instructions burned: 4 (million)
% 2.90/1.59 % (9753)------------------------------
% 2.90/1.59 % (9753)------------------------------
% 2.90/1.59 % (9754)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_9630 for (2999ds/37Mi)
% 2.90/1.60 % (9751)First to succeed.
% 2.97/1.60 % (9755)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_9630 for (2999ds/15Mi)
% 2.97/1.60 % (9751)Refutation found. Thanks to Tanya!
% 2.97/1.60 % SZS status Theorem for DTF2THF_9630
% 2.97/1.60 % SZS output start Proof for DTF2THF_9630
% 2.97/1.60 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.97/1.60 thf(type_def_7, type, dtree: $tType).
% 2.97/1.60 thf(type_def_8, type, n: $tType).
% 2.97/1.60 thf(type_def_9, type, set: ($tType) > $tType).
% 2.97/1.60 thf(type_def_10, type, t: $tType).
% 2.97/1.60 thf(func_def_0, type, dtree: $tType).
% 2.97/1.60 thf(func_def_1, type, set: ($tType) > $tType).
% 2.97/1.60 thf(func_def_2, type, t: $tType).
% 2.97/1.60 thf(func_def_3, type, n: $tType).
% 2.97/1.60 thf(func_def_4, type, itself: ($tType) > $tType).
% 2.97/1.60 thf(func_def_5, type, bounded_lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_6, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_7, type, minus: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_8, type, uminus: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_9, type, bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_10, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_11, type, neg_numeral: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_12, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_13, type, group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_14, type, lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_15, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_16, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_17, type, order_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_18, type, linordered_idom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_19, type, comm_monoid_diff: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_20, type, boolean_algebra: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_21, type, semilattice_sup: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_22, type, linordered_semidom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_23, type, ordered_ab_group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_24, type, bounded_lattice_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_25, type, cancel1352612707id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_26, type, cancel146912293up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_27, type, linord219039673up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_28, type, linord1659791738miring: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_29, type, bounde1808546759up_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_30, type, canoni770627133id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.60 thf(func_def_31, type, root: dtree > n).
% 2.97/1.60 thf(func_def_32, type, gram_L861583724lle_Fr: set @ n > dtree > set @ t).
% 2.97/1.60 thf(func_def_33, type, gram_L1333338417e_inFr: set @ n > dtree > t > $o).
% 2.97/1.60 thf(func_def_34, type, gram_L805317441_inFr2: set @ n > dtree > t > $o).
% 2.97/1.60 thf(func_def_35, type, gram_L830233218_inItr: set @ n > dtree > n > $o).
% 2.97/1.60 thf(func_def_36, type, gram_L716654942_subtr: set @ n > dtree > dtree > $o).
% 2.97/1.60 thf(func_def_37, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.97/1.60 thf(func_def_38, type, one_one: !>[X0: $tType]:(X0)).
% 2.97/1.60 thf(func_def_39, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 2.97/1.60 thf(func_def_40, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.97/1.60 thf(func_def_41, type, semilattice_neutr: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 2.97/1.60 thf(func_def_42, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.97/1.60 thf(func_def_43, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.97/1.60 thf(func_def_44, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.97/1.60 thf(func_def_45, type, order_antimono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 2.97/1.60 thf(func_def_46, type, order_strict_mono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 2.97/1.60 thf(func_def_47, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.97/1.60 thf(func_def_48, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.97/1.60 thf(func_def_49, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 2.97/1.60 thf(func_def_50, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 2.97/1.60 thf(func_def_51, type, pairwise: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > $o)).
% 2.97/1.60 thf(func_def_52, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.97/1.60 thf(func_def_53, type, ns: set @ n).
% 2.97/1.60 thf(func_def_54, type, tr1: dtree).
% 2.97/1.60 thf(func_def_55, type, tr2: dtree).
% 2.97/1.60 thf(func_def_56, type, tr3: dtree).
% 2.97/1.60 thf(func_def_70, type, sK0: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.97/1.60 thf(func_def_71, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.97/1.60 thf(func_def_72, type, sK2: !>[X0: $tType]:((X0 > $o) > set @ X0 > set @ X0 > X0)).
% 2.97/1.60 thf(func_def_73, type, sK3: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.97/1.60 thf(func_def_74, type, sK4: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.97/1.60 thf(func_def_75, type, sK5: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.97/1.60 thf(func_def_76, type, sK6: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.97/1.60 thf(func_def_77, type, sK7: !>[X0: $tType]:((X0 > $o) > set @ X0 > set @ X0 > X0)).
% 2.97/1.60 thf(func_def_79, type, ph9: !>[X0: $tType]:(X0)).
% 2.97/1.60 thf(f992,plain,(
% 2.97/1.60 $false),
% 2.97/1.60 inference(subsumption_resolution,[status(thm)],[f991,f927])).
% 2.97/1.60 thf(f927,plain,(
% 2.97/1.60 ( ! [X0 : $tType,X1 : set @ X0] : (((ord_less_eq @ set @ X0 @ X1 @ X1) = $true)) )),
% 2.97/1.60 inference(cnf_transformation,[status(thm)],[f417])).
% 2.97/1.60 thf(f417,plain,(
% 2.97/1.60 ! [X0 : $tType,X1 : set @ X0] : ((ord_less_eq @ set @ X0 @ X1 @ X1) = $true)),
% 2.97/1.60 inference(fool_elimination,[status(thm)],[f416])).
% 2.97/1.60 thf(f416,plain,(
% 2.97/1.60 ! [X0 : $tType,X1 : set @ X0] : (ord_less_eq @ set @ X0 @ X1 @ X1)),
% 2.97/1.60 inference(rectify,[status(thm)],[f27])).
% 2.97/1.60 thf(f27,axiom,(
% 2.97/1.60 ! [X0 : $tType,X10 : set @ X0] : (ord_less_eq @ set @ X0 @ X10 @ X10)),
% 2.97/1.60 file('/export/starexec/sandbox2/tmp/tmp.xTiNdgqN7t/DTF2THF_9630.p',fact_26_subset__refl)).
% 2.97/1.60 thf(f991,plain,(
% 2.97/1.60 ($true != (ord_less_eq @ set @ n @ ns @ ns))),
% 2.97/1.60 inference(forward_demodulation,[status(thm)],[f988,f963])).
% 2.97/1.60 thf(f963,plain,(
% 2.97/1.60 ( ! [X0 : $tType,X1 : set @ X0] : (((sup_sup @ set @ X0 @ X1 @ X1) = X1)) )),
% 2.97/1.60 inference(cnf_transformation,[status(thm)],[f851])).
% 2.97/1.60 thf(f851,plain,(
% 2.97/1.60 ! [X0 : $tType,X1 : set @ X0] : ((sup_sup @ set @ X0 @ X1 @ X1) = X1)),
% 2.97/1.60 inference(rectify,[status(thm)],[f57])).
% 2.97/1.60 thf(f57,axiom,(
% 2.97/1.60 ! [X0 : $tType,X10 : set @ X0] : ((sup_sup @ set @ X0 @ X10 @ X10) = X10)),
% 2.97/1.60 file('/export/starexec/sandbox2/tmp/tmp.xTiNdgqN7t/DTF2THF_9630.p',fact_56_Un__absorb)).
% 2.97/1.60 thf(f988,plain,(
% 2.97/1.60 ((ord_less_eq @ set @ n @ (sup_sup @ set @ n @ ns @ ns) @ ns) != $true)),
% 2.97/1.60 inference(trivial_inequality_removal,[status(thm)],[f985])).
% 2.97/1.60 thf(f985,plain,(
% 2.97/1.60 ($true != $true) | ((ord_less_eq @ set @ n @ (sup_sup @ set @ n @ ns @ ns) @ ns) != $true)),
% 2.97/1.60 inference(superposition,[status(thm)],[f984,f947])).
% 2.97/1.60 thf(f947,plain,(
% 2.97/1.60 ((gram_L716654942_subtr @ (sup_sup @ set @ n @ ns @ ns) @ tr1 @ tr3) = $true)),
% 2.97/1.60 inference(cnf_transformation,[status(thm)],[f697])).
% 2.97/1.60 thf(f697,plain,(
% 2.97/1.60 ((gram_L716654942_subtr @ (sup_sup @ set @ n @ ns @ ns) @ tr1 @ tr3) = $true)),
% 2.97/1.60 inference(fool_elimination,[status(thm)],[f696])).
% 2.97/1.60 thf(f696,plain,(
% 2.97/1.60 (gram_L716654942_subtr @ (sup_sup @ set @ n @ ns @ ns) @ tr1 @ tr3)),
% 2.97/1.60 inference(rectify,[status(thm)],[f1])).
% 2.97/1.60 thf(f1,axiom,(
% 2.97/1.60 (gram_L716654942_subtr @ (sup_sup @ set @ n @ ns @ ns) @ tr1 @ tr3)),
% 2.97/1.60 file('/export/starexec/sandbox2/tmp/tmp.xTiNdgqN7t/DTF2THF_9630.p',fact_0__092_060open_062subtr_A_Ins_A_092_060union_062_Ans_J_Atr1_Atr3_092_060close_062)).
% 2.97/1.60 thf(f984,plain,(
% 2.97/1.60 ( ! [X0 : set @ n] : (((gram_L716654942_subtr @ X0 @ tr1 @ tr3) != $true) | ((ord_less_eq @ set @ n @ X0 @ ns) != $true)) )),
% 2.97/1.60 inference(trivial_inequality_removal,[status(thm)],[f983])).
% 2.97/1.60 thf(f983,plain,(
% 2.97/1.60 ( ! [X0 : set @ n] : (((ord_less_eq @ set @ n @ X0 @ ns) != $true) | ($true != $true) | ((gram_L716654942_subtr @ X0 @ tr1 @ tr3) != $true)) )),
% 2.97/1.60 inference(superposition,[status(thm)],[f971,f968])).
% 2.97/1.60 thf(f968,plain,(
% 2.97/1.60 ( ! [X2 : dtree,X3 : set @ n,X0 : set @ n,X1 : dtree] : (((gram_L716654942_subtr @ X3 @ X1 @ X2) = $true) | ((gram_L716654942_subtr @ X0 @ X1 @ X2) != $true) | ($true != (ord_less_eq @ set @ n @ X0 @ X3))) )),
% 2.97/1.60 inference(cnf_transformation,[status(thm)],[f890])).
% 2.97/1.60 thf(f890,plain,(
% 2.97/1.60 ! [X0 : set @ n,X1 : dtree,X2 : dtree,X3 : set @ n] : (((gram_L716654942_subtr @ X3 @ X1 @ X2) = $true) | ($true != (ord_less_eq @ set @ n @ X0 @ X3)) | ((gram_L716654942_subtr @ X0 @ X1 @ X2) != $true))),
% 2.97/1.60 inference(flattening,[status(thm)],[f889])).
% 2.97/1.60 thf(f889,plain,(
% 2.97/1.60 ! [X0 : set @ n,X1 : dtree,X2 : dtree,X3 : set @ n] : ((((gram_L716654942_subtr @ X3 @ X1 @ X2) = $true) | ($true != (ord_less_eq @ set @ n @ X0 @ X3))) | ((gram_L716654942_subtr @ X0 @ X1 @ X2) != $true))),
% 2.97/1.60 inference(ennf_transformation,[status(thm)],[f557])).
% 2.97/1.60 thf(f557,plain,(
% 2.97/1.60 ! [X0 : set @ n,X1 : dtree,X2 : dtree,X3 : set @ n] : (((gram_L716654942_subtr @ X0 @ X1 @ X2) = $true) => (($true = (ord_less_eq @ set @ n @ X0 @ X3)) => ((gram_L716654942_subtr @ X3 @ X1 @ X2) = $true)))),
% 2.97/1.60 inference(fool_elimination,[status(thm)],[f556])).
% 2.97/1.60 thf(f556,plain,(
% 2.97/1.60 ! [X0 : set @ n,X1 : dtree,X2 : dtree,X3 : set @ n] : ((gram_L716654942_subtr @ X0 @ X1 @ X2) => ((ord_less_eq @ set @ n @ X0 @ X3) => (gram_L716654942_subtr @ X3 @ X1 @ X2)))),
% 2.97/1.60 inference(rectify,[status(thm)],[f5])).
% 2.97/1.60 thf(f5,axiom,(
% 2.97/1.60 ! [X7 : set @ n,X3 : dtree,X4 : dtree,X8 : set @ n] : ((gram_L716654942_subtr @ X7 @ X3 @ X4) => ((ord_less_eq @ set @ n @ X7 @ X8) => (gram_L716654942_subtr @ X8 @ X3 @ X4)))),
% 2.97/1.60 file('/export/starexec/sandbox2/tmp/tmp.xTiNdgqN7t/DTF2THF_9630.p',fact_4_subtr__mono)).
% 2.97/1.60 thf(f971,plain,(
% 2.97/1.60 ((gram_L716654942_subtr @ ns @ tr1 @ tr3) != $true)),
% 2.97/1.60 inference(cnf_transformation,[status(thm)],[f844])).
% 2.97/1.60 thf(f844,plain,(
% 2.97/1.60 ((gram_L716654942_subtr @ ns @ tr1 @ tr3) != $true)),
% 2.97/1.60 inference(flattening,[status(thm)],[f559])).
% 2.97/1.60 thf(f559,plain,(
% 2.97/1.60 ~((gram_L716654942_subtr @ ns @ tr1 @ tr3) = $true)),
% 2.97/1.60 inference(fool_elimination,[status(thm)],[f558])).
% 2.97/1.60 thf(f558,plain,(
% 2.97/1.60 ~(gram_L716654942_subtr @ ns @ tr1 @ tr3)),
% 2.97/1.60 inference(rectify,[status(thm)],[f298])).
% 2.97/1.60 thf(f298,negated_conjecture,(
% 2.97/1.60 ~(gram_L716654942_subtr @ ns @ tr1 @ tr3)),
% 2.97/1.60 inference(negated_conjecture,[status(cth)],[f297])).
% 2.97/1.60 thf(f297,conjecture,(
% 2.97/1.60 (gram_L716654942_subtr @ ns @ tr1 @ tr3)),
% 2.97/1.60 file('/export/starexec/sandbox2/tmp/tmp.xTiNdgqN7t/DTF2THF_9630.p',conj_0)).
% 2.97/1.60 % SZS output end Proof for DTF2THF_9630
% 2.97/1.60 % (9751)------------------------------
% 2.97/1.60 % (9751)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.60 % (9751)Termination reason: Refutation
% 2.97/1.60
% 2.97/1.60 % (9751)Memory used [KB]: 6268
% 2.97/1.60 % (9751)Time elapsed: 0.018 s
% 2.97/1.60 % (9751)Instructions burned: 45 (million)
% 2.97/1.60 % (9751)------------------------------
% 2.97/1.60 % (9751)------------------------------
% 2.97/1.60 % (9745)Success in time 0.024 s
%------------------------------------------------------------------------------