%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP123^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n008.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:44 AM UTC 2026
% Result : Theorem 2.56s 1.65s
% Output : Refutation 2.56s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP123^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n008.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 08:27:09 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
% 0.18/0.44 ---- Original DTF file ---
% 0.18/0.44 thf(spec,logic,$$dhol).
% 0.18/0.44 %------------------------------------------------------------------------------
% 0.18/0.44 % File : ITP123^2 : TPTP v9.2.1. Released v7.5.0.
% 0.18/0.44 % Domain : Interactive Theorem Proving
% 0.18/0.44 % Problem : Sledgehammer Modular_Distrib_Lattice problem prob_319__3264956_1
% 0.18/0.44 % Version : Especial.
% 0.18/0.44 % English :
% 0.18/0.44
% 0.18/0.44 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.18/0.44 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.18/0.44 % Source : [Des21]
% 0.18/0.44 % Names : Modular_Distrib_Lattice/prob_319__3264956_1 [Des21]
% 0.18/0.44
% 0.18/0.44 % Status : Theorem
% 0.18/0.44 % Rating : 0.33 v9.1.0, 0.00 v8.2.0, 0.33 v8.1.0, 0.00 v7.5.0
% 0.18/0.44 % Syntax : Number of formulae : 299 ( 129 unt; 38 typ; 0 def)
% 0.18/0.44 % Number of atoms : 599 ( 194 equ; 0 cnn)
% 0.18/0.44 % Maximal formula atoms : 6 ( 2 avg)
% 0.18/0.44 % Number of connectives : 2079 ( 51 ~; 4 |; 39 &;1756 @)
% 0.18/0.44 % ( 0 <=>; 229 =>; 0 <=; 0 <~>)
% 0.18/0.44 % Maximal formula depth : 16 ( 6 avg)
% 0.18/0.44 % Number of types : 2 ( 1 usr)
% 0.18/0.44 % Number of type conns : 396 ( 396 >; 0 *; 0 +; 0 <<)
% 0.18/0.44 % Number of symbols : 38 ( 37 usr; 3 con; 0-7 aty)
% 0.18/0.44 % Number of variables : 774 ( 25 ^; 709 !; 8 ?; 774 :)
% 0.18/0.44 % ( 32 !>; 0 ?*; 0 @-; 0 @+)
% 0.18/0.44 % SPC : TH1_THM_EQU_NAR
% 0.18/0.44
% 0.18/0.44 % Comments : This file was generated by Sledgehammer 2021-02-23 16:30:15.471
% 0.18/0.44 %------------------------------------------------------------------------------
% 0.18/0.44 % Could-be-implicit typings (2)
% 0.18/0.44 thf(ty_t_Set_Oset,type,
% 0.18/0.44 set: $tType > $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_tf_a,type,
% 0.18/0.44 a: $tType ).
% 0.18/0.44
% 0.18/0.44 % Explicit typings (36)
% 0.18/0.44 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
% 0.18/0.44 condit2040224947_above:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
% 0.18/0.44 condit1201339847_below:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Finite__Set_Ocomp__fun__idem,type,
% 0.18/0.44 finite_comp_fun_idem:
% 0.18/0.44 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Groups_Oabel__semigroup,type,
% 0.18/0.44 abel_semigroup:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Groups_Osemigroup,type,
% 0.18/0.44 semigroup:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Lattices_Oclass_Olattice,type,
% 0.18/0.44 lattice:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > ( A > A > A ) > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Lattices_Osemilattice,type,
% 0.18/0.44 semilattice:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Lattices_Osemilattice__order,type,
% 0.18/0.44 semilattice_order:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Lattices__Big_Oord_Ois__arg__max,type,
% 0.18/0.44 lattices_is_arg_max:
% 0.18/0.44 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( B > $o ) > B > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Lattices__Big_Oord_Ois__arg__min,type,
% 0.18/0.44 lattices_is_arg_min:
% 0.18/0.44 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( B > $o ) > B > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
% 0.18/0.44 lattic1693879045er_set:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Lattices__Big_Osemilattice__set,type,
% 0.18/0.44 lattic35693393ce_set:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_OM5__lattice,type,
% 0.18/0.44 modula1618706180attice:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > A ) > A > A > A > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_ON5__lattice,type,
% 0.18/0.44 modula1192512645attice:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > A ) > A > A > A > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oa__aux,type,
% 0.18/0.44 modula105138637_a_aux:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Ob__aux,type,
% 0.18/0.44 modula477145868_b_aux:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oc__aux,type,
% 0.18/0.44 modula849153099_c_aux:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Od__aux,type,
% 0.18/0.44 modula1221160330_d_aux:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oe__aux,type,
% 0.18/0.44 modula1593167561_e_aux:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oincomp,type,
% 0.18/0.44 modula1937309406incomp:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Ono__distrib,type,
% 0.18/0.44 modula141148890istrib:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > A ) > A > A > A > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord_OLeast,type,
% 0.18/0.44 least:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord_Omax,type,
% 0.18/0.44 max:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oord_Omin,type,
% 0.18/0.44 min:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oorder_OGreatest,type,
% 0.18/0.44 greatest:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Orderings_Oordering,type,
% 0.18/0.44 ordering:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Set_OCollect,type,
% 0.18/0.44 collect:
% 0.18/0.44 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan,type,
% 0.18/0.44 set_gr1161524159ssThan:
% 0.18/0.44 !>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_c_member,type,
% 0.18/0.44 member:
% 0.18/0.44 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.18/0.44
% 0.18/0.44 thf(sy_v_inf,type,
% 0.18/0.44 inf: a > a > a ).
% 0.18/0.44
% 0.18/0.44 thf(sy_v_less,type,
% 0.18/0.44 less: a > a > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_v_less__eq,type,
% 0.18/0.44 less_eq: a > a > $o ).
% 0.18/0.44
% 0.18/0.44 thf(sy_v_sup,type,
% 0.18/0.44 sup: a > a > a ).
% 0.18/0.44
% 0.18/0.44 thf(sy_v_x,type,
% 0.18/0.44 x: a ).
% 0.18/0.44
% 0.18/0.44 thf(sy_v_y,type,
% 0.18/0.44 y: a ).
% 0.18/0.44
% 0.18/0.44 thf(sy_v_z,type,
% 0.18/0.44 z: a ).
% 0.18/0.44
% 0.18/0.44 % Relevant facts (256)
% 0.18/0.44 thf(fact_0_local_Oantisym,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less_eq @ X @ Y )
% 0.18/0.44 => ( ( less_eq @ Y @ X )
% 0.18/0.44 => ( X = Y ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.antisym
% 0.18/0.44 thf(fact_1_local_Oantisym__conv,axiom,
% 0.18/0.44 ! [Y: a,X: a] :
% 0.18/0.44 ( ( less_eq @ Y @ X )
% 0.18/0.44 => ( ( less_eq @ X @ Y )
% 0.18/0.44 = ( X = Y ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.antisym_conv
% 0.18/0.44 thf(fact_2_local_Odual__order_Oantisym,axiom,
% 0.18/0.44 ! [B2: a,A2: a] :
% 0.18/0.44 ( ( less_eq @ B2 @ A2 )
% 0.18/0.44 => ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 => ( A2 = B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.dual_order.antisym
% 0.18/0.44 thf(fact_3_local_Odual__order_Oeq__iff,axiom,
% 0.18/0.44 ( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
% 0.18/0.44 = ( ^ [A3: a,B3: a] :
% 0.18/0.44 ( ( less_eq @ B3 @ A3 )
% 0.18/0.44 & ( less_eq @ A3 @ B3 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.dual_order.eq_iff
% 0.18/0.44 thf(fact_4_local_Odual__order_Otrans,axiom,
% 0.18/0.44 ! [B2: a,A2: a,C: a] :
% 0.18/0.44 ( ( less_eq @ B2 @ A2 )
% 0.18/0.44 => ( ( less_eq @ C @ B2 )
% 0.18/0.44 => ( less_eq @ C @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.dual_order.trans
% 0.18/0.44 thf(fact_5_local_Oeq__iff,axiom,
% 0.18/0.44 ( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
% 0.18/0.44 = ( ^ [X2: a,Y3: a] :
% 0.18/0.44 ( ( less_eq @ X2 @ Y3 )
% 0.18/0.44 & ( less_eq @ Y3 @ X2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.eq_iff
% 0.18/0.44 thf(fact_6_local_Oeq__refl,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( X = Y )
% 0.18/0.44 => ( less_eq @ X @ Y ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.eq_refl
% 0.18/0.44 thf(fact_7_local_Oord__eq__le__trans,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( A2 = B2 )
% 0.18/0.44 => ( ( less_eq @ B2 @ C )
% 0.18/0.44 => ( less_eq @ A2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.ord_eq_le_trans
% 0.18/0.44 thf(fact_8_local_Oord__le__eq__trans,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 => ( ( B2 = C )
% 0.18/0.44 => ( less_eq @ A2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.ord_le_eq_trans
% 0.18/0.44 thf(fact_9_local_Oorder_Oantisym,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 => ( ( less_eq @ B2 @ A2 )
% 0.18/0.44 => ( A2 = B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.order.antisym
% 0.18/0.44 thf(fact_10_local_Oorder_Oeq__iff,axiom,
% 0.18/0.44 ( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
% 0.18/0.44 = ( ^ [A3: a,B3: a] :
% 0.18/0.44 ( ( less_eq @ A3 @ B3 )
% 0.18/0.44 & ( less_eq @ B3 @ A3 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.order.eq_iff
% 0.18/0.44 thf(fact_11_local_Oorder_Otrans,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 => ( ( less_eq @ B2 @ C )
% 0.18/0.44 => ( less_eq @ A2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.order.trans
% 0.18/0.44 thf(fact_12_local_Oorder__trans,axiom,
% 0.18/0.44 ! [X: a,Y: a,Z2: a] :
% 0.18/0.44 ( ( less_eq @ X @ Y )
% 0.18/0.44 => ( ( less_eq @ Y @ Z2 )
% 0.18/0.44 => ( less_eq @ X @ Z2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.order_trans
% 0.18/0.44 thf(fact_13_local_Odual__order_Oasym,axiom,
% 0.18/0.44 ! [B2: a,A2: a] :
% 0.18/0.44 ( ( less @ B2 @ A2 )
% 0.18/0.44 => ~ ( less @ A2 @ B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.dual_order.asym
% 0.18/0.44 thf(fact_14_local_Odual__order_Ostrict__implies__not__eq,axiom,
% 0.18/0.44 ! [B2: a,A2: a] :
% 0.18/0.44 ( ( less @ B2 @ A2 )
% 0.18/0.44 => ( A2 != B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.dual_order.strict_implies_not_eq
% 0.18/0.44 thf(fact_15_local_Odual__order_Ostrict__trans,axiom,
% 0.18/0.44 ! [B2: a,A2: a,C: a] :
% 0.18/0.44 ( ( less @ B2 @ A2 )
% 0.18/0.44 => ( ( less @ C @ B2 )
% 0.18/0.44 => ( less @ C @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.dual_order.strict_trans
% 0.18/0.44 thf(fact_16_local_Oless__asym,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less @ X @ Y )
% 0.18/0.44 => ~ ( less @ Y @ X ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_asym
% 0.18/0.44 thf(fact_17_local_Oless__asym_H,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( less @ A2 @ B2 )
% 0.18/0.44 => ~ ( less @ B2 @ A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_asym'
% 0.18/0.44 thf(fact_18_local_Oless__imp__neq,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less @ X @ Y )
% 0.18/0.44 => ( X != Y ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_imp_neq
% 0.18/0.44 thf(fact_19_local_Oless__imp__not__eq,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less @ X @ Y )
% 0.18/0.44 => ( X != Y ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_imp_not_eq
% 0.18/0.44 thf(fact_20_local_Oless__imp__not__eq2,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less @ X @ Y )
% 0.18/0.44 => ( Y != X ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_imp_not_eq2
% 0.18/0.44 thf(fact_21_local_Oless__imp__not__less,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less @ X @ Y )
% 0.18/0.44 => ~ ( less @ Y @ X ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_imp_not_less
% 0.18/0.44 thf(fact_22_local_Oless__imp__triv,axiom,
% 0.18/0.44 ! [X: a,Y: a,P: $o] :
% 0.18/0.44 ( ( less @ X @ Y )
% 0.18/0.44 => ( ( less @ Y @ X )
% 0.18/0.44 => P ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_imp_triv
% 0.18/0.44 thf(fact_23_local_Oless__irrefl,axiom,
% 0.18/0.44 ! [X: a] :
% 0.18/0.44 ~ ( less @ X @ X ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_irrefl
% 0.18/0.44 thf(fact_24_local_Oless__not__sym,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less @ X @ Y )
% 0.18/0.44 => ~ ( less @ Y @ X ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_not_sym
% 0.18/0.44 thf(fact_25_local_Oless__trans,axiom,
% 0.18/0.44 ! [X: a,Y: a,Z2: a] :
% 0.18/0.44 ( ( less @ X @ Y )
% 0.18/0.44 => ( ( less @ Y @ Z2 )
% 0.18/0.44 => ( less @ X @ Z2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_trans
% 0.18/0.44 thf(fact_26_local_Oord__eq__less__trans,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( A2 = B2 )
% 0.18/0.44 => ( ( less @ B2 @ C )
% 0.18/0.44 => ( less @ A2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.ord_eq_less_trans
% 0.18/0.44 thf(fact_27_local_Oord__less__eq__trans,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( less @ A2 @ B2 )
% 0.18/0.44 => ( ( B2 = C )
% 0.18/0.44 => ( less @ A2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.ord_less_eq_trans
% 0.18/0.44 thf(fact_28_local_Oorder_Oasym,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( less @ A2 @ B2 )
% 0.18/0.44 => ~ ( less @ B2 @ A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.order.asym
% 0.18/0.44 thf(fact_29_local_Oorder_Oirrefl,axiom,
% 0.18/0.44 ! [A2: a] :
% 0.18/0.44 ~ ( less @ A2 @ A2 ) ).
% 0.18/0.44
% 0.18/0.44 % local.order.irrefl
% 0.18/0.44 thf(fact_30_local_Oorder_Ostrict__implies__not__eq,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( less @ A2 @ B2 )
% 0.18/0.44 => ( A2 != B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.order.strict_implies_not_eq
% 0.18/0.44 thf(fact_31_local_Oorder_Ostrict__trans,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( less @ A2 @ B2 )
% 0.18/0.44 => ( ( less @ B2 @ C )
% 0.18/0.44 => ( less @ A2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.order.strict_trans
% 0.18/0.44 thf(fact_32_local_Oinf_Oassoc,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( inf @ ( inf @ A2 @ B2 ) @ C )
% 0.18/0.44 = ( inf @ A2 @ ( inf @ B2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.assoc
% 0.18/0.44 thf(fact_33_local_Oinf_Ocommute,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( inf @ A2 @ B2 )
% 0.18/0.44 = ( inf @ B2 @ A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.commute
% 0.18/0.44 thf(fact_34_local_Oinf_Oleft__commute,axiom,
% 0.18/0.44 ! [B2: a,A2: a,C: a] :
% 0.18/0.44 ( ( inf @ B2 @ ( inf @ A2 @ C ) )
% 0.18/0.44 = ( inf @ A2 @ ( inf @ B2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.left_commute
% 0.18/0.44 thf(fact_35_local_Oinf__assoc,axiom,
% 0.18/0.44 ! [X: a,Y: a,Z2: a] :
% 0.18/0.44 ( ( inf @ ( inf @ X @ Y ) @ Z2 )
% 0.18/0.44 = ( inf @ X @ ( inf @ Y @ Z2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf_assoc
% 0.18/0.44 thf(fact_36_local_Oinf__commute,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( inf @ X @ Y )
% 0.18/0.44 = ( inf @ Y @ X ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf_commute
% 0.18/0.44 thf(fact_37_local_Oinf__left__commute,axiom,
% 0.18/0.44 ! [X: a,Y: a,Z2: a] :
% 0.18/0.44 ( ( inf @ X @ ( inf @ Y @ Z2 ) )
% 0.18/0.44 = ( inf @ Y @ ( inf @ X @ Z2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf_left_commute
% 0.18/0.44 thf(fact_38_local_Osup_Oassoc,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( sup @ ( sup @ A2 @ B2 ) @ C )
% 0.18/0.44 = ( sup @ A2 @ ( sup @ B2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.assoc
% 0.18/0.44 thf(fact_39_local_Osup_Ocommute,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( sup @ A2 @ B2 )
% 0.18/0.44 = ( sup @ B2 @ A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.commute
% 0.18/0.44 thf(fact_40_local_Osup_Oleft__commute,axiom,
% 0.18/0.44 ! [B2: a,A2: a,C: a] :
% 0.18/0.44 ( ( sup @ B2 @ ( sup @ A2 @ C ) )
% 0.18/0.44 = ( sup @ A2 @ ( sup @ B2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.left_commute
% 0.18/0.44 thf(fact_41_local_Osup__assoc,axiom,
% 0.18/0.44 ! [X: a,Y: a,Z2: a] :
% 0.18/0.44 ( ( sup @ ( sup @ X @ Y ) @ Z2 )
% 0.18/0.44 = ( sup @ X @ ( sup @ Y @ Z2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup_assoc
% 0.18/0.44 thf(fact_42_local_Osup__commute,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( sup @ X @ Y )
% 0.18/0.44 = ( sup @ Y @ X ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup_commute
% 0.18/0.44 thf(fact_43_local_Osup__left__commute,axiom,
% 0.18/0.44 ! [X: a,Y: a,Z2: a] :
% 0.18/0.44 ( ( sup @ X @ ( sup @ Y @ Z2 ) )
% 0.18/0.44 = ( sup @ Y @ ( sup @ X @ Z2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup_left_commute
% 0.18/0.44 thf(fact_44_local_Oantisym__conv1,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ~ ( less @ X @ Y )
% 0.18/0.44 => ( ( less_eq @ X @ Y )
% 0.18/0.44 = ( X = Y ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.antisym_conv1
% 0.18/0.44 thf(fact_45_mem__Collect__eq,axiom,
% 0.18/0.44 ! [A: $tType,A2: A,P: A > $o] :
% 0.18/0.44 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.18/0.44 = ( P @ A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % mem_Collect_eq
% 0.18/0.44 thf(fact_46_Collect__mem__eq,axiom,
% 0.18/0.44 ! [A: $tType,A4: set @ A] :
% 0.18/0.44 ( ( collect @ A
% 0.18/0.44 @ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
% 0.18/0.44 = A4 ) ).
% 0.18/0.44
% 0.18/0.44 % Collect_mem_eq
% 0.18/0.44 thf(fact_47_Collect__cong,axiom,
% 0.18/0.44 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.18/0.44 ( ! [X3: A] :
% 0.18/0.44 ( ( P @ X3 )
% 0.18/0.44 = ( Q @ X3 ) )
% 0.18/0.44 => ( ( collect @ A @ P )
% 0.18/0.44 = ( collect @ A @ Q ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % Collect_cong
% 0.18/0.44 thf(fact_48_ext,axiom,
% 0.18/0.44 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.18/0.44 ( ! [X3: A] :
% 0.18/0.44 ( ( F @ X3 )
% 0.18/0.44 = ( G @ X3 ) )
% 0.18/0.44 => ( F = G ) ) ).
% 0.18/0.44
% 0.18/0.44 % ext
% 0.18/0.44 thf(fact_49_local_Oantisym__conv2,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less_eq @ X @ Y )
% 0.18/0.44 => ( ( ~ ( less @ X @ Y ) )
% 0.18/0.44 = ( X = Y ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.antisym_conv2
% 0.18/0.44 thf(fact_50_local_Odual__order_Onot__eq__order__implies__strict,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( A2 != B2 )
% 0.18/0.44 => ( ( less_eq @ B2 @ A2 )
% 0.18/0.44 => ( less @ B2 @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.dual_order.not_eq_order_implies_strict
% 0.18/0.44 thf(fact_51_local_Odual__order_Oorder__iff__strict,axiom,
% 0.18/0.44 ! [B2: a,A2: a] :
% 0.18/0.44 ( ( less_eq @ B2 @ A2 )
% 0.18/0.44 = ( ( less @ B2 @ A2 )
% 0.18/0.44 | ( A2 = B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.dual_order.order_iff_strict
% 0.18/0.44 thf(fact_52_local_Odual__order_Ostrict__iff__order,axiom,
% 0.18/0.44 ! [B2: a,A2: a] :
% 0.18/0.44 ( ( less @ B2 @ A2 )
% 0.18/0.44 = ( ( less_eq @ B2 @ A2 )
% 0.18/0.44 & ( A2 != B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.dual_order.strict_iff_order
% 0.18/0.44 thf(fact_53_local_Odual__order_Ostrict__implies__order,axiom,
% 0.18/0.44 ! [B2: a,A2: a] :
% 0.18/0.44 ( ( less @ B2 @ A2 )
% 0.18/0.44 => ( less_eq @ B2 @ A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.dual_order.strict_implies_order
% 0.18/0.44 thf(fact_54_local_Odual__order_Ostrict__trans1,axiom,
% 0.18/0.44 ! [B2: a,A2: a,C: a] :
% 0.18/0.44 ( ( less_eq @ B2 @ A2 )
% 0.18/0.44 => ( ( less @ C @ B2 )
% 0.18/0.44 => ( less @ C @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.dual_order.strict_trans1
% 0.18/0.44 thf(fact_55_local_Odual__order_Ostrict__trans2,axiom,
% 0.18/0.44 ! [B2: a,A2: a,C: a] :
% 0.18/0.44 ( ( less @ B2 @ A2 )
% 0.18/0.44 => ( ( less_eq @ C @ B2 )
% 0.18/0.44 => ( less @ C @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.dual_order.strict_trans2
% 0.18/0.44 thf(fact_56_local_OleD,axiom,
% 0.18/0.44 ! [Y: a,X: a] :
% 0.18/0.44 ( ( less_eq @ Y @ X )
% 0.18/0.44 => ~ ( less @ X @ Y ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.leD
% 0.18/0.44 thf(fact_57_local_Ole__imp__less__or__eq,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less_eq @ X @ Y )
% 0.18/0.44 => ( ( less @ X @ Y )
% 0.18/0.44 | ( X = Y ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.le_imp_less_or_eq
% 0.18/0.44 thf(fact_58_local_Ole__less,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less_eq @ X @ Y )
% 0.18/0.44 = ( ( less @ X @ Y )
% 0.18/0.44 | ( X = Y ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.le_less
% 0.18/0.44 thf(fact_59_local_Ole__less__trans,axiom,
% 0.18/0.44 ! [X: a,Y: a,Z2: a] :
% 0.18/0.44 ( ( less_eq @ X @ Y )
% 0.18/0.44 => ( ( less @ Y @ Z2 )
% 0.18/0.44 => ( less @ X @ Z2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.le_less_trans
% 0.18/0.44 thf(fact_60_local_Ole__neq__trans,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 => ( ( A2 != B2 )
% 0.18/0.44 => ( less @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.le_neq_trans
% 0.18/0.44 thf(fact_61_local_Oless__imp__le,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less @ X @ Y )
% 0.18/0.44 => ( less_eq @ X @ Y ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_imp_le
% 0.18/0.44 thf(fact_62_local_Oless__le,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less @ X @ Y )
% 0.18/0.44 = ( ( less_eq @ X @ Y )
% 0.18/0.44 & ( X != Y ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_le
% 0.18/0.44 thf(fact_63_local_Oless__le__not__le,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less @ X @ Y )
% 0.18/0.44 = ( ( less_eq @ X @ Y )
% 0.18/0.44 & ~ ( less_eq @ Y @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_le_not_le
% 0.18/0.44 thf(fact_64_local_Oless__le__trans,axiom,
% 0.18/0.44 ! [X: a,Y: a,Z2: a] :
% 0.18/0.44 ( ( less @ X @ Y )
% 0.18/0.44 => ( ( less_eq @ Y @ Z2 )
% 0.18/0.44 => ( less @ X @ Z2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_le_trans
% 0.18/0.44 thf(fact_65_local_Oorder_Onot__eq__order__implies__strict,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( A2 != B2 )
% 0.18/0.44 => ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 => ( less @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.order.not_eq_order_implies_strict
% 0.18/0.44 thf(fact_66_local_Oorder_Oorder__iff__strict,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 = ( ( less @ A2 @ B2 )
% 0.18/0.44 | ( A2 = B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.order.order_iff_strict
% 0.18/0.44 thf(fact_67_local_Oorder_Ostrict__iff__order,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( less @ A2 @ B2 )
% 0.18/0.44 = ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 & ( A2 != B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.order.strict_iff_order
% 0.18/0.44 thf(fact_68_local_Oorder_Ostrict__implies__order,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( less @ A2 @ B2 )
% 0.18/0.44 => ( less_eq @ A2 @ B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.order.strict_implies_order
% 0.18/0.44 thf(fact_69_local_Oorder_Ostrict__trans1,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 => ( ( less @ B2 @ C )
% 0.18/0.44 => ( less @ A2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.order.strict_trans1
% 0.18/0.44 thf(fact_70_local_Oorder_Ostrict__trans2,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( less @ A2 @ B2 )
% 0.18/0.44 => ( ( less_eq @ B2 @ C )
% 0.18/0.44 => ( less @ A2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.order.strict_trans2
% 0.18/0.44 thf(fact_71_local_Oinf_Oabsorb1,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 => ( ( inf @ A2 @ B2 )
% 0.18/0.44 = A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.absorb1
% 0.18/0.44 thf(fact_72_local_Oinf_Oabsorb2,axiom,
% 0.18/0.44 ! [B2: a,A2: a] :
% 0.18/0.44 ( ( less_eq @ B2 @ A2 )
% 0.18/0.44 => ( ( inf @ A2 @ B2 )
% 0.18/0.44 = B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.absorb2
% 0.18/0.44 thf(fact_73_local_Oinf_Oabsorb__iff1,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 = ( ( inf @ A2 @ B2 )
% 0.18/0.44 = A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.absorb_iff1
% 0.18/0.44 thf(fact_74_local_Oinf_Oabsorb__iff2,axiom,
% 0.18/0.44 ! [B2: a,A2: a] :
% 0.18/0.44 ( ( less_eq @ B2 @ A2 )
% 0.18/0.44 = ( ( inf @ A2 @ B2 )
% 0.18/0.44 = B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.absorb_iff2
% 0.18/0.44 thf(fact_75_local_Oinf_OboundedE,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ ( inf @ B2 @ C ) )
% 0.18/0.44 => ~ ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 => ~ ( less_eq @ A2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.boundedE
% 0.18/0.44 thf(fact_76_local_Oinf_OboundedI,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 => ( ( less_eq @ A2 @ C )
% 0.18/0.44 => ( less_eq @ A2 @ ( inf @ B2 @ C ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.boundedI
% 0.18/0.44 thf(fact_77_local_Oinf_Ocobounded1,axiom,
% 0.18/0.44 ! [A2: a,B2: a] : ( less_eq @ ( inf @ A2 @ B2 ) @ A2 ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.cobounded1
% 0.18/0.44 thf(fact_78_local_Oinf_Ocobounded2,axiom,
% 0.18/0.44 ! [A2: a,B2: a] : ( less_eq @ ( inf @ A2 @ B2 ) @ B2 ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.cobounded2
% 0.18/0.44 thf(fact_79_local_Oinf_OcoboundedI1,axiom,
% 0.18/0.44 ! [A2: a,C: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ C )
% 0.18/0.44 => ( less_eq @ ( inf @ A2 @ B2 ) @ C ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.coboundedI1
% 0.18/0.44 thf(fact_80_local_Oinf_OcoboundedI2,axiom,
% 0.18/0.44 ! [B2: a,C: a,A2: a] :
% 0.18/0.44 ( ( less_eq @ B2 @ C )
% 0.18/0.44 => ( less_eq @ ( inf @ A2 @ B2 ) @ C ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.coboundedI2
% 0.18/0.44 thf(fact_81_local_Oinf_OorderE,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 => ( A2
% 0.18/0.44 = ( inf @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.orderE
% 0.18/0.44 thf(fact_82_local_Oinf_OorderI,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( A2
% 0.18/0.44 = ( inf @ A2 @ B2 ) )
% 0.18/0.44 => ( less_eq @ A2 @ B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.orderI
% 0.18/0.44 thf(fact_83_local_Oinf_Oorder__iff,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 = ( A2
% 0.18/0.44 = ( inf @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.order_iff
% 0.18/0.44 thf(fact_84_local_Oinf__absorb1,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less_eq @ X @ Y )
% 0.18/0.44 => ( ( inf @ X @ Y )
% 0.18/0.44 = X ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf_absorb1
% 0.18/0.44 thf(fact_85_local_Oinf__absorb2,axiom,
% 0.18/0.44 ! [Y: a,X: a] :
% 0.18/0.44 ( ( less_eq @ Y @ X )
% 0.18/0.44 => ( ( inf @ X @ Y )
% 0.18/0.44 = Y ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf_absorb2
% 0.18/0.44 thf(fact_86_local_Oinf__greatest,axiom,
% 0.18/0.44 ! [X: a,Y: a,Z2: a] :
% 0.18/0.44 ( ( less_eq @ X @ Y )
% 0.18/0.44 => ( ( less_eq @ X @ Z2 )
% 0.18/0.44 => ( less_eq @ X @ ( inf @ Y @ Z2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf_greatest
% 0.18/0.44 thf(fact_87_local_Oinf__le1,axiom,
% 0.18/0.44 ! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ X ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf_le1
% 0.18/0.44 thf(fact_88_local_Oinf__le2,axiom,
% 0.18/0.44 ! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ Y ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf_le2
% 0.18/0.44 thf(fact_89_local_Oinf__mono,axiom,
% 0.18/0.44 ! [A2: a,C: a,B2: a,D: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ C )
% 0.18/0.44 => ( ( less_eq @ B2 @ D )
% 0.18/0.44 => ( less_eq @ ( inf @ A2 @ B2 ) @ ( inf @ C @ D ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf_mono
% 0.18/0.44 thf(fact_90_local_Oinf__unique,axiom,
% 0.18/0.44 ! [F: a > a > a,X: a,Y: a] :
% 0.18/0.44 ( ! [X3: a,Y4: a] : ( less_eq @ ( F @ X3 @ Y4 ) @ X3 )
% 0.18/0.44 => ( ! [X3: a,Y4: a] : ( less_eq @ ( F @ X3 @ Y4 ) @ Y4 )
% 0.18/0.44 => ( ! [X3: a,Y4: a,Z3: a] :
% 0.18/0.44 ( ( less_eq @ X3 @ Y4 )
% 0.18/0.44 => ( ( less_eq @ X3 @ Z3 )
% 0.18/0.44 => ( less_eq @ X3 @ ( F @ Y4 @ Z3 ) ) ) )
% 0.18/0.44 => ( ( inf @ X @ Y )
% 0.18/0.44 = ( F @ X @ Y ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf_unique
% 0.18/0.44 thf(fact_91_local_Ole__iff__inf,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less_eq @ X @ Y )
% 0.18/0.44 = ( ( inf @ X @ Y )
% 0.18/0.44 = X ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.le_iff_inf
% 0.18/0.44 thf(fact_92_local_Ole__infE,axiom,
% 0.18/0.44 ! [X: a,A2: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ X @ ( inf @ A2 @ B2 ) )
% 0.18/0.44 => ~ ( ( less_eq @ X @ A2 )
% 0.18/0.44 => ~ ( less_eq @ X @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.le_infE
% 0.18/0.44 thf(fact_93_local_Ole__infI,axiom,
% 0.18/0.44 ! [X: a,A2: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ X @ A2 )
% 0.18/0.44 => ( ( less_eq @ X @ B2 )
% 0.18/0.44 => ( less_eq @ X @ ( inf @ A2 @ B2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.le_infI
% 0.18/0.44 thf(fact_94_local_Ole__infI1,axiom,
% 0.18/0.44 ! [A2: a,X: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ X )
% 0.18/0.44 => ( less_eq @ ( inf @ A2 @ B2 ) @ X ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.le_infI1
% 0.18/0.44 thf(fact_95_local_Ole__infI2,axiom,
% 0.18/0.44 ! [B2: a,X: a,A2: a] :
% 0.18/0.44 ( ( less_eq @ B2 @ X )
% 0.18/0.44 => ( less_eq @ ( inf @ A2 @ B2 ) @ X ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.le_infI2
% 0.18/0.44 thf(fact_96_local_Ole__iff__sup,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less_eq @ X @ Y )
% 0.18/0.44 = ( ( sup @ X @ Y )
% 0.18/0.44 = Y ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.le_iff_sup
% 0.18/0.44 thf(fact_97_local_Ole__supE,axiom,
% 0.18/0.44 ! [A2: a,B2: a,X: a] :
% 0.18/0.44 ( ( less_eq @ ( sup @ A2 @ B2 ) @ X )
% 0.18/0.44 => ~ ( ( less_eq @ A2 @ X )
% 0.18/0.44 => ~ ( less_eq @ B2 @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.le_supE
% 0.18/0.44 thf(fact_98_local_Ole__supI,axiom,
% 0.18/0.44 ! [A2: a,X: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ X )
% 0.18/0.44 => ( ( less_eq @ B2 @ X )
% 0.18/0.44 => ( less_eq @ ( sup @ A2 @ B2 ) @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.le_supI
% 0.18/0.44 thf(fact_99_local_Ole__supI1,axiom,
% 0.18/0.44 ! [X: a,A2: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ X @ A2 )
% 0.18/0.44 => ( less_eq @ X @ ( sup @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.le_supI1
% 0.18/0.44 thf(fact_100_local_Ole__supI2,axiom,
% 0.18/0.44 ! [X: a,B2: a,A2: a] :
% 0.18/0.44 ( ( less_eq @ X @ B2 )
% 0.18/0.44 => ( less_eq @ X @ ( sup @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.le_supI2
% 0.18/0.44 thf(fact_101_local_Osup_Oabsorb1,axiom,
% 0.18/0.44 ! [B2: a,A2: a] :
% 0.18/0.44 ( ( less_eq @ B2 @ A2 )
% 0.18/0.44 => ( ( sup @ A2 @ B2 )
% 0.18/0.44 = A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.absorb1
% 0.18/0.44 thf(fact_102_local_Osup_Oabsorb2,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 => ( ( sup @ A2 @ B2 )
% 0.18/0.44 = B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.absorb2
% 0.18/0.44 thf(fact_103_local_Osup_Oabsorb__iff1,axiom,
% 0.18/0.44 ! [B2: a,A2: a] :
% 0.18/0.44 ( ( less_eq @ B2 @ A2 )
% 0.18/0.44 = ( ( sup @ A2 @ B2 )
% 0.18/0.44 = A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.absorb_iff1
% 0.18/0.44 thf(fact_104_local_Osup_Oabsorb__iff2,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 = ( ( sup @ A2 @ B2 )
% 0.18/0.44 = B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.absorb_iff2
% 0.18/0.44 thf(fact_105_local_Osup_OboundedE,axiom,
% 0.18/0.44 ! [B2: a,C: a,A2: a] :
% 0.18/0.44 ( ( less_eq @ ( sup @ B2 @ C ) @ A2 )
% 0.18/0.44 => ~ ( ( less_eq @ B2 @ A2 )
% 0.18/0.44 => ~ ( less_eq @ C @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.boundedE
% 0.18/0.44 thf(fact_106_local_Osup_OboundedI,axiom,
% 0.18/0.44 ! [B2: a,A2: a,C: a] :
% 0.18/0.44 ( ( less_eq @ B2 @ A2 )
% 0.18/0.44 => ( ( less_eq @ C @ A2 )
% 0.18/0.44 => ( less_eq @ ( sup @ B2 @ C ) @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.boundedI
% 0.18/0.44 thf(fact_107_local_Osup_Ocobounded1,axiom,
% 0.18/0.44 ! [A2: a,B2: a] : ( less_eq @ A2 @ ( sup @ A2 @ B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.cobounded1
% 0.18/0.44 thf(fact_108_local_Osup_Ocobounded2,axiom,
% 0.18/0.44 ! [B2: a,A2: a] : ( less_eq @ B2 @ ( sup @ A2 @ B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.cobounded2
% 0.18/0.44 thf(fact_109_local_Osup_OcoboundedI1,axiom,
% 0.18/0.44 ! [C: a,A2: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ C @ A2 )
% 0.18/0.44 => ( less_eq @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.coboundedI1
% 0.18/0.44 thf(fact_110_local_Osup_OcoboundedI2,axiom,
% 0.18/0.44 ! [C: a,B2: a,A2: a] :
% 0.18/0.44 ( ( less_eq @ C @ B2 )
% 0.18/0.44 => ( less_eq @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.coboundedI2
% 0.18/0.44 thf(fact_111_local_Osup_Omono,axiom,
% 0.18/0.44 ! [C: a,A2: a,D: a,B2: a] :
% 0.18/0.44 ( ( less_eq @ C @ A2 )
% 0.18/0.44 => ( ( less_eq @ D @ B2 )
% 0.18/0.44 => ( less_eq @ ( sup @ C @ D ) @ ( sup @ A2 @ B2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.mono
% 0.18/0.44 thf(fact_112_local_Osup_OorderE,axiom,
% 0.18/0.44 ! [B2: a,A2: a] :
% 0.18/0.44 ( ( less_eq @ B2 @ A2 )
% 0.18/0.44 => ( A2
% 0.18/0.44 = ( sup @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.orderE
% 0.18/0.44 thf(fact_113_local_Osup_OorderI,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( A2
% 0.18/0.44 = ( sup @ A2 @ B2 ) )
% 0.18/0.44 => ( less_eq @ B2 @ A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.orderI
% 0.18/0.44 thf(fact_114_local_Osup_Oorder__iff,axiom,
% 0.18/0.44 ! [B2: a,A2: a] :
% 0.18/0.44 ( ( less_eq @ B2 @ A2 )
% 0.18/0.44 = ( A2
% 0.18/0.44 = ( sup @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.order_iff
% 0.18/0.44 thf(fact_115_local_Osup__absorb1,axiom,
% 0.18/0.44 ! [Y: a,X: a] :
% 0.18/0.44 ( ( less_eq @ Y @ X )
% 0.18/0.44 => ( ( sup @ X @ Y )
% 0.18/0.44 = X ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup_absorb1
% 0.18/0.44 thf(fact_116_local_Osup__absorb2,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( less_eq @ X @ Y )
% 0.18/0.44 => ( ( sup @ X @ Y )
% 0.18/0.44 = Y ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup_absorb2
% 0.18/0.44 thf(fact_117_local_Osup__ge1,axiom,
% 0.18/0.44 ! [X: a,Y: a] : ( less_eq @ X @ ( sup @ X @ Y ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup_ge1
% 0.18/0.44 thf(fact_118_local_Osup__ge2,axiom,
% 0.18/0.44 ! [Y: a,X: a] : ( less_eq @ Y @ ( sup @ X @ Y ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup_ge2
% 0.18/0.44 thf(fact_119_local_Osup__least,axiom,
% 0.18/0.44 ! [Y: a,X: a,Z2: a] :
% 0.18/0.44 ( ( less_eq @ Y @ X )
% 0.18/0.44 => ( ( less_eq @ Z2 @ X )
% 0.18/0.44 => ( less_eq @ ( sup @ Y @ Z2 ) @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup_least
% 0.18/0.44 thf(fact_120_local_Osup__mono,axiom,
% 0.18/0.44 ! [A2: a,C: a,B2: a,D: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ C )
% 0.18/0.44 => ( ( less_eq @ B2 @ D )
% 0.18/0.44 => ( less_eq @ ( sup @ A2 @ B2 ) @ ( sup @ C @ D ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup_mono
% 0.18/0.44 thf(fact_121_local_Osup__unique,axiom,
% 0.18/0.44 ! [F: a > a > a,X: a,Y: a] :
% 0.18/0.44 ( ! [X3: a,Y4: a] : ( less_eq @ X3 @ ( F @ X3 @ Y4 ) )
% 0.18/0.44 => ( ! [X3: a,Y4: a] : ( less_eq @ Y4 @ ( F @ X3 @ Y4 ) )
% 0.18/0.44 => ( ! [X3: a,Y4: a,Z3: a] :
% 0.18/0.44 ( ( less_eq @ Y4 @ X3 )
% 0.18/0.44 => ( ( less_eq @ Z3 @ X3 )
% 0.18/0.44 => ( less_eq @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
% 0.18/0.44 => ( ( sup @ X @ Y )
% 0.18/0.44 = ( F @ X @ Y ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup_unique
% 0.18/0.44 thf(fact_122_local_Oinf_Ostrict__boundedE,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( less @ A2 @ ( inf @ B2 @ C ) )
% 0.18/0.44 => ~ ( ( less @ A2 @ B2 )
% 0.18/0.44 => ~ ( less @ A2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.strict_boundedE
% 0.18/0.44 thf(fact_123_local_Oinf_Ostrict__coboundedI1,axiom,
% 0.18/0.44 ! [A2: a,C: a,B2: a] :
% 0.18/0.44 ( ( less @ A2 @ C )
% 0.18/0.44 => ( less @ ( inf @ A2 @ B2 ) @ C ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.strict_coboundedI1
% 0.18/0.44 thf(fact_124_local_Oinf_Ostrict__coboundedI2,axiom,
% 0.18/0.44 ! [B2: a,C: a,A2: a] :
% 0.18/0.44 ( ( less @ B2 @ C )
% 0.18/0.44 => ( less @ ( inf @ A2 @ B2 ) @ C ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.strict_coboundedI2
% 0.18/0.44 thf(fact_125_local_Oinf_Ostrict__order__iff,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( less @ A2 @ B2 )
% 0.18/0.44 = ( ( A2
% 0.18/0.44 = ( inf @ A2 @ B2 ) )
% 0.18/0.44 & ( A2 != B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.strict_order_iff
% 0.18/0.44 thf(fact_126_local_Oless__infI1,axiom,
% 0.18/0.44 ! [A2: a,X: a,B2: a] :
% 0.18/0.44 ( ( less @ A2 @ X )
% 0.18/0.44 => ( less @ ( inf @ A2 @ B2 ) @ X ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_infI1
% 0.18/0.44 thf(fact_127_local_Oless__infI2,axiom,
% 0.18/0.44 ! [B2: a,X: a,A2: a] :
% 0.18/0.44 ( ( less @ B2 @ X )
% 0.18/0.44 => ( less @ ( inf @ A2 @ B2 ) @ X ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_infI2
% 0.18/0.44 thf(fact_128_local_Oless__supI1,axiom,
% 0.18/0.44 ! [X: a,A2: a,B2: a] :
% 0.18/0.44 ( ( less @ X @ A2 )
% 0.18/0.44 => ( less @ X @ ( sup @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_supI1
% 0.18/0.44 thf(fact_129_local_Oless__supI2,axiom,
% 0.18/0.44 ! [X: a,B2: a,A2: a] :
% 0.18/0.44 ( ( less @ X @ B2 )
% 0.18/0.44 => ( less @ X @ ( sup @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.less_supI2
% 0.18/0.44 thf(fact_130_local_Osup_Ostrict__boundedE,axiom,
% 0.18/0.44 ! [B2: a,C: a,A2: a] :
% 0.18/0.44 ( ( less @ ( sup @ B2 @ C ) @ A2 )
% 0.18/0.44 => ~ ( ( less @ B2 @ A2 )
% 0.18/0.44 => ~ ( less @ C @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.strict_boundedE
% 0.18/0.44 thf(fact_131_local_Osup_Ostrict__coboundedI1,axiom,
% 0.18/0.44 ! [C: a,A2: a,B2: a] :
% 0.18/0.44 ( ( less @ C @ A2 )
% 0.18/0.44 => ( less @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.strict_coboundedI1
% 0.18/0.44 thf(fact_132_local_Osup_Ostrict__coboundedI2,axiom,
% 0.18/0.44 ! [C: a,B2: a,A2: a] :
% 0.18/0.44 ( ( less @ C @ B2 )
% 0.18/0.44 => ( less @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.strict_coboundedI2
% 0.18/0.44 thf(fact_133_local_Osup_Ostrict__order__iff,axiom,
% 0.18/0.44 ! [B2: a,A2: a] :
% 0.18/0.44 ( ( less @ B2 @ A2 )
% 0.18/0.44 = ( ( A2
% 0.18/0.44 = ( sup @ A2 @ B2 ) )
% 0.18/0.44 & ( A2 != B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.strict_order_iff
% 0.18/0.44 thf(fact_134_local_Odistrib__imp1,axiom,
% 0.18/0.44 ! [X: a,Y: a,Z2: a] :
% 0.18/0.44 ( ! [X3: a,Y4: a,Z3: a] :
% 0.18/0.44 ( ( inf @ X3 @ ( sup @ Y4 @ Z3 ) )
% 0.18/0.44 = ( sup @ ( inf @ X3 @ Y4 ) @ ( inf @ X3 @ Z3 ) ) )
% 0.18/0.44 => ( ( sup @ X @ ( inf @ Y @ Z2 ) )
% 0.18/0.44 = ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.distrib_imp1
% 0.18/0.44 thf(fact_135_local_Odistrib__imp2,axiom,
% 0.18/0.44 ! [X: a,Y: a,Z2: a] :
% 0.18/0.44 ( ! [X3: a,Y4: a,Z3: a] :
% 0.18/0.44 ( ( sup @ X3 @ ( inf @ Y4 @ Z3 ) )
% 0.18/0.44 = ( inf @ ( sup @ X3 @ Y4 ) @ ( sup @ X3 @ Z3 ) ) )
% 0.18/0.44 => ( ( inf @ X @ ( sup @ Y @ Z2 ) )
% 0.18/0.44 = ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.distrib_imp2
% 0.18/0.44 thf(fact_136_local_Odistrib__inf__le,axiom,
% 0.18/0.44 ! [X: a,Y: a,Z2: a] : ( less_eq @ ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z2 ) ) @ ( inf @ X @ ( sup @ Y @ Z2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.distrib_inf_le
% 0.18/0.44 thf(fact_137_local_Odistrib__sup__le,axiom,
% 0.18/0.44 ! [X: a,Y: a,Z2: a] : ( less_eq @ ( sup @ X @ ( inf @ Y @ Z2 ) ) @ ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.distrib_sup_le
% 0.18/0.44 thf(fact_138_local_Oincomp__def,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( modula1937309406incomp @ a @ less_eq @ X @ Y )
% 0.18/0.44 = ( ~ ( less_eq @ X @ Y )
% 0.18/0.44 & ~ ( less_eq @ Y @ X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.incomp_def
% 0.18/0.44 thf(fact_139_local_Oorder_Orefl,axiom,
% 0.18/0.44 ! [A2: a] : ( less_eq @ A2 @ A2 ) ).
% 0.18/0.44
% 0.18/0.44 % local.order.refl
% 0.18/0.44 thf(fact_140_local_Oorder__refl,axiom,
% 0.18/0.44 ! [X: a] : ( less_eq @ X @ X ) ).
% 0.18/0.44
% 0.18/0.44 % local.order_refl
% 0.18/0.44 thf(fact_141_local_Oinf_Oidem,axiom,
% 0.18/0.44 ! [A2: a] :
% 0.18/0.44 ( ( inf @ A2 @ A2 )
% 0.18/0.44 = A2 ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.idem
% 0.18/0.44 thf(fact_142_local_Oinf_Oleft__idem,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( inf @ A2 @ ( inf @ A2 @ B2 ) )
% 0.18/0.44 = ( inf @ A2 @ B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.left_idem
% 0.18/0.44 thf(fact_143_local_Oinf_Oright__idem,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( inf @ ( inf @ A2 @ B2 ) @ B2 )
% 0.18/0.44 = ( inf @ A2 @ B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.right_idem
% 0.18/0.44 thf(fact_144_local_Oinf__idem,axiom,
% 0.18/0.44 ! [X: a] :
% 0.18/0.44 ( ( inf @ X @ X )
% 0.18/0.44 = X ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf_idem
% 0.18/0.44 thf(fact_145_local_Oinf__left__idem,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( inf @ X @ ( inf @ X @ Y ) )
% 0.18/0.44 = ( inf @ X @ Y ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf_left_idem
% 0.18/0.44 thf(fact_146_local_Oinf__right__idem,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( inf @ ( inf @ X @ Y ) @ Y )
% 0.18/0.44 = ( inf @ X @ Y ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf_right_idem
% 0.18/0.44 thf(fact_147_local_Osup_Oidem,axiom,
% 0.18/0.44 ! [A2: a] :
% 0.18/0.44 ( ( sup @ A2 @ A2 )
% 0.18/0.44 = A2 ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.idem
% 0.18/0.44 thf(fact_148_local_Osup_Oleft__idem,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( sup @ A2 @ ( sup @ A2 @ B2 ) )
% 0.18/0.44 = ( sup @ A2 @ B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.left_idem
% 0.18/0.44 thf(fact_149_local_Osup_Oright__idem,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( sup @ ( sup @ A2 @ B2 ) @ B2 )
% 0.18/0.44 = ( sup @ A2 @ B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.right_idem
% 0.18/0.44 thf(fact_150_local_Osup__idem,axiom,
% 0.18/0.44 ! [X: a] :
% 0.18/0.44 ( ( sup @ X @ X )
% 0.18/0.44 = X ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup_idem
% 0.18/0.44 thf(fact_151_local_Osup__left__idem,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( sup @ X @ ( sup @ X @ Y ) )
% 0.18/0.44 = ( sup @ X @ Y ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup_left_idem
% 0.18/0.44 thf(fact_152_local_OGreatestI2__order,axiom,
% 0.18/0.44 ! [P: a > $o,X: a,Q: a > $o] :
% 0.18/0.44 ( ( P @ X )
% 0.18/0.44 => ( ! [Y4: a] :
% 0.18/0.44 ( ( P @ Y4 )
% 0.18/0.44 => ( less_eq @ Y4 @ X ) )
% 0.18/0.44 => ( ! [X3: a] :
% 0.18/0.44 ( ( P @ X3 )
% 0.18/0.44 => ( ! [Y5: a] :
% 0.18/0.44 ( ( P @ Y5 )
% 0.18/0.44 => ( less_eq @ Y5 @ X3 ) )
% 0.18/0.44 => ( Q @ X3 ) ) )
% 0.18/0.44 => ( Q @ ( greatest @ a @ less_eq @ P ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.GreatestI2_order
% 0.18/0.44 thf(fact_153_local_OGreatest__equality,axiom,
% 0.18/0.44 ! [P: a > $o,X: a] :
% 0.18/0.44 ( ( P @ X )
% 0.18/0.44 => ( ! [Y4: a] :
% 0.18/0.44 ( ( P @ Y4 )
% 0.18/0.44 => ( less_eq @ Y4 @ X ) )
% 0.18/0.44 => ( ( greatest @ a @ less_eq @ P )
% 0.18/0.44 = X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.Greatest_equality
% 0.18/0.44 thf(fact_154_local_Omax__def,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 => ( ( max @ a @ less_eq @ A2 @ B2 )
% 0.18/0.44 = B2 ) )
% 0.18/0.44 & ( ~ ( less_eq @ A2 @ B2 )
% 0.18/0.44 => ( ( max @ a @ less_eq @ A2 @ B2 )
% 0.18/0.44 = A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.max_def
% 0.18/0.44 thf(fact_155_local_Omin__def,axiom,
% 0.18/0.44 ! [A2: a,B2: a] :
% 0.18/0.44 ( ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 => ( ( min @ a @ less_eq @ A2 @ B2 )
% 0.18/0.44 = A2 ) )
% 0.18/0.44 & ( ~ ( less_eq @ A2 @ B2 )
% 0.18/0.44 => ( ( min @ a @ less_eq @ A2 @ B2 )
% 0.18/0.44 = B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.min_def
% 0.18/0.44 thf(fact_156_local_Ois__arg__max__def,axiom,
% 0.18/0.44 ! [B: $tType,F: B > a,P: B > $o,X: B] :
% 0.18/0.44 ( ( lattices_is_arg_max @ a @ B @ less @ F @ P @ X )
% 0.18/0.44 = ( ( P @ X )
% 0.18/0.44 & ~ ? [Y3: B] :
% 0.18/0.44 ( ( P @ Y3 )
% 0.18/0.44 & ( less @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.is_arg_max_def
% 0.18/0.44 thf(fact_157_local_Ois__arg__min__def,axiom,
% 0.18/0.44 ! [B: $tType,F: B > a,P: B > $o,X: B] :
% 0.18/0.44 ( ( lattices_is_arg_min @ a @ B @ less @ F @ P @ X )
% 0.18/0.44 = ( ( P @ X )
% 0.18/0.44 & ~ ? [Y3: B] :
% 0.18/0.44 ( ( P @ Y3 )
% 0.18/0.44 & ( less @ ( F @ Y3 ) @ ( F @ X ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.is_arg_min_def
% 0.18/0.44 thf(fact_158_local_ON5__lattice__def,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( modula1192512645attice @ a @ inf @ less @ sup @ A2 @ B2 @ C )
% 0.18/0.44 = ( ( ( inf @ A2 @ C )
% 0.18/0.44 = ( inf @ B2 @ C ) )
% 0.18/0.44 & ( less @ A2 @ B2 )
% 0.18/0.44 & ( ( sup @ A2 @ C )
% 0.18/0.44 = ( sup @ B2 @ C ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.N5_lattice_def
% 0.18/0.44 thf(fact_159_local_Ono__distrib__def,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( modula141148890istrib @ a @ inf @ less @ sup @ A2 @ B2 @ C )
% 0.18/0.44 = ( less @ ( sup @ ( inf @ A2 @ B2 ) @ ( inf @ C @ A2 ) ) @ ( inf @ A2 @ ( sup @ B2 @ C ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.no_distrib_def
% 0.18/0.44 thf(fact_160_local_Oinf_Obounded__iff,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( less_eq @ A2 @ ( inf @ B2 @ C ) )
% 0.18/0.44 = ( ( less_eq @ A2 @ B2 )
% 0.18/0.44 & ( less_eq @ A2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf.bounded_iff
% 0.18/0.44 thf(fact_161_local_Ole__inf__iff,axiom,
% 0.18/0.44 ! [X: a,Y: a,Z2: a] :
% 0.18/0.44 ( ( less_eq @ X @ ( inf @ Y @ Z2 ) )
% 0.18/0.44 = ( ( less_eq @ X @ Y )
% 0.18/0.44 & ( less_eq @ X @ Z2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.le_inf_iff
% 0.18/0.44 thf(fact_162_local_Ole__sup__iff,axiom,
% 0.18/0.44 ! [X: a,Y: a,Z2: a] :
% 0.18/0.44 ( ( less_eq @ ( sup @ X @ Y ) @ Z2 )
% 0.18/0.44 = ( ( less_eq @ X @ Z2 )
% 0.18/0.44 & ( less_eq @ Y @ Z2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.le_sup_iff
% 0.18/0.44 thf(fact_163_local_Osup_Obounded__iff,axiom,
% 0.18/0.44 ! [B2: a,C: a,A2: a] :
% 0.18/0.44 ( ( less_eq @ ( sup @ B2 @ C ) @ A2 )
% 0.18/0.44 = ( ( less_eq @ B2 @ A2 )
% 0.18/0.44 & ( less_eq @ C @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup.bounded_iff
% 0.18/0.44 thf(fact_164_local_Oinf__sup__absorb,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( inf @ X @ ( sup @ X @ Y ) )
% 0.18/0.44 = X ) ).
% 0.18/0.44
% 0.18/0.44 % local.inf_sup_absorb
% 0.18/0.44 thf(fact_165_local_Osup__inf__absorb,axiom,
% 0.18/0.44 ! [X: a,Y: a] :
% 0.18/0.44 ( ( sup @ X @ ( inf @ X @ Y ) )
% 0.18/0.44 = X ) ).
% 0.18/0.44
% 0.18/0.44 % local.sup_inf_absorb
% 0.18/0.44 thf(fact_166_local_OM5__lattice__def,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( modula1618706180attice @ a @ inf @ less @ sup @ A2 @ B2 @ C )
% 0.18/0.44 = ( ( ( inf @ A2 @ B2 )
% 0.18/0.44 = ( inf @ B2 @ C ) )
% 0.18/0.44 & ( ( inf @ C @ A2 )
% 0.18/0.44 = ( inf @ B2 @ C ) )
% 0.18/0.44 & ( ( sup @ A2 @ B2 )
% 0.18/0.44 = ( sup @ B2 @ C ) )
% 0.18/0.44 & ( ( sup @ C @ A2 )
% 0.18/0.44 = ( sup @ B2 @ C ) )
% 0.18/0.44 & ( less @ ( inf @ A2 @ B2 ) @ ( sup @ A2 @ B2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.M5_lattice_def
% 0.18/0.44 thf(fact_167_local_OLeast1I,axiom,
% 0.18/0.44 ! [P: a > $o] :
% 0.18/0.44 ( ? [X4: a] :
% 0.18/0.44 ( ( P @ X4 )
% 0.18/0.44 & ! [Y4: a] :
% 0.18/0.44 ( ( P @ Y4 )
% 0.18/0.44 => ( less_eq @ X4 @ Y4 ) )
% 0.18/0.44 & ! [Y4: a] :
% 0.18/0.44 ( ( ( P @ Y4 )
% 0.18/0.44 & ! [Ya: a] :
% 0.18/0.44 ( ( P @ Ya )
% 0.18/0.44 => ( less_eq @ Y4 @ Ya ) ) )
% 0.18/0.44 => ( Y4 = X4 ) ) )
% 0.18/0.44 => ( P @ ( least @ a @ less_eq @ P ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.Least1I
% 0.18/0.44 thf(fact_168_local_OLeast1__le,axiom,
% 0.18/0.44 ! [P: a > $o,Z2: a] :
% 0.18/0.44 ( ? [X4: a] :
% 0.18/0.44 ( ( P @ X4 )
% 0.18/0.44 & ! [Y4: a] :
% 0.18/0.44 ( ( P @ Y4 )
% 0.18/0.44 => ( less_eq @ X4 @ Y4 ) )
% 0.18/0.44 & ! [Y4: a] :
% 0.18/0.44 ( ( ( P @ Y4 )
% 0.18/0.44 & ! [Ya: a] :
% 0.18/0.44 ( ( P @ Ya )
% 0.18/0.44 => ( less_eq @ Y4 @ Ya ) ) )
% 0.18/0.44 => ( Y4 = X4 ) ) )
% 0.18/0.44 => ( ( P @ Z2 )
% 0.18/0.44 => ( less_eq @ ( least @ a @ less_eq @ P ) @ Z2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.Least1_le
% 0.18/0.44 thf(fact_169_local_OLeastI2__order,axiom,
% 0.18/0.44 ! [P: a > $o,X: a,Q: a > $o] :
% 0.18/0.44 ( ( P @ X )
% 0.18/0.44 => ( ! [Y4: a] :
% 0.18/0.44 ( ( P @ Y4 )
% 0.18/0.44 => ( less_eq @ X @ Y4 ) )
% 0.18/0.44 => ( ! [X3: a] :
% 0.18/0.44 ( ( P @ X3 )
% 0.18/0.44 => ( ! [Y5: a] :
% 0.18/0.44 ( ( P @ Y5 )
% 0.18/0.44 => ( less_eq @ X3 @ Y5 ) )
% 0.18/0.44 => ( Q @ X3 ) ) )
% 0.18/0.44 => ( Q @ ( least @ a @ less_eq @ P ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.LeastI2_order
% 0.18/0.44 thf(fact_170_local_OLeast__equality,axiom,
% 0.18/0.44 ! [P: a > $o,X: a] :
% 0.18/0.44 ( ( P @ X )
% 0.18/0.44 => ( ! [Y4: a] :
% 0.18/0.44 ( ( P @ Y4 )
% 0.18/0.44 => ( less_eq @ X @ Y4 ) )
% 0.18/0.44 => ( ( least @ a @ less_eq @ P )
% 0.18/0.44 = X ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.Least_equality
% 0.18/0.44 thf(fact_171_local_Oa__join__d,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( sup @ A2 @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 0.18/0.44 = ( sup @ A2 @ ( inf @ B2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.a_join_d
% 0.18/0.44 thf(fact_172_local_Ob__join__d,axiom,
% 0.18/0.44 ! [B2: a,A2: a,C: a] :
% 0.18/0.44 ( ( sup @ B2 @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 0.18/0.44 = ( sup @ B2 @ ( inf @ C @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.b_join_d
% 0.18/0.44 thf(fact_173_local_Od__aux__def,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 0.18/0.44 = ( sup @ ( sup @ ( inf @ A2 @ B2 ) @ ( inf @ B2 @ C ) ) @ ( inf @ C @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.d_aux_def
% 0.18/0.44 thf(fact_174_local_Od__b__c__a,axiom,
% 0.18/0.44 ! [B2: a,C: a,A2: a] :
% 0.18/0.44 ( ( modula1221160330_d_aux @ a @ inf @ sup @ B2 @ C @ A2 )
% 0.18/0.44 = ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.d_b_c_a
% 0.18/0.44 thf(fact_175_local_Od__c__a__b,axiom,
% 0.18/0.44 ! [C: a,A2: a,B2: a] :
% 0.18/0.44 ( ( modula1221160330_d_aux @ a @ inf @ sup @ C @ A2 @ B2 )
% 0.18/0.44 = ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.d_c_a_b
% 0.18/0.44 thf(fact_176_local_Oa__meet__e,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( inf @ A2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 0.18/0.44 = ( inf @ A2 @ ( sup @ B2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.a_meet_e
% 0.18/0.44 thf(fact_177_local_Ob__meet__e,axiom,
% 0.18/0.44 ! [B2: a,A2: a,C: a] :
% 0.18/0.44 ( ( inf @ B2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 0.18/0.44 = ( inf @ B2 @ ( sup @ C @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.b_meet_e
% 0.18/0.44 thf(fact_178_local_Oc__meet__e,axiom,
% 0.18/0.44 ! [C: a,A2: a,B2: a] :
% 0.18/0.44 ( ( inf @ C @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 0.18/0.44 = ( inf @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.c_meet_e
% 0.18/0.44 thf(fact_179_local_Oe__aux__def,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 0.18/0.44 = ( inf @ ( inf @ ( sup @ A2 @ B2 ) @ ( sup @ B2 @ C ) ) @ ( sup @ C @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.e_aux_def
% 0.18/0.44 thf(fact_180_local_Oe__b__c__a,axiom,
% 0.18/0.44 ! [B2: a,C: a,A2: a] :
% 0.18/0.44 ( ( modula1593167561_e_aux @ a @ inf @ sup @ B2 @ C @ A2 )
% 0.18/0.44 = ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.e_b_c_a
% 0.18/0.44 thf(fact_181_local_Oe__c__a__b,axiom,
% 0.18/0.44 ! [C: a,A2: a,B2: a] :
% 0.18/0.44 ( ( modula1593167561_e_aux @ a @ inf @ sup @ C @ A2 @ B2 )
% 0.18/0.44 = ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.e_c_a_b
% 0.18/0.44 thf(fact_182_local_OInf__fin_Osemilattice__order__set__axioms,axiom,
% 0.18/0.44 lattic1693879045er_set @ a @ inf @ less_eq @ less ).
% 0.18/0.44
% 0.18/0.44 % local.Inf_fin.semilattice_order_set_axioms
% 0.18/0.44 thf(fact_183_local_OM5__lattice__incomp,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( modula1618706180attice @ a @ inf @ less @ sup @ A2 @ B2 @ C )
% 0.18/0.44 => ( modula1937309406incomp @ a @ less_eq @ A2 @ B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.M5_lattice_incomp
% 0.18/0.44 thf(fact_184_local_Oc__aux__def,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( modula849153099_c_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 0.18/0.44 = ( sup @ ( inf @ C @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.c_aux_def
% 0.18/0.44 thf(fact_185_local_Ob__aux__def,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( modula477145868_b_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 0.18/0.44 = ( sup @ ( inf @ B2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.b_aux_def
% 0.18/0.44 thf(fact_186_local_Oa__aux__def,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( modula105138637_a_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 0.18/0.44 = ( sup @ ( inf @ A2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.a_aux_def
% 0.18/0.44 thf(fact_187_lattice_Ono__distrib_Ocong,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( modula141148890istrib @ A )
% 0.18/0.44 = ( modula141148890istrib @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 % lattice.no_distrib.cong
% 0.18/0.44 thf(fact_188_lattice_ON5__lattice_Ocong,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( modula1192512645attice @ A )
% 0.18/0.44 = ( modula1192512645attice @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 % lattice.N5_lattice.cong
% 0.18/0.44 thf(fact_189_lattice_OM5__lattice_Ocong,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( modula1618706180attice @ A )
% 0.18/0.44 = ( modula1618706180attice @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 % lattice.M5_lattice.cong
% 0.18/0.44 thf(fact_190_lattice_Oincomp_Ocong,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( modula1937309406incomp @ A )
% 0.18/0.44 = ( modula1937309406incomp @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 % lattice.incomp.cong
% 0.18/0.44 thf(fact_191_lattice_Oe__aux_Ocong,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( modula1593167561_e_aux @ A )
% 0.18/0.44 = ( modula1593167561_e_aux @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 % lattice.e_aux.cong
% 0.18/0.44 thf(fact_192_lattice_Od__aux_Ocong,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( modula1221160330_d_aux @ A )
% 0.18/0.44 = ( modula1221160330_d_aux @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 % lattice.d_aux.cong
% 0.18/0.44 thf(fact_193_local_Oinf_Osemilattice__order__axioms,axiom,
% 0.18/0.44 semilattice_order @ a @ inf @ less_eq @ less ).
% 0.18/0.44
% 0.18/0.44 % local.inf.semilattice_order_axioms
% 0.18/0.44 thf(fact_194_local_Oorder_Oordering__axioms,axiom,
% 0.18/0.44 ordering @ a @ less_eq @ less ).
% 0.18/0.44
% 0.18/0.44 % local.order.ordering_axioms
% 0.18/0.44 thf(fact_195_local_Ocomp__fun__idem__sup,axiom,
% 0.18/0.44 finite_comp_fun_idem @ a @ a @ sup ).
% 0.18/0.44
% 0.18/0.44 % local.comp_fun_idem_sup
% 0.18/0.44 thf(fact_196_local_Ocomp__fun__idem__inf,axiom,
% 0.18/0.44 finite_comp_fun_idem @ a @ a @ inf ).
% 0.18/0.44
% 0.18/0.44 % local.comp_fun_idem_inf
% 0.18/0.44 thf(fact_197_local_Ob__a,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( modula477145868_b_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 0.18/0.44 = ( modula105138637_a_aux @ a @ inf @ sup @ B2 @ C @ A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.b_a
% 0.18/0.44 thf(fact_198_local_Oc__a,axiom,
% 0.18/0.44 ! [A2: a,B2: a,C: a] :
% 0.18/0.44 ( ( modula849153099_c_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 0.18/0.44 = ( modula105138637_a_aux @ a @ inf @ sup @ C @ A2 @ B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.c_a
% 0.18/0.44 thf(fact_199_lattice_Oc__aux_Ocong,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( modula849153099_c_aux @ A )
% 0.18/0.44 = ( modula849153099_c_aux @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 % lattice.c_aux.cong
% 0.18/0.44 thf(fact_200_lattice_Ob__aux_Ocong,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( modula477145868_b_aux @ A )
% 0.18/0.44 = ( modula477145868_b_aux @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 % lattice.b_aux.cong
% 0.18/0.44 thf(fact_201_lattice_Oa__aux_Ocong,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( modula105138637_a_aux @ A )
% 0.18/0.44 = ( modula105138637_a_aux @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 % lattice.a_aux.cong
% 0.18/0.44 thf(fact_202_local_Osup_Osemigroup__axioms,axiom,
% 0.18/0.44 semigroup @ a @ sup ).
% 0.18/0.44
% 0.18/0.44 % local.sup.semigroup_axioms
% 0.18/0.44 thf(fact_203_local_Oinf_Osemigroup__axioms,axiom,
% 0.18/0.44 semigroup @ a @ inf ).
% 0.18/0.44
% 0.18/0.44 % local.inf.semigroup_axioms
% 0.18/0.44 thf(fact_204_local_Osup_Osemilattice__axioms,axiom,
% 0.18/0.44 semilattice @ a @ sup ).
% 0.18/0.44
% 0.18/0.44 % local.sup.semilattice_axioms
% 0.18/0.44 thf(fact_205_local_Oinf_Osemilattice__axioms,axiom,
% 0.18/0.44 semilattice @ a @ inf ).
% 0.18/0.44
% 0.18/0.44 % local.inf.semilattice_axioms
% 0.18/0.44 thf(fact_206_local_Osup_Oabel__semigroup__axioms,axiom,
% 0.18/0.44 abel_semigroup @ a @ sup ).
% 0.18/0.44
% 0.18/0.44 % local.sup.abel_semigroup_axioms
% 0.18/0.44 thf(fact_207_local_Oinf_Oabel__semigroup__axioms,axiom,
% 0.18/0.44 abel_semigroup @ a @ inf ).
% 0.18/0.44
% 0.18/0.44 % local.inf.abel_semigroup_axioms
% 0.18/0.44 thf(fact_208_semilattice__order__set_Oaxioms_I1_J,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.18/0.44 ( ( lattic1693879045er_set @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( semilattice_order @ A @ F @ Less_eq @ Less ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order_set.axioms(1)
% 0.18/0.44 thf(fact_209_abel__semigroup_Oaxioms_I1_J,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A] :
% 0.18/0.44 ( ( abel_semigroup @ A @ F )
% 0.18/0.44 => ( semigroup @ A @ F ) ) ).
% 0.18/0.44
% 0.18/0.44 % abel_semigroup.axioms(1)
% 0.18/0.44 thf(fact_210_abel__semigroup_Oleft__commute,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,B2: A,A2: A,C: A] :
% 0.18/0.44 ( ( abel_semigroup @ A @ F )
% 0.18/0.44 => ( ( F @ B2 @ ( F @ A2 @ C ) )
% 0.18/0.44 = ( F @ A2 @ ( F @ B2 @ C ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % abel_semigroup.left_commute
% 0.18/0.44 thf(fact_211_abel__semigroup_Ocommute,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 0.18/0.44 ( ( abel_semigroup @ A @ F )
% 0.18/0.44 => ( ( F @ A2 @ B2 )
% 0.18/0.44 = ( F @ B2 @ A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % abel_semigroup.commute
% 0.18/0.44 thf(fact_212_semigroup_Ointro,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A] :
% 0.18/0.44 ( ! [A5: A,B4: A,C2: A] :
% 0.18/0.44 ( ( F @ ( F @ A5 @ B4 ) @ C2 )
% 0.18/0.44 = ( F @ A5 @ ( F @ B4 @ C2 ) ) )
% 0.18/0.44 => ( semigroup @ A @ F ) ) ).
% 0.18/0.44
% 0.18/0.44 % semigroup.intro
% 0.18/0.44 thf(fact_213_semigroup_Oassoc,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,A2: A,B2: A,C: A] :
% 0.18/0.44 ( ( semigroup @ A @ F )
% 0.18/0.44 => ( ( F @ ( F @ A2 @ B2 ) @ C )
% 0.18/0.44 = ( F @ A2 @ ( F @ B2 @ C ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semigroup.assoc
% 0.18/0.44 thf(fact_214_semigroup__def,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( semigroup @ A )
% 0.18/0.44 = ( ^ [F2: A > A > A] :
% 0.18/0.44 ! [A3: A,B3: A,C3: A] :
% 0.18/0.44 ( ( F2 @ ( F2 @ A3 @ B3 ) @ C3 )
% 0.18/0.44 = ( F2 @ A3 @ ( F2 @ B3 @ C3 ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semigroup_def
% 0.18/0.44 thf(fact_215_ord_Ois__arg__min_Ocong,axiom,
% 0.18/0.44 ! [B: $tType,A: $tType] :
% 0.18/0.44 ( ( lattices_is_arg_min @ A @ B )
% 0.18/0.44 = ( lattices_is_arg_min @ A @ B ) ) ).
% 0.18/0.44
% 0.18/0.44 % ord.is_arg_min.cong
% 0.18/0.44 thf(fact_216_ord_Ois__arg__max_Ocong,axiom,
% 0.18/0.44 ! [B: $tType,A: $tType] :
% 0.18/0.44 ( ( lattices_is_arg_max @ A @ B )
% 0.18/0.44 = ( lattices_is_arg_max @ A @ B ) ) ).
% 0.18/0.44
% 0.18/0.44 % ord.is_arg_max.cong
% 0.18/0.44 thf(fact_217_ord_Ois__arg__min__def,axiom,
% 0.18/0.44 ! [B: $tType,A: $tType] :
% 0.18/0.44 ( ( lattices_is_arg_min @ A @ B )
% 0.18/0.44 = ( ^ [Less2: A > A > $o,F2: B > A,P2: B > $o,X2: B] :
% 0.18/0.44 ( ( P2 @ X2 )
% 0.18/0.44 & ~ ? [Y3: B] :
% 0.18/0.44 ( ( P2 @ Y3 )
% 0.18/0.44 & ( Less2 @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % ord.is_arg_min_def
% 0.18/0.44 thf(fact_218_ord_Ois__arg__max__def,axiom,
% 0.18/0.44 ! [B: $tType,A: $tType] :
% 0.18/0.44 ( ( lattices_is_arg_max @ A @ B )
% 0.18/0.44 = ( ^ [Less2: A > A > $o,F2: B > A,P2: B > $o,X2: B] :
% 0.18/0.44 ( ( P2 @ X2 )
% 0.18/0.44 & ~ ? [Y3: B] :
% 0.18/0.44 ( ( P2 @ Y3 )
% 0.18/0.44 & ( Less2 @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % ord.is_arg_max_def
% 0.18/0.44 thf(fact_219_semilattice_Oaxioms_I1_J,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A] :
% 0.18/0.44 ( ( semilattice @ A @ F )
% 0.18/0.44 => ( abel_semigroup @ A @ F ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice.axioms(1)
% 0.18/0.44 thf(fact_220_local_OSup__fin_Osemilattice__set__axioms,axiom,
% 0.18/0.44 lattic35693393ce_set @ a @ sup ).
% 0.18/0.44
% 0.18/0.44 % local.Sup_fin.semilattice_set_axioms
% 0.18/0.44 thf(fact_221_local_OInf__fin_Osemilattice__set__axioms,axiom,
% 0.18/0.44 lattic35693393ce_set @ a @ inf ).
% 0.18/0.44
% 0.18/0.44 % local.Inf_fin.semilattice_set_axioms
% 0.18/0.44 thf(fact_222_semilattice__order__set_Oaxioms_I2_J,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.18/0.44 ( ( lattic1693879045er_set @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( lattic35693393ce_set @ A @ F ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order_set.axioms(2)
% 0.18/0.44 thf(fact_223_semilattice__set__def,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( lattic35693393ce_set @ A )
% 0.18/0.44 = ( semilattice @ A ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_set_def
% 0.18/0.44 thf(fact_224_semilattice__set_Ointro,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A] :
% 0.18/0.44 ( ( semilattice @ A @ F )
% 0.18/0.44 => ( lattic35693393ce_set @ A @ F ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_set.intro
% 0.18/0.44 thf(fact_225_semilattice__set_Oaxioms,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A] :
% 0.18/0.44 ( ( lattic35693393ce_set @ A @ F )
% 0.18/0.44 => ( semilattice @ A @ F ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_set.axioms
% 0.18/0.44 thf(fact_226_semilattice__order__set__def,axiom,
% 0.18/0.44 ! [A: $tType] :
% 0.18/0.44 ( ( lattic1693879045er_set @ A )
% 0.18/0.44 = ( ^ [F2: A > A > A,Less_eq2: A > A > $o,Less2: A > A > $o] :
% 0.18/0.44 ( ( semilattice_order @ A @ F2 @ Less_eq2 @ Less2 )
% 0.18/0.44 & ( lattic35693393ce_set @ A @ F2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order_set_def
% 0.18/0.44 thf(fact_227_semilattice__order__set_Ointro,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( lattic35693393ce_set @ A @ F )
% 0.18/0.44 => ( lattic1693879045er_set @ A @ F @ Less_eq @ Less ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order_set.intro
% 0.18/0.44 thf(fact_228_semilattice__order_Omono,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,C: A,B2: A,D: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( Less_eq @ A2 @ C )
% 0.18/0.44 => ( ( Less_eq @ B2 @ D )
% 0.18/0.44 => ( Less_eq @ ( F @ A2 @ B2 ) @ ( F @ C @ D ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.mono
% 0.18/0.44 thf(fact_229_semilattice__order_OorderE,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( Less_eq @ A2 @ B2 )
% 0.18/0.44 => ( A2
% 0.18/0.44 = ( F @ A2 @ B2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.orderE
% 0.18/0.44 thf(fact_230_semilattice__order_OorderI,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( A2
% 0.18/0.44 = ( F @ A2 @ B2 ) )
% 0.18/0.44 => ( Less_eq @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.orderI
% 0.18/0.44 thf(fact_231_semilattice__order_Oabsorb1,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( Less_eq @ A2 @ B2 )
% 0.18/0.44 => ( ( F @ A2 @ B2 )
% 0.18/0.44 = A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.absorb1
% 0.18/0.44 thf(fact_232_semilattice__order_Oabsorb2,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A2: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( Less_eq @ B2 @ A2 )
% 0.18/0.44 => ( ( F @ A2 @ B2 )
% 0.18/0.44 = B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.absorb2
% 0.18/0.44 thf(fact_233_semilattice__order_OboundedE,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( Less_eq @ A2 @ ( F @ B2 @ C ) )
% 0.18/0.44 => ~ ( ( Less_eq @ A2 @ B2 )
% 0.18/0.44 => ~ ( Less_eq @ A2 @ C ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.boundedE
% 0.18/0.44 thf(fact_234_semilattice__order_OboundedI,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( Less_eq @ A2 @ B2 )
% 0.18/0.44 => ( ( Less_eq @ A2 @ C )
% 0.18/0.44 => ( Less_eq @ A2 @ ( F @ B2 @ C ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.boundedI
% 0.18/0.44 thf(fact_235_semilattice__order_Oorder__iff,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( Less_eq @ A2 @ B2 )
% 0.18/0.44 = ( A2
% 0.18/0.44 = ( F @ A2 @ B2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.order_iff
% 0.18/0.44 thf(fact_236_semilattice__order_Ocobounded1,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( Less_eq @ ( F @ A2 @ B2 ) @ A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.cobounded1
% 0.18/0.44 thf(fact_237_semilattice__order_Ocobounded2,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( Less_eq @ ( F @ A2 @ B2 ) @ B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.cobounded2
% 0.18/0.44 thf(fact_238_semilattice__order_Oabsorb__iff1,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( Less_eq @ A2 @ B2 )
% 0.18/0.44 = ( ( F @ A2 @ B2 )
% 0.18/0.44 = A2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.absorb_iff1
% 0.18/0.44 thf(fact_239_semilattice__order_Oabsorb__iff2,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A2: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( Less_eq @ B2 @ A2 )
% 0.18/0.44 = ( ( F @ A2 @ B2 )
% 0.18/0.44 = B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.absorb_iff2
% 0.18/0.44 thf(fact_240_semilattice__order_Obounded__iff,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( Less_eq @ A2 @ ( F @ B2 @ C ) )
% 0.18/0.44 = ( ( Less_eq @ A2 @ B2 )
% 0.18/0.44 & ( Less_eq @ A2 @ C ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.bounded_iff
% 0.18/0.44 thf(fact_241_semilattice__order_OcoboundedI1,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,C: A,B2: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( Less_eq @ A2 @ C )
% 0.18/0.44 => ( Less_eq @ ( F @ A2 @ B2 ) @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.coboundedI1
% 0.18/0.44 thf(fact_242_semilattice__order_OcoboundedI2,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C: A,A2: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( Less_eq @ B2 @ C )
% 0.18/0.44 => ( Less_eq @ ( F @ A2 @ B2 ) @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.coboundedI2
% 0.18/0.44 thf(fact_243_semilattice__order_Ostrict__boundedE,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( Less @ A2 @ ( F @ B2 @ C ) )
% 0.18/0.44 => ~ ( ( Less @ A2 @ B2 )
% 0.18/0.44 => ~ ( Less @ A2 @ C ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.strict_boundedE
% 0.18/0.44 thf(fact_244_semilattice__order_Ostrict__order__iff,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( Less @ A2 @ B2 )
% 0.18/0.44 = ( ( A2
% 0.18/0.44 = ( F @ A2 @ B2 ) )
% 0.18/0.44 & ( A2 != B2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.strict_order_iff
% 0.18/0.44 thf(fact_245_semilattice__order_Ostrict__coboundedI1,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,C: A,B2: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( Less @ A2 @ C )
% 0.18/0.44 => ( Less @ ( F @ A2 @ B2 ) @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.strict_coboundedI1
% 0.18/0.44 thf(fact_246_semilattice__order_Ostrict__coboundedI2,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C: A,A2: A] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( ( Less @ B2 @ C )
% 0.18/0.44 => ( Less @ ( F @ A2 @ B2 ) @ C ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.strict_coboundedI2
% 0.18/0.44 thf(fact_247_semilattice_Oidem,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,A2: A] :
% 0.18/0.44 ( ( semilattice @ A @ F )
% 0.18/0.44 => ( ( F @ A2 @ A2 )
% 0.18/0.44 = A2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice.idem
% 0.18/0.44 thf(fact_248_semilattice_Oleft__idem,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 0.18/0.44 ( ( semilattice @ A @ F )
% 0.18/0.44 => ( ( F @ A2 @ ( F @ A2 @ B2 ) )
% 0.18/0.44 = ( F @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice.left_idem
% 0.18/0.44 thf(fact_249_semilattice_Oright__idem,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 0.18/0.44 ( ( semilattice @ A @ F )
% 0.18/0.44 => ( ( F @ ( F @ A2 @ B2 ) @ B2 )
% 0.18/0.44 = ( F @ A2 @ B2 ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice.right_idem
% 0.18/0.44 thf(fact_250_semilattice__order_Oaxioms_I1_J,axiom,
% 0.18/0.44 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 0.18/0.44 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 0.18/0.44 => ( semilattice @ A @ F ) ) ).
% 0.18/0.44
% 0.18/0.44 % semilattice_order.axioms(1)
% 0.18/0.44 thf(fact_251_local_Obdd__below__def,axiom,
% 0.18/0.44 ! [A4: set @ a] :
% 0.18/0.44 ( ( condit1201339847_below @ a @ less_eq @ A4 )
% 0.18/0.44 = ( ? [M: a] :
% 0.18/0.44 ! [X2: a] :
% 0.18/0.44 ( ( member @ a @ X2 @ A4 )
% 0.18/0.44 => ( less_eq @ M @ X2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.bdd_below_def
% 0.18/0.44 thf(fact_252_local_Obdd__above__def,axiom,
% 0.18/0.44 ! [A4: set @ a] :
% 0.18/0.44 ( ( condit2040224947_above @ a @ less_eq @ A4 )
% 0.18/0.44 = ( ? [M2: a] :
% 0.18/0.44 ! [X2: a] :
% 0.18/0.44 ( ( member @ a @ X2 @ A4 )
% 0.18/0.44 => ( less_eq @ X2 @ M2 ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.bdd_above_def
% 0.18/0.44 thf(fact_253_local_Obdd__belowI,axiom,
% 0.18/0.44 ! [A4: set @ a,M3: a] :
% 0.18/0.44 ( ! [X3: a] :
% 0.18/0.44 ( ( member @ a @ X3 @ A4 )
% 0.18/0.44 => ( less_eq @ M3 @ X3 ) )
% 0.18/0.44 => ( condit1201339847_below @ a @ less_eq @ A4 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.bdd_belowI
% 0.18/0.44 thf(fact_254_local_Obdd__aboveI,axiom,
% 0.18/0.44 ! [A4: set @ a,M4: a] :
% 0.18/0.44 ( ! [X3: a] :
% 0.18/0.44 ( ( member @ a @ X3 @ A4 )
% 0.18/0.44 => ( less_eq @ X3 @ M4 ) )
% 0.18/0.44 => ( condit2040224947_above @ a @ less_eq @ A4 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.bdd_aboveI
% 0.18/0.44 thf(fact_255_local_Obdd__below__Ioo,axiom,
% 0.18/0.44 ! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_gr1161524159ssThan @ a @ less @ A2 @ B2 ) ) ).
% 0.18/0.44
% 0.18/0.44 % local.bdd_below_Ioo
% 0.18/0.44
% 0.18/0.44 % Conjectures (5)
% 0.18/0.44 thf(conj_0,hypothesis,
% 0.18/0.44 lattice @ a @ inf @ less_eq @ less @ sup ).
% 0.18/0.44
% 0.18/0.44 thf(conj_1,hypothesis,
% 0.18/0.44 less_eq @ x @ y ).
% 0.18/0.44
% 0.18/0.44 thf(conj_2,hypothesis,
% 0.18/0.44 ( ( sup @ x @ ( inf @ y @ z ) )
% 0.18/0.44 != ( inf @ y @ ( sup @ x @ z ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(conj_3,hypothesis,
% 0.18/0.44 less @ ( sup @ x @ ( inf @ y @ z ) ) @ ( inf @ y @ ( sup @ x @ z ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(conj_4,conjecture,
% 0.18/0.44 less_eq @ x @ ( sup @ ( inf @ y @ ( sup @ x @ z ) ) @ z ) ).
% 0.18/0.44
% 0.18/0.44 %------------------------------------------------------------------------------
% 0.18/0.44 ------------------------
% 2.38/1.47 ---- Embedded in THF ---
% 2.38/1.47 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.38/1.47 %%% Generated on Tue Feb 24 08:27:10 EST 2026
% 2.38/1.47 %%% using '$$dhol' embedding, version 1.3.0.
% 2.38/1.47 %%% Logic specification used:
% 2.38/1.47 %%% thf(spec, logic, $$dhol).
% 2.38/1.47
% 2.38/1.47 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.Vy3zEj9nqo/DTF2DTF5351.p
% 2.38/1.47 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.38/1.47 thf(ty_tf_a, type, a: $tType).
% 2.38/1.47 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above, type, condit2040224947_above: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > $o))))).
% 2.38/1.47 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below, type, condit1201339847_below: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > $o))))).
% 2.38/1.47 thf(sy_c_Finite__Set_Ocomp__fun__idem, type, finite_comp_fun_idem: (!> [A:$tType,B:$tType]: (((A > (B > B)) > $o)))).
% 2.38/1.47 thf(sy_c_Groups_Oabel__semigroup, type, abel_semigroup: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 2.38/1.47 thf(sy_c_Groups_Osemigroup, type, semigroup: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 2.38/1.47 thf(sy_c_Lattices_Oclass_Olattice, type, lattice: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > $o)) > ((A > (A > $o)) > ((A > (A > A)) > $o))))))).
% 2.38/1.47 thf(sy_c_Lattices_Osemilattice, type, semilattice: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 2.38/1.47 thf(sy_c_Lattices_Osemilattice__order, type, semilattice_order: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > $o)) > ((A > (A > $o)) > $o)))))).
% 2.38/1.47 thf(sy_c_Lattices__Big_Oord_Ois__arg__max, type, lattices_is_arg_max: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((B > A) > ((B > $o) > (B > $o))))))).
% 2.38/1.47 thf(sy_c_Lattices__Big_Oord_Ois__arg__min, type, lattices_is_arg_min: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((B > A) > ((B > $o) > (B > $o))))))).
% 2.38/1.47 thf(sy_c_Lattices__Big_Osemilattice__order__set, type, lattic1693879045er_set: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > $o)) > ((A > (A > $o)) > $o)))))).
% 2.38/1.47 thf(sy_c_Lattices__Big_Osemilattice__set, type, lattic35693393ce_set: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 2.38/1.47 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_OM5__lattice, type, modula1618706180attice: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > $o)) > ((A > (A > A)) > (A > (A > (A > $o))))))))).
% 2.38/1.47 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_ON5__lattice, type, modula1192512645attice: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > $o)) > ((A > (A > A)) > (A > (A > (A > $o))))))))).
% 2.38/1.47 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oa__aux, type, modula105138637_a_aux: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > (A > (A > (A > A)))))))).
% 2.38/1.47 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Ob__aux, type, modula477145868_b_aux: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > (A > (A > (A > A)))))))).
% 2.38/1.47 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oc__aux, type, modula849153099_c_aux: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > (A > (A > (A > A)))))))).
% 2.38/1.47 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Od__aux, type, modula1221160330_d_aux: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > (A > (A > (A > A)))))))).
% 2.38/1.47 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oe__aux, type, modula1593167561_e_aux: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > (A > (A > (A > A)))))))).
% 2.38/1.47 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oincomp, type, modula1937309406incomp: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > $o)))))).
% 2.38/1.47 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Ono__distrib, type, modula141148890istrib: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > $o)) > ((A > (A > A)) > (A > (A > (A > $o))))))))).
% 2.38/1.47 thf(sy_c_Orderings_Oord_OLeast, type, least: (!> [A:$tType]: (((A > (A > $o)) > ((A > $o) > A))))).
% 2.38/1.47 thf(sy_c_Orderings_Oord_Omax, type, max: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > A)))))).
% 2.38/1.47 thf(sy_c_Orderings_Oord_Omin, type, min: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > A)))))).
% 2.38/1.47 thf(sy_c_Orderings_Oorder_OGreatest, type, greatest: (!> [A:$tType]: (((A > (A > $o)) > ((A > $o) > A))))).
% 2.38/1.47 thf(sy_c_Orderings_Oordering, type, ordering: (!> [A:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > $o))))).
% 2.38/1.47 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.38/1.47 thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan, type, set_gr1161524159ssThan: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > (set @ A))))))).
% 2.38/1.47 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.38/1.47 thf(sy_v_inf, type, inf: (a > (a > a))).
% 2.38/1.47 thf(sy_v_less, type, less: (a > (a > $o))).
% 2.38/1.47 thf(sy_v_less__eq, type, less_eq: (a > (a > $o))).
% 2.38/1.47 thf(sy_v_sup, type, sup: (a > (a > a))).
% 2.38/1.47 thf(sy_v_x, type, x: a).
% 2.38/1.47 thf(sy_v_y, type, y: a).
% 2.38/1.47 thf(sy_v_z, type, z: a).
% 2.38/1.47 thf(fact_0_local_Oantisym, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) => (((less_eq @ Y) @ X) => (X = Y)))))).
% 2.38/1.47 thf(fact_1_local_Oantisym__conv, axiom, (! [Y:a,X:a]: ((((less_eq @ Y) @ X) => (((less_eq @ X) @ Y) = (X = Y)))))).
% 2.38/1.47 thf(fact_2_local_Odual__order_Oantisym, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) => (((less_eq @ A2) @ B2) => (A2 = B2)))))).
% 2.38/1.47 thf(fact_3_local_Odual__order_Oeq__iff, axiom, ((^ [Y2:a,Z:a]: ((Y2 = Z))) = (^ [A3:a,B3:a]: ((((less_eq @ B3) @ A3) & ((less_eq @ A3) @ B3)))))).
% 2.38/1.47 thf(fact_4_local_Odual__order_Otrans, axiom, (! [B2:a,A2:a,C:a]: ((((less_eq @ B2) @ A2) => (((less_eq @ C) @ B2) => ((less_eq @ C) @ A2)))))).
% 2.38/1.47 thf(fact_5_local_Oeq__iff, axiom, ((^ [Y2:a,Z:a]: ((Y2 = Z))) = (^ [X2:a,Y3:a]: ((((less_eq @ X2) @ Y3) & ((less_eq @ Y3) @ X2)))))).
% 2.38/1.47 thf(fact_6_local_Oeq__refl, axiom, (! [X:a,Y:a]: (((X = Y) => ((less_eq @ X) @ Y))))).
% 2.38/1.47 thf(fact_7_local_Oord__eq__le__trans, axiom, (! [A2:a,B2:a,C:a]: (((A2 = B2) => (((less_eq @ B2) @ C) => ((less_eq @ A2) @ C)))))).
% 2.38/1.47 thf(fact_8_local_Oord__le__eq__trans, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ B2) => ((B2 = C) => ((less_eq @ A2) @ C)))))).
% 2.38/1.47 thf(fact_9_local_Oorder_Oantisym, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) => (((less_eq @ B2) @ A2) => (A2 = B2)))))).
% 2.38/1.47 thf(fact_10_local_Oorder_Oeq__iff, axiom, ((^ [Y2:a,Z:a]: ((Y2 = Z))) = (^ [A3:a,B3:a]: ((((less_eq @ A3) @ B3) & ((less_eq @ B3) @ A3)))))).
% 2.38/1.47 thf(fact_11_local_Oorder_Otrans, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ B2) => (((less_eq @ B2) @ C) => ((less_eq @ A2) @ C)))))).
% 2.38/1.47 thf(fact_12_local_Oorder__trans, axiom, (! [X:a,Y:a,Z2:a]: ((((less_eq @ X) @ Y) => (((less_eq @ Y) @ Z2) => ((less_eq @ X) @ Z2)))))).
% 2.38/1.47 thf(fact_13_local_Odual__order_Oasym, axiom, (! [B2:a,A2:a]: ((((less @ B2) @ A2) => (~ (((less @ A2) @ B2))))))).
% 2.38/1.47 thf(fact_14_local_Odual__order_Ostrict__implies__not__eq, axiom, (! [B2:a,A2:a]: ((((less @ B2) @ A2) => (A2 != B2))))).
% 2.38/1.47 thf(fact_15_local_Odual__order_Ostrict__trans, axiom, (! [B2:a,A2:a,C:a]: ((((less @ B2) @ A2) => (((less @ C) @ B2) => ((less @ C) @ A2)))))).
% 2.38/1.47 thf(fact_16_local_Oless__asym, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (~ (((less @ Y) @ X))))))).
% 2.38/1.47 thf(fact_17_local_Oless__asym_H, axiom, (! [A2:a,B2:a]: ((((less @ A2) @ B2) => (~ (((less @ B2) @ A2))))))).
% 2.38/1.47 thf(fact_18_local_Oless__imp__neq, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (X != Y))))).
% 2.38/1.47 thf(fact_19_local_Oless__imp__not__eq, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (X != Y))))).
% 2.38/1.47 thf(fact_20_local_Oless__imp__not__eq2, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (Y != X))))).
% 2.38/1.47 thf(fact_21_local_Oless__imp__not__less, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (~ (((less @ Y) @ X))))))).
% 2.38/1.47 thf(fact_22_local_Oless__imp__triv, axiom, (! [X:a,Y:a,P:$o]: ((((less @ X) @ Y) => (((less @ Y) @ X) => P))))).
% 2.38/1.47 thf(fact_23_local_Oless__irrefl, axiom, (! [X:a]: ((~ (((less @ X) @ X)))))).
% 2.38/1.47 thf(fact_24_local_Oless__not__sym, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (~ (((less @ Y) @ X))))))).
% 2.38/1.47 thf(fact_25_local_Oless__trans, axiom, (! [X:a,Y:a,Z2:a]: ((((less @ X) @ Y) => (((less @ Y) @ Z2) => ((less @ X) @ Z2)))))).
% 2.38/1.47 thf(fact_26_local_Oord__eq__less__trans, axiom, (! [A2:a,B2:a,C:a]: (((A2 = B2) => (((less @ B2) @ C) => ((less @ A2) @ C)))))).
% 2.38/1.47 thf(fact_27_local_Oord__less__eq__trans, axiom, (! [A2:a,B2:a,C:a]: ((((less @ A2) @ B2) => ((B2 = C) => ((less @ A2) @ C)))))).
% 2.38/1.47 thf(fact_28_local_Oorder_Oasym, axiom, (! [A2:a,B2:a]: ((((less @ A2) @ B2) => (~ (((less @ B2) @ A2))))))).
% 2.38/1.47 thf(fact_29_local_Oorder_Oirrefl, axiom, (! [A2:a]: ((~ (((less @ A2) @ A2)))))).
% 2.38/1.47 thf(fact_30_local_Oorder_Ostrict__implies__not__eq, axiom, (! [A2:a,B2:a]: ((((less @ A2) @ B2) => (A2 != B2))))).
% 2.38/1.47 thf(fact_31_local_Oorder_Ostrict__trans, axiom, (! [A2:a,B2:a,C:a]: ((((less @ A2) @ B2) => (((less @ B2) @ C) => ((less @ A2) @ C)))))).
% 2.38/1.47 thf(fact_32_local_Oinf_Oassoc, axiom, (! [A2:a,B2:a,C:a]: ((((inf @ ((inf @ A2) @ B2)) @ C) = ((inf @ A2) @ ((inf @ B2) @ C)))))).
% 2.38/1.47 thf(fact_33_local_Oinf_Ocommute, axiom, (! [A2:a,B2:a]: ((((inf @ A2) @ B2) = ((inf @ B2) @ A2))))).
% 2.38/1.47 thf(fact_34_local_Oinf_Oleft__commute, axiom, (! [B2:a,A2:a,C:a]: ((((inf @ B2) @ ((inf @ A2) @ C)) = ((inf @ A2) @ ((inf @ B2) @ C)))))).
% 2.38/1.47 thf(fact_35_local_Oinf__assoc, axiom, (! [X:a,Y:a,Z2:a]: ((((inf @ ((inf @ X) @ Y)) @ Z2) = ((inf @ X) @ ((inf @ Y) @ Z2)))))).
% 2.38/1.47 thf(fact_36_local_Oinf__commute, axiom, (! [X:a,Y:a]: ((((inf @ X) @ Y) = ((inf @ Y) @ X))))).
% 2.38/1.47 thf(fact_37_local_Oinf__left__commute, axiom, (! [X:a,Y:a,Z2:a]: ((((inf @ X) @ ((inf @ Y) @ Z2)) = ((inf @ Y) @ ((inf @ X) @ Z2)))))).
% 2.38/1.47 thf(fact_38_local_Osup_Oassoc, axiom, (! [A2:a,B2:a,C:a]: ((((sup @ ((sup @ A2) @ B2)) @ C) = ((sup @ A2) @ ((sup @ B2) @ C)))))).
% 2.38/1.47 thf(fact_39_local_Osup_Ocommute, axiom, (! [A2:a,B2:a]: ((((sup @ A2) @ B2) = ((sup @ B2) @ A2))))).
% 2.38/1.47 thf(fact_40_local_Osup_Oleft__commute, axiom, (! [B2:a,A2:a,C:a]: ((((sup @ B2) @ ((sup @ A2) @ C)) = ((sup @ A2) @ ((sup @ B2) @ C)))))).
% 2.38/1.47 thf(fact_41_local_Osup__assoc, axiom, (! [X:a,Y:a,Z2:a]: ((((sup @ ((sup @ X) @ Y)) @ Z2) = ((sup @ X) @ ((sup @ Y) @ Z2)))))).
% 2.38/1.47 thf(fact_42_local_Osup__commute, axiom, (! [X:a,Y:a]: ((((sup @ X) @ Y) = ((sup @ Y) @ X))))).
% 2.38/1.47 thf(fact_43_local_Osup__left__commute, axiom, (! [X:a,Y:a,Z2:a]: ((((sup @ X) @ ((sup @ Y) @ Z2)) = ((sup @ Y) @ ((sup @ X) @ Z2)))))).
% 2.38/1.47 thf(fact_44_local_Oantisym__conv1, axiom, (! [X:a,Y:a]: (((~ (((less @ X) @ Y))) => (((less_eq @ X) @ Y) = (X = Y)))))).
% 2.38/1.47 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.38/1.47 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A4)))) = A4)))).
% 2.38/1.47 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.38/1.47 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.38/1.47 thf(fact_49_local_Oantisym__conv2, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) => ((~ (((less @ X) @ Y))) = (X = Y)))))).
% 2.38/1.47 thf(fact_50_local_Odual__order_Onot__eq__order__implies__strict, axiom, (! [A2:a,B2:a]: (((A2 != B2) => (((less_eq @ B2) @ A2) => ((less @ B2) @ A2)))))).
% 2.38/1.47 thf(fact_51_local_Odual__order_Oorder__iff__strict, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) = (((less @ B2) @ A2) | (A2 = B2)))))).
% 2.38/1.47 thf(fact_52_local_Odual__order_Ostrict__iff__order, axiom, (! [B2:a,A2:a]: ((((less @ B2) @ A2) = (((less_eq @ B2) @ A2) & (A2 != B2)))))).
% 2.38/1.47 thf(fact_53_local_Odual__order_Ostrict__implies__order, axiom, (! [B2:a,A2:a]: ((((less @ B2) @ A2) => ((less_eq @ B2) @ A2))))).
% 2.38/1.47 thf(fact_54_local_Odual__order_Ostrict__trans1, axiom, (! [B2:a,A2:a,C:a]: ((((less_eq @ B2) @ A2) => (((less @ C) @ B2) => ((less @ C) @ A2)))))).
% 2.38/1.47 thf(fact_55_local_Odual__order_Ostrict__trans2, axiom, (! [B2:a,A2:a,C:a]: ((((less @ B2) @ A2) => (((less_eq @ C) @ B2) => ((less @ C) @ A2)))))).
% 2.38/1.47 thf(fact_56_local_OleD, axiom, (! [Y:a,X:a]: ((((less_eq @ Y) @ X) => (~ (((less @ X) @ Y))))))).
% 2.38/1.47 thf(fact_57_local_Ole__imp__less__or__eq, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) => (((less @ X) @ Y) | (X = Y)))))).
% 2.38/1.47 thf(fact_58_local_Ole__less, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) = (((less @ X) @ Y) | (X = Y)))))).
% 2.38/1.47 thf(fact_59_local_Ole__less__trans, axiom, (! [X:a,Y:a,Z2:a]: ((((less_eq @ X) @ Y) => (((less @ Y) @ Z2) => ((less @ X) @ Z2)))))).
% 2.38/1.47 thf(fact_60_local_Ole__neq__trans, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) => ((A2 != B2) => ((less @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_61_local_Oless__imp__le, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => ((less_eq @ X) @ Y))))).
% 2.38/1.47 thf(fact_62_local_Oless__le, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) = (((less_eq @ X) @ Y) & (X != Y)))))).
% 2.38/1.47 thf(fact_63_local_Oless__le__not__le, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) = (((less_eq @ X) @ Y) & (~ (((less_eq @ Y) @ X)))))))).
% 2.38/1.47 thf(fact_64_local_Oless__le__trans, axiom, (! [X:a,Y:a,Z2:a]: ((((less @ X) @ Y) => (((less_eq @ Y) @ Z2) => ((less @ X) @ Z2)))))).
% 2.38/1.47 thf(fact_65_local_Oorder_Onot__eq__order__implies__strict, axiom, (! [A2:a,B2:a]: (((A2 != B2) => (((less_eq @ A2) @ B2) => ((less @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_66_local_Oorder_Oorder__iff__strict, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) = (((less @ A2) @ B2) | (A2 = B2)))))).
% 2.38/1.47 thf(fact_67_local_Oorder_Ostrict__iff__order, axiom, (! [A2:a,B2:a]: ((((less @ A2) @ B2) = (((less_eq @ A2) @ B2) & (A2 != B2)))))).
% 2.38/1.47 thf(fact_68_local_Oorder_Ostrict__implies__order, axiom, (! [A2:a,B2:a]: ((((less @ A2) @ B2) => ((less_eq @ A2) @ B2))))).
% 2.38/1.47 thf(fact_69_local_Oorder_Ostrict__trans1, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ B2) => (((less @ B2) @ C) => ((less @ A2) @ C)))))).
% 2.38/1.47 thf(fact_70_local_Oorder_Ostrict__trans2, axiom, (! [A2:a,B2:a,C:a]: ((((less @ A2) @ B2) => (((less_eq @ B2) @ C) => ((less @ A2) @ C)))))).
% 2.38/1.47 thf(fact_71_local_Oinf_Oabsorb1, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) => (((inf @ A2) @ B2) = A2))))).
% 2.38/1.47 thf(fact_72_local_Oinf_Oabsorb2, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) => (((inf @ A2) @ B2) = B2))))).
% 2.38/1.47 thf(fact_73_local_Oinf_Oabsorb__iff1, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) = (((inf @ A2) @ B2) = A2))))).
% 2.38/1.47 thf(fact_74_local_Oinf_Oabsorb__iff2, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) = (((inf @ A2) @ B2) = B2))))).
% 2.38/1.47 thf(fact_75_local_Oinf_OboundedE, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ ((inf @ B2) @ C)) => (~ ((((less_eq @ A2) @ B2) => (~ (((less_eq @ A2) @ C)))))))))).
% 2.38/1.47 thf(fact_76_local_Oinf_OboundedI, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ B2) => (((less_eq @ A2) @ C) => ((less_eq @ A2) @ ((inf @ B2) @ C))))))).
% 2.38/1.47 thf(fact_77_local_Oinf_Ocobounded1, axiom, (! [A2:a,B2:a]: (((less_eq @ ((inf @ A2) @ B2)) @ A2)))).
% 2.38/1.47 thf(fact_78_local_Oinf_Ocobounded2, axiom, (! [A2:a,B2:a]: (((less_eq @ ((inf @ A2) @ B2)) @ B2)))).
% 2.38/1.47 thf(fact_79_local_Oinf_OcoboundedI1, axiom, (! [A2:a,C:a,B2:a]: ((((less_eq @ A2) @ C) => ((less_eq @ ((inf @ A2) @ B2)) @ C))))).
% 2.38/1.47 thf(fact_80_local_Oinf_OcoboundedI2, axiom, (! [B2:a,C:a,A2:a]: ((((less_eq @ B2) @ C) => ((less_eq @ ((inf @ A2) @ B2)) @ C))))).
% 2.38/1.47 thf(fact_81_local_Oinf_OorderE, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) => (A2 = ((inf @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_82_local_Oinf_OorderI, axiom, (! [A2:a,B2:a]: (((A2 = ((inf @ A2) @ B2)) => ((less_eq @ A2) @ B2))))).
% 2.38/1.47 thf(fact_83_local_Oinf_Oorder__iff, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) = (A2 = ((inf @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_84_local_Oinf__absorb1, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) => (((inf @ X) @ Y) = X))))).
% 2.38/1.47 thf(fact_85_local_Oinf__absorb2, axiom, (! [Y:a,X:a]: ((((less_eq @ Y) @ X) => (((inf @ X) @ Y) = Y))))).
% 2.38/1.47 thf(fact_86_local_Oinf__greatest, axiom, (! [X:a,Y:a,Z2:a]: ((((less_eq @ X) @ Y) => (((less_eq @ X) @ Z2) => ((less_eq @ X) @ ((inf @ Y) @ Z2))))))).
% 2.38/1.47 thf(fact_87_local_Oinf__le1, axiom, (! [X:a,Y:a]: (((less_eq @ ((inf @ X) @ Y)) @ X)))).
% 2.38/1.47 thf(fact_88_local_Oinf__le2, axiom, (! [X:a,Y:a]: (((less_eq @ ((inf @ X) @ Y)) @ Y)))).
% 2.38/1.47 thf(fact_89_local_Oinf__mono, axiom, (! [A2:a,C:a,B2:a,D:a]: ((((less_eq @ A2) @ C) => (((less_eq @ B2) @ D) => ((less_eq @ ((inf @ A2) @ B2)) @ ((inf @ C) @ D))))))).
% 2.38/1.47 thf(fact_90_local_Oinf__unique, axiom, (! [F:(a > (a > a)),X:a,Y:a]: (((! [X3:a,Y4:a]: (((less_eq @ ((F @ X3) @ Y4)) @ X3))) => ((! [X3:a,Y4:a]: (((less_eq @ ((F @ X3) @ Y4)) @ Y4))) => ((! [X3:a,Y4:a,Z3:a]: ((((less_eq @ X3) @ Y4) => (((less_eq @ X3) @ Z3) => ((less_eq @ X3) @ ((F @ Y4) @ Z3)))))) => (((inf @ X) @ Y) = ((F @ X) @ Y)))))))).
% 2.38/1.47 thf(fact_91_local_Ole__iff__inf, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) = (((inf @ X) @ Y) = X))))).
% 2.38/1.47 thf(fact_92_local_Ole__infE, axiom, (! [X:a,A2:a,B2:a]: ((((less_eq @ X) @ ((inf @ A2) @ B2)) => (~ ((((less_eq @ X) @ A2) => (~ (((less_eq @ X) @ B2)))))))))).
% 2.38/1.47 thf(fact_93_local_Ole__infI, axiom, (! [X:a,A2:a,B2:a]: ((((less_eq @ X) @ A2) => (((less_eq @ X) @ B2) => ((less_eq @ X) @ ((inf @ A2) @ B2))))))).
% 2.38/1.47 thf(fact_94_local_Ole__infI1, axiom, (! [A2:a,X:a,B2:a]: ((((less_eq @ A2) @ X) => ((less_eq @ ((inf @ A2) @ B2)) @ X))))).
% 2.38/1.47 thf(fact_95_local_Ole__infI2, axiom, (! [B2:a,X:a,A2:a]: ((((less_eq @ B2) @ X) => ((less_eq @ ((inf @ A2) @ B2)) @ X))))).
% 2.38/1.47 thf(fact_96_local_Ole__iff__sup, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) = (((sup @ X) @ Y) = Y))))).
% 2.38/1.47 thf(fact_97_local_Ole__supE, axiom, (! [A2:a,B2:a,X:a]: ((((less_eq @ ((sup @ A2) @ B2)) @ X) => (~ ((((less_eq @ A2) @ X) => (~ (((less_eq @ B2) @ X)))))))))).
% 2.38/1.47 thf(fact_98_local_Ole__supI, axiom, (! [A2:a,X:a,B2:a]: ((((less_eq @ A2) @ X) => (((less_eq @ B2) @ X) => ((less_eq @ ((sup @ A2) @ B2)) @ X)))))).
% 2.38/1.47 thf(fact_99_local_Ole__supI1, axiom, (! [X:a,A2:a,B2:a]: ((((less_eq @ X) @ A2) => ((less_eq @ X) @ ((sup @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_100_local_Ole__supI2, axiom, (! [X:a,B2:a,A2:a]: ((((less_eq @ X) @ B2) => ((less_eq @ X) @ ((sup @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_101_local_Osup_Oabsorb1, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) => (((sup @ A2) @ B2) = A2))))).
% 2.38/1.47 thf(fact_102_local_Osup_Oabsorb2, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) => (((sup @ A2) @ B2) = B2))))).
% 2.38/1.47 thf(fact_103_local_Osup_Oabsorb__iff1, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) = (((sup @ A2) @ B2) = A2))))).
% 2.38/1.47 thf(fact_104_local_Osup_Oabsorb__iff2, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) = (((sup @ A2) @ B2) = B2))))).
% 2.38/1.47 thf(fact_105_local_Osup_OboundedE, axiom, (! [B2:a,C:a,A2:a]: ((((less_eq @ ((sup @ B2) @ C)) @ A2) => (~ ((((less_eq @ B2) @ A2) => (~ (((less_eq @ C) @ A2)))))))))).
% 2.38/1.47 thf(fact_106_local_Osup_OboundedI, axiom, (! [B2:a,A2:a,C:a]: ((((less_eq @ B2) @ A2) => (((less_eq @ C) @ A2) => ((less_eq @ ((sup @ B2) @ C)) @ A2)))))).
% 2.38/1.47 thf(fact_107_local_Osup_Ocobounded1, axiom, (! [A2:a,B2:a]: (((less_eq @ A2) @ ((sup @ A2) @ B2))))).
% 2.38/1.47 thf(fact_108_local_Osup_Ocobounded2, axiom, (! [B2:a,A2:a]: (((less_eq @ B2) @ ((sup @ A2) @ B2))))).
% 2.38/1.47 thf(fact_109_local_Osup_OcoboundedI1, axiom, (! [C:a,A2:a,B2:a]: ((((less_eq @ C) @ A2) => ((less_eq @ C) @ ((sup @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_110_local_Osup_OcoboundedI2, axiom, (! [C:a,B2:a,A2:a]: ((((less_eq @ C) @ B2) => ((less_eq @ C) @ ((sup @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_111_local_Osup_Omono, axiom, (! [C:a,A2:a,D:a,B2:a]: ((((less_eq @ C) @ A2) => (((less_eq @ D) @ B2) => ((less_eq @ ((sup @ C) @ D)) @ ((sup @ A2) @ B2))))))).
% 2.38/1.47 thf(fact_112_local_Osup_OorderE, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) => (A2 = ((sup @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_113_local_Osup_OorderI, axiom, (! [A2:a,B2:a]: (((A2 = ((sup @ A2) @ B2)) => ((less_eq @ B2) @ A2))))).
% 2.38/1.47 thf(fact_114_local_Osup_Oorder__iff, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) = (A2 = ((sup @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_115_local_Osup__absorb1, axiom, (! [Y:a,X:a]: ((((less_eq @ Y) @ X) => (((sup @ X) @ Y) = X))))).
% 2.38/1.47 thf(fact_116_local_Osup__absorb2, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) => (((sup @ X) @ Y) = Y))))).
% 2.38/1.47 thf(fact_117_local_Osup__ge1, axiom, (! [X:a,Y:a]: (((less_eq @ X) @ ((sup @ X) @ Y))))).
% 2.38/1.47 thf(fact_118_local_Osup__ge2, axiom, (! [Y:a,X:a]: (((less_eq @ Y) @ ((sup @ X) @ Y))))).
% 2.38/1.47 thf(fact_119_local_Osup__least, axiom, (! [Y:a,X:a,Z2:a]: ((((less_eq @ Y) @ X) => (((less_eq @ Z2) @ X) => ((less_eq @ ((sup @ Y) @ Z2)) @ X)))))).
% 2.38/1.47 thf(fact_120_local_Osup__mono, axiom, (! [A2:a,C:a,B2:a,D:a]: ((((less_eq @ A2) @ C) => (((less_eq @ B2) @ D) => ((less_eq @ ((sup @ A2) @ B2)) @ ((sup @ C) @ D))))))).
% 2.38/1.47 thf(fact_121_local_Osup__unique, axiom, (! [F:(a > (a > a)),X:a,Y:a]: (((! [X3:a,Y4:a]: (((less_eq @ X3) @ ((F @ X3) @ Y4)))) => ((! [X3:a,Y4:a]: (((less_eq @ Y4) @ ((F @ X3) @ Y4)))) => ((! [X3:a,Y4:a,Z3:a]: ((((less_eq @ Y4) @ X3) => (((less_eq @ Z3) @ X3) => ((less_eq @ ((F @ Y4) @ Z3)) @ X3))))) => (((sup @ X) @ Y) = ((F @ X) @ Y)))))))).
% 2.38/1.47 thf(fact_122_local_Oinf_Ostrict__boundedE, axiom, (! [A2:a,B2:a,C:a]: ((((less @ A2) @ ((inf @ B2) @ C)) => (~ ((((less @ A2) @ B2) => (~ (((less @ A2) @ C)))))))))).
% 2.38/1.47 thf(fact_123_local_Oinf_Ostrict__coboundedI1, axiom, (! [A2:a,C:a,B2:a]: ((((less @ A2) @ C) => ((less @ ((inf @ A2) @ B2)) @ C))))).
% 2.38/1.47 thf(fact_124_local_Oinf_Ostrict__coboundedI2, axiom, (! [B2:a,C:a,A2:a]: ((((less @ B2) @ C) => ((less @ ((inf @ A2) @ B2)) @ C))))).
% 2.38/1.47 thf(fact_125_local_Oinf_Ostrict__order__iff, axiom, (! [A2:a,B2:a]: ((((less @ A2) @ B2) = ((A2 = ((inf @ A2) @ B2)) & (A2 != B2)))))).
% 2.38/1.47 thf(fact_126_local_Oless__infI1, axiom, (! [A2:a,X:a,B2:a]: ((((less @ A2) @ X) => ((less @ ((inf @ A2) @ B2)) @ X))))).
% 2.38/1.47 thf(fact_127_local_Oless__infI2, axiom, (! [B2:a,X:a,A2:a]: ((((less @ B2) @ X) => ((less @ ((inf @ A2) @ B2)) @ X))))).
% 2.38/1.47 thf(fact_128_local_Oless__supI1, axiom, (! [X:a,A2:a,B2:a]: ((((less @ X) @ A2) => ((less @ X) @ ((sup @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_129_local_Oless__supI2, axiom, (! [X:a,B2:a,A2:a]: ((((less @ X) @ B2) => ((less @ X) @ ((sup @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_130_local_Osup_Ostrict__boundedE, axiom, (! [B2:a,C:a,A2:a]: ((((less @ ((sup @ B2) @ C)) @ A2) => (~ ((((less @ B2) @ A2) => (~ (((less @ C) @ A2)))))))))).
% 2.38/1.47 thf(fact_131_local_Osup_Ostrict__coboundedI1, axiom, (! [C:a,A2:a,B2:a]: ((((less @ C) @ A2) => ((less @ C) @ ((sup @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_132_local_Osup_Ostrict__coboundedI2, axiom, (! [C:a,B2:a,A2:a]: ((((less @ C) @ B2) => ((less @ C) @ ((sup @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_133_local_Osup_Ostrict__order__iff, axiom, (! [B2:a,A2:a]: ((((less @ B2) @ A2) = ((A2 = ((sup @ A2) @ B2)) & (A2 != B2)))))).
% 2.38/1.47 thf(fact_134_local_Odistrib__imp1, axiom, (! [X:a,Y:a,Z2:a]: (((! [X3:a,Y4:a,Z3:a]: ((((inf @ X3) @ ((sup @ Y4) @ Z3)) = ((sup @ ((inf @ X3) @ Y4)) @ ((inf @ X3) @ Z3))))) => (((sup @ X) @ ((inf @ Y) @ Z2)) = ((inf @ ((sup @ X) @ Y)) @ ((sup @ X) @ Z2))))))).
% 2.38/1.47 thf(fact_135_local_Odistrib__imp2, axiom, (! [X:a,Y:a,Z2:a]: (((! [X3:a,Y4:a,Z3:a]: ((((sup @ X3) @ ((inf @ Y4) @ Z3)) = ((inf @ ((sup @ X3) @ Y4)) @ ((sup @ X3) @ Z3))))) => (((inf @ X) @ ((sup @ Y) @ Z2)) = ((sup @ ((inf @ X) @ Y)) @ ((inf @ X) @ Z2))))))).
% 2.38/1.47 thf(fact_136_local_Odistrib__inf__le, axiom, (! [X:a,Y:a,Z2:a]: (((less_eq @ ((sup @ ((inf @ X) @ Y)) @ ((inf @ X) @ Z2))) @ ((inf @ X) @ ((sup @ Y) @ Z2)))))).
% 2.38/1.47 thf(fact_137_local_Odistrib__sup__le, axiom, (! [X:a,Y:a,Z2:a]: (((less_eq @ ((sup @ X) @ ((inf @ Y) @ Z2))) @ ((inf @ ((sup @ X) @ Y)) @ ((sup @ X) @ Z2)))))).
% 2.38/1.47 thf(fact_138_local_Oincomp__def, axiom, (! [X:a,Y:a]: ((((((modula1937309406incomp @ a) @ less_eq) @ X) @ Y) = ((~ (((less_eq @ X) @ Y))) & (~ (((less_eq @ Y) @ X)))))))).
% 2.38/1.47 thf(fact_139_local_Oorder_Orefl, axiom, (! [A2:a]: (((less_eq @ A2) @ A2)))).
% 2.38/1.47 thf(fact_140_local_Oorder__refl, axiom, (! [X:a]: (((less_eq @ X) @ X)))).
% 2.38/1.47 thf(fact_141_local_Oinf_Oidem, axiom, (! [A2:a]: ((((inf @ A2) @ A2) = A2)))).
% 2.38/1.47 thf(fact_142_local_Oinf_Oleft__idem, axiom, (! [A2:a,B2:a]: ((((inf @ A2) @ ((inf @ A2) @ B2)) = ((inf @ A2) @ B2))))).
% 2.38/1.47 thf(fact_143_local_Oinf_Oright__idem, axiom, (! [A2:a,B2:a]: ((((inf @ ((inf @ A2) @ B2)) @ B2) = ((inf @ A2) @ B2))))).
% 2.38/1.47 thf(fact_144_local_Oinf__idem, axiom, (! [X:a]: ((((inf @ X) @ X) = X)))).
% 2.38/1.47 thf(fact_145_local_Oinf__left__idem, axiom, (! [X:a,Y:a]: ((((inf @ X) @ ((inf @ X) @ Y)) = ((inf @ X) @ Y))))).
% 2.38/1.47 thf(fact_146_local_Oinf__right__idem, axiom, (! [X:a,Y:a]: ((((inf @ ((inf @ X) @ Y)) @ Y) = ((inf @ X) @ Y))))).
% 2.38/1.47 thf(fact_147_local_Osup_Oidem, axiom, (! [A2:a]: ((((sup @ A2) @ A2) = A2)))).
% 2.38/1.47 thf(fact_148_local_Osup_Oleft__idem, axiom, (! [A2:a,B2:a]: ((((sup @ A2) @ ((sup @ A2) @ B2)) = ((sup @ A2) @ B2))))).
% 2.38/1.47 thf(fact_149_local_Osup_Oright__idem, axiom, (! [A2:a,B2:a]: ((((sup @ ((sup @ A2) @ B2)) @ B2) = ((sup @ A2) @ B2))))).
% 2.38/1.47 thf(fact_150_local_Osup__idem, axiom, (! [X:a]: ((((sup @ X) @ X) = X)))).
% 2.38/1.47 thf(fact_151_local_Osup__left__idem, axiom, (! [X:a,Y:a]: ((((sup @ X) @ ((sup @ X) @ Y)) = ((sup @ X) @ Y))))).
% 2.38/1.47 thf(fact_152_local_OGreatestI2__order, axiom, (! [P:(a > $o),X:a,Q:(a > $o)]: (((P @ X) => ((! [Y4:a]: (((P @ Y4) => ((less_eq @ Y4) @ X)))) => ((! [X3:a]: (((P @ X3) => ((! [Y5:a]: (((P @ Y5) => ((less_eq @ Y5) @ X3)))) => (Q @ X3))))) => (Q @ (((greatest @ a) @ less_eq) @ P)))))))).
% 2.38/1.47 thf(fact_153_local_OGreatest__equality, axiom, (! [P:(a > $o),X:a]: (((P @ X) => ((! [Y4:a]: (((P @ Y4) => ((less_eq @ Y4) @ X)))) => ((((greatest @ a) @ less_eq) @ P) = X)))))).
% 2.38/1.47 thf(fact_154_local_Omax__def, axiom, (! [A2:a,B2:a]: (((((less_eq @ A2) @ B2) => (((((max @ a) @ less_eq) @ A2) @ B2) = B2)) & ((~ (((less_eq @ A2) @ B2))) => (((((max @ a) @ less_eq) @ A2) @ B2) = A2)))))).
% 2.38/1.47 thf(fact_155_local_Omin__def, axiom, (! [A2:a,B2:a]: (((((less_eq @ A2) @ B2) => (((((min @ a) @ less_eq) @ A2) @ B2) = A2)) & ((~ (((less_eq @ A2) @ B2))) => (((((min @ a) @ less_eq) @ A2) @ B2) = B2)))))).
% 2.38/1.47 thf(fact_156_local_Ois__arg__max__def, axiom, (! [B:$tType,F:(B > a),P:(B > $o),X:B]: ((((((((lattices_is_arg_max @ a) @ B) @ less) @ F) @ P) @ X) = ((P @ X) & (~ ((? [Y3:B]: (((P @ Y3) & ((less @ (F @ X)) @ (F @ Y3)))))))))))).
% 2.38/1.47 thf(fact_157_local_Ois__arg__min__def, axiom, (! [B:$tType,F:(B > a),P:(B > $o),X:B]: ((((((((lattices_is_arg_min @ a) @ B) @ less) @ F) @ P) @ X) = ((P @ X) & (~ ((? [Y3:B]: (((P @ Y3) & ((less @ (F @ Y3)) @ (F @ X)))))))))))).
% 2.38/1.47 thf(fact_158_local_ON5__lattice__def, axiom, (! [A2:a,B2:a,C:a]: (((((((((modula1192512645attice @ a) @ inf) @ less) @ sup) @ A2) @ B2) @ C) = ((((inf @ A2) @ C) = ((inf @ B2) @ C)) & (((less @ A2) @ B2) & (((sup @ A2) @ C) = ((sup @ B2) @ C)))))))).
% 2.38/1.47 thf(fact_159_local_Ono__distrib__def, axiom, (! [A2:a,B2:a,C:a]: (((((((((modula141148890istrib @ a) @ inf) @ less) @ sup) @ A2) @ B2) @ C) = ((less @ ((sup @ ((inf @ A2) @ B2)) @ ((inf @ C) @ A2))) @ ((inf @ A2) @ ((sup @ B2) @ C))))))).
% 2.38/1.47 thf(fact_160_local_Oinf_Obounded__iff, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ ((inf @ B2) @ C)) = (((less_eq @ A2) @ B2) & ((less_eq @ A2) @ C)))))).
% 2.38/1.47 thf(fact_161_local_Ole__inf__iff, axiom, (! [X:a,Y:a,Z2:a]: ((((less_eq @ X) @ ((inf @ Y) @ Z2)) = (((less_eq @ X) @ Y) & ((less_eq @ X) @ Z2)))))).
% 2.38/1.47 thf(fact_162_local_Ole__sup__iff, axiom, (! [X:a,Y:a,Z2:a]: ((((less_eq @ ((sup @ X) @ Y)) @ Z2) = (((less_eq @ X) @ Z2) & ((less_eq @ Y) @ Z2)))))).
% 2.38/1.47 thf(fact_163_local_Osup_Obounded__iff, axiom, (! [B2:a,C:a,A2:a]: ((((less_eq @ ((sup @ B2) @ C)) @ A2) = (((less_eq @ B2) @ A2) & ((less_eq @ C) @ A2)))))).
% 2.38/1.47 thf(fact_164_local_Oinf__sup__absorb, axiom, (! [X:a,Y:a]: ((((inf @ X) @ ((sup @ X) @ Y)) = X)))).
% 2.38/1.47 thf(fact_165_local_Osup__inf__absorb, axiom, (! [X:a,Y:a]: ((((sup @ X) @ ((inf @ X) @ Y)) = X)))).
% 2.38/1.47 thf(fact_166_local_OM5__lattice__def, axiom, (! [A2:a,B2:a,C:a]: (((((((((modula1618706180attice @ a) @ inf) @ less) @ sup) @ A2) @ B2) @ C) = ((((inf @ A2) @ B2) = ((inf @ B2) @ C)) & ((((inf @ C) @ A2) = ((inf @ B2) @ C)) & ((((sup @ A2) @ B2) = ((sup @ B2) @ C)) & ((((sup @ C) @ A2) = ((sup @ B2) @ C)) & ((less @ ((inf @ A2) @ B2)) @ ((sup @ A2) @ B2)))))))))).
% 2.38/1.47 thf(fact_167_local_OLeast1I, axiom, (! [P:(a > $o)]: (((? [X4:a]: (((P @ X4) & ((! [Y4:a]: (((P @ Y4) => ((less_eq @ X4) @ Y4)))) & (! [Y4:a]: ((((P @ Y4) & (! [Ya:a]: (((P @ Ya) => ((less_eq @ Y4) @ Ya))))) => (Y4 = X4)))))))) => (P @ (((least @ a) @ less_eq) @ P)))))).
% 2.38/1.47 thf(fact_168_local_OLeast1__le, axiom, (! [P:(a > $o),Z2:a]: (((? [X4:a]: (((P @ X4) & ((! [Y4:a]: (((P @ Y4) => ((less_eq @ X4) @ Y4)))) & (! [Y4:a]: ((((P @ Y4) & (! [Ya:a]: (((P @ Ya) => ((less_eq @ Y4) @ Ya))))) => (Y4 = X4)))))))) => ((P @ Z2) => ((less_eq @ (((least @ a) @ less_eq) @ P)) @ Z2)))))).
% 2.38/1.47 thf(fact_169_local_OLeastI2__order, axiom, (! [P:(a > $o),X:a,Q:(a > $o)]: (((P @ X) => ((! [Y4:a]: (((P @ Y4) => ((less_eq @ X) @ Y4)))) => ((! [X3:a]: (((P @ X3) => ((! [Y5:a]: (((P @ Y5) => ((less_eq @ X3) @ Y5)))) => (Q @ X3))))) => (Q @ (((least @ a) @ less_eq) @ P)))))))).
% 2.38/1.47 thf(fact_170_local_OLeast__equality, axiom, (! [P:(a > $o),X:a]: (((P @ X) => ((! [Y4:a]: (((P @ Y4) => ((less_eq @ X) @ Y4)))) => ((((least @ a) @ less_eq) @ P) = X)))))).
% 2.38/1.47 thf(fact_171_local_Oa__join__d, axiom, (! [A2:a,B2:a,C:a]: ((((sup @ A2) @ ((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C)) = ((sup @ A2) @ ((inf @ B2) @ C)))))).
% 2.38/1.47 thf(fact_172_local_Ob__join__d, axiom, (! [B2:a,A2:a,C:a]: ((((sup @ B2) @ ((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C)) = ((sup @ B2) @ ((inf @ C) @ A2)))))).
% 2.38/1.47 thf(fact_173_local_Od__aux__def, axiom, (! [A2:a,B2:a,C:a]: ((((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C) = ((sup @ ((sup @ ((inf @ A2) @ B2)) @ ((inf @ B2) @ C))) @ ((inf @ C) @ A2)))))).
% 2.38/1.47 thf(fact_174_local_Od__b__c__a, axiom, (! [B2:a,C:a,A2:a]: ((((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ B2) @ C) @ A2) = ((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C))))).
% 2.38/1.47 thf(fact_175_local_Od__c__a__b, axiom, (! [C:a,A2:a,B2:a]: ((((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ C) @ A2) @ B2) = ((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C))))).
% 2.38/1.47 thf(fact_176_local_Oa__meet__e, axiom, (! [A2:a,B2:a,C:a]: ((((inf @ A2) @ ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C)) = ((inf @ A2) @ ((sup @ B2) @ C)))))).
% 2.38/1.47 thf(fact_177_local_Ob__meet__e, axiom, (! [B2:a,A2:a,C:a]: ((((inf @ B2) @ ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C)) = ((inf @ B2) @ ((sup @ C) @ A2)))))).
% 2.38/1.47 thf(fact_178_local_Oc__meet__e, axiom, (! [C:a,A2:a,B2:a]: ((((inf @ C) @ ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C)) = ((inf @ C) @ ((sup @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_179_local_Oe__aux__def, axiom, (! [A2:a,B2:a,C:a]: ((((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C) = ((inf @ ((inf @ ((sup @ A2) @ B2)) @ ((sup @ B2) @ C))) @ ((sup @ C) @ A2)))))).
% 2.38/1.47 thf(fact_180_local_Oe__b__c__a, axiom, (! [B2:a,C:a,A2:a]: ((((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ B2) @ C) @ A2) = ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C))))).
% 2.38/1.47 thf(fact_181_local_Oe__c__a__b, axiom, (! [C:a,A2:a,B2:a]: ((((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ C) @ A2) @ B2) = ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C))))).
% 2.38/1.47 thf(fact_182_local_OInf__fin_Osemilattice__order__set__axioms, axiom, ((((lattic1693879045er_set @ a) @ inf) @ less_eq) @ less)).
% 2.38/1.47 thf(fact_183_local_OM5__lattice__incomp, axiom, (! [A2:a,B2:a,C:a]: (((((((((modula1618706180attice @ a) @ inf) @ less) @ sup) @ A2) @ B2) @ C) => ((((modula1937309406incomp @ a) @ less_eq) @ A2) @ B2))))).
% 2.38/1.47 thf(fact_184_local_Oc__aux__def, axiom, (! [A2:a,B2:a,C:a]: ((((((((modula849153099_c_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C) = ((sup @ ((inf @ C) @ ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C))) @ ((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C)))))).
% 2.38/1.47 thf(fact_185_local_Ob__aux__def, axiom, (! [A2:a,B2:a,C:a]: ((((((((modula477145868_b_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C) = ((sup @ ((inf @ B2) @ ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C))) @ ((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C)))))).
% 2.38/1.47 thf(fact_186_local_Oa__aux__def, axiom, (! [A2:a,B2:a,C:a]: ((((((((modula105138637_a_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C) = ((sup @ ((inf @ A2) @ ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C))) @ ((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C)))))).
% 2.38/1.47 thf(fact_187_lattice_Ono__distrib_Ocong, axiom, (! [A:$tType]: (((modula141148890istrib @ A) = (modula141148890istrib @ A))))).
% 2.38/1.47 thf(fact_188_lattice_ON5__lattice_Ocong, axiom, (! [A:$tType]: (((modula1192512645attice @ A) = (modula1192512645attice @ A))))).
% 2.38/1.47 thf(fact_189_lattice_OM5__lattice_Ocong, axiom, (! [A:$tType]: (((modula1618706180attice @ A) = (modula1618706180attice @ A))))).
% 2.38/1.47 thf(fact_190_lattice_Oincomp_Ocong, axiom, (! [A:$tType]: (((modula1937309406incomp @ A) = (modula1937309406incomp @ A))))).
% 2.38/1.47 thf(fact_191_lattice_Oe__aux_Ocong, axiom, (! [A:$tType]: (((modula1593167561_e_aux @ A) = (modula1593167561_e_aux @ A))))).
% 2.38/1.47 thf(fact_192_lattice_Od__aux_Ocong, axiom, (! [A:$tType]: (((modula1221160330_d_aux @ A) = (modula1221160330_d_aux @ A))))).
% 2.38/1.47 thf(fact_193_local_Oinf_Osemilattice__order__axioms, axiom, ((((semilattice_order @ a) @ inf) @ less_eq) @ less)).
% 2.38/1.47 thf(fact_194_local_Oorder_Oordering__axioms, axiom, (((ordering @ a) @ less_eq) @ less)).
% 2.38/1.47 thf(fact_195_local_Ocomp__fun__idem__sup, axiom, (((finite_comp_fun_idem @ a) @ a) @ sup)).
% 2.38/1.47 thf(fact_196_local_Ocomp__fun__idem__inf, axiom, (((finite_comp_fun_idem @ a) @ a) @ inf)).
% 2.38/1.47 thf(fact_197_local_Ob__a, axiom, (! [A2:a,B2:a,C:a]: ((((((((modula477145868_b_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C) = ((((((modula105138637_a_aux @ a) @ inf) @ sup) @ B2) @ C) @ A2))))).
% 2.38/1.47 thf(fact_198_local_Oc__a, axiom, (! [A2:a,B2:a,C:a]: ((((((((modula849153099_c_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C) = ((((((modula105138637_a_aux @ a) @ inf) @ sup) @ C) @ A2) @ B2))))).
% 2.38/1.47 thf(fact_199_lattice_Oc__aux_Ocong, axiom, (! [A:$tType]: (((modula849153099_c_aux @ A) = (modula849153099_c_aux @ A))))).
% 2.38/1.47 thf(fact_200_lattice_Ob__aux_Ocong, axiom, (! [A:$tType]: (((modula477145868_b_aux @ A) = (modula477145868_b_aux @ A))))).
% 2.38/1.47 thf(fact_201_lattice_Oa__aux_Ocong, axiom, (! [A:$tType]: (((modula105138637_a_aux @ A) = (modula105138637_a_aux @ A))))).
% 2.38/1.47 thf(fact_202_local_Osup_Osemigroup__axioms, axiom, ((semigroup @ a) @ sup)).
% 2.38/1.47 thf(fact_203_local_Oinf_Osemigroup__axioms, axiom, ((semigroup @ a) @ inf)).
% 2.38/1.47 thf(fact_204_local_Osup_Osemilattice__axioms, axiom, ((semilattice @ a) @ sup)).
% 2.38/1.47 thf(fact_205_local_Oinf_Osemilattice__axioms, axiom, ((semilattice @ a) @ inf)).
% 2.38/1.47 thf(fact_206_local_Osup_Oabel__semigroup__axioms, axiom, ((abel_semigroup @ a) @ sup)).
% 2.38/1.47 thf(fact_207_local_Oinf_Oabel__semigroup__axioms, axiom, ((abel_semigroup @ a) @ inf)).
% 2.38/1.47 thf(fact_208_semilattice__order__set_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: ((((((lattic1693879045er_set @ A) @ F) @ Less_eq) @ Less) => ((((semilattice_order @ A) @ F) @ Less_eq) @ Less))))).
% 2.38/1.47 thf(fact_209_abel__semigroup_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A))]: ((((abel_semigroup @ A) @ F) => ((semigroup @ A) @ F))))).
% 2.38/1.47 thf(fact_210_abel__semigroup_Oleft__commute, axiom, (! [A:$tType,F:(A > (A > A)),B2:A,A2:A,C:A]: ((((abel_semigroup @ A) @ F) => (((F @ B2) @ ((F @ A2) @ C)) = ((F @ A2) @ ((F @ B2) @ C))))))).
% 2.38/1.47 thf(fact_211_abel__semigroup_Ocommute, axiom, (! [A:$tType,F:(A > (A > A)),A2:A,B2:A]: ((((abel_semigroup @ A) @ F) => (((F @ A2) @ B2) = ((F @ B2) @ A2)))))).
% 2.38/1.47 thf(fact_212_semigroup_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: (((! [A5:A,B4:A,C2:A]: ((((F @ ((F @ A5) @ B4)) @ C2) = ((F @ A5) @ ((F @ B4) @ C2))))) => ((semigroup @ A) @ F))))).
% 2.38/1.47 thf(fact_213_semigroup_Oassoc, axiom, (! [A:$tType,F:(A > (A > A)),A2:A,B2:A,C:A]: ((((semigroup @ A) @ F) => (((F @ ((F @ A2) @ B2)) @ C) = ((F @ A2) @ ((F @ B2) @ C))))))).
% 2.38/1.47 thf(fact_214_semigroup__def, axiom, (! [A:$tType]: (((semigroup @ A) = (^ [F2:(A > (A > A))]: ((! [A3:A,B3:A,C3:A]: ((((F2 @ ((F2 @ A3) @ B3)) @ C3) = ((F2 @ A3) @ ((F2 @ B3) @ C3))))))))))).
% 2.38/1.47 thf(fact_215_ord_Ois__arg__min_Ocong, axiom, (! [B:$tType,A:$tType]: ((((lattices_is_arg_min @ A) @ B) = ((lattices_is_arg_min @ A) @ B))))).
% 2.38/1.47 thf(fact_216_ord_Ois__arg__max_Ocong, axiom, (! [B:$tType,A:$tType]: ((((lattices_is_arg_max @ A) @ B) = ((lattices_is_arg_max @ A) @ B))))).
% 2.38/1.47 thf(fact_217_ord_Ois__arg__min__def, axiom, (! [B:$tType,A:$tType]: ((((lattices_is_arg_min @ A) @ B) = (^ [Less2:(A > (A > $o)),F2:(B > A),P2:(B > $o),X2:B]: (((P2 @ X2) & (~ ((? [Y3:B]: (((P2 @ Y3) & ((Less2 @ (F2 @ Y3)) @ (F2 @ X2)))))))))))))).
% 2.38/1.47 thf(fact_218_ord_Ois__arg__max__def, axiom, (! [B:$tType,A:$tType]: ((((lattices_is_arg_max @ A) @ B) = (^ [Less2:(A > (A > $o)),F2:(B > A),P2:(B > $o),X2:B]: (((P2 @ X2) & (~ ((? [Y3:B]: (((P2 @ Y3) & ((Less2 @ (F2 @ X2)) @ (F2 @ Y3)))))))))))))).
% 2.38/1.47 thf(fact_219_semilattice_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A))]: ((((semilattice @ A) @ F) => ((abel_semigroup @ A) @ F))))).
% 2.38/1.47 thf(fact_220_local_OSup__fin_Osemilattice__set__axioms, axiom, ((lattic35693393ce_set @ a) @ sup)).
% 2.38/1.47 thf(fact_221_local_OInf__fin_Osemilattice__set__axioms, axiom, ((lattic35693393ce_set @ a) @ inf)).
% 2.38/1.47 thf(fact_222_semilattice__order__set_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: ((((((lattic1693879045er_set @ A) @ F) @ Less_eq) @ Less) => ((lattic35693393ce_set @ A) @ F))))).
% 2.38/1.47 thf(fact_223_semilattice__set__def, axiom, (! [A:$tType]: (((lattic35693393ce_set @ A) = (semilattice @ A))))).
% 2.38/1.47 thf(fact_224_semilattice__set_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: ((((semilattice @ A) @ F) => ((lattic35693393ce_set @ A) @ F))))).
% 2.38/1.47 thf(fact_225_semilattice__set_Oaxioms, axiom, (! [A:$tType,F:(A > (A > A))]: ((((lattic35693393ce_set @ A) @ F) => ((semilattice @ A) @ F))))).
% 2.38/1.47 thf(fact_226_semilattice__order__set__def, axiom, (! [A:$tType]: (((lattic1693879045er_set @ A) = (^ [F2:(A > (A > A)),Less_eq2:(A > (A > $o)),Less2:(A > (A > $o))]: ((((((semilattice_order @ A) @ F2) @ Less_eq2) @ Less2) & ((lattic35693393ce_set @ A) @ F2)))))))).
% 2.38/1.47 thf(fact_227_semilattice__order__set_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((lattic35693393ce_set @ A) @ F) => ((((lattic1693879045er_set @ A) @ F) @ Less_eq) @ Less)))))).
% 2.38/1.47 thf(fact_228_semilattice__order_Omono, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,C:A,B2:A,D:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A2) @ C) => (((Less_eq @ B2) @ D) => ((Less_eq @ ((F @ A2) @ B2)) @ ((F @ C) @ D)))))))).
% 2.38/1.47 thf(fact_229_semilattice__order_OorderE, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A2) @ B2) => (A2 = ((F @ A2) @ B2))))))).
% 2.38/1.47 thf(fact_230_semilattice__order_OorderI, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => ((A2 = ((F @ A2) @ B2)) => ((Less_eq @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_231_semilattice__order_Oabsorb1, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A2) @ B2) => (((F @ A2) @ B2) = A2)))))).
% 2.38/1.47 thf(fact_232_semilattice__order_Oabsorb2, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),B2:A,A2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ B2) @ A2) => (((F @ A2) @ B2) = B2)))))).
% 2.38/1.47 thf(fact_233_semilattice__order_OboundedE, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A,C:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A2) @ ((F @ B2) @ C)) => (~ ((((Less_eq @ A2) @ B2) => (~ (((Less_eq @ A2) @ C))))))))))).
% 2.38/1.47 thf(fact_234_semilattice__order_OboundedI, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A,C:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A2) @ B2) => (((Less_eq @ A2) @ C) => ((Less_eq @ A2) @ ((F @ B2) @ C)))))))).
% 2.38/1.47 thf(fact_235_semilattice__order_Oorder__iff, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A2) @ B2) = (A2 = ((F @ A2) @ B2))))))).
% 2.38/1.47 thf(fact_236_semilattice__order_Ocobounded1, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => ((Less_eq @ ((F @ A2) @ B2)) @ A2))))).
% 2.38/1.47 thf(fact_237_semilattice__order_Ocobounded2, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => ((Less_eq @ ((F @ A2) @ B2)) @ B2))))).
% 2.38/1.47 thf(fact_238_semilattice__order_Oabsorb__iff1, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A2) @ B2) = (((F @ A2) @ B2) = A2)))))).
% 2.38/1.47 thf(fact_239_semilattice__order_Oabsorb__iff2, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),B2:A,A2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ B2) @ A2) = (((F @ A2) @ B2) = B2)))))).
% 2.38/1.47 thf(fact_240_semilattice__order_Obounded__iff, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A,C:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A2) @ ((F @ B2) @ C)) = (((Less_eq @ A2) @ B2) & ((Less_eq @ A2) @ C))))))).
% 2.38/1.47 thf(fact_241_semilattice__order_OcoboundedI1, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,C:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ A2) @ C) => ((Less_eq @ ((F @ A2) @ B2)) @ C)))))).
% 2.38/1.47 thf(fact_242_semilattice__order_OcoboundedI2, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),B2:A,C:A,A2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less_eq @ B2) @ C) => ((Less_eq @ ((F @ A2) @ B2)) @ C)))))).
% 2.38/1.47 thf(fact_243_semilattice__order_Ostrict__boundedE, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A,C:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less @ A2) @ ((F @ B2) @ C)) => (~ ((((Less @ A2) @ B2) => (~ (((Less @ A2) @ C))))))))))).
% 2.38/1.47 thf(fact_244_semilattice__order_Ostrict__order__iff, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less @ A2) @ B2) = ((A2 = ((F @ A2) @ B2)) & (A2 != B2))))))).
% 2.38/1.47 thf(fact_245_semilattice__order_Ostrict__coboundedI1, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,C:A,B2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less @ A2) @ C) => ((Less @ ((F @ A2) @ B2)) @ C)))))).
% 2.38/1.47 thf(fact_246_semilattice__order_Ostrict__coboundedI2, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o)),B2:A,C:A,A2:A]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => (((Less @ B2) @ C) => ((Less @ ((F @ A2) @ B2)) @ C)))))).
% 2.38/1.47 thf(fact_247_semilattice_Oidem, axiom, (! [A:$tType,F:(A > (A > A)),A2:A]: ((((semilattice @ A) @ F) => (((F @ A2) @ A2) = A2))))).
% 2.38/1.47 thf(fact_248_semilattice_Oleft__idem, axiom, (! [A:$tType,F:(A > (A > A)),A2:A,B2:A]: ((((semilattice @ A) @ F) => (((F @ A2) @ ((F @ A2) @ B2)) = ((F @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_249_semilattice_Oright__idem, axiom, (! [A:$tType,F:(A > (A > A)),A2:A,B2:A]: ((((semilattice @ A) @ F) => (((F @ ((F @ A2) @ B2)) @ B2) = ((F @ A2) @ B2)))))).
% 2.38/1.47 thf(fact_250_semilattice__order_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A)),Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: ((((((semilattice_order @ A) @ F) @ Less_eq) @ Less) => ((semilattice @ A) @ F))))).
% 2.38/1.47 thf(fact_251_local_Obdd__below__def, axiom, (! [A4:(set @ a)]: (((((condit1201339847_below @ a) @ less_eq) @ A4) = (? [M:a]: ((! [X2:a]: (((((member @ a) @ X2) @ A4) => ((less_eq @ M) @ X2)))))))))).
% 2.38/1.47 thf(fact_252_local_Obdd__above__def, axiom, (! [A4:(set @ a)]: (((((condit2040224947_above @ a) @ less_eq) @ A4) = (? [M2:a]: ((! [X2:a]: (((((member @ a) @ X2) @ A4) => ((less_eq @ X2) @ M2)))))))))).
% 2.38/1.47 thf(fact_253_local_Obdd__belowI, axiom, (! [A4:(set @ a),M3:a]: (((! [X3:a]: (((((member @ a) @ X3) @ A4) => ((less_eq @ M3) @ X3)))) => (((condit1201339847_below @ a) @ less_eq) @ A4))))).
% 2.38/1.47 thf(fact_254_local_Obdd__aboveI, axiom, (! [A4:(set @ a),M4:a]: (((! [X3:a]: (((((member @ a) @ X3) @ A4) => ((less_eq @ X3) @ M4)))) => (((condit2040224947_above @ a) @ less_eq) @ A4))))).
% 2.38/1.47 thf(fact_255_local_Obdd__below__Ioo, axiom, (! [A2:a,B2:a]: ((((condit1201339847_below @ a) @ less_eq) @ ((((set_gr1161524159ssThan @ a) @ less) @ A2) @ B2))))).
% 2.38/1.47 thf(conj_0, hypothesis, (((((lattice @ a) @ inf) @ less_eq) @ less) @ sup)).
% 2.38/1.47 thf(conj_1, hypothesis, ((less_eq @ x) @ y)).
% 2.38/1.47 thf(conj_2, hypothesis, (((sup @ x) @ ((inf @ y) @ z)) != ((inf @ y) @ ((sup @ x) @ z)))).
% 2.38/1.47 thf(conj_3, hypothesis, ((less @ ((sup @ x) @ ((inf @ y) @ z))) @ ((inf @ y) @ ((sup @ x) @ z)))).
% 2.38/1.47 thf(conj_4, conjecture, ((less_eq @ x) @ ((sup @ ((inf @ y) @ ((sup @ x) @ z))) @ z))).
% 2.38/1.47 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.Vy3zEj9nqo/DTF2DTF5351.p
% 2.38/1.47 ------------------------
% 2.38/1.49 ---- Cleaned THF ---
% 2.38/1.49 thf(ty_t_Set_Oset,type,
% 2.38/1.49 set: $tType > $tType ).
% 2.38/1.49
% 2.38/1.49 thf(ty_tf_a,type,
% 2.38/1.49 a: $tType ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
% 2.38/1.49 condit2040224947_above:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
% 2.38/1.49 condit1201339847_below:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Finite__Set_Ocomp__fun__idem,type,
% 2.38/1.49 finite_comp_fun_idem:
% 2.38/1.49 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Groups_Oabel__semigroup,type,
% 2.38/1.49 abel_semigroup:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Groups_Osemigroup,type,
% 2.38/1.49 semigroup:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Lattices_Oclass_Olattice,type,
% 2.38/1.49 lattice:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > ( A > A > A ) > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Lattices_Osemilattice,type,
% 2.38/1.49 semilattice:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Lattices_Osemilattice__order,type,
% 2.38/1.49 semilattice_order:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Lattices__Big_Oord_Ois__arg__max,type,
% 2.38/1.49 lattices_is_arg_max:
% 2.38/1.49 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( B > $o ) > B > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Lattices__Big_Oord_Ois__arg__min,type,
% 2.38/1.49 lattices_is_arg_min:
% 2.38/1.49 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( B > $o ) > B > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
% 2.38/1.49 lattic1693879045er_set:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Lattices__Big_Osemilattice__set,type,
% 2.38/1.49 lattic35693393ce_set:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_OM5__lattice,type,
% 2.38/1.49 modula1618706180attice:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > A ) > A > A > A > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_ON5__lattice,type,
% 2.38/1.49 modula1192512645attice:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > A ) > A > A > A > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oa__aux,type,
% 2.38/1.49 modula105138637_a_aux:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Ob__aux,type,
% 2.38/1.49 modula477145868_b_aux:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oc__aux,type,
% 2.38/1.49 modula849153099_c_aux:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Od__aux,type,
% 2.38/1.49 modula1221160330_d_aux:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oe__aux,type,
% 2.38/1.49 modula1593167561_e_aux:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oincomp,type,
% 2.38/1.49 modula1937309406incomp:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Ono__distrib,type,
% 2.38/1.49 modula141148890istrib:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > A ) > A > A > A > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Orderings_Oord_OLeast,type,
% 2.38/1.49 least:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Orderings_Oord_Omax,type,
% 2.38/1.49 max:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Orderings_Oord_Omin,type,
% 2.38/1.49 min:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Orderings_Oorder_OGreatest,type,
% 2.38/1.49 greatest:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Orderings_Oordering,type,
% 2.38/1.49 ordering:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Set_OCollect,type,
% 2.38/1.49 collect:
% 2.38/1.49 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan,type,
% 2.38/1.49 set_gr1161524159ssThan:
% 2.38/1.49 !>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_c_member,type,
% 2.38/1.49 member:
% 2.38/1.49 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.38/1.49
% 2.38/1.49 thf(sy_v_inf,type,
% 2.38/1.49 inf: a > a > a ).
% 2.38/1.49
% 2.38/1.49 thf(sy_v_less,type,
% 2.38/1.49 less: a > a > $o ).
% 2.38/1.49
% 2.38/1.49 thf(sy_v_less__eq,type,
% 2.38/1.49 less_eq: a > a > $o ).
% 2.38/1.49
% 2.38/1.49 thf(sy_v_sup,type,
% 2.38/1.49 sup: a > a > a ).
% 2.38/1.49
% 2.38/1.49 thf(sy_v_x,type,
% 2.38/1.49 x: a ).
% 2.38/1.49
% 2.38/1.49 thf(sy_v_y,type,
% 2.38/1.49 y: a ).
% 2.38/1.49
% 2.38/1.49 thf(sy_v_z,type,
% 2.38/1.49 z: a ).
% 2.38/1.49
% 2.38/1.49 thf(fact_0_local_Oantisym,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less_eq @ X @ Y )
% 2.38/1.49 => ( ( less_eq @ Y @ X )
% 2.38/1.49 => ( X = Y ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_1_local_Oantisym__conv,axiom,
% 2.38/1.49 ! [Y: a,X: a] :
% 2.38/1.49 ( ( less_eq @ Y @ X )
% 2.38/1.49 => ( ( less_eq @ X @ Y )
% 2.38/1.49 = ( X = Y ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_2_local_Odual__order_Oantisym,axiom,
% 2.38/1.49 ! [B2: a,A2: a] :
% 2.38/1.49 ( ( less_eq @ B2 @ A2 )
% 2.38/1.49 => ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 => ( A2 = B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_3_local_Odual__order_Oeq__iff,axiom,
% 2.38/1.49 ( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
% 2.38/1.49 = ( ^ [A3: a,B3: a] :
% 2.38/1.49 ( ( less_eq @ B3 @ A3 )
% 2.38/1.49 & ( less_eq @ A3 @ B3 ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_4_local_Odual__order_Otrans,axiom,
% 2.38/1.49 ! [B2: a,A2: a,C: a] :
% 2.38/1.49 ( ( less_eq @ B2 @ A2 )
% 2.38/1.49 => ( ( less_eq @ C @ B2 )
% 2.38/1.49 => ( less_eq @ C @ A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_5_local_Oeq__iff,axiom,
% 2.38/1.49 ( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
% 2.38/1.49 = ( ^ [X2: a,Y3: a] :
% 2.38/1.49 ( ( less_eq @ X2 @ Y3 )
% 2.38/1.49 & ( less_eq @ Y3 @ X2 ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_6_local_Oeq__refl,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( X = Y )
% 2.38/1.49 => ( less_eq @ X @ Y ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_7_local_Oord__eq__le__trans,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( A2 = B2 )
% 2.38/1.49 => ( ( less_eq @ B2 @ C )
% 2.38/1.49 => ( less_eq @ A2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_8_local_Oord__le__eq__trans,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 => ( ( B2 = C )
% 2.38/1.49 => ( less_eq @ A2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_9_local_Oorder_Oantisym,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 => ( ( less_eq @ B2 @ A2 )
% 2.38/1.49 => ( A2 = B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_10_local_Oorder_Oeq__iff,axiom,
% 2.38/1.49 ( ( ^ [Y2: a,Z: a] : ( Y2 = Z ) )
% 2.38/1.49 = ( ^ [A3: a,B3: a] :
% 2.38/1.49 ( ( less_eq @ A3 @ B3 )
% 2.38/1.49 & ( less_eq @ B3 @ A3 ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_11_local_Oorder_Otrans,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 => ( ( less_eq @ B2 @ C )
% 2.38/1.49 => ( less_eq @ A2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_12_local_Oorder__trans,axiom,
% 2.38/1.49 ! [X: a,Y: a,Z2: a] :
% 2.38/1.49 ( ( less_eq @ X @ Y )
% 2.38/1.49 => ( ( less_eq @ Y @ Z2 )
% 2.38/1.49 => ( less_eq @ X @ Z2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_13_local_Odual__order_Oasym,axiom,
% 2.38/1.49 ! [B2: a,A2: a] :
% 2.38/1.49 ( ( less @ B2 @ A2 )
% 2.38/1.49 => ~ ( less @ A2 @ B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_14_local_Odual__order_Ostrict__implies__not__eq,axiom,
% 2.38/1.49 ! [B2: a,A2: a] :
% 2.38/1.49 ( ( less @ B2 @ A2 )
% 2.38/1.49 => ( A2 != B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_15_local_Odual__order_Ostrict__trans,axiom,
% 2.38/1.49 ! [B2: a,A2: a,C: a] :
% 2.38/1.49 ( ( less @ B2 @ A2 )
% 2.38/1.49 => ( ( less @ C @ B2 )
% 2.38/1.49 => ( less @ C @ A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_16_local_Oless__asym,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less @ X @ Y )
% 2.38/1.49 => ~ ( less @ Y @ X ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_17_local_Oless__asym_H,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( less @ A2 @ B2 )
% 2.38/1.49 => ~ ( less @ B2 @ A2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_18_local_Oless__imp__neq,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less @ X @ Y )
% 2.38/1.49 => ( X != Y ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_19_local_Oless__imp__not__eq,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less @ X @ Y )
% 2.38/1.49 => ( X != Y ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_20_local_Oless__imp__not__eq2,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less @ X @ Y )
% 2.38/1.49 => ( Y != X ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_21_local_Oless__imp__not__less,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less @ X @ Y )
% 2.38/1.49 => ~ ( less @ Y @ X ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_22_local_Oless__imp__triv,axiom,
% 2.38/1.49 ! [X: a,Y: a,P: $o] :
% 2.38/1.49 ( ( less @ X @ Y )
% 2.38/1.49 => ( ( less @ Y @ X )
% 2.38/1.49 => P ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_23_local_Oless__irrefl,axiom,
% 2.38/1.49 ! [X: a] :
% 2.38/1.49 ~ ( less @ X @ X ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_24_local_Oless__not__sym,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less @ X @ Y )
% 2.38/1.49 => ~ ( less @ Y @ X ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_25_local_Oless__trans,axiom,
% 2.38/1.49 ! [X: a,Y: a,Z2: a] :
% 2.38/1.49 ( ( less @ X @ Y )
% 2.38/1.49 => ( ( less @ Y @ Z2 )
% 2.38/1.49 => ( less @ X @ Z2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_26_local_Oord__eq__less__trans,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( A2 = B2 )
% 2.38/1.49 => ( ( less @ B2 @ C )
% 2.38/1.49 => ( less @ A2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_27_local_Oord__less__eq__trans,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( less @ A2 @ B2 )
% 2.38/1.49 => ( ( B2 = C )
% 2.38/1.49 => ( less @ A2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_28_local_Oorder_Oasym,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( less @ A2 @ B2 )
% 2.38/1.49 => ~ ( less @ B2 @ A2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_29_local_Oorder_Oirrefl,axiom,
% 2.38/1.49 ! [A2: a] :
% 2.38/1.49 ~ ( less @ A2 @ A2 ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_30_local_Oorder_Ostrict__implies__not__eq,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( less @ A2 @ B2 )
% 2.38/1.49 => ( A2 != B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_31_local_Oorder_Ostrict__trans,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( less @ A2 @ B2 )
% 2.38/1.49 => ( ( less @ B2 @ C )
% 2.38/1.49 => ( less @ A2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_32_local_Oinf_Oassoc,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( inf @ ( inf @ A2 @ B2 ) @ C )
% 2.38/1.49 = ( inf @ A2 @ ( inf @ B2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_33_local_Oinf_Ocommute,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( inf @ A2 @ B2 )
% 2.38/1.49 = ( inf @ B2 @ A2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_34_local_Oinf_Oleft__commute,axiom,
% 2.38/1.49 ! [B2: a,A2: a,C: a] :
% 2.38/1.49 ( ( inf @ B2 @ ( inf @ A2 @ C ) )
% 2.38/1.49 = ( inf @ A2 @ ( inf @ B2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_35_local_Oinf__assoc,axiom,
% 2.38/1.49 ! [X: a,Y: a,Z2: a] :
% 2.38/1.49 ( ( inf @ ( inf @ X @ Y ) @ Z2 )
% 2.38/1.49 = ( inf @ X @ ( inf @ Y @ Z2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_36_local_Oinf__commute,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( inf @ X @ Y )
% 2.38/1.49 = ( inf @ Y @ X ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_37_local_Oinf__left__commute,axiom,
% 2.38/1.49 ! [X: a,Y: a,Z2: a] :
% 2.38/1.49 ( ( inf @ X @ ( inf @ Y @ Z2 ) )
% 2.38/1.49 = ( inf @ Y @ ( inf @ X @ Z2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_38_local_Osup_Oassoc,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( sup @ ( sup @ A2 @ B2 ) @ C )
% 2.38/1.49 = ( sup @ A2 @ ( sup @ B2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_39_local_Osup_Ocommute,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( sup @ A2 @ B2 )
% 2.38/1.49 = ( sup @ B2 @ A2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_40_local_Osup_Oleft__commute,axiom,
% 2.38/1.49 ! [B2: a,A2: a,C: a] :
% 2.38/1.49 ( ( sup @ B2 @ ( sup @ A2 @ C ) )
% 2.38/1.49 = ( sup @ A2 @ ( sup @ B2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_41_local_Osup__assoc,axiom,
% 2.38/1.49 ! [X: a,Y: a,Z2: a] :
% 2.38/1.49 ( ( sup @ ( sup @ X @ Y ) @ Z2 )
% 2.38/1.49 = ( sup @ X @ ( sup @ Y @ Z2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_42_local_Osup__commute,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( sup @ X @ Y )
% 2.38/1.49 = ( sup @ Y @ X ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_43_local_Osup__left__commute,axiom,
% 2.38/1.49 ! [X: a,Y: a,Z2: a] :
% 2.38/1.49 ( ( sup @ X @ ( sup @ Y @ Z2 ) )
% 2.38/1.49 = ( sup @ Y @ ( sup @ X @ Z2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_44_local_Oantisym__conv1,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ~ ( less @ X @ Y )
% 2.38/1.49 => ( ( less_eq @ X @ Y )
% 2.38/1.49 = ( X = Y ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_45_mem__Collect__eq,axiom,
% 2.38/1.49 ! [A: $tType,A2: A,P: A > $o] :
% 2.38/1.49 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.38/1.49 = ( P @ A2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_46_Collect__mem__eq,axiom,
% 2.38/1.49 ! [A: $tType,A4: set @ A] :
% 2.38/1.49 ( ( collect @ A
% 2.38/1.49 @ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
% 2.38/1.49 = A4 ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_47_Collect__cong,axiom,
% 2.38/1.49 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.38/1.49 ( ! [X3: A] :
% 2.38/1.49 ( ( P @ X3 )
% 2.38/1.49 = ( Q @ X3 ) )
% 2.38/1.49 => ( ( collect @ A @ P )
% 2.38/1.49 = ( collect @ A @ Q ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_48_ext,axiom,
% 2.38/1.49 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.38/1.49 ( ! [X3: A] :
% 2.38/1.49 ( ( F @ X3 )
% 2.38/1.49 = ( G @ X3 ) )
% 2.38/1.49 => ( F = G ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_49_local_Oantisym__conv2,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less_eq @ X @ Y )
% 2.38/1.49 => ( ( ~ ( less @ X @ Y ) )
% 2.38/1.49 = ( X = Y ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_50_local_Odual__order_Onot__eq__order__implies__strict,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( A2 != B2 )
% 2.38/1.49 => ( ( less_eq @ B2 @ A2 )
% 2.38/1.49 => ( less @ B2 @ A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_51_local_Odual__order_Oorder__iff__strict,axiom,
% 2.38/1.49 ! [B2: a,A2: a] :
% 2.38/1.49 ( ( less_eq @ B2 @ A2 )
% 2.38/1.49 = ( ( less @ B2 @ A2 )
% 2.38/1.49 | ( A2 = B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_52_local_Odual__order_Ostrict__iff__order,axiom,
% 2.38/1.49 ! [B2: a,A2: a] :
% 2.38/1.49 ( ( less @ B2 @ A2 )
% 2.38/1.49 = ( ( less_eq @ B2 @ A2 )
% 2.38/1.49 & ( A2 != B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_53_local_Odual__order_Ostrict__implies__order,axiom,
% 2.38/1.49 ! [B2: a,A2: a] :
% 2.38/1.49 ( ( less @ B2 @ A2 )
% 2.38/1.49 => ( less_eq @ B2 @ A2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_54_local_Odual__order_Ostrict__trans1,axiom,
% 2.38/1.49 ! [B2: a,A2: a,C: a] :
% 2.38/1.49 ( ( less_eq @ B2 @ A2 )
% 2.38/1.49 => ( ( less @ C @ B2 )
% 2.38/1.49 => ( less @ C @ A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_55_local_Odual__order_Ostrict__trans2,axiom,
% 2.38/1.49 ! [B2: a,A2: a,C: a] :
% 2.38/1.49 ( ( less @ B2 @ A2 )
% 2.38/1.49 => ( ( less_eq @ C @ B2 )
% 2.38/1.49 => ( less @ C @ A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_56_local_OleD,axiom,
% 2.38/1.49 ! [Y: a,X: a] :
% 2.38/1.49 ( ( less_eq @ Y @ X )
% 2.38/1.49 => ~ ( less @ X @ Y ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_57_local_Ole__imp__less__or__eq,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less_eq @ X @ Y )
% 2.38/1.49 => ( ( less @ X @ Y )
% 2.38/1.49 | ( X = Y ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_58_local_Ole__less,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less_eq @ X @ Y )
% 2.38/1.49 = ( ( less @ X @ Y )
% 2.38/1.49 | ( X = Y ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_59_local_Ole__less__trans,axiom,
% 2.38/1.49 ! [X: a,Y: a,Z2: a] :
% 2.38/1.49 ( ( less_eq @ X @ Y )
% 2.38/1.49 => ( ( less @ Y @ Z2 )
% 2.38/1.49 => ( less @ X @ Z2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_60_local_Ole__neq__trans,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 => ( ( A2 != B2 )
% 2.38/1.49 => ( less @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_61_local_Oless__imp__le,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less @ X @ Y )
% 2.38/1.49 => ( less_eq @ X @ Y ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_62_local_Oless__le,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less @ X @ Y )
% 2.38/1.49 = ( ( less_eq @ X @ Y )
% 2.38/1.49 & ( X != Y ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_63_local_Oless__le__not__le,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less @ X @ Y )
% 2.38/1.49 = ( ( less_eq @ X @ Y )
% 2.38/1.49 & ~ ( less_eq @ Y @ X ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_64_local_Oless__le__trans,axiom,
% 2.38/1.49 ! [X: a,Y: a,Z2: a] :
% 2.38/1.49 ( ( less @ X @ Y )
% 2.38/1.49 => ( ( less_eq @ Y @ Z2 )
% 2.38/1.49 => ( less @ X @ Z2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_65_local_Oorder_Onot__eq__order__implies__strict,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( A2 != B2 )
% 2.38/1.49 => ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 => ( less @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_66_local_Oorder_Oorder__iff__strict,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 = ( ( less @ A2 @ B2 )
% 2.38/1.49 | ( A2 = B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_67_local_Oorder_Ostrict__iff__order,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( less @ A2 @ B2 )
% 2.38/1.49 = ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 & ( A2 != B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_68_local_Oorder_Ostrict__implies__order,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( less @ A2 @ B2 )
% 2.38/1.49 => ( less_eq @ A2 @ B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_69_local_Oorder_Ostrict__trans1,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 => ( ( less @ B2 @ C )
% 2.38/1.49 => ( less @ A2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_70_local_Oorder_Ostrict__trans2,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( less @ A2 @ B2 )
% 2.38/1.49 => ( ( less_eq @ B2 @ C )
% 2.38/1.49 => ( less @ A2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_71_local_Oinf_Oabsorb1,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 => ( ( inf @ A2 @ B2 )
% 2.38/1.49 = A2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_72_local_Oinf_Oabsorb2,axiom,
% 2.38/1.49 ! [B2: a,A2: a] :
% 2.38/1.49 ( ( less_eq @ B2 @ A2 )
% 2.38/1.49 => ( ( inf @ A2 @ B2 )
% 2.38/1.49 = B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_73_local_Oinf_Oabsorb__iff1,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 = ( ( inf @ A2 @ B2 )
% 2.38/1.49 = A2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_74_local_Oinf_Oabsorb__iff2,axiom,
% 2.38/1.49 ! [B2: a,A2: a] :
% 2.38/1.49 ( ( less_eq @ B2 @ A2 )
% 2.38/1.49 = ( ( inf @ A2 @ B2 )
% 2.38/1.49 = B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_75_local_Oinf_OboundedE,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ ( inf @ B2 @ C ) )
% 2.38/1.49 => ~ ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 => ~ ( less_eq @ A2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_76_local_Oinf_OboundedI,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 => ( ( less_eq @ A2 @ C )
% 2.38/1.49 => ( less_eq @ A2 @ ( inf @ B2 @ C ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_77_local_Oinf_Ocobounded1,axiom,
% 2.38/1.49 ! [A2: a,B2: a] : ( less_eq @ ( inf @ A2 @ B2 ) @ A2 ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_78_local_Oinf_Ocobounded2,axiom,
% 2.38/1.49 ! [A2: a,B2: a] : ( less_eq @ ( inf @ A2 @ B2 ) @ B2 ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_79_local_Oinf_OcoboundedI1,axiom,
% 2.38/1.49 ! [A2: a,C: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ C )
% 2.38/1.49 => ( less_eq @ ( inf @ A2 @ B2 ) @ C ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_80_local_Oinf_OcoboundedI2,axiom,
% 2.38/1.49 ! [B2: a,C: a,A2: a] :
% 2.38/1.49 ( ( less_eq @ B2 @ C )
% 2.38/1.49 => ( less_eq @ ( inf @ A2 @ B2 ) @ C ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_81_local_Oinf_OorderE,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 => ( A2
% 2.38/1.49 = ( inf @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_82_local_Oinf_OorderI,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( A2
% 2.38/1.49 = ( inf @ A2 @ B2 ) )
% 2.38/1.49 => ( less_eq @ A2 @ B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_83_local_Oinf_Oorder__iff,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 = ( A2
% 2.38/1.49 = ( inf @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_84_local_Oinf__absorb1,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less_eq @ X @ Y )
% 2.38/1.49 => ( ( inf @ X @ Y )
% 2.38/1.49 = X ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_85_local_Oinf__absorb2,axiom,
% 2.38/1.49 ! [Y: a,X: a] :
% 2.38/1.49 ( ( less_eq @ Y @ X )
% 2.38/1.49 => ( ( inf @ X @ Y )
% 2.38/1.49 = Y ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_86_local_Oinf__greatest,axiom,
% 2.38/1.49 ! [X: a,Y: a,Z2: a] :
% 2.38/1.49 ( ( less_eq @ X @ Y )
% 2.38/1.49 => ( ( less_eq @ X @ Z2 )
% 2.38/1.49 => ( less_eq @ X @ ( inf @ Y @ Z2 ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_87_local_Oinf__le1,axiom,
% 2.38/1.49 ! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ X ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_88_local_Oinf__le2,axiom,
% 2.38/1.49 ! [X: a,Y: a] : ( less_eq @ ( inf @ X @ Y ) @ Y ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_89_local_Oinf__mono,axiom,
% 2.38/1.49 ! [A2: a,C: a,B2: a,D: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ C )
% 2.38/1.49 => ( ( less_eq @ B2 @ D )
% 2.38/1.49 => ( less_eq @ ( inf @ A2 @ B2 ) @ ( inf @ C @ D ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_90_local_Oinf__unique,axiom,
% 2.38/1.49 ! [F: a > a > a,X: a,Y: a] :
% 2.38/1.49 ( ! [X3: a,Y4: a] : ( less_eq @ ( F @ X3 @ Y4 ) @ X3 )
% 2.38/1.49 => ( ! [X3: a,Y4: a] : ( less_eq @ ( F @ X3 @ Y4 ) @ Y4 )
% 2.38/1.49 => ( ! [X3: a,Y4: a,Z3: a] :
% 2.38/1.49 ( ( less_eq @ X3 @ Y4 )
% 2.38/1.49 => ( ( less_eq @ X3 @ Z3 )
% 2.38/1.49 => ( less_eq @ X3 @ ( F @ Y4 @ Z3 ) ) ) )
% 2.38/1.49 => ( ( inf @ X @ Y )
% 2.38/1.49 = ( F @ X @ Y ) ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_91_local_Ole__iff__inf,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less_eq @ X @ Y )
% 2.38/1.49 = ( ( inf @ X @ Y )
% 2.38/1.49 = X ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_92_local_Ole__infE,axiom,
% 2.38/1.49 ! [X: a,A2: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ X @ ( inf @ A2 @ B2 ) )
% 2.38/1.49 => ~ ( ( less_eq @ X @ A2 )
% 2.38/1.49 => ~ ( less_eq @ X @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_93_local_Ole__infI,axiom,
% 2.38/1.49 ! [X: a,A2: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ X @ A2 )
% 2.38/1.49 => ( ( less_eq @ X @ B2 )
% 2.38/1.49 => ( less_eq @ X @ ( inf @ A2 @ B2 ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_94_local_Ole__infI1,axiom,
% 2.38/1.49 ! [A2: a,X: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ X )
% 2.38/1.49 => ( less_eq @ ( inf @ A2 @ B2 ) @ X ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_95_local_Ole__infI2,axiom,
% 2.38/1.49 ! [B2: a,X: a,A2: a] :
% 2.38/1.49 ( ( less_eq @ B2 @ X )
% 2.38/1.49 => ( less_eq @ ( inf @ A2 @ B2 ) @ X ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_96_local_Ole__iff__sup,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less_eq @ X @ Y )
% 2.38/1.49 = ( ( sup @ X @ Y )
% 2.38/1.49 = Y ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_97_local_Ole__supE,axiom,
% 2.38/1.49 ! [A2: a,B2: a,X: a] :
% 2.38/1.49 ( ( less_eq @ ( sup @ A2 @ B2 ) @ X )
% 2.38/1.49 => ~ ( ( less_eq @ A2 @ X )
% 2.38/1.49 => ~ ( less_eq @ B2 @ X ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_98_local_Ole__supI,axiom,
% 2.38/1.49 ! [A2: a,X: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ X )
% 2.38/1.49 => ( ( less_eq @ B2 @ X )
% 2.38/1.49 => ( less_eq @ ( sup @ A2 @ B2 ) @ X ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_99_local_Ole__supI1,axiom,
% 2.38/1.49 ! [X: a,A2: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ X @ A2 )
% 2.38/1.49 => ( less_eq @ X @ ( sup @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_100_local_Ole__supI2,axiom,
% 2.38/1.49 ! [X: a,B2: a,A2: a] :
% 2.38/1.49 ( ( less_eq @ X @ B2 )
% 2.38/1.49 => ( less_eq @ X @ ( sup @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_101_local_Osup_Oabsorb1,axiom,
% 2.38/1.49 ! [B2: a,A2: a] :
% 2.38/1.49 ( ( less_eq @ B2 @ A2 )
% 2.38/1.49 => ( ( sup @ A2 @ B2 )
% 2.38/1.49 = A2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_102_local_Osup_Oabsorb2,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 => ( ( sup @ A2 @ B2 )
% 2.38/1.49 = B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_103_local_Osup_Oabsorb__iff1,axiom,
% 2.38/1.49 ! [B2: a,A2: a] :
% 2.38/1.49 ( ( less_eq @ B2 @ A2 )
% 2.38/1.49 = ( ( sup @ A2 @ B2 )
% 2.38/1.49 = A2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_104_local_Osup_Oabsorb__iff2,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 = ( ( sup @ A2 @ B2 )
% 2.38/1.49 = B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_105_local_Osup_OboundedE,axiom,
% 2.38/1.49 ! [B2: a,C: a,A2: a] :
% 2.38/1.49 ( ( less_eq @ ( sup @ B2 @ C ) @ A2 )
% 2.38/1.49 => ~ ( ( less_eq @ B2 @ A2 )
% 2.38/1.49 => ~ ( less_eq @ C @ A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_106_local_Osup_OboundedI,axiom,
% 2.38/1.49 ! [B2: a,A2: a,C: a] :
% 2.38/1.49 ( ( less_eq @ B2 @ A2 )
% 2.38/1.49 => ( ( less_eq @ C @ A2 )
% 2.38/1.49 => ( less_eq @ ( sup @ B2 @ C ) @ A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_107_local_Osup_Ocobounded1,axiom,
% 2.38/1.49 ! [A2: a,B2: a] : ( less_eq @ A2 @ ( sup @ A2 @ B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_108_local_Osup_Ocobounded2,axiom,
% 2.38/1.49 ! [B2: a,A2: a] : ( less_eq @ B2 @ ( sup @ A2 @ B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_109_local_Osup_OcoboundedI1,axiom,
% 2.38/1.49 ! [C: a,A2: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ C @ A2 )
% 2.38/1.49 => ( less_eq @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_110_local_Osup_OcoboundedI2,axiom,
% 2.38/1.49 ! [C: a,B2: a,A2: a] :
% 2.38/1.49 ( ( less_eq @ C @ B2 )
% 2.38/1.49 => ( less_eq @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_111_local_Osup_Omono,axiom,
% 2.38/1.49 ! [C: a,A2: a,D: a,B2: a] :
% 2.38/1.49 ( ( less_eq @ C @ A2 )
% 2.38/1.49 => ( ( less_eq @ D @ B2 )
% 2.38/1.49 => ( less_eq @ ( sup @ C @ D ) @ ( sup @ A2 @ B2 ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_112_local_Osup_OorderE,axiom,
% 2.38/1.49 ! [B2: a,A2: a] :
% 2.38/1.49 ( ( less_eq @ B2 @ A2 )
% 2.38/1.49 => ( A2
% 2.38/1.49 = ( sup @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_113_local_Osup_OorderI,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( A2
% 2.38/1.49 = ( sup @ A2 @ B2 ) )
% 2.38/1.49 => ( less_eq @ B2 @ A2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_114_local_Osup_Oorder__iff,axiom,
% 2.38/1.49 ! [B2: a,A2: a] :
% 2.38/1.49 ( ( less_eq @ B2 @ A2 )
% 2.38/1.49 = ( A2
% 2.38/1.49 = ( sup @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_115_local_Osup__absorb1,axiom,
% 2.38/1.49 ! [Y: a,X: a] :
% 2.38/1.49 ( ( less_eq @ Y @ X )
% 2.38/1.49 => ( ( sup @ X @ Y )
% 2.38/1.49 = X ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_116_local_Osup__absorb2,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( less_eq @ X @ Y )
% 2.38/1.49 => ( ( sup @ X @ Y )
% 2.38/1.49 = Y ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_117_local_Osup__ge1,axiom,
% 2.38/1.49 ! [X: a,Y: a] : ( less_eq @ X @ ( sup @ X @ Y ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_118_local_Osup__ge2,axiom,
% 2.38/1.49 ! [Y: a,X: a] : ( less_eq @ Y @ ( sup @ X @ Y ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_119_local_Osup__least,axiom,
% 2.38/1.49 ! [Y: a,X: a,Z2: a] :
% 2.38/1.49 ( ( less_eq @ Y @ X )
% 2.38/1.49 => ( ( less_eq @ Z2 @ X )
% 2.38/1.49 => ( less_eq @ ( sup @ Y @ Z2 ) @ X ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_120_local_Osup__mono,axiom,
% 2.38/1.49 ! [A2: a,C: a,B2: a,D: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ C )
% 2.38/1.49 => ( ( less_eq @ B2 @ D )
% 2.38/1.49 => ( less_eq @ ( sup @ A2 @ B2 ) @ ( sup @ C @ D ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_121_local_Osup__unique,axiom,
% 2.38/1.49 ! [F: a > a > a,X: a,Y: a] :
% 2.38/1.49 ( ! [X3: a,Y4: a] : ( less_eq @ X3 @ ( F @ X3 @ Y4 ) )
% 2.38/1.49 => ( ! [X3: a,Y4: a] : ( less_eq @ Y4 @ ( F @ X3 @ Y4 ) )
% 2.38/1.49 => ( ! [X3: a,Y4: a,Z3: a] :
% 2.38/1.49 ( ( less_eq @ Y4 @ X3 )
% 2.38/1.49 => ( ( less_eq @ Z3 @ X3 )
% 2.38/1.49 => ( less_eq @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
% 2.38/1.49 => ( ( sup @ X @ Y )
% 2.38/1.49 = ( F @ X @ Y ) ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_122_local_Oinf_Ostrict__boundedE,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( less @ A2 @ ( inf @ B2 @ C ) )
% 2.38/1.49 => ~ ( ( less @ A2 @ B2 )
% 2.38/1.49 => ~ ( less @ A2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_123_local_Oinf_Ostrict__coboundedI1,axiom,
% 2.38/1.49 ! [A2: a,C: a,B2: a] :
% 2.38/1.49 ( ( less @ A2 @ C )
% 2.38/1.49 => ( less @ ( inf @ A2 @ B2 ) @ C ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_124_local_Oinf_Ostrict__coboundedI2,axiom,
% 2.38/1.49 ! [B2: a,C: a,A2: a] :
% 2.38/1.49 ( ( less @ B2 @ C )
% 2.38/1.49 => ( less @ ( inf @ A2 @ B2 ) @ C ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_125_local_Oinf_Ostrict__order__iff,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( less @ A2 @ B2 )
% 2.38/1.49 = ( ( A2
% 2.38/1.49 = ( inf @ A2 @ B2 ) )
% 2.38/1.49 & ( A2 != B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_126_local_Oless__infI1,axiom,
% 2.38/1.49 ! [A2: a,X: a,B2: a] :
% 2.38/1.49 ( ( less @ A2 @ X )
% 2.38/1.49 => ( less @ ( inf @ A2 @ B2 ) @ X ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_127_local_Oless__infI2,axiom,
% 2.38/1.49 ! [B2: a,X: a,A2: a] :
% 2.38/1.49 ( ( less @ B2 @ X )
% 2.38/1.49 => ( less @ ( inf @ A2 @ B2 ) @ X ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_128_local_Oless__supI1,axiom,
% 2.38/1.49 ! [X: a,A2: a,B2: a] :
% 2.38/1.49 ( ( less @ X @ A2 )
% 2.38/1.49 => ( less @ X @ ( sup @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_129_local_Oless__supI2,axiom,
% 2.38/1.49 ! [X: a,B2: a,A2: a] :
% 2.38/1.49 ( ( less @ X @ B2 )
% 2.38/1.49 => ( less @ X @ ( sup @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_130_local_Osup_Ostrict__boundedE,axiom,
% 2.38/1.49 ! [B2: a,C: a,A2: a] :
% 2.38/1.49 ( ( less @ ( sup @ B2 @ C ) @ A2 )
% 2.38/1.49 => ~ ( ( less @ B2 @ A2 )
% 2.38/1.49 => ~ ( less @ C @ A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_131_local_Osup_Ostrict__coboundedI1,axiom,
% 2.38/1.49 ! [C: a,A2: a,B2: a] :
% 2.38/1.49 ( ( less @ C @ A2 )
% 2.38/1.49 => ( less @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_132_local_Osup_Ostrict__coboundedI2,axiom,
% 2.38/1.49 ! [C: a,B2: a,A2: a] :
% 2.38/1.49 ( ( less @ C @ B2 )
% 2.38/1.49 => ( less @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_133_local_Osup_Ostrict__order__iff,axiom,
% 2.38/1.49 ! [B2: a,A2: a] :
% 2.38/1.49 ( ( less @ B2 @ A2 )
% 2.38/1.49 = ( ( A2
% 2.38/1.49 = ( sup @ A2 @ B2 ) )
% 2.38/1.49 & ( A2 != B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_134_local_Odistrib__imp1,axiom,
% 2.38/1.49 ! [X: a,Y: a,Z2: a] :
% 2.38/1.49 ( ! [X3: a,Y4: a,Z3: a] :
% 2.38/1.49 ( ( inf @ X3 @ ( sup @ Y4 @ Z3 ) )
% 2.38/1.49 = ( sup @ ( inf @ X3 @ Y4 ) @ ( inf @ X3 @ Z3 ) ) )
% 2.38/1.49 => ( ( sup @ X @ ( inf @ Y @ Z2 ) )
% 2.38/1.49 = ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z2 ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_135_local_Odistrib__imp2,axiom,
% 2.38/1.49 ! [X: a,Y: a,Z2: a] :
% 2.38/1.49 ( ! [X3: a,Y4: a,Z3: a] :
% 2.38/1.49 ( ( sup @ X3 @ ( inf @ Y4 @ Z3 ) )
% 2.38/1.49 = ( inf @ ( sup @ X3 @ Y4 ) @ ( sup @ X3 @ Z3 ) ) )
% 2.38/1.49 => ( ( inf @ X @ ( sup @ Y @ Z2 ) )
% 2.38/1.49 = ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z2 ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_136_local_Odistrib__inf__le,axiom,
% 2.38/1.49 ! [X: a,Y: a,Z2: a] : ( less_eq @ ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z2 ) ) @ ( inf @ X @ ( sup @ Y @ Z2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_137_local_Odistrib__sup__le,axiom,
% 2.38/1.49 ! [X: a,Y: a,Z2: a] : ( less_eq @ ( sup @ X @ ( inf @ Y @ Z2 ) ) @ ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_138_local_Oincomp__def,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( modula1937309406incomp @ a @ less_eq @ X @ Y )
% 2.38/1.49 = ( ~ ( less_eq @ X @ Y )
% 2.38/1.49 & ~ ( less_eq @ Y @ X ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_139_local_Oorder_Orefl,axiom,
% 2.38/1.49 ! [A2: a] : ( less_eq @ A2 @ A2 ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_140_local_Oorder__refl,axiom,
% 2.38/1.49 ! [X: a] : ( less_eq @ X @ X ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_141_local_Oinf_Oidem,axiom,
% 2.38/1.49 ! [A2: a] :
% 2.38/1.49 ( ( inf @ A2 @ A2 )
% 2.38/1.49 = A2 ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_142_local_Oinf_Oleft__idem,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( inf @ A2 @ ( inf @ A2 @ B2 ) )
% 2.38/1.49 = ( inf @ A2 @ B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_143_local_Oinf_Oright__idem,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( inf @ ( inf @ A2 @ B2 ) @ B2 )
% 2.38/1.49 = ( inf @ A2 @ B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_144_local_Oinf__idem,axiom,
% 2.38/1.49 ! [X: a] :
% 2.38/1.49 ( ( inf @ X @ X )
% 2.38/1.49 = X ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_145_local_Oinf__left__idem,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( inf @ X @ ( inf @ X @ Y ) )
% 2.38/1.49 = ( inf @ X @ Y ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_146_local_Oinf__right__idem,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( inf @ ( inf @ X @ Y ) @ Y )
% 2.38/1.49 = ( inf @ X @ Y ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_147_local_Osup_Oidem,axiom,
% 2.38/1.49 ! [A2: a] :
% 2.38/1.49 ( ( sup @ A2 @ A2 )
% 2.38/1.49 = A2 ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_148_local_Osup_Oleft__idem,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( sup @ A2 @ ( sup @ A2 @ B2 ) )
% 2.38/1.49 = ( sup @ A2 @ B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_149_local_Osup_Oright__idem,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( sup @ ( sup @ A2 @ B2 ) @ B2 )
% 2.38/1.49 = ( sup @ A2 @ B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_150_local_Osup__idem,axiom,
% 2.38/1.49 ! [X: a] :
% 2.38/1.49 ( ( sup @ X @ X )
% 2.38/1.49 = X ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_151_local_Osup__left__idem,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( sup @ X @ ( sup @ X @ Y ) )
% 2.38/1.49 = ( sup @ X @ Y ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_152_local_OGreatestI2__order,axiom,
% 2.38/1.49 ! [P: a > $o,X: a,Q: a > $o] :
% 2.38/1.49 ( ( P @ X )
% 2.38/1.49 => ( ! [Y4: a] :
% 2.38/1.49 ( ( P @ Y4 )
% 2.38/1.49 => ( less_eq @ Y4 @ X ) )
% 2.38/1.49 => ( ! [X3: a] :
% 2.38/1.49 ( ( P @ X3 )
% 2.38/1.49 => ( ! [Y5: a] :
% 2.38/1.49 ( ( P @ Y5 )
% 2.38/1.49 => ( less_eq @ Y5 @ X3 ) )
% 2.38/1.49 => ( Q @ X3 ) ) )
% 2.38/1.49 => ( Q @ ( greatest @ a @ less_eq @ P ) ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_153_local_OGreatest__equality,axiom,
% 2.38/1.49 ! [P: a > $o,X: a] :
% 2.38/1.49 ( ( P @ X )
% 2.38/1.49 => ( ! [Y4: a] :
% 2.38/1.49 ( ( P @ Y4 )
% 2.38/1.49 => ( less_eq @ Y4 @ X ) )
% 2.38/1.49 => ( ( greatest @ a @ less_eq @ P )
% 2.38/1.49 = X ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_154_local_Omax__def,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 => ( ( max @ a @ less_eq @ A2 @ B2 )
% 2.38/1.49 = B2 ) )
% 2.38/1.49 & ( ~ ( less_eq @ A2 @ B2 )
% 2.38/1.49 => ( ( max @ a @ less_eq @ A2 @ B2 )
% 2.38/1.49 = A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_155_local_Omin__def,axiom,
% 2.38/1.49 ! [A2: a,B2: a] :
% 2.38/1.49 ( ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 => ( ( min @ a @ less_eq @ A2 @ B2 )
% 2.38/1.49 = A2 ) )
% 2.38/1.49 & ( ~ ( less_eq @ A2 @ B2 )
% 2.38/1.49 => ( ( min @ a @ less_eq @ A2 @ B2 )
% 2.38/1.49 = B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_156_local_Ois__arg__max__def,axiom,
% 2.38/1.49 ! [B: $tType,F: B > a,P: B > $o,X: B] :
% 2.38/1.49 ( ( lattices_is_arg_max @ a @ B @ less @ F @ P @ X )
% 2.38/1.49 = ( ( P @ X )
% 2.38/1.49 & ~ ? [Y3: B] :
% 2.38/1.49 ( ( P @ Y3 )
% 2.38/1.49 & ( less @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_157_local_Ois__arg__min__def,axiom,
% 2.38/1.49 ! [B: $tType,F: B > a,P: B > $o,X: B] :
% 2.38/1.49 ( ( lattices_is_arg_min @ a @ B @ less @ F @ P @ X )
% 2.38/1.49 = ( ( P @ X )
% 2.38/1.49 & ~ ? [Y3: B] :
% 2.38/1.49 ( ( P @ Y3 )
% 2.38/1.49 & ( less @ ( F @ Y3 ) @ ( F @ X ) ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_158_local_ON5__lattice__def,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( modula1192512645attice @ a @ inf @ less @ sup @ A2 @ B2 @ C )
% 2.38/1.49 = ( ( ( inf @ A2 @ C )
% 2.38/1.49 = ( inf @ B2 @ C ) )
% 2.38/1.49 & ( less @ A2 @ B2 )
% 2.38/1.49 & ( ( sup @ A2 @ C )
% 2.38/1.49 = ( sup @ B2 @ C ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_159_local_Ono__distrib__def,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( modula141148890istrib @ a @ inf @ less @ sup @ A2 @ B2 @ C )
% 2.38/1.49 = ( less @ ( sup @ ( inf @ A2 @ B2 ) @ ( inf @ C @ A2 ) ) @ ( inf @ A2 @ ( sup @ B2 @ C ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_160_local_Oinf_Obounded__iff,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( less_eq @ A2 @ ( inf @ B2 @ C ) )
% 2.38/1.49 = ( ( less_eq @ A2 @ B2 )
% 2.38/1.49 & ( less_eq @ A2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_161_local_Ole__inf__iff,axiom,
% 2.38/1.49 ! [X: a,Y: a,Z2: a] :
% 2.38/1.49 ( ( less_eq @ X @ ( inf @ Y @ Z2 ) )
% 2.38/1.49 = ( ( less_eq @ X @ Y )
% 2.38/1.49 & ( less_eq @ X @ Z2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_162_local_Ole__sup__iff,axiom,
% 2.38/1.49 ! [X: a,Y: a,Z2: a] :
% 2.38/1.49 ( ( less_eq @ ( sup @ X @ Y ) @ Z2 )
% 2.38/1.49 = ( ( less_eq @ X @ Z2 )
% 2.38/1.49 & ( less_eq @ Y @ Z2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_163_local_Osup_Obounded__iff,axiom,
% 2.38/1.49 ! [B2: a,C: a,A2: a] :
% 2.38/1.49 ( ( less_eq @ ( sup @ B2 @ C ) @ A2 )
% 2.38/1.49 = ( ( less_eq @ B2 @ A2 )
% 2.38/1.49 & ( less_eq @ C @ A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_164_local_Oinf__sup__absorb,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( inf @ X @ ( sup @ X @ Y ) )
% 2.38/1.49 = X ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_165_local_Osup__inf__absorb,axiom,
% 2.38/1.49 ! [X: a,Y: a] :
% 2.38/1.49 ( ( sup @ X @ ( inf @ X @ Y ) )
% 2.38/1.49 = X ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_166_local_OM5__lattice__def,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( modula1618706180attice @ a @ inf @ less @ sup @ A2 @ B2 @ C )
% 2.38/1.49 = ( ( ( inf @ A2 @ B2 )
% 2.38/1.49 = ( inf @ B2 @ C ) )
% 2.38/1.49 & ( ( inf @ C @ A2 )
% 2.38/1.49 = ( inf @ B2 @ C ) )
% 2.38/1.49 & ( ( sup @ A2 @ B2 )
% 2.38/1.49 = ( sup @ B2 @ C ) )
% 2.38/1.49 & ( ( sup @ C @ A2 )
% 2.38/1.49 = ( sup @ B2 @ C ) )
% 2.38/1.49 & ( less @ ( inf @ A2 @ B2 ) @ ( sup @ A2 @ B2 ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_167_local_OLeast1I,axiom,
% 2.38/1.49 ! [P: a > $o] :
% 2.38/1.49 ( ? [X4: a] :
% 2.38/1.49 ( ( P @ X4 )
% 2.38/1.49 & ! [Y4: a] :
% 2.38/1.49 ( ( P @ Y4 )
% 2.38/1.49 => ( less_eq @ X4 @ Y4 ) )
% 2.38/1.49 & ! [Y4: a] :
% 2.38/1.49 ( ( ( P @ Y4 )
% 2.38/1.49 & ! [Ya: a] :
% 2.38/1.49 ( ( P @ Ya )
% 2.38/1.49 => ( less_eq @ Y4 @ Ya ) ) )
% 2.38/1.49 => ( Y4 = X4 ) ) )
% 2.38/1.49 => ( P @ ( least @ a @ less_eq @ P ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_168_local_OLeast1__le,axiom,
% 2.38/1.49 ! [P: a > $o,Z2: a] :
% 2.38/1.49 ( ? [X4: a] :
% 2.38/1.49 ( ( P @ X4 )
% 2.38/1.49 & ! [Y4: a] :
% 2.38/1.49 ( ( P @ Y4 )
% 2.38/1.49 => ( less_eq @ X4 @ Y4 ) )
% 2.38/1.49 & ! [Y4: a] :
% 2.38/1.49 ( ( ( P @ Y4 )
% 2.38/1.49 & ! [Ya: a] :
% 2.38/1.49 ( ( P @ Ya )
% 2.38/1.49 => ( less_eq @ Y4 @ Ya ) ) )
% 2.38/1.49 => ( Y4 = X4 ) ) )
% 2.38/1.49 => ( ( P @ Z2 )
% 2.38/1.49 => ( less_eq @ ( least @ a @ less_eq @ P ) @ Z2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_169_local_OLeastI2__order,axiom,
% 2.38/1.49 ! [P: a > $o,X: a,Q: a > $o] :
% 2.38/1.49 ( ( P @ X )
% 2.38/1.49 => ( ! [Y4: a] :
% 2.38/1.49 ( ( P @ Y4 )
% 2.38/1.49 => ( less_eq @ X @ Y4 ) )
% 2.38/1.49 => ( ! [X3: a] :
% 2.38/1.49 ( ( P @ X3 )
% 2.38/1.49 => ( ! [Y5: a] :
% 2.38/1.49 ( ( P @ Y5 )
% 2.38/1.49 => ( less_eq @ X3 @ Y5 ) )
% 2.38/1.49 => ( Q @ X3 ) ) )
% 2.38/1.49 => ( Q @ ( least @ a @ less_eq @ P ) ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_170_local_OLeast__equality,axiom,
% 2.38/1.49 ! [P: a > $o,X: a] :
% 2.38/1.49 ( ( P @ X )
% 2.38/1.49 => ( ! [Y4: a] :
% 2.38/1.49 ( ( P @ Y4 )
% 2.38/1.49 => ( less_eq @ X @ Y4 ) )
% 2.38/1.49 => ( ( least @ a @ less_eq @ P )
% 2.38/1.49 = X ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_171_local_Oa__join__d,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( sup @ A2 @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 2.38/1.49 = ( sup @ A2 @ ( inf @ B2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_172_local_Ob__join__d,axiom,
% 2.38/1.49 ! [B2: a,A2: a,C: a] :
% 2.38/1.49 ( ( sup @ B2 @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 2.38/1.49 = ( sup @ B2 @ ( inf @ C @ A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_173_local_Od__aux__def,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 2.38/1.49 = ( sup @ ( sup @ ( inf @ A2 @ B2 ) @ ( inf @ B2 @ C ) ) @ ( inf @ C @ A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_174_local_Od__b__c__a,axiom,
% 2.38/1.49 ! [B2: a,C: a,A2: a] :
% 2.38/1.49 ( ( modula1221160330_d_aux @ a @ inf @ sup @ B2 @ C @ A2 )
% 2.38/1.49 = ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_175_local_Od__c__a__b,axiom,
% 2.38/1.49 ! [C: a,A2: a,B2: a] :
% 2.38/1.49 ( ( modula1221160330_d_aux @ a @ inf @ sup @ C @ A2 @ B2 )
% 2.38/1.49 = ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_176_local_Oa__meet__e,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( inf @ A2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 2.38/1.49 = ( inf @ A2 @ ( sup @ B2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_177_local_Ob__meet__e,axiom,
% 2.38/1.49 ! [B2: a,A2: a,C: a] :
% 2.38/1.49 ( ( inf @ B2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 2.38/1.49 = ( inf @ B2 @ ( sup @ C @ A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_178_local_Oc__meet__e,axiom,
% 2.38/1.49 ! [C: a,A2: a,B2: a] :
% 2.38/1.49 ( ( inf @ C @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 2.38/1.49 = ( inf @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_179_local_Oe__aux__def,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 2.38/1.49 = ( inf @ ( inf @ ( sup @ A2 @ B2 ) @ ( sup @ B2 @ C ) ) @ ( sup @ C @ A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_180_local_Oe__b__c__a,axiom,
% 2.38/1.49 ! [B2: a,C: a,A2: a] :
% 2.38/1.49 ( ( modula1593167561_e_aux @ a @ inf @ sup @ B2 @ C @ A2 )
% 2.38/1.49 = ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_181_local_Oe__c__a__b,axiom,
% 2.38/1.49 ! [C: a,A2: a,B2: a] :
% 2.38/1.49 ( ( modula1593167561_e_aux @ a @ inf @ sup @ C @ A2 @ B2 )
% 2.38/1.49 = ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_182_local_OInf__fin_Osemilattice__order__set__axioms,axiom,
% 2.38/1.49 lattic1693879045er_set @ a @ inf @ less_eq @ less ).
% 2.38/1.49
% 2.38/1.49 thf(fact_183_local_OM5__lattice__incomp,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( modula1618706180attice @ a @ inf @ less @ sup @ A2 @ B2 @ C )
% 2.38/1.49 => ( modula1937309406incomp @ a @ less_eq @ A2 @ B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_184_local_Oc__aux__def,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( modula849153099_c_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 2.38/1.49 = ( sup @ ( inf @ C @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_185_local_Ob__aux__def,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( modula477145868_b_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 2.38/1.49 = ( sup @ ( inf @ B2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_186_local_Oa__aux__def,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( modula105138637_a_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 2.38/1.49 = ( sup @ ( inf @ A2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_187_lattice_Ono__distrib_Ocong,axiom,
% 2.38/1.49 ! [A: $tType] :
% 2.38/1.49 ( ( modula141148890istrib @ A )
% 2.38/1.49 = ( modula141148890istrib @ A ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_188_lattice_ON5__lattice_Ocong,axiom,
% 2.38/1.49 ! [A: $tType] :
% 2.38/1.49 ( ( modula1192512645attice @ A )
% 2.38/1.49 = ( modula1192512645attice @ A ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_189_lattice_OM5__lattice_Ocong,axiom,
% 2.38/1.49 ! [A: $tType] :
% 2.38/1.49 ( ( modula1618706180attice @ A )
% 2.38/1.49 = ( modula1618706180attice @ A ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_190_lattice_Oincomp_Ocong,axiom,
% 2.38/1.49 ! [A: $tType] :
% 2.38/1.49 ( ( modula1937309406incomp @ A )
% 2.38/1.49 = ( modula1937309406incomp @ A ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_191_lattice_Oe__aux_Ocong,axiom,
% 2.38/1.49 ! [A: $tType] :
% 2.38/1.49 ( ( modula1593167561_e_aux @ A )
% 2.38/1.49 = ( modula1593167561_e_aux @ A ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_192_lattice_Od__aux_Ocong,axiom,
% 2.38/1.49 ! [A: $tType] :
% 2.38/1.49 ( ( modula1221160330_d_aux @ A )
% 2.38/1.49 = ( modula1221160330_d_aux @ A ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_193_local_Oinf_Osemilattice__order__axioms,axiom,
% 2.38/1.49 semilattice_order @ a @ inf @ less_eq @ less ).
% 2.38/1.49
% 2.38/1.49 thf(fact_194_local_Oorder_Oordering__axioms,axiom,
% 2.38/1.49 ordering @ a @ less_eq @ less ).
% 2.38/1.49
% 2.38/1.49 thf(fact_195_local_Ocomp__fun__idem__sup,axiom,
% 2.38/1.49 finite_comp_fun_idem @ a @ a @ sup ).
% 2.38/1.49
% 2.38/1.49 thf(fact_196_local_Ocomp__fun__idem__inf,axiom,
% 2.38/1.49 finite_comp_fun_idem @ a @ a @ inf ).
% 2.38/1.49
% 2.38/1.49 thf(fact_197_local_Ob__a,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( modula477145868_b_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 2.38/1.49 = ( modula105138637_a_aux @ a @ inf @ sup @ B2 @ C @ A2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_198_local_Oc__a,axiom,
% 2.38/1.49 ! [A2: a,B2: a,C: a] :
% 2.38/1.49 ( ( modula849153099_c_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 2.38/1.49 = ( modula105138637_a_aux @ a @ inf @ sup @ C @ A2 @ B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_199_lattice_Oc__aux_Ocong,axiom,
% 2.38/1.49 ! [A: $tType] :
% 2.38/1.49 ( ( modula849153099_c_aux @ A )
% 2.38/1.49 = ( modula849153099_c_aux @ A ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_200_lattice_Ob__aux_Ocong,axiom,
% 2.38/1.49 ! [A: $tType] :
% 2.38/1.49 ( ( modula477145868_b_aux @ A )
% 2.38/1.49 = ( modula477145868_b_aux @ A ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_201_lattice_Oa__aux_Ocong,axiom,
% 2.38/1.49 ! [A: $tType] :
% 2.38/1.49 ( ( modula105138637_a_aux @ A )
% 2.38/1.49 = ( modula105138637_a_aux @ A ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_202_local_Osup_Osemigroup__axioms,axiom,
% 2.38/1.49 semigroup @ a @ sup ).
% 2.38/1.49
% 2.38/1.49 thf(fact_203_local_Oinf_Osemigroup__axioms,axiom,
% 2.38/1.49 semigroup @ a @ inf ).
% 2.38/1.49
% 2.38/1.49 thf(fact_204_local_Osup_Osemilattice__axioms,axiom,
% 2.38/1.49 semilattice @ a @ sup ).
% 2.38/1.49
% 2.38/1.49 thf(fact_205_local_Oinf_Osemilattice__axioms,axiom,
% 2.38/1.49 semilattice @ a @ inf ).
% 2.38/1.49
% 2.38/1.49 thf(fact_206_local_Osup_Oabel__semigroup__axioms,axiom,
% 2.38/1.49 abel_semigroup @ a @ sup ).
% 2.38/1.49
% 2.38/1.49 thf(fact_207_local_Oinf_Oabel__semigroup__axioms,axiom,
% 2.38/1.49 abel_semigroup @ a @ inf ).
% 2.38/1.49
% 2.38/1.49 thf(fact_208_semilattice__order__set_Oaxioms_I1_J,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 2.38/1.49 ( ( lattic1693879045er_set @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( semilattice_order @ A @ F @ Less_eq @ Less ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_209_abel__semigroup_Oaxioms_I1_J,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A] :
% 2.38/1.49 ( ( abel_semigroup @ A @ F )
% 2.38/1.49 => ( semigroup @ A @ F ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_210_abel__semigroup_Oleft__commute,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,B2: A,A2: A,C: A] :
% 2.38/1.49 ( ( abel_semigroup @ A @ F )
% 2.38/1.49 => ( ( F @ B2 @ ( F @ A2 @ C ) )
% 2.38/1.49 = ( F @ A2 @ ( F @ B2 @ C ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_211_abel__semigroup_Ocommute,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 2.38/1.49 ( ( abel_semigroup @ A @ F )
% 2.38/1.49 => ( ( F @ A2 @ B2 )
% 2.38/1.49 = ( F @ B2 @ A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_212_semigroup_Ointro,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A] :
% 2.38/1.49 ( ! [A5: A,B4: A,C2: A] :
% 2.38/1.49 ( ( F @ ( F @ A5 @ B4 ) @ C2 )
% 2.38/1.49 = ( F @ A5 @ ( F @ B4 @ C2 ) ) )
% 2.38/1.49 => ( semigroup @ A @ F ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_213_semigroup_Oassoc,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,A2: A,B2: A,C: A] :
% 2.38/1.49 ( ( semigroup @ A @ F )
% 2.38/1.49 => ( ( F @ ( F @ A2 @ B2 ) @ C )
% 2.38/1.49 = ( F @ A2 @ ( F @ B2 @ C ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_214_semigroup__def,axiom,
% 2.38/1.49 ! [A: $tType] :
% 2.38/1.49 ( ( semigroup @ A )
% 2.38/1.49 = ( ^ [F2: A > A > A] :
% 2.38/1.49 ! [A3: A,B3: A,C3: A] :
% 2.38/1.49 ( ( F2 @ ( F2 @ A3 @ B3 ) @ C3 )
% 2.38/1.49 = ( F2 @ A3 @ ( F2 @ B3 @ C3 ) ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_215_ord_Ois__arg__min_Ocong,axiom,
% 2.38/1.49 ! [B: $tType,A: $tType] :
% 2.38/1.49 ( ( lattices_is_arg_min @ A @ B )
% 2.38/1.49 = ( lattices_is_arg_min @ A @ B ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_216_ord_Ois__arg__max_Ocong,axiom,
% 2.38/1.49 ! [B: $tType,A: $tType] :
% 2.38/1.49 ( ( lattices_is_arg_max @ A @ B )
% 2.38/1.49 = ( lattices_is_arg_max @ A @ B ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_217_ord_Ois__arg__min__def,axiom,
% 2.38/1.49 ! [B: $tType,A: $tType] :
% 2.38/1.49 ( ( lattices_is_arg_min @ A @ B )
% 2.38/1.49 = ( ^ [Less2: A > A > $o,F2: B > A,P2: B > $o,X2: B] :
% 2.38/1.49 ( ( P2 @ X2 )
% 2.38/1.49 & ~ ? [Y3: B] :
% 2.38/1.49 ( ( P2 @ Y3 )
% 2.38/1.49 & ( Less2 @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_218_ord_Ois__arg__max__def,axiom,
% 2.38/1.49 ! [B: $tType,A: $tType] :
% 2.38/1.49 ( ( lattices_is_arg_max @ A @ B )
% 2.38/1.49 = ( ^ [Less2: A > A > $o,F2: B > A,P2: B > $o,X2: B] :
% 2.38/1.49 ( ( P2 @ X2 )
% 2.38/1.49 & ~ ? [Y3: B] :
% 2.38/1.49 ( ( P2 @ Y3 )
% 2.38/1.49 & ( Less2 @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_219_semilattice_Oaxioms_I1_J,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A] :
% 2.38/1.49 ( ( semilattice @ A @ F )
% 2.38/1.49 => ( abel_semigroup @ A @ F ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_220_local_OSup__fin_Osemilattice__set__axioms,axiom,
% 2.38/1.49 lattic35693393ce_set @ a @ sup ).
% 2.38/1.49
% 2.38/1.49 thf(fact_221_local_OInf__fin_Osemilattice__set__axioms,axiom,
% 2.38/1.49 lattic35693393ce_set @ a @ inf ).
% 2.38/1.49
% 2.38/1.49 thf(fact_222_semilattice__order__set_Oaxioms_I2_J,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 2.38/1.49 ( ( lattic1693879045er_set @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( lattic35693393ce_set @ A @ F ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_223_semilattice__set__def,axiom,
% 2.38/1.49 ! [A: $tType] :
% 2.38/1.49 ( ( lattic35693393ce_set @ A )
% 2.38/1.49 = ( semilattice @ A ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_224_semilattice__set_Ointro,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A] :
% 2.38/1.49 ( ( semilattice @ A @ F )
% 2.38/1.49 => ( lattic35693393ce_set @ A @ F ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_225_semilattice__set_Oaxioms,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A] :
% 2.38/1.49 ( ( lattic35693393ce_set @ A @ F )
% 2.38/1.49 => ( semilattice @ A @ F ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_226_semilattice__order__set__def,axiom,
% 2.38/1.49 ! [A: $tType] :
% 2.38/1.49 ( ( lattic1693879045er_set @ A )
% 2.38/1.49 = ( ^ [F2: A > A > A,Less_eq2: A > A > $o,Less2: A > A > $o] :
% 2.38/1.49 ( ( semilattice_order @ A @ F2 @ Less_eq2 @ Less2 )
% 2.38/1.49 & ( lattic35693393ce_set @ A @ F2 ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_227_semilattice__order__set_Ointro,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( lattic35693393ce_set @ A @ F )
% 2.38/1.49 => ( lattic1693879045er_set @ A @ F @ Less_eq @ Less ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_228_semilattice__order_Omono,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,C: A,B2: A,D: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( Less_eq @ A2 @ C )
% 2.38/1.49 => ( ( Less_eq @ B2 @ D )
% 2.38/1.49 => ( Less_eq @ ( F @ A2 @ B2 ) @ ( F @ C @ D ) ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_229_semilattice__order_OorderE,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( Less_eq @ A2 @ B2 )
% 2.38/1.49 => ( A2
% 2.38/1.49 = ( F @ A2 @ B2 ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_230_semilattice__order_OorderI,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( A2
% 2.38/1.49 = ( F @ A2 @ B2 ) )
% 2.38/1.49 => ( Less_eq @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_231_semilattice__order_Oabsorb1,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( Less_eq @ A2 @ B2 )
% 2.38/1.49 => ( ( F @ A2 @ B2 )
% 2.38/1.49 = A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_232_semilattice__order_Oabsorb2,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A2: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( Less_eq @ B2 @ A2 )
% 2.38/1.49 => ( ( F @ A2 @ B2 )
% 2.38/1.49 = B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_233_semilattice__order_OboundedE,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( Less_eq @ A2 @ ( F @ B2 @ C ) )
% 2.38/1.49 => ~ ( ( Less_eq @ A2 @ B2 )
% 2.38/1.49 => ~ ( Less_eq @ A2 @ C ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_234_semilattice__order_OboundedI,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( Less_eq @ A2 @ B2 )
% 2.38/1.49 => ( ( Less_eq @ A2 @ C )
% 2.38/1.49 => ( Less_eq @ A2 @ ( F @ B2 @ C ) ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_235_semilattice__order_Oorder__iff,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( Less_eq @ A2 @ B2 )
% 2.38/1.49 = ( A2
% 2.38/1.49 = ( F @ A2 @ B2 ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_236_semilattice__order_Ocobounded1,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( Less_eq @ ( F @ A2 @ B2 ) @ A2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_237_semilattice__order_Ocobounded2,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( Less_eq @ ( F @ A2 @ B2 ) @ B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_238_semilattice__order_Oabsorb__iff1,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( Less_eq @ A2 @ B2 )
% 2.38/1.49 = ( ( F @ A2 @ B2 )
% 2.38/1.49 = A2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_239_semilattice__order_Oabsorb__iff2,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A2: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( Less_eq @ B2 @ A2 )
% 2.38/1.49 = ( ( F @ A2 @ B2 )
% 2.38/1.49 = B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_240_semilattice__order_Obounded__iff,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( Less_eq @ A2 @ ( F @ B2 @ C ) )
% 2.38/1.49 = ( ( Less_eq @ A2 @ B2 )
% 2.38/1.49 & ( Less_eq @ A2 @ C ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_241_semilattice__order_OcoboundedI1,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,C: A,B2: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( Less_eq @ A2 @ C )
% 2.38/1.49 => ( Less_eq @ ( F @ A2 @ B2 ) @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_242_semilattice__order_OcoboundedI2,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C: A,A2: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( Less_eq @ B2 @ C )
% 2.38/1.49 => ( Less_eq @ ( F @ A2 @ B2 ) @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_243_semilattice__order_Ostrict__boundedE,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( Less @ A2 @ ( F @ B2 @ C ) )
% 2.38/1.49 => ~ ( ( Less @ A2 @ B2 )
% 2.38/1.49 => ~ ( Less @ A2 @ C ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_244_semilattice__order_Ostrict__order__iff,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( Less @ A2 @ B2 )
% 2.38/1.49 = ( ( A2
% 2.38/1.49 = ( F @ A2 @ B2 ) )
% 2.38/1.49 & ( A2 != B2 ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_245_semilattice__order_Ostrict__coboundedI1,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,C: A,B2: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( Less @ A2 @ C )
% 2.38/1.49 => ( Less @ ( F @ A2 @ B2 ) @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_246_semilattice__order_Ostrict__coboundedI2,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C: A,A2: A] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( ( Less @ B2 @ C )
% 2.38/1.49 => ( Less @ ( F @ A2 @ B2 ) @ C ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_247_semilattice_Oidem,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,A2: A] :
% 2.38/1.49 ( ( semilattice @ A @ F )
% 2.38/1.49 => ( ( F @ A2 @ A2 )
% 2.38/1.49 = A2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_248_semilattice_Oleft__idem,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 2.38/1.49 ( ( semilattice @ A @ F )
% 2.38/1.49 => ( ( F @ A2 @ ( F @ A2 @ B2 ) )
% 2.38/1.49 = ( F @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_249_semilattice_Oright__idem,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 2.38/1.49 ( ( semilattice @ A @ F )
% 2.38/1.49 => ( ( F @ ( F @ A2 @ B2 ) @ B2 )
% 2.38/1.49 = ( F @ A2 @ B2 ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_250_semilattice__order_Oaxioms_I1_J,axiom,
% 2.38/1.49 ! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
% 2.38/1.49 ( ( semilattice_order @ A @ F @ Less_eq @ Less )
% 2.38/1.49 => ( semilattice @ A @ F ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_251_local_Obdd__below__def,axiom,
% 2.38/1.49 ! [A4: set @ a] :
% 2.38/1.49 ( ( condit1201339847_below @ a @ less_eq @ A4 )
% 2.38/1.49 = ( ? [M: a] :
% 2.38/1.49 ! [X2: a] :
% 2.38/1.49 ( ( member @ a @ X2 @ A4 )
% 2.38/1.49 => ( less_eq @ M @ X2 ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_252_local_Obdd__above__def,axiom,
% 2.38/1.49 ! [A4: set @ a] :
% 2.38/1.49 ( ( condit2040224947_above @ a @ less_eq @ A4 )
% 2.38/1.49 = ( ? [M2: a] :
% 2.38/1.49 ! [X2: a] :
% 2.38/1.49 ( ( member @ a @ X2 @ A4 )
% 2.38/1.49 => ( less_eq @ X2 @ M2 ) ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_253_local_Obdd__belowI,axiom,
% 2.38/1.49 ! [A4: set @ a,M3: a] :
% 2.38/1.49 ( ! [X3: a] :
% 2.38/1.49 ( ( member @ a @ X3 @ A4 )
% 2.38/1.49 => ( less_eq @ M3 @ X3 ) )
% 2.38/1.49 => ( condit1201339847_below @ a @ less_eq @ A4 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_254_local_Obdd__aboveI,axiom,
% 2.38/1.49 ! [A4: set @ a,M4: a] :
% 2.38/1.49 ( ! [X3: a] :
% 2.38/1.49 ( ( member @ a @ X3 @ A4 )
% 2.38/1.49 => ( less_eq @ X3 @ M4 ) )
% 2.38/1.49 => ( condit2040224947_above @ a @ less_eq @ A4 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(fact_255_local_Obdd__below__Ioo,axiom,
% 2.38/1.49 ! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_gr1161524159ssThan @ a @ less @ A2 @ B2 ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(conj_0,hypothesis,
% 2.38/1.49 lattice @ a @ inf @ less_eq @ less @ sup ).
% 2.38/1.49
% 2.38/1.49 thf(conj_1,hypothesis,
% 2.38/1.49 less_eq @ x @ y ).
% 2.38/1.49
% 2.38/1.49 thf(conj_2,hypothesis,
% 2.38/1.49 ( ( sup @ x @ ( inf @ y @ z ) )
% 2.38/1.49 != ( inf @ y @ ( sup @ x @ z ) ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(conj_3,hypothesis,
% 2.38/1.49 less @ ( sup @ x @ ( inf @ y @ z ) ) @ ( inf @ y @ ( sup @ x @ z ) ) ).
% 2.38/1.49
% 2.38/1.49 thf(conj_4,conjecture,
% 2.38/1.49 less_eq @ x @ ( sup @ ( inf @ y @ ( sup @ x @ z ) ) @ z ) ).
% 2.38/1.49 ------------------------
% 2.56/1.51 % (5501)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_5351 for (2999ds/4Mi)
% 2.56/1.51 % (5504)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_5351 for (2999ds/2Mi)
% 2.56/1.51 % (5500)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/183Mi)
% 2.56/1.51 % (5503)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/2Mi)
% 2.56/1.51 % (5504)Instruction limit reached!
% 2.56/1.51 % (5504)------------------------------
% 2.56/1.51 % (5504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.51 % (5505)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_5351 for (2999ds/275Mi)
% 2.56/1.51 % (5504)Termination reason: Unknown
% 2.56/1.51 % (5504)Termination phase: shuffling
% 2.56/1.51
% 2.56/1.51 % (5504)Memory used [KB]: 1279
% 2.56/1.51 % (5504)Time elapsed: 0.002 s
% 2.56/1.51 % (5504)Instructions burned: 2 (million)
% 2.56/1.51 % (5504)------------------------------
% 2.56/1.51 % (5504)------------------------------
% 2.56/1.51 % (5501)Instruction limit reached!
% 2.56/1.51 % (5501)------------------------------
% 2.56/1.51 % (5501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.51 % (5501)Termination reason: Unknown
% 2.56/1.51 % (5501)Termination phase: shuffling
% 2.56/1.51
% 2.56/1.51 % (5501)Memory used [KB]: 1279
% 2.56/1.51 % (5501)Time elapsed: 0.003 s
% 2.56/1.51 % (5501)Instructions burned: 6 (million)
% 2.56/1.51 % (5501)------------------------------
% 2.56/1.51 % (5501)------------------------------
% 2.56/1.51 % (5503)Instruction limit reached!
% 2.56/1.51 % (5503)------------------------------
% 2.56/1.51 % (5503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.51 % (5503)Termination reason: Unknown
% 2.56/1.51 % (5503)Termination phase: shuffling
% 2.56/1.51
% 2.56/1.51 % (5503)Memory used [KB]: 1279
% 2.56/1.51 % (5503)Time elapsed: 0.002 s
% 2.56/1.51 % (5503)Instructions burned: 3 (million)
% 2.56/1.51 % (5503)------------------------------
% 2.56/1.51 % (5503)------------------------------
% 2.56/1.51 % (5502)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_5351 for (2999ds/27Mi)
% 2.56/1.51 % (5506)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/18Mi)
% 2.56/1.52 % (5506)Instruction limit reached!
% 2.56/1.52 % (5506)------------------------------
% 2.56/1.52 % (5506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.52 % (5506)Termination reason: Unknown
% 2.56/1.52 % (5511)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_5351 for (2999ds/15Mi)
% 2.56/1.52 % (5506)Termination phase: Property scanning
% 2.56/1.52
% 2.56/1.52 % (5506)Memory used [KB]: 1407
% 2.56/1.52 % (5506)Time elapsed: 0.010 s
% 2.56/1.52 % (5506)Instructions burned: 18 (million)
% 2.56/1.52 % (5506)------------------------------
% 2.56/1.52 % (5506)------------------------------
% 2.56/1.52 % (5509)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/3Mi)
% 2.56/1.52 % (5510)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_5351 for (2999ds/37Mi)
% 2.56/1.52 % (5509)Instruction limit reached!
% 2.56/1.52 % (5509)------------------------------
% 2.56/1.52 % (5509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.52 % (5509)Termination reason: Unknown
% 2.56/1.52 % (5509)Termination phase: shuffling
% 2.56/1.52
% 2.56/1.52 % (5509)Memory used [KB]: 1279
% 2.56/1.52 % (5509)Time elapsed: 0.003 s
% 2.56/1.52 % (5509)Instructions burned: 4 (million)
% 2.56/1.52 % (5509)------------------------------
% 2.56/1.52 % (5509)------------------------------
% 2.56/1.53 % (5502)Instruction limit reached!
% 2.56/1.53 % (5502)------------------------------
% 2.56/1.53 % (5502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.53 % (5502)Termination reason: Unknown
% 2.56/1.53 % (5502)Termination phase: Preprocessing 1
% 2.56/1.53
% 2.56/1.53 % (5502)Memory used [KB]: 1535
% 2.56/1.53 % (5502)Time elapsed: 0.016 s
% 2.56/1.53 % (5502)Instructions burned: 27 (million)
% 2.56/1.53 % (5502)------------------------------
% 2.56/1.53 % (5502)------------------------------
% 2.56/1.53 % (5511)Instruction limit reached!
% 2.56/1.53 % (5511)------------------------------
% 2.56/1.53 % (5511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.53 % (5511)Termination reason: Unknown
% 2.56/1.53 % (5511)Termination phase: Property scanning
% 2.56/1.53
% 2.56/1.53 % (5511)Memory used [KB]: 1407
% 2.56/1.53 % (5511)Time elapsed: 0.008 s
% 2.56/1.53 % (5511)Instructions burned: 18 (million)
% 2.56/1.53 % (5511)------------------------------
% 2.56/1.53 % (5511)------------------------------
% 2.56/1.54 % (5519)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/3Mi)
% 2.56/1.54 % (5522)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/16Mi)
% 2.56/1.54 % (5510)Instruction limit reached!
% 2.56/1.54 % (5510)------------------------------
% 2.56/1.54 % (5510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.54 % (5510)Termination reason: Unknown
% 2.56/1.54 % (5510)Termination phase: Property scanning
% 2.56/1.54
% 2.56/1.54 % (5510)Memory used [KB]: 1791
% 2.56/1.54 % (5520)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/1041Mi)
% 2.56/1.54 % (5510)Time elapsed: 0.018 s
% 2.56/1.54 % (5510)Instructions burned: 37 (million)
% 2.56/1.54 % (5510)------------------------------
% 2.56/1.54 % (5510)------------------------------
% 2.56/1.54 % (5519)Instruction limit reached!
% 2.56/1.54 % (5519)------------------------------
% 2.56/1.54 % (5519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.54 % (5519)Termination reason: Unknown
% 2.56/1.54 % (5519)Termination phase: shuffling
% 2.56/1.54
% 2.56/1.54 % (5519)Memory used [KB]: 1279
% 2.56/1.54 % (5519)Time elapsed: 0.003 s
% 2.56/1.54 % (5519)Instructions burned: 5 (million)
% 2.56/1.54 % (5519)------------------------------
% 2.56/1.54 % (5519)------------------------------
% 2.56/1.55 % (5521)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/7Mi)
% 2.56/1.55 % (5522)Instruction limit reached!
% 2.56/1.55 % (5522)------------------------------
% 2.56/1.55 % (5522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.55 % (5522)Termination reason: Unknown
% 2.56/1.55 % (5522)Termination phase: Property scanning
% 2.56/1.55
% 2.56/1.55 % (5522)Memory used [KB]: 1535
% 2.56/1.55 % (5522)Time elapsed: 0.011 s
% 2.56/1.55 % (5522)Instructions burned: 17 (million)
% 2.56/1.55 % (5522)------------------------------
% 2.56/1.55 % (5522)------------------------------
% 2.56/1.55 % (5521)Instruction limit reached!
% 2.56/1.55 % (5521)------------------------------
% 2.56/1.55 % (5521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.55 % (5521)Termination reason: Unknown
% 2.56/1.55 % (5521)Termination phase: shuffling
% 2.56/1.55
% 2.56/1.55 % (5521)Memory used [KB]: 1407
% 2.56/1.55 % (5521)Time elapsed: 0.006 s
% 2.56/1.55 % (5521)Instructions burned: 8 (million)
% 2.56/1.55 % (5521)------------------------------
% 2.56/1.55 % (5521)------------------------------
% 2.56/1.55 % (5527)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/3Mi)
% 2.56/1.55 % (5527)Instruction limit reached!
% 2.56/1.55 % (5527)------------------------------
% 2.56/1.55 % (5527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.55 % (5527)Termination reason: Unknown
% 2.56/1.55 % (5527)Termination phase: shuffling
% 2.56/1.55
% 2.56/1.55 % (5527)Memory used [KB]: 1279
% 2.56/1.55 % (5527)Time elapsed: 0.002 s
% 2.56/1.55 % (5527)Instructions burned: 3 (million)
% 2.56/1.55 % (5527)------------------------------
% 2.56/1.55 % (5527)------------------------------
% 2.56/1.56 % (5526)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/3Mi)
% 2.56/1.56 % (5526)Instruction limit reached!
% 2.56/1.56 % (5526)------------------------------
% 2.56/1.56 % (5526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.56 % (5526)Termination reason: Unknown
% 2.56/1.56 % (5526)Termination phase: shuffling
% 2.56/1.56
% 2.56/1.56 % (5526)Memory used [KB]: 1279
% 2.56/1.56 % (5526)Time elapsed: 0.003 s
% 2.56/1.56 % (5526)Instructions burned: 3 (million)
% 2.56/1.56 % (5526)------------------------------
% 2.56/1.56 % (5526)------------------------------
% 2.56/1.56 % (5531)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/3Mi)
% 2.56/1.56 % (5531)Instruction limit reached!
% 2.56/1.56 % (5531)------------------------------
% 2.56/1.56 % (5531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.56 % (5531)Termination reason: Unknown
% 2.56/1.56 % (5531)Termination phase: shuffling
% 2.56/1.56
% 2.56/1.56 % (5531)Memory used [KB]: 1279
% 2.56/1.56 % (5531)Time elapsed: 0.003 s
% 2.56/1.56 % (5531)Instructions burned: 5 (million)
% 2.56/1.56 % (5531)------------------------------
% 2.56/1.56 % (5531)------------------------------
% 2.56/1.56 % (5530)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/7Mi)
% 2.56/1.57 % (5532)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/4Mi)
% 2.56/1.57 % (5530)Instruction limit reached!
% 2.56/1.57 % (5530)------------------------------
% 2.56/1.57 % (5530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.57 % (5530)Termination reason: Unknown
% 2.56/1.57 % (5530)Termination phase: shuffling
% 2.56/1.57
% 2.56/1.57 % (5530)Memory used [KB]: 1279
% 2.56/1.57 % (5530)Time elapsed: 0.005 s
% 2.56/1.57 % (5530)Instructions burned: 7 (million)
% 2.56/1.57 % (5530)------------------------------
% 2.56/1.57 % (5530)------------------------------
% 2.56/1.57 % (5532)Instruction limit reached!
% 2.56/1.57 % (5532)------------------------------
% 2.56/1.57 % (5532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.57 % (5532)Termination reason: Unknown
% 2.56/1.57 % (5532)Termination phase: shuffling
% 2.56/1.57
% 2.56/1.57 % (5532)Memory used [KB]: 1279
% 2.56/1.57 % (5532)Time elapsed: 0.004 s
% 2.56/1.57 % (5532)Instructions burned: 5 (million)
% 2.56/1.57 % (5532)------------------------------
% 2.56/1.57 % (5532)------------------------------
% 2.56/1.57 % (5537)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/710Mi)
% 2.56/1.57 % (5535)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/18Mi)
% 2.56/1.58 % (5539)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/6Mi)
% 2.56/1.58 % (5535)Instruction limit reached!
% 2.56/1.58 % (5535)------------------------------
% 2.56/1.58 % (5535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.58 % (5535)Termination reason: Unknown
% 2.56/1.58 % (5535)Termination phase: Property scanning
% 2.56/1.58
% 2.56/1.58 % (5535)Memory used [KB]: 1407
% 2.56/1.58 % (5535)Time elapsed: 0.011 s
% 2.56/1.58 % (5535)Instructions burned: 20 (million)
% 2.56/1.58 % (5535)------------------------------
% 2.56/1.58 % (5535)------------------------------
% 2.56/1.58 % (5539)Instruction limit reached!
% 2.56/1.58 % (5539)------------------------------
% 2.56/1.58 % (5539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.58 % (5539)Termination reason: Unknown
% 2.56/1.58 % (5539)Termination phase: shuffling
% 2.56/1.58
% 2.56/1.58 % (5539)Memory used [KB]: 1279
% 2.56/1.58 % (5539)Time elapsed: 0.003 s
% 2.56/1.58 % (5539)Instructions burned: 6 (million)
% 2.56/1.58 % (5539)------------------------------
% 2.56/1.58 % (5539)------------------------------
% 2.56/1.59 % (5540)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/902Mi)
% 2.56/1.59 % (5546)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/5Mi)
% 2.56/1.60 % (5546)Instruction limit reached!
% 2.56/1.60 % (5546)------------------------------
% 2.56/1.60 % (5546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.60 % (5546)Termination reason: Unknown
% 2.56/1.60 % (5546)Termination phase: shuffling
% 2.56/1.60
% 2.56/1.60 % (5546)Memory used [KB]: 1279
% 2.56/1.60 % (5546)Time elapsed: 0.004 s
% 2.56/1.60 % (5546)Instructions burned: 6 (million)
% 2.56/1.60 % (5546)------------------------------
% 2.56/1.60 % (5546)------------------------------
% 2.56/1.60 % (5500)Instruction limit reached!
% 2.56/1.60 % (5500)------------------------------
% 2.56/1.60 % (5500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.60 % (5500)Termination reason: Unknown
% 2.56/1.60 % (5500)Termination phase: Saturation
% 2.56/1.60
% 2.56/1.60 % (5500)Memory used [KB]: 6652
% 2.56/1.60 % (5500)Time elapsed: 0.088 s
% 2.56/1.60 % (5500)Instructions burned: 183 (million)
% 2.56/1.60 % (5500)------------------------------
% 2.56/1.60 % (5500)------------------------------
% 2.56/1.60 % (5545)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_5351 for (2999ds/21Mi)
% 2.56/1.61 % (5545)Instruction limit reached!
% 2.56/1.61 % (5545)------------------------------
% 2.56/1.61 % (5545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.61 % (5545)Termination reason: Unknown
% 2.56/1.61 % (5545)Termination phase: Property scanning
% 2.56/1.61
% 2.56/1.61 % (5545)Memory used [KB]: 1535
% 2.56/1.61 % (5545)Time elapsed: 0.009 s
% 2.56/1.61 % (5545)Instructions burned: 22 (million)
% 2.56/1.61 % (5545)------------------------------
% 2.56/1.61 % (5545)------------------------------
% 2.56/1.61 % (5549)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_5351 for (2998ds/6Mi)
% 2.56/1.61 % (5549)Instruction limit reached!
% 2.56/1.61 % (5549)------------------------------
% 2.56/1.61 % (5549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.61 % (5549)Termination reason: Unknown
% 2.56/1.61 % (5549)Termination phase: shuffling
% 2.56/1.61
% 2.56/1.61 % (5549)Memory used [KB]: 1279
% 2.56/1.61 % (5549)Time elapsed: 0.003 s
% 2.56/1.61 % (5549)Instructions burned: 6 (million)
% 2.56/1.61 % (5549)------------------------------
% 2.56/1.61 % (5549)------------------------------
% 2.56/1.62 % (5553)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_5351 for (2998ds/779Mi)
% 2.56/1.62 % (5550)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_5351 for (2998ds/377Mi)
% 2.56/1.62 % (5555)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_5351 for (2998ds/19Mi)
% 2.56/1.63 % (5555)Instruction limit reached!
% 2.56/1.63 % (5555)------------------------------
% 2.56/1.63 % (5555)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.63 % (5555)Termination reason: Unknown
% 2.56/1.63 % (5555)Termination phase: shuffling
% 2.56/1.63
% 2.56/1.63 % (5555)Memory used [KB]: 1663
% 2.56/1.63 % (5555)Time elapsed: 0.010 s
% 2.56/1.63 % (5555)Instructions burned: 19 (million)
% 2.56/1.63 % (5555)------------------------------
% 2.56/1.63 % (5555)------------------------------
% 2.56/1.65 % (5505)First to succeed.
% 2.56/1.65 % (5505)Refutation found. Thanks to Tanya!
% 2.56/1.65 % SZS status Theorem for DTF2THF_5351
% 2.56/1.65 % SZS output start Proof for DTF2THF_5351
% 2.56/1.65 thf(type_def_6, type, set: ($tType) > $tType).
% 2.56/1.65 thf(type_def_7, type, a: $tType).
% 2.56/1.65 thf(func_def_0, type, set: ($tType) > $tType).
% 2.56/1.65 thf(func_def_1, type, a: $tType).
% 2.56/1.65 thf(func_def_2, type, condit2040224947_above: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > $o)).
% 2.56/1.65 thf(func_def_3, type, condit1201339847_below: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > $o)).
% 2.56/1.65 thf(func_def_4, type, finite_comp_fun_idem: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > $o)).
% 2.56/1.65 thf(func_def_5, type, abel_semigroup: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 2.56/1.65 thf(func_def_6, type, semigroup: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 2.56/1.65 thf(func_def_7, type, lattice: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > $o) > (X0 > X0 > $o) > (X0 > X0 > X0) > $o)).
% 2.56/1.65 thf(func_def_8, type, semilattice: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 2.56/1.65 thf(func_def_9, type, semilattice_order: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 2.56/1.65 thf(func_def_10, type, lattices_is_arg_max: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > (X1 > $o) > X1 > $o)).
% 2.56/1.65 thf(func_def_11, type, lattices_is_arg_min: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > (X1 > $o) > X1 > $o)).
% 2.56/1.65 thf(func_def_12, type, lattic1693879045er_set: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 2.56/1.65 thf(func_def_13, type, lattic35693393ce_set: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 2.56/1.65 thf(func_def_14, type, modula1618706180attice: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > $o) > (X0 > X0 > X0) > X0 > X0 > X0 > $o)).
% 2.56/1.65 thf(func_def_15, type, modula1192512645attice: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > $o) > (X0 > X0 > X0) > X0 > X0 > X0 > $o)).
% 2.56/1.65 thf(func_def_16, type, modula105138637_a_aux: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > X0 > X0 > X0 > X0)).
% 2.56/1.65 thf(func_def_17, type, modula477145868_b_aux: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > X0 > X0 > X0 > X0)).
% 2.56/1.65 thf(func_def_18, type, modula849153099_c_aux: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > X0 > X0 > X0 > X0)).
% 2.56/1.65 thf(func_def_19, type, modula1221160330_d_aux: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > X0 > X0 > X0 > X0)).
% 2.56/1.65 thf(func_def_20, type, modula1593167561_e_aux: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > X0 > X0 > X0 > X0)).
% 2.56/1.65 thf(func_def_21, type, modula1937309406incomp: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > $o)).
% 2.56/1.65 thf(func_def_22, type, modula141148890istrib: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > $o) > (X0 > X0 > X0) > X0 > X0 > X0 > $o)).
% 2.56/1.65 thf(func_def_23, type, least: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > $o) > X0)).
% 2.56/1.65 thf(func_def_24, type, max: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > X0)).
% 2.56/1.65 thf(func_def_25, type, min: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > X0)).
% 2.56/1.65 thf(func_def_26, type, greatest: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > $o) > X0)).
% 2.56/1.65 thf(func_def_27, type, ordering: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 2.56/1.65 thf(func_def_28, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.56/1.65 thf(func_def_29, type, set_gr1161524159ssThan: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > set @ X0)).
% 2.56/1.65 thf(func_def_30, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.56/1.65 thf(func_def_31, type, inf: a > a > a).
% 2.56/1.65 thf(func_def_32, type, less: a > a > $o).
% 2.56/1.65 thf(func_def_33, type, less_eq: a > a > $o).
% 2.56/1.65 thf(func_def_34, type, sup: a > a > a).
% 2.56/1.65 thf(func_def_35, type, x: a).
% 2.56/1.65 thf(func_def_36, type, y: a).
% 2.56/1.65 thf(func_def_37, type, z: a).
% 2.56/1.65 thf(func_def_52, type, sK0: (a > a > a) > a).
% 2.56/1.65 thf(func_def_53, type, sK1: (a > a > a) > a).
% 2.56/1.65 thf(func_def_54, type, sK2: (a > a > a) > a).
% 2.56/1.65 thf(func_def_55, type, sK3: (a > a > a) > a).
% 2.56/1.65 thf(func_def_56, type, sK4: (a > a > a) > a).
% 2.56/1.65 thf(func_def_57, type, sK5: (a > a > a) > a).
% 2.56/1.65 thf(func_def_58, type, sK6: (a > a > a) > a).
% 2.56/1.65 thf(func_def_59, type, sK7: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.56/1.65 thf(func_def_60, type, sK8: (a > a > a) > a).
% 2.56/1.65 thf(func_def_61, type, sK9: (a > a > a) > a).
% 2.56/1.65 thf(func_def_62, type, sK10: (a > a > a) > a).
% 2.56/1.65 thf(func_def_63, type, sK11: (a > a > a) > a).
% 2.56/1.65 thf(func_def_64, type, sK12: (a > a > a) > a).
% 2.56/1.65 thf(func_def_65, type, sK13: (a > a > a) > a).
% 2.56/1.65 thf(func_def_66, type, sK14: (a > a > a) > a).
% 2.56/1.65 thf(func_def_67, type, sK15: a).
% 2.56/1.65 thf(func_def_68, type, sK16: a).
% 2.56/1.65 thf(func_def_69, type, sK17: a).
% 2.56/1.65 thf(func_def_70, type, sK18: a).
% 2.56/1.65 thf(func_def_71, type, sK19: a).
% 2.56/1.65 thf(func_def_72, type, sK20: a).
% 2.56/1.65 thf(func_def_74, type, ph22: !>[X0: $tType]:(X0)).
% 2.56/1.65 thf(f2527,plain,(
% 2.56/1.65 $false),
% 2.56/1.65 inference(avatar_sat_refutation,[status(thm)],[f1568,f2524])).
% 2.56/1.65 thf(f2524,plain,(
% 2.56/1.65 ~spl21_25),
% 2.56/1.65 inference(avatar_contradiction_clause,[status(thm)],[f2523])).
% 2.56/1.65 thf(f2523,plain,(
% 2.56/1.65 $false | ~spl21_25),
% 2.56/1.65 inference(subsumption_resolution,[status(thm)],[f2500,f1564])).
% 2.56/1.65 thf(f1564,plain,(
% 2.56/1.65 ($true = (less @ x @ (inf @ y @ (sup @ x @ z)))) | ~spl21_25),
% 2.56/1.65 inference(avatar_component_clause,[status(thm)],[f1562])).
% 2.56/1.65 thf(f1562,definition,(
% 2.56/1.65 spl21_25 <=> ($true = (less @ x @ (inf @ y @ (sup @ x @ z))))),
% 2.56/1.65 introduced(definition,[new_symbols(naming,[spl21_25])],[avatar_definition])).
% 2.56/1.65 thf(f2500,plain,(
% 2.56/1.65 ($true != (less @ x @ (inf @ y @ (sup @ x @ z))))),
% 2.56/1.65 inference(trivial_inequality_removal,[status(thm)],[f2466])).
% 2.56/1.65 thf(f2466,plain,(
% 2.56/1.65 ($true != $true) | ($true != (less @ x @ (inf @ y @ (sup @ x @ z))))),
% 2.56/1.65 inference(superposition,[status(thm)],[f1223,f912])).
% 2.56/1.65 thf(f912,plain,(
% 2.56/1.65 ( ! [X2 : a,X0 : a,X1 : a] : (($true = (less @ X0 @ (sup @ X2 @ X1))) | ((less @ X0 @ X1) != $true)) )),
% 2.56/1.65 inference(cnf_transformation,[status(thm)],[f783])).
% 2.56/1.65 thf(f783,plain,(
% 2.56/1.65 ! [X0 : a,X1 : a,X2 : a] : (($true = (less @ X0 @ (sup @ X2 @ X1))) | ((less @ X0 @ X1) != $true))),
% 2.56/1.65 inference(ennf_transformation,[status(thm)],[f535])).
% 2.56/1.65 thf(f535,plain,(
% 2.56/1.65 ! [X0 : a,X1 : a,X2 : a] : (((less @ X0 @ X1) = $true) => ($true = (less @ X0 @ (sup @ X2 @ X1))))),
% 2.56/1.65 inference(fool_elimination,[status(thm)],[f534])).
% 2.56/1.65 thf(f534,plain,(
% 2.56/1.65 ! [X0 : a,X1 : a,X2 : a] : ((less @ X0 @ X1) => (less @ X0 @ (sup @ X2 @ X1)))),
% 2.56/1.65 inference(rectify,[status(thm)],[f133])).
% 2.56/1.65 thf(f133,axiom,(
% 2.56/1.65 ! [X10 : a,X4 : a,X5 : a] : ((less @ X10 @ X4) => (less @ X10 @ (sup @ X5 @ X4)))),
% 2.56/1.65 file('/export/starexec/sandbox2/tmp/tmp.Vy3zEj9nqo/DTF2THF_5351.p',fact_132_local_Osup_Ostrict__coboundedI2)).
% 2.56/1.65 thf(f1223,plain,(
% 2.56/1.65 ((less @ x @ (sup @ z @ (inf @ y @ (sup @ x @ z)))) != $true)),
% 2.56/1.65 inference(trivial_inequality_removal,[status(thm)],[f1189])).
% 2.56/1.65 thf(f1189,plain,(
% 2.56/1.65 ((less @ x @ (sup @ z @ (inf @ y @ (sup @ x @ z)))) != $true) | ($true != $true)),
% 2.56/1.65 inference(superposition,[status(thm)],[f1115,f1001])).
% 2.56/1.65 thf(f1001,plain,(
% 2.56/1.65 ( ! [X0 : a,X1 : a] : (((less_eq @ X0 @ X1) = $true) | ((less @ X0 @ X1) != $true)) )),
% 2.56/1.65 inference(cnf_transformation,[status(thm)],[f726])).
% 2.56/1.65 thf(f726,plain,(
% 2.56/1.65 ! [X0 : a,X1 : a] : (((less @ X0 @ X1) != $true) | ((less_eq @ X0 @ X1) = $true))),
% 2.56/1.65 inference(ennf_transformation,[status(thm)],[f595])).
% 2.56/1.65 thf(f595,plain,(
% 2.56/1.65 ! [X0 : a,X1 : a] : (((less @ X0 @ X1) = $true) => ((less_eq @ X0 @ X1) = $true))),
% 2.56/1.65 inference(fool_elimination,[status(thm)],[f594])).
% 2.56/1.65 thf(f594,plain,(
% 2.56/1.65 ! [X0 : a,X1 : a] : ((less @ X0 @ X1) => (less_eq @ X0 @ X1))),
% 2.56/1.65 inference(rectify,[status(thm)],[f54])).
% 2.56/1.65 thf(f54,axiom,(
% 2.56/1.65 ! [X4 : a,X5 : a] : ((less @ X4 @ X5) => (less_eq @ X4 @ X5))),
% 2.56/1.65 file('/export/starexec/sandbox2/tmp/tmp.Vy3zEj9nqo/DTF2THF_5351.p',fact_53_local_Odual__order_Ostrict__implies__order)).
% 2.56/1.65 thf(f1115,plain,(
% 2.56/1.65 ((less_eq @ x @ (sup @ z @ (inf @ y @ (sup @ x @ z)))) != $true)),
% 2.56/1.65 inference(forward_demodulation,[status(thm)],[f1010,f902])).
% 2.56/1.65 thf(f902,plain,(
% 2.56/1.65 ( ! [X0 : a,X1 : a] : (((sup @ X1 @ X0) = (sup @ X0 @ X1))) )),
% 2.56/1.65 inference(cnf_transformation,[status(thm)],[f684])).
% 2.56/1.65 thf(f684,plain,(
% 2.56/1.65 ! [X0 : a,X1 : a] : ((sup @ X1 @ X0) = (sup @ X0 @ X1))),
% 2.56/1.65 inference(rectify,[status(thm)],[f43])).
% 2.56/1.65 thf(f43,axiom,(
% 2.56/1.65 ! [X2 : a,X3 : a] : ((sup @ X2 @ X3) = (sup @ X3 @ X2))),
% 2.56/1.65 file('/export/starexec/sandbox2/tmp/tmp.Vy3zEj9nqo/DTF2THF_5351.p',fact_42_local_Osup__commute)).
% 2.56/1.65 thf(f1010,plain,(
% 2.56/1.65 ((less_eq @ x @ (sup @ (inf @ y @ (sup @ x @ z)) @ z)) != $true)),
% 2.56/1.65 inference(cnf_transformation,[status(thm)],[f666])).
% 2.56/1.65 thf(f666,plain,(
% 2.56/1.65 ((less_eq @ x @ (sup @ (inf @ y @ (sup @ x @ z)) @ z)) != $true)),
% 2.56/1.65 inference(flattening,[status(thm)],[f633])).
% 2.56/1.65 thf(f633,plain,(
% 2.56/1.65 ~((less_eq @ x @ (sup @ (inf @ y @ (sup @ x @ z)) @ z)) = $true)),
% 2.56/1.65 inference(fool_elimination,[status(thm)],[f632])).
% 2.56/1.65 thf(f632,plain,(
% 2.56/1.65 ~(less_eq @ x @ (sup @ (inf @ y @ (sup @ x @ z)) @ z))),
% 2.56/1.65 inference(rectify,[status(thm)],[f262])).
% 2.56/1.65 thf(f262,negated_conjecture,(
% 2.56/1.65 ~(less_eq @ x @ (sup @ (inf @ y @ (sup @ x @ z)) @ z))),
% 2.56/1.65 inference(negated_conjecture,[status(cth)],[f261])).
% 2.56/1.65 thf(f261,conjecture,(
% 2.56/1.65 (less_eq @ x @ (sup @ (inf @ y @ (sup @ x @ z)) @ z))),
% 2.56/1.65 file('/export/starexec/sandbox2/tmp/tmp.Vy3zEj9nqo/DTF2THF_5351.p',conj_4)).
% 2.56/1.65 thf(f1568,plain,(
% 2.56/1.65 spl21_25),
% 2.56/1.65 inference(avatar_split_clause,[status(thm)],[f1549,f1562])).
% 2.56/1.65 thf(f1549,plain,(
% 2.56/1.65 ($true = (less @ x @ (inf @ y @ (sup @ x @ z))))),
% 2.56/1.65 inference(trivial_inequality_removal,[status(thm)],[f1527])).
% 2.56/1.65 thf(f1527,plain,(
% 2.56/1.65 ($true != $true) | ($true = (less @ x @ (inf @ y @ (sup @ x @ z))))),
% 2.56/1.65 inference(superposition,[status(thm)],[f976,f906])).
% 2.56/1.65 thf(f906,plain,(
% 2.56/1.65 ((less @ (sup @ x @ (inf @ y @ z)) @ (inf @ y @ (sup @ x @ z))) = $true)),
% 2.56/1.65 inference(cnf_transformation,[status(thm)],[f619])).
% 2.56/1.65 thf(f619,plain,(
% 2.56/1.65 ((less @ (sup @ x @ (inf @ y @ z)) @ (inf @ y @ (sup @ x @ z))) = $true)),
% 2.56/1.65 inference(fool_elimination,[status(thm)],[f618])).
% 2.56/1.65 thf(f618,plain,(
% 2.56/1.65 (less @ (sup @ x @ (inf @ y @ z)) @ (inf @ y @ (sup @ x @ z)))),
% 2.56/1.65 inference(rectify,[status(thm)],[f260])).
% 2.56/1.65 thf(f260,axiom,(
% 2.56/1.65 (less @ (sup @ x @ (inf @ y @ z)) @ (inf @ y @ (sup @ x @ z)))),
% 2.56/1.65 file('/export/starexec/sandbox2/tmp/tmp.Vy3zEj9nqo/DTF2THF_5351.p',conj_3)).
% 2.56/1.65 thf(f976,plain,(
% 2.56/1.65 ( ! [X2 : a,X0 : a,X1 : a] : (((less @ (sup @ X0 @ X1) @ X2) != $true) | ((less @ X0 @ X2) = $true)) )),
% 2.56/1.65 inference(cnf_transformation,[status(thm)],[f717])).
% 2.56/1.65 thf(f717,plain,(
% 2.56/1.65 ! [X0 : a,X1 : a,X2 : a] : (((less @ (sup @ X0 @ X1) @ X2) != $true) | (((less @ X1 @ X2) = $true) & ((less @ X0 @ X2) = $true)))),
% 2.56/1.65 inference(ennf_transformation,[status(thm)],[f699])).
% 2.56/1.65 thf(f699,plain,(
% 2.56/1.65 ! [X0 : a,X1 : a,X2 : a] : (((less @ (sup @ X0 @ X1) @ X2) = $true) => ~(((less @ X0 @ X2) = $true) => ((less @ X1 @ X2) != $true)))),
% 2.56/1.65 inference(flattening,[status(thm)],[f665])).
% 2.56/1.65 thf(f665,plain,(
% 2.56/1.65 ! [X0 : a,X1 : a,X2 : a] : (((less @ (sup @ X0 @ X1) @ X2) = $true) => ~(((less @ X0 @ X2) = $true) => ~((less @ X1 @ X2) = $true)))),
% 2.56/1.65 inference(fool_elimination,[status(thm)],[f664])).
% 2.56/1.65 thf(f664,plain,(
% 2.56/1.65 ! [X0 : a,X1 : a,X2 : a] : ((less @ (sup @ X0 @ X1) @ X2) => ~((less @ X0 @ X2) => ~(less @ X1 @ X2)))),
% 2.56/1.65 inference(rectify,[status(thm)],[f131])).
% 2.56/1.65 thf(f131,axiom,(
% 2.56/1.65 ! [X4 : a,X10 : a,X5 : a] : ((less @ (sup @ X4 @ X10) @ X5) => ~((less @ X4 @ X5) => ~(less @ X10 @ X5)))),
% 2.56/1.65 file('/export/starexec/sandbox2/tmp/tmp.Vy3zEj9nqo/DTF2THF_5351.p',fact_130_local_Osup_Ostrict__boundedE)).
% 2.56/1.65 % SZS output end Proof for DTF2THF_5351
% 2.56/1.65 % (5505)------------------------------
% 2.56/1.65 % (5505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.65 % (5505)Termination reason: Refutation
% 2.56/1.65
% 2.56/1.65 % (5505)Memory used [KB]: 6524
% 2.56/1.65 % (5505)Time elapsed: 0.137 s
% 2.56/1.65 % (5505)Instructions burned: 272 (million)
% 2.56/1.65 % (5505)------------------------------
% 2.56/1.65 % (5505)------------------------------
% 2.56/1.65 % (5499)Success in time 0.161 s
%------------------------------------------------------------------------------