%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM195^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n009.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:41:55 AM UTC 2026
% Result : Theorem 2.82s 1.62s
% Output : Refutation 2.82s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : COM195^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n009.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 04:17:10 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.28/0.45 ---- Original DTF file ---
% 0.28/0.45 thf(spec,logic,$$dhol).
% 0.28/0.45 %------------------------------------------------------------------------------
% 0.28/0.45 % File : COM195^1 : TPTP v9.2.1. Released v7.0.0.
% 0.28/0.45 % Domain : Computing Theory
% 0.28/0.45 % Problem : Grammars and languages 1319
% 0.28/0.45 % Version : [Bla16] axioms : Especial.
% 0.28/0.45 % English :
% 0.28/0.45
% 0.28/0.45 % Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% 0.28/0.45 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.28/0.45 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.28/0.45 % Source : [Bla16]
% 0.28/0.45 % Names : gram_lang__1319.p [Bla16]
% 0.28/0.45
% 0.28/0.45 % Status : Theorem
% 0.28/0.45 % Rating : 0.00 v7.2.0, 0.25 v7.1.0
% 0.28/0.45 % Syntax : Number of formulae : 320 ( 37 unt; 43 typ; 0 def)
% 0.28/0.45 % Number of atoms : 1012 ( 202 equ; 0 cnn)
% 0.28/0.45 % Maximal formula atoms : 13 ( 3 avg)
% 0.28/0.45 % Number of connectives : 3896 ( 78 ~; 25 |; 65 &;3228 @)
% 0.28/0.45 % ( 0 <=>; 500 =>; 0 <=; 0 <~>)
% 0.28/0.45 % Maximal formula depth : 21 ( 8 avg)
% 0.28/0.45 % Number of types : 2 ( 1 usr)
% 0.28/0.45 % Number of type conns : 146 ( 146 >; 0 *; 0 +; 0 <<)
% 0.28/0.45 % Number of symbols : 43 ( 42 usr; 2 con; 0-3 aty)
% 0.28/0.45 % Number of variables : 1060 ( 52 ^; 936 !; 32 ?;1060 :)
% 0.28/0.45 % ( 40 !>; 0 ?*; 0 @-; 0 @+)
% 0.28/0.45 % SPC : TH1_THM_EQU_NAR
% 0.28/0.45
% 0.28/0.45 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.28/0.45 % 2016-07-13 14:48:25.615
% 0.28/0.45 %------------------------------------------------------------------------------
% 0.28/0.45 %----Could-be-implicit typings (3)
% 0.28/0.45 thf(ty_t_Set_Oset,type,
% 0.28/0.45 set: $tType > $tType ).
% 0.28/0.45
% 0.28/0.45 thf(ty_t_itself,type,
% 0.28/0.45 itself: $tType > $tType ).
% 0.28/0.45
% 0.28/0.45 thf(ty_tf_a,type,
% 0.28/0.45 a: $tType ).
% 0.28/0.45
% 0.28/0.45 %----Explicit typings (40)
% 0.28/0.45 thf(sy_cl_HOL_Otype,type,
% 0.28/0.45 type:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_cl_Groups_Ouminus,type,
% 0.28/0.45 uminus:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_cl_Orderings_Oord,type,
% 0.28/0.45 ord:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_cl_Orderings_Oorder,type,
% 0.28/0.45 order:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_cl_Groups_Ogroup__add,type,
% 0.28/0.45 group_add:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_cl_Orderings_Ono__bot,type,
% 0.28/0.45 no_bot:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_cl_Orderings_Ono__top,type,
% 0.28/0.45 no_top:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_cl_Orderings_Olinorder,type,
% 0.28/0.45 linorder:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_cl_Orderings_Opreorder,type,
% 0.28/0.45 preorder:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_cl_Orderings_Owellorder,type,
% 0.28/0.45 wellorder:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_cl_Orderings_Odense__order,type,
% 0.28/0.45 dense_order:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 0.28/0.45 boolean_algebra:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.28/0.45 semilattice_sup:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.28/0.45 dense_linorder:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.28/0.45 ordered_ab_group_add:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.28/0.45 condit1656338222tinuum:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.28/0.45 condit1037483654norder:
% 0.28/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Oleqv,type,
% 0.28/0.45 gram_L1456083582e_leqv:
% 0.28/0.45 !>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ A ) ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Osubs,type,
% 0.28/0.45 gram_L608943123e_subs:
% 0.28/0.45 !>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ A ) ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.28/0.45 uminus_uminus:
% 0.28/0.45 !>[A: $tType] : ( A > A ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.28/0.45 sup_sup:
% 0.28/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.28/0.45 ord_less:
% 0.28/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.28/0.45 ord_less_eq:
% 0.28/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 0.28/0.45 order_antimono:
% 0.28/0.45 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 0.28/0.45 order_strict_mono:
% 0.28/0.45 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Pure_Otype,type,
% 0.28/0.45 type2:
% 0.28/0.45 !>[A: $tType] : ( itself @ A ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Set_OCollect,type,
% 0.28/0.45 collect:
% 0.28/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Set_OPow,type,
% 0.28/0.45 pow:
% 0.28/0.45 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Set_Opairwise,type,
% 0.28/0.45 pairwise:
% 0.28/0.45 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
% 0.28/0.45 set_ord_atLeast:
% 0.28/0.45 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
% 0.28/0.45 set_or331188842AtMost:
% 0.28/0.45 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
% 0.28/0.45 set_or1433837966ssThan:
% 0.28/0.45 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Set__Interval_Oord__class_OatMost,type,
% 0.28/0.45 set_ord_atMost:
% 0.28/0.45 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
% 0.28/0.45 set_ord_greaterThan:
% 0.28/0.45 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
% 0.28/0.45 set_or1361889807AtMost:
% 0.28/0.45 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
% 0.28/0.45 set_or578182835ssThan:
% 0.28/0.45 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
% 0.28/0.45 set_ord_lessThan:
% 0.28/0.45 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_c_member,type,
% 0.28/0.45 member:
% 0.28/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_v_L1,type,
% 0.28/0.45 l1: set @ ( set @ a ) ).
% 0.28/0.45
% 0.28/0.45 thf(sy_v_L2,type,
% 0.28/0.45 l2: set @ ( set @ a ) ).
% 0.28/0.45
% 0.28/0.45 %----Relevant facts (256)
% 0.28/0.45 thf(fact_0_subs__refl,axiom,
% 0.28/0.45 ! [A: $tType,L1: set @ ( set @ A )] : ( gram_L608943123e_subs @ A @ L1 @ L1 ) ).
% 0.28/0.45
% 0.28/0.45 % subs_refl
% 0.28/0.45 thf(fact_1_subs__trans,axiom,
% 0.28/0.45 ! [A: $tType,L1: set @ ( set @ A ),L2: set @ ( set @ A ),L3: set @ ( set @ A )] :
% 0.28/0.45 ( ( gram_L608943123e_subs @ A @ L1 @ L2 )
% 0.28/0.45 => ( ( gram_L608943123e_subs @ A @ L2 @ L3 )
% 0.28/0.45 => ( gram_L608943123e_subs @ A @ L1 @ L3 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % subs_trans
% 0.28/0.45 thf(fact_2_leqv__def,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( gram_L1456083582e_leqv @ A )
% 0.28/0.45 = ( ^ [L12: set @ ( set @ A ),L22: set @ ( set @ A )] :
% 0.28/0.45 ( ( gram_L608943123e_subs @ A @ L12 @ L22 )
% 0.28/0.45 & ( gram_L608943123e_subs @ A @ L22 @ L12 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % leqv_def
% 0.28/0.45 thf(fact_3_incl__subs,axiom,
% 0.28/0.45 ! [A: $tType,L2: set @ ( set @ A ),L1: set @ ( set @ A )] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ L2 @ L1 )
% 0.28/0.45 => ( gram_L608943123e_subs @ A @ L1 @ L2 ) ) ).
% 0.28/0.45
% 0.28/0.45 % incl_subs
% 0.28/0.45 thf(fact_4_subs__def,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( gram_L608943123e_subs @ A )
% 0.28/0.45 = ( ^ [L12: set @ ( set @ A ),L22: set @ ( set @ A )] :
% 0.28/0.45 ! [X: set @ A] :
% 0.28/0.45 ( ( member @ ( set @ A ) @ X @ L22 )
% 0.28/0.45 => ? [Y: set @ A] :
% 0.28/0.45 ( ( member @ ( set @ A ) @ Y @ L12 )
% 0.28/0.45 & ( ord_less_eq @ ( set @ A ) @ Y @ X ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % subs_def
% 0.28/0.45 thf(fact_5_subsetI,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.28/0.45 ( ! [X2: A] :
% 0.28/0.45 ( ( member @ A @ X2 @ A2 )
% 0.28/0.45 => ( member @ A @ X2 @ B2 ) )
% 0.28/0.45 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.28/0.45
% 0.28/0.45 % subsetI
% 0.28/0.45 thf(fact_6_subset__antisym,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.28/0.45 => ( A2 = B2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % subset_antisym
% 0.28/0.45 thf(fact_7_order__refl,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 0.28/0.45
% 0.28/0.45 % order_refl
% 0.28/0.45 thf(fact_8_set__mp,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.45 => ( ( member @ A @ X3 @ A2 )
% 0.28/0.45 => ( member @ A @ X3 @ B2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % set_mp
% 0.28/0.45 thf(fact_9_in__mono,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.45 => ( ( member @ A @ X3 @ A2 )
% 0.28/0.45 => ( member @ A @ X3 @ B2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % in_mono
% 0.28/0.45 thf(fact_10_subsetD,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.45 => ( ( member @ A @ C @ A2 )
% 0.28/0.45 => ( member @ A @ C @ B2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % subsetD
% 0.28/0.45 thf(fact_11_subsetCE,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.45 => ( ( member @ A @ C @ A2 )
% 0.28/0.45 => ( member @ A @ C @ B2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % subsetCE
% 0.28/0.45 thf(fact_12_equalityE,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.28/0.45 ( ( A2 = B2 )
% 0.28/0.45 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.45 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % equalityE
% 0.28/0.45 thf(fact_13_subset__eq,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) )
% 0.28/0.45 = ( ^ [A3: set @ A,B3: set @ A] :
% 0.28/0.45 ! [X: A] :
% 0.28/0.45 ( ( member @ A @ X @ A3 )
% 0.28/0.45 => ( member @ A @ X @ B3 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % subset_eq
% 0.28/0.45 thf(fact_14_equalityD1,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.28/0.45 ( ( A2 = B2 )
% 0.28/0.45 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.28/0.45
% 0.28/0.45 % equalityD1
% 0.28/0.45 thf(fact_15_equalityD2,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.28/0.45 ( ( A2 = B2 )
% 0.28/0.45 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.28/0.45
% 0.28/0.45 % equalityD2
% 0.28/0.45 thf(fact_16_dual__order_Oantisym,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [B4: A,A4: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.28/0.45 => ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.28/0.45 => ( A4 = B4 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % dual_order.antisym
% 0.28/0.45 thf(fact_17_dual__order_Otrans,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [B4: A,A4: A,C: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.28/0.45 => ( ( ord_less_eq @ A @ C @ B4 )
% 0.28/0.45 => ( ord_less_eq @ A @ C @ A4 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % dual_order.trans
% 0.28/0.45 thf(fact_18_linorder__wlog,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [P: A > A > $o,A4: A,B4: A] :
% 0.28/0.45 ( ! [A5: A,B5: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.28/0.45 => ( P @ A5 @ B5 ) )
% 0.28/0.45 => ( ! [A5: A,B5: A] :
% 0.28/0.45 ( ( P @ B5 @ A5 )
% 0.28/0.45 => ( P @ A5 @ B5 ) )
% 0.28/0.45 => ( P @ A4 @ B4 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % linorder_wlog
% 0.28/0.45 thf(fact_19_dual__order_Orefl,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 0.28/0.45
% 0.28/0.45 % dual_order.refl
% 0.28/0.45 thf(fact_20_order__trans,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A,Z: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/0.45 => ( ( ord_less_eq @ A @ Y2 @ Z )
% 0.28/0.45 => ( ord_less_eq @ A @ X3 @ Z ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % order_trans
% 0.28/0.45 thf(fact_21_order__class_Oorder_Oantisym,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.28/0.45 => ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.28/0.45 => ( A4 = B4 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % order_class.order.antisym
% 0.28/0.45 thf(fact_22_ord__le__eq__trans,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A,C: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.28/0.45 => ( ( B4 = C )
% 0.28/0.45 => ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % ord_le_eq_trans
% 0.28/0.45 thf(fact_23_ord__eq__le__trans,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A,C: A] :
% 0.28/0.45 ( ( A4 = B4 )
% 0.28/0.45 => ( ( ord_less_eq @ A @ B4 @ C )
% 0.28/0.45 => ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % ord_eq_le_trans
% 0.28/0.45 thf(fact_24_antisym__conv,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [Y2: A,X3: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ Y2 @ X3 )
% 0.28/0.45 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/0.45 = ( X3 = Y2 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % antisym_conv
% 0.28/0.45 thf(fact_25_le__cases3,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A,Z: A] :
% 0.28/0.45 ( ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/0.45 => ~ ( ord_less_eq @ A @ Y2 @ Z ) )
% 0.28/0.45 => ( ( ( ord_less_eq @ A @ Y2 @ X3 )
% 0.28/0.45 => ~ ( ord_less_eq @ A @ X3 @ Z ) )
% 0.28/0.45 => ( ( ( ord_less_eq @ A @ X3 @ Z )
% 0.28/0.45 => ~ ( ord_less_eq @ A @ Z @ Y2 ) )
% 0.28/0.45 => ( ( ( ord_less_eq @ A @ Z @ Y2 )
% 0.28/0.45 => ~ ( ord_less_eq @ A @ Y2 @ X3 ) )
% 0.28/0.45 => ( ( ( ord_less_eq @ A @ Y2 @ Z )
% 0.28/0.45 => ~ ( ord_less_eq @ A @ Z @ X3 ) )
% 0.28/0.45 => ~ ( ( ord_less_eq @ A @ Z @ X3 )
% 0.28/0.45 => ~ ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % le_cases3
% 0.28/0.45 thf(fact_26_order_Otrans,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A,C: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.28/0.45 => ( ( ord_less_eq @ A @ B4 @ C )
% 0.28/0.45 => ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % order.trans
% 0.28/0.45 thf(fact_27_le__cases,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ~ ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/0.45 => ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % le_cases
% 0.28/0.45 thf(fact_28_eq__refl,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( X3 = Y2 )
% 0.28/0.45 => ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % eq_refl
% 0.28/0.45 thf(fact_29_linear,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/0.45 | ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % linear
% 0.28/0.45 thf(fact_30_antisym,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/0.45 => ( ( ord_less_eq @ A @ Y2 @ X3 )
% 0.28/0.45 => ( X3 = Y2 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % antisym
% 0.28/0.45 thf(fact_31_eq__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ( ( ^ [Y3: A,Z2: A] : ( Y3 = Z2 ) )
% 0.28/0.45 = ( ^ [X: A,Y: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.28/0.45 & ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % eq_iff
% 0.28/0.45 thf(fact_32_ord__le__eq__subst,axiom,
% 0.28/0.45 ! [A: $tType,B: $tType] :
% 0.28/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.28/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.28/0.45 => ! [A4: A,B4: A,F: A > B,C: B] :
% 0.28/0.45 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.28/0.45 => ( ( ( F @ B4 )
% 0.28/0.45 = C )
% 0.28/0.45 => ( ! [X2: A,Y4: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 0.28/0.45 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.28/0.45 => ( ord_less_eq @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % ord_le_eq_subst
% 0.28/0.45 thf(fact_33_ord__eq__le__subst,axiom,
% 0.28/0.45 ! [A: $tType,B: $tType] :
% 0.28/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.28/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.28/0.45 => ! [A4: A,F: B > A,B4: B,C: B] :
% 0.28/0.45 ( ( A4
% 0.28/0.45 = ( F @ B4 ) )
% 0.28/0.45 => ( ( ord_less_eq @ B @ B4 @ C )
% 0.28/0.45 => ( ! [X2: B,Y4: B] :
% 0.28/0.45 ( ( ord_less_eq @ B @ X2 @ Y4 )
% 0.28/0.45 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.28/0.45 => ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % ord_eq_le_subst
% 0.28/0.45 thf(fact_34_order__subst2,axiom,
% 0.28/0.45 ! [A: $tType,C2: $tType] :
% 0.28/0.45 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.28/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.28/0.45 => ! [A4: A,B4: A,F: A > C2,C: C2] :
% 0.28/0.45 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.28/0.45 => ( ( ord_less_eq @ C2 @ ( F @ B4 ) @ C )
% 0.28/0.45 => ( ! [X2: A,Y4: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 0.28/0.45 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.28/0.45 => ( ord_less_eq @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % order_subst2
% 0.28/0.45 thf(fact_35_order__subst1,axiom,
% 0.28/0.45 ! [A: $tType,B: $tType] :
% 0.28/0.45 ( ( ( order @ B @ ( type2 @ B ) )
% 0.28/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.28/0.45 => ! [A4: A,F: B > A,B4: B,C: B] :
% 0.28/0.45 ( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
% 0.28/0.45 => ( ( ord_less_eq @ B @ B4 @ C )
% 0.28/0.45 => ( ! [X2: B,Y4: B] :
% 0.28/0.45 ( ( ord_less_eq @ B @ X2 @ Y4 )
% 0.28/0.45 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.28/0.45 => ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % order_subst1
% 0.28/0.45 thf(fact_36_le__fun__def,axiom,
% 0.28/0.45 ! [B: $tType,A: $tType] :
% 0.28/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.28/0.45 => ( ( ord_less_eq @ ( A > B ) )
% 0.28/0.45 = ( ^ [F2: A > B,G: A > B] :
% 0.28/0.45 ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % le_fun_def
% 0.28/0.45 thf(fact_37_le__funI,axiom,
% 0.28/0.45 ! [B: $tType,A: $tType] :
% 0.28/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.28/0.45 => ! [F: A > B,G2: A > B] :
% 0.28/0.45 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G2 @ X2 ) )
% 0.28/0.45 => ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % le_funI
% 0.28/0.45 thf(fact_38_le__funE,axiom,
% 0.28/0.45 ! [B: $tType,A: $tType] :
% 0.28/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.28/0.45 => ! [F: A > B,G2: A > B,X3: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 0.28/0.45 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % le_funE
% 0.28/0.45 thf(fact_39_le__funD,axiom,
% 0.28/0.45 ! [B: $tType,A: $tType] :
% 0.28/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.28/0.45 => ! [F: A > B,G2: A > B,X3: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 0.28/0.45 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % le_funD
% 0.28/0.45 thf(fact_40_Collect__mono__iff,axiom,
% 0.28/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.28/0.45 = ( ! [X: A] :
% 0.28/0.45 ( ( P @ X )
% 0.28/0.45 => ( Q @ X ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % Collect_mono_iff
% 0.28/0.45 thf(fact_41_contra__subsetD,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.45 => ( ~ ( member @ A @ C @ B2 )
% 0.28/0.45 => ~ ( member @ A @ C @ A2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % contra_subsetD
% 0.28/0.45 thf(fact_42_set__eq__subset,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( ^ [Y3: set @ A,Z2: set @ A] : ( Y3 = Z2 ) )
% 0.28/0.45 = ( ^ [A3: set @ A,B3: set @ A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.28/0.45 & ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % set_eq_subset
% 0.28/0.45 thf(fact_43_subset__trans,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.28/0.45 => ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % subset_trans
% 0.28/0.45 thf(fact_44_Collect__mono,axiom,
% 0.28/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.45 ( ! [X2: A] :
% 0.28/0.45 ( ( P @ X2 )
% 0.28/0.45 => ( Q @ X2 ) )
% 0.28/0.45 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % Collect_mono
% 0.28/0.45 thf(fact_45_mem__Collect__eq,axiom,
% 0.28/0.45 ! [A: $tType,A4: A,P: A > $o] :
% 0.28/0.45 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 0.28/0.45 = ( P @ A4 ) ) ).
% 0.28/0.45
% 0.28/0.45 % mem_Collect_eq
% 0.28/0.45 thf(fact_46_Collect__mem__eq,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A] :
% 0.28/0.45 ( ( collect @ A
% 0.28/0.45 @ ^ [X: A] : ( member @ A @ X @ A2 ) )
% 0.28/0.45 = A2 ) ).
% 0.28/0.45
% 0.28/0.45 % Collect_mem_eq
% 0.28/0.45 thf(fact_47_Collect__cong,axiom,
% 0.28/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.45 ( ! [X2: A] :
% 0.28/0.45 ( ( P @ X2 )
% 0.28/0.45 = ( Q @ X2 ) )
% 0.28/0.45 => ( ( collect @ A @ P )
% 0.28/0.45 = ( collect @ A @ Q ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % Collect_cong
% 0.28/0.45 thf(fact_48_ext,axiom,
% 0.28/0.45 ! [B: $tType,A: $tType,F: A > B,G2: A > B] :
% 0.28/0.45 ( ! [X2: A] :
% 0.28/0.45 ( ( F @ X2 )
% 0.28/0.45 = ( G2 @ X2 ) )
% 0.28/0.45 => ( F = G2 ) ) ).
% 0.28/0.45
% 0.28/0.45 % ext
% 0.28/0.45 thf(fact_49_subset__refl,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.28/0.45
% 0.28/0.45 % subset_refl
% 0.28/0.45 thf(fact_50_rev__subsetD,axiom,
% 0.28/0.45 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 0.28/0.45 ( ( member @ A @ C @ A2 )
% 0.28/0.45 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.45 => ( member @ A @ C @ B2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % rev_subsetD
% 0.28/0.45 thf(fact_51_subset__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) )
% 0.28/0.45 = ( ^ [A3: set @ A,B3: set @ A] :
% 0.28/0.45 ! [T: A] :
% 0.28/0.45 ( ( member @ A @ T @ A3 )
% 0.28/0.45 => ( member @ A @ T @ B3 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % subset_iff
% 0.28/0.45 thf(fact_52_set__rev__mp,axiom,
% 0.28/0.45 ! [A: $tType,X3: A,A2: set @ A,B2: set @ A] :
% 0.28/0.45 ( ( member @ A @ X3 @ A2 )
% 0.28/0.45 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.45 => ( member @ A @ X3 @ B2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % set_rev_mp
% 0.28/0.45 thf(fact_53_antimono__def,axiom,
% 0.28/0.45 ! [B: $tType,A: $tType] :
% 0.28/0.45 ( ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.28/0.45 => ( ( order_antimono @ A @ B )
% 0.28/0.45 = ( ^ [F2: A > B] :
% 0.28/0.45 ! [X: A,Y: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.28/0.45 => ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % antimono_def
% 0.28/0.45 thf(fact_54_antimonoI,axiom,
% 0.28/0.45 ! [B: $tType,A: $tType] :
% 0.28/0.45 ( ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.28/0.45 => ! [F: A > B] :
% 0.28/0.45 ( ! [X2: A,Y4: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 0.28/0.45 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X2 ) ) )
% 0.28/0.45 => ( order_antimono @ A @ B @ F ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % antimonoI
% 0.28/0.45 thf(fact_55_antimonoE,axiom,
% 0.28/0.45 ! [B: $tType,A: $tType] :
% 0.28/0.45 ( ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.28/0.45 => ! [F: A > B,X3: A,Y2: A] :
% 0.28/0.45 ( ( order_antimono @ A @ B @ F )
% 0.28/0.45 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/0.45 => ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % antimonoE
% 0.28/0.45 thf(fact_56_antimonoD,axiom,
% 0.28/0.45 ! [B: $tType,A: $tType] :
% 0.28/0.45 ( ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.28/0.45 => ! [F: A > B,X3: A,Y2: A] :
% 0.28/0.45 ( ( order_antimono @ A @ B @ F )
% 0.28/0.45 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/0.45 => ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % antimonoD
% 0.28/0.45 thf(fact_57_Pow__mono,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.45 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A2 ) @ ( pow @ A @ B2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % Pow_mono
% 0.28/0.45 thf(fact_58_strict__mono__less__eq,axiom,
% 0.28/0.45 ! [B: $tType,A: $tType] :
% 0.28/0.45 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.28/0.45 => ! [F: A > B,X3: A,Y2: A] :
% 0.28/0.45 ( ( order_strict_mono @ A @ B @ F )
% 0.28/0.45 => ( ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) )
% 0.28/0.45 = ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % strict_mono_less_eq
% 0.28/0.45 thf(fact_59_pairwise__subset,axiom,
% 0.28/0.45 ! [A: $tType,P: A > A > $o,S: set @ A,T2: set @ A] :
% 0.28/0.45 ( ( pairwise @ A @ P @ S )
% 0.28/0.45 => ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
% 0.28/0.45 => ( pairwise @ A @ P @ T2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % pairwise_subset
% 0.28/0.45 thf(fact_60_greaterThan__subset__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X3 ) @ ( set_ord_greaterThan @ A @ Y2 ) )
% 0.28/0.45 = ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % greaterThan_subset_iff
% 0.28/0.45 thf(fact_61_lessThan__subset__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X3 ) @ ( set_ord_lessThan @ A @ Y2 ) )
% 0.28/0.45 = ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % lessThan_subset_iff
% 0.28/0.45 thf(fact_62_atLeast__subset__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X3 ) @ ( set_ord_atLeast @ A @ Y2 ) )
% 0.28/0.45 = ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % atLeast_subset_iff
% 0.28/0.45 thf(fact_63_lessThan__eq__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ( set_ord_lessThan @ A @ X3 )
% 0.28/0.45 = ( set_ord_lessThan @ A @ Y2 ) )
% 0.28/0.45 = ( X3 = Y2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % lessThan_eq_iff
% 0.28/0.45 thf(fact_64_atLeast__eq__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ( set_ord_atLeast @ A @ X3 )
% 0.28/0.45 = ( set_ord_atLeast @ A @ Y2 ) )
% 0.28/0.45 = ( X3 = Y2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % atLeast_eq_iff
% 0.28/0.45 thf(fact_65_greaterThan__eq__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ( set_ord_greaterThan @ A @ X3 )
% 0.28/0.45 = ( set_ord_greaterThan @ A @ Y2 ) )
% 0.28/0.45 = ( X3 = Y2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % greaterThan_eq_iff
% 0.28/0.45 thf(fact_66_atLeast__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [I: A,K: A] :
% 0.28/0.45 ( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) )
% 0.28/0.45 = ( ord_less_eq @ A @ K @ I ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % atLeast_iff
% 0.28/0.45 thf(fact_67_PowI,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.45 => ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % PowI
% 0.28/0.45 thf(fact_68_Pow__iff,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.28/0.45 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 0.28/0.45 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.28/0.45
% 0.28/0.45 % Pow_iff
% 0.28/0.45 thf(fact_69_Ioi__le__Ico,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A4 ) @ ( set_ord_atLeast @ A @ A4 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % Ioi_le_Ico
% 0.28/0.45 thf(fact_70_pairwise__def,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( pairwise @ A )
% 0.28/0.45 = ( ^ [R: A > A > $o,S2: set @ A] :
% 0.28/0.45 ! [X: A] :
% 0.28/0.45 ( ( member @ A @ X @ S2 )
% 0.28/0.45 => ! [Y: A] :
% 0.28/0.45 ( ( member @ A @ Y @ S2 )
% 0.28/0.45 => ( ( X != Y )
% 0.28/0.45 => ( R @ X @ Y ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % pairwise_def
% 0.28/0.45 thf(fact_71_Pow__top,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A] : ( member @ ( set @ A ) @ A2 @ ( pow @ A @ A2 ) ) ).
% 0.28/0.45
% 0.28/0.45 % Pow_top
% 0.28/0.45 thf(fact_72_strict__mono__eq,axiom,
% 0.28/0.45 ! [B: $tType,A: $tType] :
% 0.28/0.45 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 & ( order @ B @ ( type2 @ B ) ) )
% 0.28/0.45 => ! [F: A > B,X3: A,Y2: A] :
% 0.28/0.45 ( ( order_strict_mono @ A @ B @ F )
% 0.28/0.45 => ( ( ( F @ X3 )
% 0.28/0.45 = ( F @ Y2 ) )
% 0.28/0.45 = ( X3 = Y2 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % strict_mono_eq
% 0.28/0.45 thf(fact_73_PowD,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.28/0.45 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 0.28/0.45 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.28/0.45
% 0.28/0.45 % PowD
% 0.28/0.45 thf(fact_74_Icc__subset__Ici__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [L: A,H: A,L4: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ H ) @ ( set_ord_atLeast @ A @ L4 ) )
% 0.28/0.45 = ( ~ ( ord_less_eq @ A @ L @ H )
% 0.28/0.45 | ( ord_less_eq @ A @ L4 @ L ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % Icc_subset_Ici_iff
% 0.28/0.45 thf(fact_75_Ici__subset__Ioi__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A4 ) @ ( set_ord_greaterThan @ A @ B4 ) )
% 0.28/0.45 = ( ord_less @ A @ B4 @ A4 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % Ici_subset_Ioi_iff
% 0.28/0.45 thf(fact_76_Compl__atLeast,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [K: A] :
% 0.28/0.45 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atLeast @ A @ K ) )
% 0.28/0.45 = ( set_ord_lessThan @ A @ K ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % Compl_atLeast
% 0.28/0.45 thf(fact_77_Compl__lessThan,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [K: A] :
% 0.28/0.45 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) )
% 0.28/0.45 = ( set_ord_atLeast @ A @ K ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % Compl_lessThan
% 0.28/0.45 thf(fact_78_atMost__subset__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X3 ) @ ( set_ord_atMost @ A @ Y2 ) )
% 0.28/0.45 = ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % atMost_subset_iff
% 0.28/0.45 thf(fact_79_ivl__subset,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [I: A,J: A,M: A,N: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1433837966ssThan @ A @ I @ J ) @ ( set_or1433837966ssThan @ A @ M @ N ) )
% 0.28/0.45 = ( ( ord_less_eq @ A @ J @ I )
% 0.28/0.45 | ( ( ord_less_eq @ A @ M @ I )
% 0.28/0.45 & ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % ivl_subset
% 0.28/0.45 thf(fact_80_not__Iic__le__Ici,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [H: A,L4: A] :
% 0.28/0.45 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_ord_atLeast @ A @ L4 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % not_Iic_le_Ici
% 0.28/0.45 thf(fact_81_psubsetI,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.45 => ( ( A2 != B2 )
% 0.28/0.45 => ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % psubsetI
% 0.28/0.45 thf(fact_82_ComplI,axiom,
% 0.28/0.45 ! [A: $tType,C: A,A2: set @ A] :
% 0.28/0.45 ( ~ ( member @ A @ C @ A2 )
% 0.28/0.45 => ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % ComplI
% 0.28/0.45 thf(fact_83_Compl__iff,axiom,
% 0.28/0.45 ! [A: $tType,C: A,A2: set @ A] :
% 0.28/0.45 ( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.28/0.45 = ( ~ ( member @ A @ C @ A2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % Compl_iff
% 0.28/0.45 thf(fact_84_Compl__eq__Compl__iff,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.28/0.45 ( ( ( uminus_uminus @ ( set @ A ) @ A2 )
% 0.28/0.45 = ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 0.28/0.45 = ( A2 = B2 ) ) ).
% 0.28/0.45
% 0.28/0.45 % Compl_eq_Compl_iff
% 0.28/0.45 thf(fact_85_atMost__eq__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ( set_ord_atMost @ A @ X3 )
% 0.28/0.45 = ( set_ord_atMost @ A @ Y2 ) )
% 0.28/0.45 = ( X3 = Y2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % atMost_eq_iff
% 0.28/0.45 thf(fact_86_atLeastAtMost__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [I: A,L: A,U: A] :
% 0.28/0.45 ( ( member @ A @ I @ ( set_or331188842AtMost @ A @ L @ U ) )
% 0.28/0.45 = ( ( ord_less_eq @ A @ L @ I )
% 0.28/0.45 & ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % atLeastAtMost_iff
% 0.28/0.45 thf(fact_87_Icc__eq__Icc,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [L: A,H: A,L4: A,H2: A] :
% 0.28/0.45 ( ( ( set_or331188842AtMost @ A @ L @ H )
% 0.28/0.45 = ( set_or331188842AtMost @ A @ L4 @ H2 ) )
% 0.28/0.45 = ( ( ( L = L4 )
% 0.28/0.45 & ( H = H2 ) )
% 0.28/0.45 | ( ~ ( ord_less_eq @ A @ L @ H )
% 0.28/0.45 & ~ ( ord_less_eq @ A @ L4 @ H2 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % Icc_eq_Icc
% 0.28/0.45 thf(fact_88_lessThan__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [I: A,K: A] :
% 0.28/0.45 ( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
% 0.28/0.45 = ( ord_less @ A @ I @ K ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % lessThan_iff
% 0.28/0.45 thf(fact_89_atMost__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [I: A,K: A] :
% 0.28/0.45 ( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) )
% 0.28/0.45 = ( ord_less_eq @ A @ I @ K ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % atMost_iff
% 0.28/0.45 thf(fact_90_Compl__subset__Compl__iff,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 0.28/0.45 = ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.28/0.45
% 0.28/0.45 % Compl_subset_Compl_iff
% 0.28/0.45 thf(fact_91_Compl__anti__mono,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.45 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % Compl_anti_mono
% 0.28/0.45 thf(fact_92_greaterThan__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [I: A,K: A] :
% 0.28/0.45 ( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K ) )
% 0.28/0.45 = ( ord_less @ A @ K @ I ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % greaterThan_iff
% 0.28/0.45 thf(fact_93_atLeastLessThan__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [I: A,L: A,U: A] :
% 0.28/0.45 ( ( member @ A @ I @ ( set_or1433837966ssThan @ A @ L @ U ) )
% 0.28/0.45 = ( ( ord_less_eq @ A @ L @ I )
% 0.28/0.45 & ( ord_less @ A @ I @ U ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % atLeastLessThan_iff
% 0.28/0.45 thf(fact_94_atLeastatMost__subset__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A,C: A,D: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A4 @ B4 ) @ ( set_or331188842AtMost @ A @ C @ D ) )
% 0.28/0.45 = ( ~ ( ord_less_eq @ A @ A4 @ B4 )
% 0.28/0.45 | ( ( ord_less_eq @ A @ C @ A4 )
% 0.28/0.45 & ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % atLeastatMost_subset_iff
% 0.28/0.45 thf(fact_95_Compl__atMost,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [K: A] :
% 0.28/0.45 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atMost @ A @ K ) )
% 0.28/0.45 = ( set_ord_greaterThan @ A @ K ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % Compl_atMost
% 0.28/0.45 thf(fact_96_Compl__greaterThan,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [K: A] :
% 0.28/0.45 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_greaterThan @ A @ K ) )
% 0.28/0.45 = ( set_ord_atMost @ A @ K ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % Compl_greaterThan
% 0.28/0.45 thf(fact_97_Icc__subset__Iic__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [L: A,H: A,H2: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ H ) @ ( set_ord_atMost @ A @ H2 ) )
% 0.28/0.45 = ( ~ ( ord_less_eq @ A @ L @ H )
% 0.28/0.45 | ( ord_less_eq @ A @ H @ H2 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % Icc_subset_Iic_iff
% 0.28/0.45 thf(fact_98_not__Iic__le__Icc,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [H: A,L4: A,H2: A] :
% 0.28/0.45 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_or331188842AtMost @ A @ L4 @ H2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % not_Iic_le_Icc
% 0.28/0.45 thf(fact_99_atLeastatMost__psubset__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A,C: A,D: A] :
% 0.28/0.45 ( ( ord_less @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A4 @ B4 ) @ ( set_or331188842AtMost @ A @ C @ D ) )
% 0.28/0.45 = ( ( ~ ( ord_less_eq @ A @ A4 @ B4 )
% 0.28/0.45 | ( ( ord_less_eq @ A @ C @ A4 )
% 0.28/0.45 & ( ord_less_eq @ A @ B4 @ D )
% 0.28/0.45 & ( ( ord_less @ A @ C @ A4 )
% 0.28/0.45 | ( ord_less @ A @ B4 @ D ) ) ) )
% 0.28/0.45 & ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % atLeastatMost_psubset_iff
% 0.28/0.45 thf(fact_100_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A,C: A,D: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A4 @ B4 ) @ ( set_or1433837966ssThan @ A @ C @ D ) )
% 0.28/0.45 = ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.28/0.45 => ( ( ord_less_eq @ A @ C @ A4 )
% 0.28/0.45 & ( ord_less @ A @ B4 @ D ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % atLeastAtMost_subseteq_atLeastLessThan_iff
% 0.28/0.45 thf(fact_101_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A,C: A,D: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1433837966ssThan @ A @ A4 @ B4 ) @ ( set_or331188842AtMost @ A @ C @ D ) )
% 0.28/0.45 = ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.45 => ( ( ord_less_eq @ A @ C @ A4 )
% 0.28/0.45 & ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % atLeastLessThan_subseteq_atLeastAtMost_iff
% 0.28/0.45 thf(fact_102_atLeastLessThan__inj_I2_J,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A,C: A,D: A] :
% 0.28/0.45 ( ( ( set_or1433837966ssThan @ A @ A4 @ B4 )
% 0.28/0.45 = ( set_or1433837966ssThan @ A @ C @ D ) )
% 0.28/0.45 => ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.45 => ( ( ord_less @ A @ C @ D )
% 0.28/0.45 => ( B4 = D ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % atLeastLessThan_inj(2)
% 0.28/0.45 thf(fact_103_atLeastLessThan__inj_I1_J,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A,C: A,D: A] :
% 0.28/0.45 ( ( ( set_or1433837966ssThan @ A @ A4 @ B4 )
% 0.28/0.45 = ( set_or1433837966ssThan @ A @ C @ D ) )
% 0.28/0.45 => ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.45 => ( ( ord_less @ A @ C @ D )
% 0.28/0.45 => ( A4 = C ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % atLeastLessThan_inj(1)
% 0.28/0.45 thf(fact_104_atLeastLessThan__eq__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A,C: A,D: A] :
% 0.28/0.45 ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.45 => ( ( ord_less @ A @ C @ D )
% 0.28/0.45 => ( ( ( set_or1433837966ssThan @ A @ A4 @ B4 )
% 0.28/0.45 = ( set_or1433837966ssThan @ A @ C @ D ) )
% 0.28/0.45 = ( ( A4 = C )
% 0.28/0.45 & ( B4 = D ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % atLeastLessThan_eq_iff
% 0.28/0.45 thf(fact_105_lessThan__strict__subset__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [M: A,N: A] :
% 0.28/0.45 ( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
% 0.28/0.45 = ( ord_less @ A @ M @ N ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % lessThan_strict_subset_iff
% 0.28/0.45 thf(fact_106_not__Iic__eq__Icc,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [H2: A,L: A,H: A] :
% 0.28/0.45 ( ( set_ord_atMost @ A @ H2 )
% 0.28/0.45 != ( set_or331188842AtMost @ A @ L @ H ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % not_Iic_eq_Icc
% 0.28/0.45 thf(fact_107_ComplD,axiom,
% 0.28/0.45 ! [A: $tType,C: A,A2: set @ A] :
% 0.28/0.45 ( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.28/0.45 => ~ ( member @ A @ C @ A2 ) ) ).
% 0.28/0.45
% 0.28/0.45 % ComplD
% 0.28/0.45 thf(fact_108_double__complement,axiom,
% 0.28/0.45 ! [A: $tType,A2: set @ A] :
% 0.28/0.45 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.28/0.45 = A2 ) ).
% 0.28/0.45
% 0.28/0.45 % double_complement
% 0.28/0.45 thf(fact_109_ord__eq__less__subst,axiom,
% 0.28/0.45 ! [A: $tType,B: $tType] :
% 0.28/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.28/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.28/0.45 => ! [A4: A,F: B > A,B4: B,C: B] :
% 0.28/0.45 ( ( A4
% 0.28/0.45 = ( F @ B4 ) )
% 0.28/0.45 => ( ( ord_less @ B @ B4 @ C )
% 0.28/0.45 => ( ! [X2: B,Y4: B] :
% 0.28/0.45 ( ( ord_less @ B @ X2 @ Y4 )
% 0.28/0.45 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.28/0.45 => ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % ord_eq_less_subst
% 0.28/0.45 thf(fact_110_ord__less__eq__subst,axiom,
% 0.28/0.45 ! [A: $tType,B: $tType] :
% 0.28/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.28/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.28/0.45 => ! [A4: A,B4: A,F: A > B,C: B] :
% 0.28/0.45 ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.45 => ( ( ( F @ B4 )
% 0.28/0.45 = C )
% 0.28/0.45 => ( ! [X2: A,Y4: A] :
% 0.28/0.45 ( ( ord_less @ A @ X2 @ Y4 )
% 0.28/0.45 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.28/0.45 => ( ord_less @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % ord_less_eq_subst
% 0.28/0.45 thf(fact_111_order__less__subst1,axiom,
% 0.28/0.45 ! [A: $tType,B: $tType] :
% 0.28/0.45 ( ( ( order @ B @ ( type2 @ B ) )
% 0.28/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.28/0.45 => ! [A4: A,F: B > A,B4: B,C: B] :
% 0.28/0.45 ( ( ord_less @ A @ A4 @ ( F @ B4 ) )
% 0.28/0.45 => ( ( ord_less @ B @ B4 @ C )
% 0.28/0.45 => ( ! [X2: B,Y4: B] :
% 0.28/0.45 ( ( ord_less @ B @ X2 @ Y4 )
% 0.28/0.45 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.28/0.45 => ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % order_less_subst1
% 0.28/0.45 thf(fact_112_order__less__subst2,axiom,
% 0.28/0.45 ! [A: $tType,C2: $tType] :
% 0.28/0.45 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.28/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.28/0.45 => ! [A4: A,B4: A,F: A > C2,C: C2] :
% 0.28/0.45 ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.45 => ( ( ord_less @ C2 @ ( F @ B4 ) @ C )
% 0.28/0.45 => ( ! [X2: A,Y4: A] :
% 0.28/0.45 ( ( ord_less @ A @ X2 @ Y4 )
% 0.28/0.45 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.28/0.45 => ( ord_less @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % order_less_subst2
% 0.28/0.45 thf(fact_113_lt__ex,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A] :
% 0.28/0.45 ? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ).
% 0.28/0.45
% 0.28/0.45 % lt_ex
% 0.28/0.45 thf(fact_114_gt__ex,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( no_top @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A] :
% 0.28/0.45 ? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% 0.28/0.45
% 0.28/0.45 % gt_ex
% 0.28/0.45 thf(fact_115_neqE,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( X3 != Y2 )
% 0.28/0.45 => ( ~ ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.45 => ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % neqE
% 0.28/0.45 thf(fact_116_neq__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( X3 != Y2 )
% 0.28/0.45 = ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.45 | ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % neq_iff
% 0.28/0.45 thf(fact_117_order_Oasym,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A] :
% 0.28/0.45 ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.45 => ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % order.asym
% 0.28/0.45 thf(fact_118_dense,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.45 => ? [Z3: A] :
% 0.28/0.45 ( ( ord_less @ A @ X3 @ Z3 )
% 0.28/0.45 & ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % dense
% 0.28/0.45 thf(fact_119_less__imp__neq,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.45 => ( X3 != Y2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % less_imp_neq
% 0.28/0.45 thf(fact_120_less__asym,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.45 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % less_asym
% 0.28/0.45 thf(fact_121_less__asym_H,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A] :
% 0.28/0.45 ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.45 => ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % less_asym'
% 0.28/0.45 thf(fact_122_less__trans,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A,Z: A] :
% 0.28/0.45 ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.45 => ( ( ord_less @ A @ Y2 @ Z )
% 0.28/0.45 => ( ord_less @ A @ X3 @ Z ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % less_trans
% 0.28/0.45 thf(fact_123_less__linear,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.45 | ( X3 = Y2 )
% 0.28/0.45 | ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % less_linear
% 0.28/0.45 thf(fact_124_less__irrefl,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A] :
% 0.28/0.45 ~ ( ord_less @ A @ X3 @ X3 ) ) ).
% 0.28/0.45
% 0.28/0.45 % less_irrefl
% 0.28/0.45 thf(fact_125_ord__eq__less__trans,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A,C: A] :
% 0.28/0.45 ( ( A4 = B4 )
% 0.28/0.45 => ( ( ord_less @ A @ B4 @ C )
% 0.28/0.45 => ( ord_less @ A @ A4 @ C ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % ord_eq_less_trans
% 0.28/0.45 thf(fact_126_ord__less__eq__trans,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A,C: A] :
% 0.28/0.45 ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.45 => ( ( B4 = C )
% 0.28/0.45 => ( ord_less @ A @ A4 @ C ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % ord_less_eq_trans
% 0.28/0.45 thf(fact_127_dual__order_Oasym,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [B4: A,A4: A] :
% 0.28/0.45 ( ( ord_less @ A @ B4 @ A4 )
% 0.28/0.45 => ~ ( ord_less @ A @ A4 @ B4 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % dual_order.asym
% 0.28/0.45 thf(fact_128_less__imp__not__eq,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.45 => ( X3 != Y2 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % less_imp_not_eq
% 0.28/0.45 thf(fact_129_less__not__sym,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.45 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % less_not_sym
% 0.28/0.45 thf(fact_130_less__induct,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [P: A > $o,A4: A] :
% 0.28/0.45 ( ! [X2: A] :
% 0.28/0.45 ( ! [Y5: A] :
% 0.28/0.45 ( ( ord_less @ A @ Y5 @ X2 )
% 0.28/0.45 => ( P @ Y5 ) )
% 0.28/0.45 => ( P @ X2 ) )
% 0.28/0.45 => ( P @ A4 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % less_induct
% 0.28/0.45 thf(fact_131_antisym__conv3,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [Y2: A,X3: A] :
% 0.28/0.45 ( ~ ( ord_less @ A @ Y2 @ X3 )
% 0.28/0.45 => ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 0.28/0.45 = ( X3 = Y2 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % antisym_conv3
% 0.28/0.45 thf(fact_132_less__imp__not__eq2,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.45 => ( Y2 != X3 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % less_imp_not_eq2
% 0.28/0.45 thf(fact_133_less__imp__triv,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A,P: $o] :
% 0.28/0.45 ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.45 => ( ( ord_less @ A @ Y2 @ X3 )
% 0.28/0.45 => P ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % less_imp_triv
% 0.28/0.45 thf(fact_134_linorder__cases,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.45 => ( ( X3 != Y2 )
% 0.28/0.45 => ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % linorder_cases
% 0.28/0.45 thf(fact_135_dual__order_Oirrefl,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A] :
% 0.28/0.45 ~ ( ord_less @ A @ A4 @ A4 ) ) ).
% 0.28/0.45
% 0.28/0.45 % dual_order.irrefl
% 0.28/0.45 thf(fact_136_order_Ostrict__trans,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A,C: A] :
% 0.28/0.45 ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.45 => ( ( ord_less @ A @ B4 @ C )
% 0.28/0.45 => ( ord_less @ A @ A4 @ C ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % order.strict_trans
% 0.28/0.45 thf(fact_137_less__imp__not__less,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.45 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % less_imp_not_less
% 0.28/0.45 thf(fact_138_dual__order_Ostrict__trans,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [B4: A,A4: A,C: A] :
% 0.28/0.45 ( ( ord_less @ A @ B4 @ A4 )
% 0.28/0.45 => ( ( ord_less @ A @ C @ B4 )
% 0.28/0.45 => ( ord_less @ A @ C @ A4 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % dual_order.strict_trans
% 0.28/0.45 thf(fact_139_not__less__iff__gr__or__eq,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A] :
% 0.28/0.45 ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 0.28/0.45 = ( ( ord_less @ A @ Y2 @ X3 )
% 0.28/0.45 | ( X3 = Y2 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % not_less_iff_gr_or_eq
% 0.28/0.45 thf(fact_140_order_Ostrict__implies__not__eq,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A] :
% 0.28/0.45 ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.45 => ( A4 != B4 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % order.strict_implies_not_eq
% 0.28/0.45 thf(fact_141_dual__order_Ostrict__implies__not__eq,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [B4: A,A4: A] :
% 0.28/0.45 ( ( ord_less @ A @ B4 @ A4 )
% 0.28/0.45 => ( A4 != B4 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % dual_order.strict_implies_not_eq
% 0.28/0.45 thf(fact_142_Iic__subset__Iio__iff,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A] :
% 0.28/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A4 ) @ ( set_ord_lessThan @ A @ B4 ) )
% 0.28/0.45 = ( ord_less @ A @ A4 @ B4 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % Iic_subset_Iio_iff
% 0.28/0.45 thf(fact_143_order_Onot__eq__order__implies__strict,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A] :
% 0.28/0.45 ( ( A4 != B4 )
% 0.28/0.45 => ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.28/0.45 => ( ord_less @ A @ A4 @ B4 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % order.not_eq_order_implies_strict
% 0.28/0.45 thf(fact_144_dual__order_Ostrict__implies__order,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [B4: A,A4: A] :
% 0.28/0.45 ( ( ord_less @ A @ B4 @ A4 )
% 0.28/0.45 => ( ord_less_eq @ A @ B4 @ A4 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % dual_order.strict_implies_order
% 0.28/0.45 thf(fact_145_dual__order_Ostrict__iff__order,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ( ( ord_less @ A )
% 0.28/0.45 = ( ^ [B6: A,A6: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ B6 @ A6 )
% 0.28/0.45 & ( A6 != B6 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % dual_order.strict_iff_order
% 0.28/0.45 thf(fact_146_dual__order_Oorder__iff__strict,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ( ( ord_less_eq @ A )
% 0.28/0.45 = ( ^ [B6: A,A6: A] :
% 0.28/0.45 ( ( ord_less @ A @ B6 @ A6 )
% 0.28/0.45 | ( A6 = B6 ) ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % dual_order.order_iff_strict
% 0.28/0.45 thf(fact_147_order_Ostrict__implies__order,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [A4: A,B4: A] :
% 0.28/0.45 ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.45 => ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % order.strict_implies_order
% 0.28/0.45 thf(fact_148_dense__le__bounded,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [X3: A,Y2: A,Z: A] :
% 0.28/0.45 ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.45 => ( ! [W: A] :
% 0.28/0.45 ( ( ord_less @ A @ X3 @ W )
% 0.28/0.45 => ( ( ord_less @ A @ W @ Y2 )
% 0.28/0.45 => ( ord_less_eq @ A @ W @ Z ) ) )
% 0.28/0.45 => ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % dense_le_bounded
% 0.28/0.45 thf(fact_149_dense__ge__bounded,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [Z: A,X3: A,Y2: A] :
% 0.28/0.45 ( ( ord_less @ A @ Z @ X3 )
% 0.28/0.45 => ( ! [W: A] :
% 0.28/0.45 ( ( ord_less @ A @ Z @ W )
% 0.28/0.45 => ( ( ord_less @ A @ W @ X3 )
% 0.28/0.45 => ( ord_less_eq @ A @ Y2 @ W ) ) )
% 0.28/0.45 => ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % dense_ge_bounded
% 0.28/0.45 thf(fact_150_dual__order_Ostrict__trans2,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [B4: A,A4: A,C: A] :
% 0.28/0.45 ( ( ord_less @ A @ B4 @ A4 )
% 0.28/0.45 => ( ( ord_less_eq @ A @ C @ B4 )
% 0.28/0.45 => ( ord_less @ A @ C @ A4 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % dual_order.strict_trans2
% 0.28/0.45 thf(fact_151_dual__order_Ostrict__trans1,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ! [B4: A,A4: A,C: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.28/0.45 => ( ( ord_less @ A @ C @ B4 )
% 0.28/0.45 => ( ord_less @ A @ C @ A4 ) ) ) ) ).
% 0.28/0.45
% 0.28/0.45 % dual_order.strict_trans1
% 0.28/0.45 thf(fact_152_order_Ostrict__iff__order,axiom,
% 0.28/0.45 ! [A: $tType] :
% 0.28/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.45 => ( ( ord_less @ A )
% 0.28/0.45 = ( ^ [A6: A,B6: A] :
% 0.28/0.45 ( ( ord_less_eq @ A @ A6 @ B6 )
% 0.28/0.46 & ( A6 != B6 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % order.strict_iff_order
% 0.28/0.46 thf(fact_153_order_Oorder__iff__strict,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.46 => ( ( ord_less_eq @ A )
% 0.28/0.46 = ( ^ [A6: A,B6: A] :
% 0.28/0.46 ( ( ord_less @ A @ A6 @ B6 )
% 0.28/0.46 | ( A6 = B6 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % order.order_iff_strict
% 0.28/0.46 thf(fact_154_order_Ostrict__trans2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A,C: A] :
% 0.28/0.46 ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.46 => ( ( ord_less_eq @ A @ B4 @ C )
% 0.28/0.46 => ( ord_less @ A @ A4 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % order.strict_trans2
% 0.28/0.46 thf(fact_155_order_Ostrict__trans1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A,C: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.28/0.46 => ( ( ord_less @ A @ B4 @ C )
% 0.28/0.46 => ( ord_less @ A @ A4 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % order.strict_trans1
% 0.28/0.46 thf(fact_156_not__le__imp__less,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [Y2: A,X3: A] :
% 0.28/0.46 ( ~ ( ord_less_eq @ A @ Y2 @ X3 )
% 0.28/0.46 => ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % not_le_imp_less
% 0.28/0.46 thf(fact_157_less__le__not__le,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ( ( ord_less @ A )
% 0.28/0.46 = ( ^ [X: A,Y: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.28/0.46 & ~ ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_le_not_le
% 0.28/0.46 thf(fact_158_le__imp__less__or__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A,Y2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/0.46 => ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.46 | ( X3 = Y2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_imp_less_or_eq
% 0.28/0.46 thf(fact_159_le__less__linear,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A,Y2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/0.46 | ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_less_linear
% 0.28/0.46 thf(fact_160_dense__le,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [Y2: A,Z: A] :
% 0.28/0.46 ( ! [X2: A] :
% 0.28/0.46 ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.46 => ( ord_less_eq @ A @ X2 @ Z ) )
% 0.28/0.46 => ( ord_less_eq @ A @ Y2 @ Z ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % dense_le
% 0.28/0.46 thf(fact_161_dense__ge,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [Z: A,Y2: A] :
% 0.28/0.46 ( ! [X2: A] :
% 0.28/0.46 ( ( ord_less @ A @ Z @ X2 )
% 0.28/0.46 => ( ord_less_eq @ A @ Y2 @ X2 ) )
% 0.28/0.46 => ( ord_less_eq @ A @ Y2 @ Z ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % dense_ge
% 0.28/0.46 thf(fact_162_less__le__trans,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A,Y2: A,Z: A] :
% 0.28/0.46 ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.46 => ( ( ord_less_eq @ A @ Y2 @ Z )
% 0.28/0.46 => ( ord_less @ A @ X3 @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_le_trans
% 0.28/0.46 thf(fact_163_le__less__trans,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A,Y2: A,Z: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/0.46 => ( ( ord_less @ A @ Y2 @ Z )
% 0.28/0.46 => ( ord_less @ A @ X3 @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_less_trans
% 0.28/0.46 thf(fact_164_antisym__conv2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A,Y2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/0.46 => ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 0.28/0.46 = ( X3 = Y2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % antisym_conv2
% 0.28/0.46 thf(fact_165_antisym__conv1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A,Y2: A] :
% 0.28/0.46 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.46 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/0.46 = ( X3 = Y2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % antisym_conv1
% 0.28/0.46 thf(fact_166_less__imp__le,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A,Y2: A] :
% 0.28/0.46 ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.46 => ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_imp_le
% 0.28/0.46 thf(fact_167_le__neq__trans,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.28/0.46 => ( ( A4 != B4 )
% 0.28/0.46 => ( ord_less @ A @ A4 @ B4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_neq_trans
% 0.28/0.46 thf(fact_168_not__less,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A,Y2: A] :
% 0.28/0.46 ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 0.28/0.46 = ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % not_less
% 0.28/0.46 thf(fact_169_not__le,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A,Y2: A] :
% 0.28/0.46 ( ( ~ ( ord_less_eq @ A @ X3 @ Y2 ) )
% 0.28/0.46 = ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % not_le
% 0.28/0.46 thf(fact_170_order__less__le__subst2,axiom,
% 0.28/0.46 ! [A: $tType,C2: $tType] :
% 0.28/0.46 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.28/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.28/0.46 => ! [A4: A,B4: A,F: A > C2,C: C2] :
% 0.28/0.46 ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.46 => ( ( ord_less_eq @ C2 @ ( F @ B4 ) @ C )
% 0.28/0.46 => ( ! [X2: A,Y4: A] :
% 0.28/0.46 ( ( ord_less @ A @ X2 @ Y4 )
% 0.28/0.46 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.28/0.46 => ( ord_less @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % order_less_le_subst2
% 0.28/0.46 thf(fact_171_order__less__le__subst1,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType] :
% 0.28/0.46 ( ( ( order @ B @ ( type2 @ B ) )
% 0.28/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.28/0.46 => ! [A4: A,F: B > A,B4: B,C: B] :
% 0.28/0.46 ( ( ord_less @ A @ A4 @ ( F @ B4 ) )
% 0.28/0.46 => ( ( ord_less_eq @ B @ B4 @ C )
% 0.28/0.46 => ( ! [X2: B,Y4: B] :
% 0.28/0.46 ( ( ord_less_eq @ B @ X2 @ Y4 )
% 0.28/0.46 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.28/0.46 => ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % order_less_le_subst1
% 0.28/0.46 thf(fact_172_order__le__less__subst2,axiom,
% 0.28/0.46 ! [A: $tType,C2: $tType] :
% 0.28/0.46 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.28/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.28/0.46 => ! [A4: A,B4: A,F: A > C2,C: C2] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.28/0.46 => ( ( ord_less @ C2 @ ( F @ B4 ) @ C )
% 0.28/0.46 => ( ! [X2: A,Y4: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 0.28/0.46 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.28/0.46 => ( ord_less @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % order_le_less_subst2
% 0.28/0.46 thf(fact_173_order__le__less__subst1,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType] :
% 0.28/0.46 ( ( ( order @ B @ ( type2 @ B ) )
% 0.28/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.28/0.46 => ! [A4: A,F: B > A,B4: B,C: B] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
% 0.28/0.46 => ( ( ord_less @ B @ B4 @ C )
% 0.28/0.46 => ( ! [X2: B,Y4: B] :
% 0.28/0.46 ( ( ord_less @ B @ X2 @ Y4 )
% 0.28/0.46 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.28/0.46 => ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % order_le_less_subst1
% 0.28/0.46 thf(fact_174_less__le,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.46 => ( ( ord_less @ A )
% 0.28/0.46 = ( ^ [X: A,Y: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.28/0.46 & ( X != Y ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_le
% 0.28/0.46 thf(fact_175_le__less,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.46 => ( ( ord_less_eq @ A )
% 0.28/0.46 = ( ^ [X: A,Y: A] :
% 0.28/0.46 ( ( ord_less @ A @ X @ Y )
% 0.28/0.46 | ( X = Y ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_less
% 0.28/0.46 thf(fact_176_leI,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A,Y2: A] :
% 0.28/0.46 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.46 => ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % leI
% 0.28/0.46 thf(fact_177_leD,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [Y2: A,X3: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ Y2 @ X3 )
% 0.28/0.46 => ~ ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % leD
% 0.28/0.46 thf(fact_178_subset__iff__psubset__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.28/0.46 = ( ^ [A3: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.28/0.46 | ( A3 = B3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % subset_iff_psubset_eq
% 0.28/0.46 thf(fact_179_subset__psubset__trans,axiom,
% 0.28/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.28/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.46 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 0.28/0.46 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % subset_psubset_trans
% 0.28/0.46 thf(fact_180_subset__not__subset__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ord_less @ ( set @ A ) )
% 0.28/0.46 = ( ^ [A3: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.28/0.46 & ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % subset_not_subset_eq
% 0.28/0.46 thf(fact_181_psubset__subset__trans,axiom,
% 0.28/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.28/0.46 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.28/0.46 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % psubset_subset_trans
% 0.28/0.46 thf(fact_182_psubset__imp__subset,axiom,
% 0.28/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.28/0.46 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % psubset_imp_subset
% 0.28/0.46 thf(fact_183_psubset__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ord_less @ ( set @ A ) )
% 0.28/0.46 = ( ^ [A3: set @ A,B3: set @ A] :
% 0.28/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.28/0.46 & ( A3 != B3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % psubset_eq
% 0.28/0.46 thf(fact_184_psubsetE,axiom,
% 0.28/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.28/0.46 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.46 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % psubsetE
% 0.28/0.46 thf(fact_185_not__Ici__eq__Icc,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( no_top @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [L4: A,L: A,H: A] :
% 0.28/0.46 ( ( set_ord_atLeast @ A @ L4 )
% 0.28/0.46 != ( set_or331188842AtMost @ A @ L @ H ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % not_Ici_eq_Icc
% 0.28/0.46 thf(fact_186_less__fun__def,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.28/0.46 => ( ( ord_less @ ( A > B ) )
% 0.28/0.46 = ( ^ [F2: A > B,G: A > B] :
% 0.28/0.46 ( ( ord_less_eq @ ( A > B ) @ F2 @ G )
% 0.28/0.46 & ~ ( ord_less_eq @ ( A > B ) @ G @ F2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_fun_def
% 0.28/0.46 thf(fact_187_not__Iic__eq__Ici,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( no_top @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [H: A,L4: A] :
% 0.28/0.46 ( ( set_ord_atMost @ A @ H )
% 0.28/0.46 != ( set_ord_atLeast @ A @ L4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % not_Iic_eq_Ici
% 0.28/0.46 thf(fact_188_strict__monoD,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.46 & ( order @ B @ ( type2 @ B ) ) )
% 0.28/0.46 => ! [F: A > B,X3: A,Y2: A] :
% 0.28/0.46 ( ( order_strict_mono @ A @ B @ F )
% 0.28/0.46 => ( ( ord_less @ A @ X3 @ Y2 )
% 0.28/0.46 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % strict_monoD
% 0.28/0.46 thf(fact_189_strict__monoI,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.46 & ( order @ B @ ( type2 @ B ) ) )
% 0.28/0.46 => ! [F: A > B] :
% 0.28/0.46 ( ! [X2: A,Y4: A] :
% 0.28/0.46 ( ( ord_less @ A @ X2 @ Y4 )
% 0.28/0.46 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.28/0.46 => ( order_strict_mono @ A @ B @ F ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % strict_monoI
% 0.28/0.46 thf(fact_190_strict__mono__def,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.46 & ( order @ B @ ( type2 @ B ) ) )
% 0.28/0.46 => ( ( order_strict_mono @ A @ B )
% 0.28/0.46 = ( ^ [F2: A > B] :
% 0.28/0.46 ! [X: A,Y: A] :
% 0.28/0.46 ( ( ord_less @ A @ X @ Y )
% 0.28/0.46 => ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % strict_mono_def
% 0.28/0.46 thf(fact_191_strict__mono__less,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 & ( order @ B @ ( type2 @ B ) ) )
% 0.28/0.46 => ! [F: A > B,X3: A,Y2: A] :
% 0.28/0.46 ( ( order_strict_mono @ A @ B @ F )
% 0.28/0.46 => ( ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) )
% 0.28/0.46 = ( ord_less @ A @ X3 @ Y2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % strict_mono_less
% 0.28/0.46 thf(fact_192_not__Ici__le__Icc,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( no_top @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [L: A,L4: A,H2: A] :
% 0.28/0.46 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or331188842AtMost @ A @ L4 @ H2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % not_Ici_le_Icc
% 0.28/0.46 thf(fact_193_atLeastLessThan__subset__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A,C: A,D: A] :
% 0.28/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1433837966ssThan @ A @ A4 @ B4 ) @ ( set_or1433837966ssThan @ A @ C @ D ) )
% 0.28/0.46 => ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.28/0.46 | ( ( ord_less_eq @ A @ C @ A4 )
% 0.28/0.46 & ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % atLeastLessThan_subset_iff
% 0.28/0.46 thf(fact_194_not__Ici__le__Iic,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( no_top @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [L: A,H2: A] :
% 0.28/0.46 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_ord_atMost @ A @ H2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % not_Ici_le_Iic
% 0.28/0.46 thf(fact_195_neg__less__iff__less,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [B4: A,A4: A] :
% 0.28/0.46 ( ( ord_less @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A4 ) )
% 0.28/0.46 = ( ord_less @ A @ A4 @ B4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_less_iff_less
% 0.28/0.46 thf(fact_196_neg__le__iff__le,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [B4: A,A4: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A4 ) )
% 0.28/0.46 = ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_le_iff_le
% 0.28/0.46 thf(fact_197_compl__le__compl__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A,Y2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y2 ) )
% 0.28/0.46 = ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % compl_le_compl_iff
% 0.28/0.46 thf(fact_198_compl__eq__compl__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A,Y2: A] :
% 0.28/0.46 ( ( ( uminus_uminus @ A @ X3 )
% 0.28/0.46 = ( uminus_uminus @ A @ Y2 ) )
% 0.28/0.46 = ( X3 = Y2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % compl_eq_compl_iff
% 0.28/0.46 thf(fact_199_double__compl,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A] :
% 0.28/0.46 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X3 ) )
% 0.28/0.46 = X3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % double_compl
% 0.28/0.46 thf(fact_200_neg__equal__iff__equal,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A] :
% 0.28/0.46 ( ( ( uminus_uminus @ A @ A4 )
% 0.28/0.46 = ( uminus_uminus @ A @ B4 ) )
% 0.28/0.46 = ( A4 = B4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % neg_equal_iff_equal
% 0.28/0.46 thf(fact_201_uminus__apply,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( uminus @ B @ ( type2 @ B ) )
% 0.28/0.46 => ( ( uminus_uminus @ ( A > B ) )
% 0.28/0.46 = ( ^ [A3: A > B,X: A] : ( uminus_uminus @ B @ ( A3 @ X ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % uminus_apply
% 0.28/0.46 thf(fact_202_add_Oinverse__inverse,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A] :
% 0.28/0.46 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A4 ) )
% 0.28/0.46 = A4 ) ) ).
% 0.28/0.46
% 0.28/0.46 % add.inverse_inverse
% 0.28/0.46 thf(fact_203_psubsetD,axiom,
% 0.28/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
% 0.28/0.46 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.46 => ( ( member @ A @ C @ A2 )
% 0.28/0.46 => ( member @ A @ C @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % psubsetD
% 0.28/0.46 thf(fact_204_psubset__trans,axiom,
% 0.28/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.28/0.46 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 0.28/0.46 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 0.28/0.46 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % psubset_trans
% 0.28/0.46 thf(fact_205_fun__Compl__def,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( uminus @ B @ ( type2 @ B ) )
% 0.28/0.46 => ( ( uminus_uminus @ ( A > B ) )
% 0.28/0.46 = ( ^ [A3: A > B,X: A] : ( uminus_uminus @ B @ ( A3 @ X ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % fun_Compl_def
% 0.28/0.46 thf(fact_206_equation__minus__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A] :
% 0.28/0.46 ( ( A4
% 0.28/0.46 = ( uminus_uminus @ A @ B4 ) )
% 0.28/0.46 = ( B4
% 0.28/0.46 = ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % equation_minus_iff
% 0.28/0.46 thf(fact_207_minus__equation__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( group_add @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A] :
% 0.28/0.46 ( ( ( uminus_uminus @ A @ A4 )
% 0.28/0.46 = B4 )
% 0.28/0.46 = ( ( uminus_uminus @ A @ B4 )
% 0.28/0.46 = A4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % minus_equation_iff
% 0.28/0.46 thf(fact_208_le__imp__neg__le,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.28/0.46 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_imp_neg_le
% 0.28/0.46 thf(fact_209_minus__le__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B4 )
% 0.28/0.46 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ A4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % minus_le_iff
% 0.28/0.46 thf(fact_210_le__minus__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ B4 ) )
% 0.28/0.46 = ( ord_less_eq @ A @ B4 @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_minus_iff
% 0.28/0.46 thf(fact_211_compl__le__swap2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [Y2: A,X3: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ X3 )
% 0.28/0.46 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % compl_le_swap2
% 0.28/0.46 thf(fact_212_compl__le__swap1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [Y2: A,X3: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ Y2 @ ( uminus_uminus @ A @ X3 ) )
% 0.28/0.46 => ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % compl_le_swap1
% 0.28/0.46 thf(fact_213_compl__mono,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A,Y2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/0.46 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % compl_mono
% 0.28/0.46 thf(fact_214_compl__less__swap1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [Y2: A,X3: A] :
% 0.28/0.46 ( ( ord_less @ A @ Y2 @ ( uminus_uminus @ A @ X3 ) )
% 0.28/0.46 => ( ord_less @ A @ X3 @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % compl_less_swap1
% 0.28/0.46 thf(fact_215_compl__less__swap2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [Y2: A,X3: A] :
% 0.28/0.46 ( ( ord_less @ A @ ( uminus_uminus @ A @ Y2 ) @ X3 )
% 0.28/0.46 => ( ord_less @ A @ ( uminus_uminus @ A @ X3 ) @ Y2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % compl_less_swap2
% 0.28/0.46 thf(fact_216_less__minus__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A] :
% 0.28/0.46 ( ( ord_less @ A @ A4 @ ( uminus_uminus @ A @ B4 ) )
% 0.28/0.46 = ( ord_less @ A @ B4 @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_minus_iff
% 0.28/0.46 thf(fact_217_minus__less__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A] :
% 0.28/0.46 ( ( ord_less @ A @ ( uminus_uminus @ A @ A4 ) @ B4 )
% 0.28/0.46 = ( ord_less @ A @ ( uminus_uminus @ A @ B4 ) @ A4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % minus_less_iff
% 0.28/0.46 thf(fact_218_compl__less__compl__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A,Y2: A] :
% 0.28/0.46 ( ( ord_less @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y2 ) )
% 0.28/0.46 = ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % compl_less_compl_iff
% 0.28/0.46 thf(fact_219_complete__interval,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A,P: A > $o] :
% 0.28/0.46 ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.46 => ( ( P @ A4 )
% 0.28/0.46 => ( ~ ( P @ B4 )
% 0.28/0.46 => ? [C4: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A4 @ C4 )
% 0.28/0.46 & ( ord_less_eq @ A @ C4 @ B4 )
% 0.28/0.46 & ! [X4: A] :
% 0.28/0.46 ( ( ( ord_less_eq @ A @ A4 @ X4 )
% 0.28/0.46 & ( ord_less @ A @ X4 @ C4 ) )
% 0.28/0.46 => ( P @ X4 ) )
% 0.28/0.46 & ! [D2: A] :
% 0.28/0.46 ( ! [X2: A] :
% 0.28/0.46 ( ( ( ord_less_eq @ A @ A4 @ X2 )
% 0.28/0.46 & ( ord_less @ A @ X2 @ D2 ) )
% 0.28/0.46 => ( P @ X2 ) )
% 0.28/0.46 => ( ord_less_eq @ A @ D2 @ C4 ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % complete_interval
% 0.28/0.46 thf(fact_220_pinf_I6_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [T3: A] :
% 0.28/0.46 ? [Z3: A] :
% 0.28/0.46 ! [X4: A] :
% 0.28/0.46 ( ( ord_less @ A @ Z3 @ X4 )
% 0.28/0.46 => ~ ( ord_less_eq @ A @ X4 @ T3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % pinf(6)
% 0.28/0.46 thf(fact_221_ex__gt__or__lt,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A] :
% 0.28/0.46 ? [B5: A] :
% 0.28/0.46 ( ( ord_less @ A @ A4 @ B5 )
% 0.28/0.46 | ( ord_less @ A @ B5 @ A4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ex_gt_or_lt
% 0.28/0.46 thf(fact_222_pinf_I1_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.28/0.46 ( ? [Z4: A] :
% 0.28/0.46 ! [X2: A] :
% 0.28/0.46 ( ( ord_less @ A @ Z4 @ X2 )
% 0.28/0.46 => ( ( P @ X2 )
% 0.28/0.46 = ( P2 @ X2 ) ) )
% 0.28/0.46 => ( ? [Z4: A] :
% 0.28/0.46 ! [X2: A] :
% 0.28/0.46 ( ( ord_less @ A @ Z4 @ X2 )
% 0.28/0.46 => ( ( Q @ X2 )
% 0.28/0.46 = ( Q2 @ X2 ) ) )
% 0.28/0.46 => ? [Z3: A] :
% 0.28/0.46 ! [X4: A] :
% 0.28/0.46 ( ( ord_less @ A @ Z3 @ X4 )
% 0.28/0.46 => ( ( ( P @ X4 )
% 0.28/0.46 & ( Q @ X4 ) )
% 0.28/0.46 = ( ( P2 @ X4 )
% 0.28/0.46 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % pinf(1)
% 0.28/0.46 thf(fact_223_pinf_I2_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.28/0.46 ( ? [Z4: A] :
% 0.28/0.46 ! [X2: A] :
% 0.28/0.46 ( ( ord_less @ A @ Z4 @ X2 )
% 0.28/0.46 => ( ( P @ X2 )
% 0.28/0.46 = ( P2 @ X2 ) ) )
% 0.28/0.46 => ( ? [Z4: A] :
% 0.28/0.46 ! [X2: A] :
% 0.28/0.46 ( ( ord_less @ A @ Z4 @ X2 )
% 0.28/0.46 => ( ( Q @ X2 )
% 0.28/0.46 = ( Q2 @ X2 ) ) )
% 0.28/0.46 => ? [Z3: A] :
% 0.28/0.46 ! [X4: A] :
% 0.28/0.46 ( ( ord_less @ A @ Z3 @ X4 )
% 0.28/0.46 => ( ( ( P @ X4 )
% 0.28/0.46 | ( Q @ X4 ) )
% 0.28/0.46 = ( ( P2 @ X4 )
% 0.28/0.46 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % pinf(2)
% 0.28/0.46 thf(fact_224_pinf_I3_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [T3: A] :
% 0.28/0.46 ? [Z3: A] :
% 0.28/0.46 ! [X4: A] :
% 0.28/0.46 ( ( ord_less @ A @ Z3 @ X4 )
% 0.28/0.46 => ( X4 != T3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % pinf(3)
% 0.28/0.46 thf(fact_225_pinf_I4_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [T3: A] :
% 0.28/0.46 ? [Z3: A] :
% 0.28/0.46 ! [X4: A] :
% 0.28/0.46 ( ( ord_less @ A @ Z3 @ X4 )
% 0.28/0.46 => ( X4 != T3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % pinf(4)
% 0.28/0.46 thf(fact_226_pinf_I5_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [T3: A] :
% 0.28/0.46 ? [Z3: A] :
% 0.28/0.46 ! [X4: A] :
% 0.28/0.46 ( ( ord_less @ A @ Z3 @ X4 )
% 0.28/0.46 => ~ ( ord_less @ A @ X4 @ T3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % pinf(5)
% 0.28/0.46 thf(fact_227_pinf_I7_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [T3: A] :
% 0.28/0.46 ? [Z3: A] :
% 0.28/0.46 ! [X4: A] :
% 0.28/0.46 ( ( ord_less @ A @ Z3 @ X4 )
% 0.28/0.46 => ( ord_less @ A @ T3 @ X4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % pinf(7)
% 0.28/0.46 thf(fact_228_pinf_I11_J,axiom,
% 0.28/0.46 ! [C2: $tType,D3: $tType] :
% 0.28/0.46 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.28/0.46 => ! [F3: D3] :
% 0.28/0.46 ? [Z3: C2] :
% 0.28/0.46 ! [X4: C2] :
% 0.28/0.46 ( ( ord_less @ C2 @ Z3 @ X4 )
% 0.28/0.46 => ( F3 = F3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % pinf(11)
% 0.28/0.46 thf(fact_229_minf_I1_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.28/0.46 ( ? [Z4: A] :
% 0.28/0.46 ! [X2: A] :
% 0.28/0.46 ( ( ord_less @ A @ X2 @ Z4 )
% 0.28/0.46 => ( ( P @ X2 )
% 0.28/0.46 = ( P2 @ X2 ) ) )
% 0.28/0.46 => ( ? [Z4: A] :
% 0.28/0.46 ! [X2: A] :
% 0.28/0.46 ( ( ord_less @ A @ X2 @ Z4 )
% 0.28/0.46 => ( ( Q @ X2 )
% 0.28/0.46 = ( Q2 @ X2 ) ) )
% 0.28/0.46 => ? [Z3: A] :
% 0.28/0.46 ! [X4: A] :
% 0.28/0.46 ( ( ord_less @ A @ X4 @ Z3 )
% 0.28/0.46 => ( ( ( P @ X4 )
% 0.28/0.46 & ( Q @ X4 ) )
% 0.28/0.46 = ( ( P2 @ X4 )
% 0.28/0.46 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % minf(1)
% 0.28/0.46 thf(fact_230_minf_I2_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.28/0.46 ( ? [Z4: A] :
% 0.28/0.46 ! [X2: A] :
% 0.28/0.46 ( ( ord_less @ A @ X2 @ Z4 )
% 0.28/0.46 => ( ( P @ X2 )
% 0.28/0.46 = ( P2 @ X2 ) ) )
% 0.28/0.46 => ( ? [Z4: A] :
% 0.28/0.46 ! [X2: A] :
% 0.28/0.46 ( ( ord_less @ A @ X2 @ Z4 )
% 0.28/0.46 => ( ( Q @ X2 )
% 0.28/0.46 = ( Q2 @ X2 ) ) )
% 0.28/0.46 => ? [Z3: A] :
% 0.28/0.46 ! [X4: A] :
% 0.28/0.46 ( ( ord_less @ A @ X4 @ Z3 )
% 0.28/0.46 => ( ( ( P @ X4 )
% 0.28/0.46 | ( Q @ X4 ) )
% 0.28/0.46 = ( ( P2 @ X4 )
% 0.28/0.46 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % minf(2)
% 0.28/0.46 thf(fact_231_minf_I3_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [T3: A] :
% 0.28/0.46 ? [Z3: A] :
% 0.28/0.46 ! [X4: A] :
% 0.28/0.46 ( ( ord_less @ A @ X4 @ Z3 )
% 0.28/0.46 => ( X4 != T3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % minf(3)
% 0.28/0.46 thf(fact_232_minf_I4_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [T3: A] :
% 0.28/0.46 ? [Z3: A] :
% 0.28/0.46 ! [X4: A] :
% 0.28/0.46 ( ( ord_less @ A @ X4 @ Z3 )
% 0.28/0.46 => ( X4 != T3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % minf(4)
% 0.28/0.46 thf(fact_233_minf_I5_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [T3: A] :
% 0.28/0.46 ? [Z3: A] :
% 0.28/0.46 ! [X4: A] :
% 0.28/0.46 ( ( ord_less @ A @ X4 @ Z3 )
% 0.28/0.46 => ( ord_less @ A @ X4 @ T3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % minf(5)
% 0.28/0.46 thf(fact_234_minf_I7_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [T3: A] :
% 0.28/0.46 ? [Z3: A] :
% 0.28/0.46 ! [X4: A] :
% 0.28/0.46 ( ( ord_less @ A @ X4 @ Z3 )
% 0.28/0.46 => ~ ( ord_less @ A @ T3 @ X4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % minf(7)
% 0.28/0.46 thf(fact_235_minf_I11_J,axiom,
% 0.28/0.46 ! [C2: $tType,D3: $tType] :
% 0.28/0.46 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.28/0.46 => ! [F3: D3] :
% 0.28/0.46 ? [Z3: C2] :
% 0.28/0.46 ! [X4: C2] :
% 0.28/0.46 ( ( ord_less @ C2 @ X4 @ Z3 )
% 0.28/0.46 => ( F3 = F3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % minf(11)
% 0.28/0.46 thf(fact_236_minf_I8_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [T3: A] :
% 0.28/0.46 ? [Z3: A] :
% 0.28/0.46 ! [X4: A] :
% 0.28/0.46 ( ( ord_less @ A @ X4 @ Z3 )
% 0.28/0.46 => ~ ( ord_less_eq @ A @ T3 @ X4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % minf(8)
% 0.28/0.46 thf(fact_237_minf_I6_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [T3: A] :
% 0.28/0.46 ? [Z3: A] :
% 0.28/0.46 ! [X4: A] :
% 0.28/0.46 ( ( ord_less @ A @ X4 @ Z3 )
% 0.28/0.46 => ( ord_less_eq @ A @ X4 @ T3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % minf(6)
% 0.28/0.46 thf(fact_238_pinf_I8_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [T3: A] :
% 0.28/0.46 ? [Z3: A] :
% 0.28/0.46 ! [X4: A] :
% 0.28/0.46 ( ( ord_less @ A @ Z3 @ X4 )
% 0.28/0.46 => ( ord_less_eq @ A @ T3 @ X4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % pinf(8)
% 0.28/0.46 thf(fact_239_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A,C: A,D: A] :
% 0.28/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( set_or578182835ssThan @ A @ A4 @ B4 ) @ ( set_or1433837966ssThan @ A @ C @ D ) )
% 0.28/0.46 = ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.46 => ( ( ord_less_eq @ A @ C @ A4 )
% 0.28/0.46 & ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % greaterThanLessThan_subseteq_atLeastLessThan_iff
% 0.28/0.46 thf(fact_240_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A,C: A,D: A] :
% 0.28/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( set_or578182835ssThan @ A @ A4 @ B4 ) @ ( set_or331188842AtMost @ A @ C @ D ) )
% 0.28/0.46 = ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.46 => ( ( ord_less_eq @ A @ C @ A4 )
% 0.28/0.46 & ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % greaterThanLessThan_subseteq_atLeastAtMost_iff
% 0.28/0.46 thf(fact_241_greaterThanLessThan__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [I: A,L: A,U: A] :
% 0.28/0.46 ( ( member @ A @ I @ ( set_or578182835ssThan @ A @ L @ U ) )
% 0.28/0.46 = ( ( ord_less @ A @ L @ I )
% 0.28/0.46 & ( ord_less @ A @ I @ U ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % greaterThanLessThan_iff
% 0.28/0.46 thf(fact_242_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A,C: A,D: A] :
% 0.28/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( set_or578182835ssThan @ A @ A4 @ B4 ) @ ( set_or578182835ssThan @ A @ C @ D ) )
% 0.28/0.46 = ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.46 => ( ( ord_less_eq @ A @ C @ A4 )
% 0.28/0.46 & ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % greaterThanLessThan_subseteq_greaterThanLessThan
% 0.28/0.46 thf(fact_243_ivl__disj__un__two_I4_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [L: A,M: A,U: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ L @ M )
% 0.28/0.46 => ( ( ord_less @ A @ M @ U )
% 0.28/0.46 => ( ( sup_sup @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ M ) @ ( set_or578182835ssThan @ A @ M @ U ) )
% 0.28/0.46 = ( set_or1433837966ssThan @ A @ L @ U ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ivl_disj_un_two(4)
% 0.28/0.46 thf(fact_244_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A,C: A,D: A] :
% 0.28/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( set_or578182835ssThan @ A @ A4 @ B4 ) @ ( set_or1361889807AtMost @ A @ C @ D ) )
% 0.28/0.46 = ( ( ord_less @ A @ A4 @ B4 )
% 0.28/0.46 => ( ( ord_less_eq @ A @ C @ A4 )
% 0.28/0.46 & ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % greaterThanLessThan_subseteq_greaterThanAtMost_iff
% 0.28/0.46 thf(fact_245_sup__apply,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 0.28/0.46 => ( ( sup_sup @ ( A > B ) )
% 0.28/0.46 = ( ^ [F2: A > B,G: A > B,X: A] : ( sup_sup @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_apply
% 0.28/0.46 thf(fact_246_sup_Oidem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A] :
% 0.28/0.46 ( ( sup_sup @ A @ A4 @ A4 )
% 0.28/0.46 = A4 ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup.idem
% 0.28/0.46 thf(fact_247_sup__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A] :
% 0.28/0.46 ( ( sup_sup @ A @ X3 @ X3 )
% 0.28/0.46 = X3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_idem
% 0.28/0.46 thf(fact_248_sup_Oleft__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A] :
% 0.28/0.46 ( ( sup_sup @ A @ A4 @ ( sup_sup @ A @ A4 @ B4 ) )
% 0.28/0.46 = ( sup_sup @ A @ A4 @ B4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup.left_idem
% 0.28/0.46 thf(fact_249_sup__left__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A,Y2: A] :
% 0.28/0.46 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
% 0.28/0.46 = ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup_left_idem
% 0.28/0.46 thf(fact_250_sup_Oright__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [A4: A,B4: A] :
% 0.28/0.46 ( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B4 ) @ B4 )
% 0.28/0.46 = ( sup_sup @ A @ A4 @ B4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup.right_idem
% 0.28/0.46 thf(fact_251_UnCI,axiom,
% 0.28/0.46 ! [A: $tType,C: A,B2: set @ A,A2: set @ A] :
% 0.28/0.46 ( ( ~ ( member @ A @ C @ B2 )
% 0.28/0.46 => ( member @ A @ C @ A2 ) )
% 0.28/0.46 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % UnCI
% 0.28/0.46 thf(fact_252_Un__iff,axiom,
% 0.28/0.46 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 0.28/0.46 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.28/0.46 = ( ( member @ A @ C @ A2 )
% 0.28/0.46 | ( member @ A @ C @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Un_iff
% 0.28/0.46 thf(fact_253_sup_Obounded__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [B4: A,C: A,A4: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B4 @ C ) @ A4 )
% 0.28/0.46 = ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.28/0.46 & ( ord_less_eq @ A @ C @ A4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % sup.bounded_iff
% 0.28/0.46 thf(fact_254_le__sup__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.28/0.46 => ! [X3: A,Y2: A,Z: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z )
% 0.28/0.46 = ( ( ord_less_eq @ A @ X3 @ Z )
% 0.28/0.46 & ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_sup_iff
% 0.28/0.46 thf(fact_255_Un__subset__iff,axiom,
% 0.28/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.28/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 )
% 0.28/0.46 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 0.28/0.46 & ( ord_less_eq @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Un_subset_iff
% 0.28/0.46
% 0.28/0.46 %----Type constructors (19)
% 0.28/0.46 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( semilattice_sup @ A8 @ ( type2 @ A8 ) )
% 0.28/0.46 => ( semilattice_sup @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( boolean_algebra @ A8 @ ( type2 @ A8 ) )
% 0.28/0.46 => ( boolean_algebra @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 0.28/0.46 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( order @ A8 @ ( type2 @ A8 ) )
% 0.28/0.46 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 0.28/0.46 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Groups_Ouminus,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( uminus @ A8 @ ( type2 @ A8 ) )
% 0.28/0.46 => ( uminus @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
% 0.28/0.46 ! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_2,axiom,
% 0.28/0.46 ! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
% 0.28/0.46 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Orderings_Oorder_4,axiom,
% 0.28/0.46 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
% 0.28/0.46 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Groups_Ouminus_6,axiom,
% 0.28/0.46 ! [A7: $tType] : ( uminus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_7,axiom,
% 0.28/0.46 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_8,axiom,
% 0.28/0.46 boolean_algebra @ $o @ ( type2 @ $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_9,axiom,
% 0.28/0.46 preorder @ $o @ ( type2 @ $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.28/0.46 linorder @ $o @ ( type2 @ $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_10,axiom,
% 0.28/0.46 order @ $o @ ( type2 @ $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Orderings_Oord_11,axiom,
% 0.28/0.46 ord @ $o @ ( type2 @ $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Groups_Ouminus_12,axiom,
% 0.28/0.46 uminus @ $o @ ( type2 @ $o ) ).
% 0.28/0.46
% 0.28/0.46 %----Conjectures (2)
% 0.28/0.46 thf(conj_0,hypothesis,
% 0.28/0.46 ( ( gram_L608943123e_subs @ a @ l1 @ l2 )
% 0.28/0.46 & ( gram_L608943123e_subs @ a @ l2 @ l1 ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(conj_1,conjecture,
% 0.28/0.46 gram_L608943123e_subs @ a @ l1 @ l2 ).
% 0.28/0.46
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 ------------------------
% 2.71/1.54 ---- Embedded in THF ---
% 2.71/1.54 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.71/1.54 %%% Generated on Tue Feb 24 04:17:11 EST 2026
% 2.71/1.54 %%% using '$$dhol' embedding, version 1.3.0.
% 2.71/1.54 %%% Logic specification used:
% 2.71/1.54 %%% thf(spec, logic, $$dhol).
% 2.71/1.54
% 2.71/1.54 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.LeYYwvo4TN/DTF2DTF12112.p
% 2.71/1.54 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.71/1.54 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.71/1.54 thf(ty_tf_a, type, a: $tType).
% 2.71/1.54 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_cl_Groups_Ouminus, type, uminus: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_cl_Lattices_Oboolean__algebra, type, boolean_algebra: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.71/1.54 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Oleqv, type, gram_L1456083582e_leqv: (!> [A:$tType]: (((set @ (set @ A)) > ((set @ (set @ A)) > $o))))).
% 2.71/1.54 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Osubs, type, gram_L608943123e_subs: (!> [A:$tType]: (((set @ (set @ A)) > ((set @ (set @ A)) > $o))))).
% 2.71/1.54 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.71/1.54 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 2.71/1.54 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.71/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.71/1.54 thf(sy_c_Orderings_Oorder__class_Oantimono, type, order_antimono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.71/1.54 thf(sy_c_Orderings_Oorder__class_Ostrict__mono, type, order_strict_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.71/1.54 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.71/1.54 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.71/1.54 thf(sy_c_Set_OPow, type, pow: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))).
% 2.71/1.54 thf(sy_c_Set_Opairwise, type, pairwise: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > $o))))).
% 2.71/1.54 thf(sy_c_Set__Interval_Oord__class_OatLeast, type, set_ord_atLeast: (!> [A:$tType]: ((A > (set @ A))))).
% 2.71/1.54 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost, type, set_or331188842AtMost: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 2.71/1.54 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan, type, set_or1433837966ssThan: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 2.71/1.54 thf(sy_c_Set__Interval_Oord__class_OatMost, type, set_ord_atMost: (!> [A:$tType]: ((A > (set @ A))))).
% 2.71/1.54 thf(sy_c_Set__Interval_Oord__class_OgreaterThan, type, set_ord_greaterThan: (!> [A:$tType]: ((A > (set @ A))))).
% 2.71/1.54 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost, type, set_or1361889807AtMost: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 2.71/1.54 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan, type, set_or578182835ssThan: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 2.71/1.54 thf(sy_c_Set__Interval_Oord__class_OlessThan, type, set_ord_lessThan: (!> [A:$tType]: ((A > (set @ A))))).
% 2.71/1.54 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.71/1.54 thf(sy_v_L1, type, l1: (set @ (set @ a))).
% 2.71/1.54 thf(sy_v_L2, type, l2: (set @ (set @ a))).
% 2.71/1.54 thf(fact_0_subs__refl, axiom, (! [A:$tType,L1:(set @ (set @ A))]: ((((gram_L608943123e_subs @ A) @ L1) @ L1)))).
% 2.71/1.54 thf(fact_1_subs__trans, axiom, (! [A:$tType,L1:(set @ (set @ A)),L2:(set @ (set @ A)),L3:(set @ (set @ A))]: (((((gram_L608943123e_subs @ A) @ L1) @ L2) => ((((gram_L608943123e_subs @ A) @ L2) @ L3) => (((gram_L608943123e_subs @ A) @ L1) @ L3)))))).
% 2.71/1.54 thf(fact_2_leqv__def, axiom, (! [A:$tType]: (((gram_L1456083582e_leqv @ A) = (^ [L12:(set @ (set @ A)),L22:(set @ (set @ A))]: (((((gram_L608943123e_subs @ A) @ L12) @ L22) & (((gram_L608943123e_subs @ A) @ L22) @ L12)))))))).
% 2.71/1.54 thf(fact_3_incl__subs, axiom, (! [A:$tType,L2:(set @ (set @ A)),L1:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ L2) @ L1) => (((gram_L608943123e_subs @ A) @ L1) @ L2))))).
% 2.71/1.54 thf(fact_4_subs__def, axiom, (! [A:$tType]: (((gram_L608943123e_subs @ A) = (^ [L12:(set @ (set @ A)),L22:(set @ (set @ A))]: ((! [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ L22) => (? [Y:(set @ A)]: (((((member @ (set @ A)) @ Y) @ L12) & (((ord_less_eq @ (set @ A)) @ Y) @ X))))))))))))).
% 2.71/1.54 thf(fact_5_subsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ A2) => (((member @ A) @ X2) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.71/1.54 thf(fact_6_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => (A2 = B2)))))).
% 2.71/1.54 thf(fact_7_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A]: ((((ord_less_eq @ A) @ X3) @ X3))))))).
% 2.71/1.54 thf(fact_8_set__mp, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X3) @ A2) => (((member @ A) @ X3) @ B2)))))).
% 2.71/1.54 thf(fact_9_in__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X3) @ A2) => (((member @ A) @ X3) @ B2)))))).
% 2.71/1.54 thf(fact_10_subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C) @ A2) => (((member @ A) @ C) @ B2)))))).
% 2.71/1.54 thf(fact_11_subsetCE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C) @ A2) => (((member @ A) @ C) @ B2)))))).
% 2.71/1.54 thf(fact_12_equalityE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A2)))))))))).
% 2.71/1.54 thf(fact_13_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B3:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ A3) => (((member @ A) @ X) @ B3)))))))))).
% 2.71/1.54 thf(fact_14_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.71/1.54 thf(fact_15_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 2.71/1.54 thf(fact_16_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A4:A]: (((((ord_less_eq @ A) @ B4) @ A4) => ((((ord_less_eq @ A) @ A4) @ B4) => (A4 = B4))))))))).
% 2.71/1.54 thf(fact_17_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A4:A,C:A]: (((((ord_less_eq @ A) @ B4) @ A4) => ((((ord_less_eq @ A) @ C) @ B4) => (((ord_less_eq @ A) @ C) @ A4))))))))).
% 2.71/1.54 thf(fact_18_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A4:A,B4:A]: (((! [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((P @ A5) @ B5)))) => ((! [A5:A,B5:A]: ((((P @ B5) @ A5) => ((P @ A5) @ B5)))) => ((P @ A4) @ B4))))))))).
% 2.71/1.54 thf(fact_19_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A]: ((((ord_less_eq @ A) @ A4) @ A4))))))).
% 2.71/1.54 thf(fact_20_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A,Z:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z) => (((ord_less_eq @ A) @ X3) @ Z))))))))).
% 2.71/1.54 thf(fact_21_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ B4) @ A4) => (A4 = B4))))))))).
% 2.71/1.54 thf(fact_22_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((B4 = C) => (((ord_less_eq @ A) @ A4) @ C))))))))).
% 2.71/1.54 thf(fact_23_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A]: (((A4 = B4) => ((((ord_less_eq @ A) @ B4) @ C) => (((ord_less_eq @ A) @ A4) @ C))))))))).
% 2.71/1.54 thf(fact_24_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y2:A,X3:A]: (((((ord_less_eq @ A) @ Y2) @ X3) => ((((ord_less_eq @ A) @ X3) @ Y2) = (X3 = Y2))))))))).
% 2.71/1.54 thf(fact_25_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A,Z:A]: ((((((ord_less_eq @ A) @ X3) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ Z)))) => (((((ord_less_eq @ A) @ Y2) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Z)))) => (((((ord_less_eq @ A) @ X3) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y2)))) => (((((ord_less_eq @ A) @ Z) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ X3)))) => (((((ord_less_eq @ A) @ Y2) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X3)))) => (~ (((((ord_less_eq @ A) @ Z) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Y2))))))))))))))))).
% 2.71/1.54 thf(fact_26_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ B4) @ C) => (((ord_less_eq @ A) @ A4) @ C))))))))).
% 2.71/1.54 thf(fact_27_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X3)))))))).
% 2.71/1.54 thf(fact_28_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((X3 = Y2) => (((ord_less_eq @ A) @ X3) @ Y2)))))))).
% 2.71/1.54 thf(fact_29_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) | (((ord_less_eq @ A) @ Y2) @ X3)))))))).
% 2.71/1.54 thf(fact_30_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ X3) => (X3 = Y2))))))))).
% 2.71/1.54 thf(fact_31_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y3:A,Z2:A]: ((Y3 = Z2))) = (^ [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) & (((ord_less_eq @ A) @ Y) @ X))))))))).
% 2.71/1.54 thf(fact_32_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A4:A,B4:A,F:(A > B),C:B]: (((((ord_less_eq @ A) @ A4) @ B4) => (((F @ B4) = C) => ((! [X2:A,Y4:A]: (((((ord_less_eq @ A) @ X2) @ Y4) => (((ord_less_eq @ B) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less_eq @ B) @ (F @ A4)) @ C)))))))))).
% 2.71/1.54 thf(fact_33_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B4:B,C:B]: (((A4 = (F @ B4)) => ((((ord_less_eq @ B) @ B4) @ C) => ((! [X2:B,Y4:B]: (((((ord_less_eq @ B) @ X2) @ Y4) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A4) @ (F @ C))))))))))).
% 2.71/1.54 thf(fact_34_order__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,B4:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ C2) @ (F @ B4)) @ C) => ((! [X2:A,Y4:A]: (((((ord_less_eq @ A) @ X2) @ Y4) => (((ord_less_eq @ C2) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less_eq @ C2) @ (F @ A4)) @ C)))))))))).
% 2.71/1.54 thf(fact_35_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B4:B,C:B]: (((((ord_less_eq @ A) @ A4) @ (F @ B4)) => ((((ord_less_eq @ B) @ B4) @ C) => ((! [X2:B,Y4:B]: (((((ord_less_eq @ B) @ X2) @ Y4) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A4) @ (F @ C))))))))))).
% 2.71/1.54 thf(fact_36_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G:(A > B)]: ((! [X:A]: ((((ord_less_eq @ B) @ (F2 @ X)) @ (G @ X))))))))))).
% 2.71/1.54 thf(fact_37_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G2:(A > B)]: (((! [X2:A]: ((((ord_less_eq @ B) @ (F @ X2)) @ (G2 @ X2)))) => (((ord_less_eq @ (A > B)) @ F) @ G2)))))))).
% 2.71/1.54 thf(fact_38_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G2:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F) @ G2) => (((ord_less_eq @ B) @ (F @ X3)) @ (G2 @ X3))))))))).
% 2.71/1.54 thf(fact_39_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G2:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F) @ G2) => (((ord_less_eq @ B) @ (F @ X3)) @ (G2 @ X3))))))))).
% 2.71/1.54 thf(fact_40_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X:A]: (((P @ X) => (Q @ X)))))))).
% 2.71/1.54 thf(fact_41_contra__subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((~ ((((member @ A) @ C) @ B2))) => (~ ((((member @ A) @ C) @ A2)))))))).
% 2.71/1.54 thf(fact_42_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y3:(set @ A),Z2:(set @ A)]: ((Y3 = Z2))) = (^ [A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) & (((ord_less_eq @ (set @ A)) @ B3) @ A3)))))))).
% 2.71/1.54 thf(fact_43_subset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C3) => (((ord_less_eq @ (set @ A)) @ A2) @ C3)))))).
% 2.71/1.54 thf(fact_44_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) => (Q @ X2)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.71/1.54 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A4:A,P:(A > $o)]: (((((member @ A) @ A4) @ ((collect @ A) @ P)) = (P @ A4))))).
% 2.71/1.54 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A2)))) = A2)))).
% 2.71/1.54 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.71/1.54 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G2:(A > B)]: (((! [X2:A]: (((F @ X2) = (G2 @ X2)))) => (F = G2))))).
% 2.71/1.54 thf(fact_49_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 2.71/1.54 thf(fact_50_rev__subsetD, axiom, (! [A:$tType,C:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ A2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((member @ A) @ C) @ B2)))))).
% 2.71/1.54 thf(fact_51_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B3:(set @ A)]: ((! [T:A]: (((((member @ A) @ T) @ A3) => (((member @ A) @ T) @ B3)))))))))).
% 2.71/1.54 thf(fact_52_set__rev__mp, axiom, (! [A:$tType,X3:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ X3) @ A2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((member @ A) @ X3) @ B2)))))).
% 2.71/1.54 thf(fact_53_antimono__def, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (((order_antimono @ A) @ B) = (^ [F2:(A > B)]: ((! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ B) @ (F2 @ Y)) @ (F2 @ X)))))))))))).
% 2.71/1.54 thf(fact_54_antimonoI, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B)]: (((! [X2:A,Y4:A]: (((((ord_less_eq @ A) @ X2) @ Y4) => (((ord_less_eq @ B) @ (F @ Y4)) @ (F @ X2))))) => (((order_antimono @ A) @ B) @ F)))))))).
% 2.71/1.54 thf(fact_55_antimonoE, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X3:A,Y2:A]: (((((order_antimono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ B) @ (F @ Y2)) @ (F @ X3)))))))))).
% 2.71/1.54 thf(fact_56_antimonoD, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X3:A,Y2:A]: (((((order_antimono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ B) @ (F @ Y2)) @ (F @ X3)))))))))).
% 2.71/1.54 thf(fact_57_Pow__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ (set @ A))) @ ((pow @ A) @ A2)) @ ((pow @ A) @ B2)))))).
% 2.71/1.54 thf(fact_58_strict__mono__less__eq, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X3:A,Y2:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y2)) = (((ord_less_eq @ A) @ X3) @ Y2))))))))).
% 2.71/1.54 thf(fact_59_pairwise__subset, axiom, (! [A:$tType,P:(A > (A > $o)),S:(set @ A),T2:(set @ A)]: (((((pairwise @ A) @ P) @ S) => ((((ord_less_eq @ (set @ A)) @ T2) @ S) => (((pairwise @ A) @ P) @ T2)))))).
% 2.71/1.54 thf(fact_60_greaterThan__subset__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_greaterThan @ A) @ X3)) @ ((set_ord_greaterThan @ A) @ Y2)) = (((ord_less_eq @ A) @ Y2) @ X3)))))))).
% 2.71/1.54 thf(fact_61_lessThan__subset__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_lessThan @ A) @ X3)) @ ((set_ord_lessThan @ A) @ Y2)) = (((ord_less_eq @ A) @ X3) @ Y2)))))))).
% 2.71/1.54 thf(fact_62_atLeast__subset__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ X3)) @ ((set_ord_atLeast @ A) @ Y2)) = (((ord_less_eq @ A) @ Y2) @ X3)))))))).
% 2.71/1.54 thf(fact_63_lessThan__eq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((set_ord_lessThan @ A) @ X3) = ((set_ord_lessThan @ A) @ Y2)) = (X3 = Y2)))))))).
% 2.71/1.54 thf(fact_64_atLeast__eq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((set_ord_atLeast @ A) @ X3) = ((set_ord_atLeast @ A) @ Y2)) = (X3 = Y2)))))))).
% 2.71/1.54 thf(fact_65_greaterThan__eq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((set_ord_greaterThan @ A) @ X3) = ((set_ord_greaterThan @ A) @ Y2)) = (X3 = Y2)))))))).
% 2.71/1.54 thf(fact_66_atLeast__iff, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_atLeast @ A) @ K)) = (((ord_less_eq @ A) @ K) @ I)))))))).
% 2.71/1.54 thf(fact_67_PowI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((member @ (set @ A)) @ A2) @ ((pow @ A) @ B2)))))).
% 2.71/1.54 thf(fact_68_Pow__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((member @ (set @ A)) @ A2) @ ((pow @ A) @ B2)) = (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.71/1.54 thf(fact_69_Ioi__le__Ico, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A4:A]: ((((ord_less_eq @ (set @ A)) @ ((set_ord_greaterThan @ A) @ A4)) @ ((set_ord_atLeast @ A) @ A4)))))))).
% 2.71/1.54 thf(fact_70_pairwise__def, axiom, (! [A:$tType]: (((pairwise @ A) = (^ [R:(A > (A > $o)),S2:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ S2) => (! [Y:A]: (((((member @ A) @ Y) @ S2) => ((X != Y) => ((R @ X) @ Y)))))))))))))).
% 2.71/1.54 thf(fact_71_Pow__top, axiom, (! [A:$tType,A2:(set @ A)]: ((((member @ (set @ A)) @ A2) @ ((pow @ A) @ A2))))).
% 2.71/1.54 thf(fact_72_strict__mono__eq, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X3:A,Y2:A]: (((((order_strict_mono @ A) @ B) @ F) => (((F @ X3) = (F @ Y2)) = (X3 = Y2))))))))).
% 2.71/1.54 thf(fact_73_PowD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((member @ (set @ A)) @ A2) @ ((pow @ A) @ B2)) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.71/1.54 thf(fact_74_Icc__subset__Ici__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [L:A,H:A,L4:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or331188842AtMost @ A) @ L) @ H)) @ ((set_ord_atLeast @ A) @ L4)) = ((~ ((((ord_less_eq @ A) @ L) @ H))) | (((ord_less_eq @ A) @ L4) @ L))))))))).
% 2.71/1.54 thf(fact_75_Ici__subset__Ioi__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ A4)) @ ((set_ord_greaterThan @ A) @ B4)) = (((ord_less @ A) @ B4) @ A4)))))))).
% 2.71/1.54 thf(fact_76_Compl__atLeast, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [K:A]: ((((uminus_uminus @ (set @ A)) @ ((set_ord_atLeast @ A) @ K)) = ((set_ord_lessThan @ A) @ K)))))))).
% 2.71/1.54 thf(fact_77_Compl__lessThan, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [K:A]: ((((uminus_uminus @ (set @ A)) @ ((set_ord_lessThan @ A) @ K)) = ((set_ord_atLeast @ A) @ K)))))))).
% 2.71/1.54 thf(fact_78_atMost__subset__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ X3)) @ ((set_ord_atMost @ A) @ Y2)) = (((ord_less_eq @ A) @ X3) @ Y2)))))))).
% 2.71/1.54 thf(fact_79_ivl__subset, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [I:A,J:A,M:A,N:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or1433837966ssThan @ A) @ I) @ J)) @ (((set_or1433837966ssThan @ A) @ M) @ N)) = ((((ord_less_eq @ A) @ J) @ I) | ((((ord_less_eq @ A) @ M) @ I) & (((ord_less_eq @ A) @ J) @ N)))))))))).
% 2.71/1.54 thf(fact_80_not__Iic__le__Ici, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [H:A,L4:A]: ((~ ((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ H)) @ ((set_ord_atLeast @ A) @ L4)))))))))).
% 2.71/1.54 thf(fact_81_psubsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((A2 != B2) => (((ord_less @ (set @ A)) @ A2) @ B2)))))).
% 2.71/1.54 thf(fact_82_ComplI, axiom, (! [A:$tType,C:A,A2:(set @ A)]: (((~ ((((member @ A) @ C) @ A2))) => (((member @ A) @ C) @ ((uminus_uminus @ (set @ A)) @ A2)))))).
% 2.71/1.54 thf(fact_83_Compl__iff, axiom, (! [A:$tType,C:A,A2:(set @ A)]: (((((member @ A) @ C) @ ((uminus_uminus @ (set @ A)) @ A2)) = (~ ((((member @ A) @ C) @ A2))))))).
% 2.71/1.54 thf(fact_84_Compl__eq__Compl__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((uminus_uminus @ (set @ A)) @ A2) = ((uminus_uminus @ (set @ A)) @ B2)) = (A2 = B2))))).
% 2.71/1.54 thf(fact_85_atMost__eq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((set_ord_atMost @ A) @ X3) = ((set_ord_atMost @ A) @ Y2)) = (X3 = Y2)))))))).
% 2.71/1.54 thf(fact_86_atLeastAtMost__iff, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [I:A,L:A,U:A]: (((((member @ A) @ I) @ (((set_or331188842AtMost @ A) @ L) @ U)) = ((((ord_less_eq @ A) @ L) @ I) & (((ord_less_eq @ A) @ I) @ U))))))))).
% 2.71/1.54 thf(fact_87_Icc__eq__Icc, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [L:A,H:A,L4:A,H2:A]: ((((((set_or331188842AtMost @ A) @ L) @ H) = (((set_or331188842AtMost @ A) @ L4) @ H2)) = (((L = L4) & (H = H2)) | ((~ ((((ord_less_eq @ A) @ L) @ H))) & (~ ((((ord_less_eq @ A) @ L4) @ H2)))))))))))).
% 2.71/1.54 thf(fact_88_lessThan__iff, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_lessThan @ A) @ K)) = (((ord_less @ A) @ I) @ K)))))))).
% 2.71/1.54 thf(fact_89_atMost__iff, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_atMost @ A) @ K)) = (((ord_less_eq @ A) @ I) @ K)))))))).
% 2.71/1.54 thf(fact_90_Compl__subset__Compl__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A2)) @ ((uminus_uminus @ (set @ A)) @ B2)) = (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 2.71/1.54 thf(fact_91_Compl__anti__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ B2)) @ ((uminus_uminus @ (set @ A)) @ A2)))))).
% 2.71/1.54 thf(fact_92_greaterThan__iff, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_greaterThan @ A) @ K)) = (((ord_less @ A) @ K) @ I)))))))).
% 2.71/1.54 thf(fact_93_atLeastLessThan__iff, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [I:A,L:A,U:A]: (((((member @ A) @ I) @ (((set_or1433837966ssThan @ A) @ L) @ U)) = ((((ord_less_eq @ A) @ L) @ I) & (((ord_less @ A) @ I) @ U))))))))).
% 2.71/1.54 thf(fact_94_atLeastatMost__subset__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A,D:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or331188842AtMost @ A) @ A4) @ B4)) @ (((set_or331188842AtMost @ A) @ C) @ D)) = ((~ ((((ord_less_eq @ A) @ A4) @ B4))) | ((((ord_less_eq @ A) @ C) @ A4) & (((ord_less_eq @ A) @ B4) @ D)))))))))).
% 2.71/1.54 thf(fact_95_Compl__atMost, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [K:A]: ((((uminus_uminus @ (set @ A)) @ ((set_ord_atMost @ A) @ K)) = ((set_ord_greaterThan @ A) @ K)))))))).
% 2.71/1.54 thf(fact_96_Compl__greaterThan, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [K:A]: ((((uminus_uminus @ (set @ A)) @ ((set_ord_greaterThan @ A) @ K)) = ((set_ord_atMost @ A) @ K)))))))).
% 2.71/1.54 thf(fact_97_Icc__subset__Iic__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [L:A,H:A,H2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or331188842AtMost @ A) @ L) @ H)) @ ((set_ord_atMost @ A) @ H2)) = ((~ ((((ord_less_eq @ A) @ L) @ H))) | (((ord_less_eq @ A) @ H) @ H2))))))))).
% 2.71/1.54 thf(fact_98_not__Iic__le__Icc, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [H:A,L4:A,H2:A]: ((~ ((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ H)) @ (((set_or331188842AtMost @ A) @ L4) @ H2)))))))))).
% 2.71/1.54 thf(fact_99_atLeastatMost__psubset__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A,D:A]: (((((ord_less @ (set @ A)) @ (((set_or331188842AtMost @ A) @ A4) @ B4)) @ (((set_or331188842AtMost @ A) @ C) @ D)) = (((~ ((((ord_less_eq @ A) @ A4) @ B4))) | ((((ord_less_eq @ A) @ C) @ A4) & ((((ord_less_eq @ A) @ B4) @ D) & ((((ord_less @ A) @ C) @ A4) | (((ord_less @ A) @ B4) @ D))))) & (((ord_less_eq @ A) @ C) @ D))))))))).
% 2.71/1.54 thf(fact_100_atLeastAtMost__subseteq__atLeastLessThan__iff, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A,D:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or331188842AtMost @ A) @ A4) @ B4)) @ (((set_or1433837966ssThan @ A) @ C) @ D)) = ((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ C) @ A4) & (((ord_less @ A) @ B4) @ D)))))))))).
% 2.71/1.54 thf(fact_101_atLeastLessThan__subseteq__atLeastAtMost__iff, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A,D:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or1433837966ssThan @ A) @ A4) @ B4)) @ (((set_or331188842AtMost @ A) @ C) @ D)) = ((((ord_less @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ C) @ A4) & (((ord_less_eq @ A) @ B4) @ D)))))))))).
% 2.71/1.54 thf(fact_102_atLeastLessThan__inj_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A,D:A]: ((((((set_or1433837966ssThan @ A) @ A4) @ B4) = (((set_or1433837966ssThan @ A) @ C) @ D)) => ((((ord_less @ A) @ A4) @ B4) => ((((ord_less @ A) @ C) @ D) => (B4 = D)))))))))).
% 2.71/1.54 thf(fact_103_atLeastLessThan__inj_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A,D:A]: ((((((set_or1433837966ssThan @ A) @ A4) @ B4) = (((set_or1433837966ssThan @ A) @ C) @ D)) => ((((ord_less @ A) @ A4) @ B4) => ((((ord_less @ A) @ C) @ D) => (A4 = C)))))))))).
% 2.71/1.54 thf(fact_104_atLeastLessThan__eq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A,D:A]: (((((ord_less @ A) @ A4) @ B4) => ((((ord_less @ A) @ C) @ D) => (((((set_or1433837966ssThan @ A) @ A4) @ B4) = (((set_or1433837966ssThan @ A) @ C) @ D)) = ((A4 = C) & (B4 = D))))))))))).
% 2.71/1.54 thf(fact_105_lessThan__strict__subset__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [M:A,N:A]: (((((ord_less @ (set @ A)) @ ((set_ord_lessThan @ A) @ M)) @ ((set_ord_lessThan @ A) @ N)) = (((ord_less @ A) @ M) @ N)))))))).
% 2.71/1.54 thf(fact_106_not__Iic__eq__Icc, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [H2:A,L:A,H:A]: ((((set_ord_atMost @ A) @ H2) != (((set_or331188842AtMost @ A) @ L) @ H)))))))).
% 2.71/1.54 thf(fact_107_ComplD, axiom, (! [A:$tType,C:A,A2:(set @ A)]: (((((member @ A) @ C) @ ((uminus_uminus @ (set @ A)) @ A2)) => (~ ((((member @ A) @ C) @ A2))))))).
% 2.71/1.54 thf(fact_108_double__complement, axiom, (! [A:$tType,A2:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A2)) = A2)))).
% 2.71/1.54 thf(fact_109_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B4:B,C:B]: (((A4 = (F @ B4)) => ((((ord_less @ B) @ B4) @ C) => ((! [X2:B,Y4:B]: (((((ord_less @ B) @ X2) @ Y4) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less @ A) @ A4) @ (F @ C))))))))))).
% 2.71/1.54 thf(fact_110_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A4:A,B4:A,F:(A > B),C:B]: (((((ord_less @ A) @ A4) @ B4) => (((F @ B4) = C) => ((! [X2:A,Y4:A]: (((((ord_less @ A) @ X2) @ Y4) => (((ord_less @ B) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less @ B) @ (F @ A4)) @ C)))))))))).
% 2.71/1.54 thf(fact_111_order__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B4:B,C:B]: (((((ord_less @ A) @ A4) @ (F @ B4)) => ((((ord_less @ B) @ B4) @ C) => ((! [X2:B,Y4:B]: (((((ord_less @ B) @ X2) @ Y4) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less @ A) @ A4) @ (F @ C))))))))))).
% 2.71/1.54 thf(fact_112_order__less__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,B4:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A4) @ B4) => ((((ord_less @ C2) @ (F @ B4)) @ C) => ((! [X2:A,Y4:A]: (((((ord_less @ A) @ X2) @ Y4) => (((ord_less @ C2) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less @ C2) @ (F @ A4)) @ C)))))))))).
% 2.71/1.54 thf(fact_113_lt__ex, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [X3:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X3))))))))).
% 2.71/1.54 thf(fact_114_gt__ex, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [X3:A]: ((? [X1:A]: ((((ord_less @ A) @ X3) @ X1))))))))).
% 2.71/1.54 thf(fact_115_neqE, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((X3 != Y2) => ((~ ((((ord_less @ A) @ X3) @ Y2))) => (((ord_less @ A) @ Y2) @ X3))))))))).
% 2.71/1.54 thf(fact_116_neq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((X3 != Y2) = ((((ord_less @ A) @ X3) @ Y2) | (((ord_less @ A) @ Y2) @ X3))))))))).
% 2.71/1.54 thf(fact_117_order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) => (~ ((((ord_less @ A) @ B4) @ A4)))))))))).
% 2.71/1.54 thf(fact_118_dense, axiom, (! [A:$tType]: ((((dense_order @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (? [Z3:A]: (((((ord_less @ A) @ X3) @ Z3) & (((ord_less @ A) @ Z3) @ Y2))))))))))).
% 2.71/1.54 thf(fact_119_less__imp__neq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (X3 != Y2)))))))).
% 2.71/1.54 thf(fact_120_less__asym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X3)))))))))).
% 2.71/1.54 thf(fact_121_less__asym_H, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) => (~ ((((ord_less @ A) @ B4) @ A4)))))))))).
% 2.71/1.54 thf(fact_122_less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A,Z:A]: (((((ord_less @ A) @ X3) @ Y2) => ((((ord_less @ A) @ Y2) @ Z) => (((ord_less @ A) @ X3) @ Z))))))))).
% 2.71/1.54 thf(fact_123_less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) | ((X3 = Y2) | (((ord_less @ A) @ Y2) @ X3))))))))).
% 2.71/1.54 thf(fact_124_less__irrefl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A]: ((~ ((((ord_less @ A) @ X3) @ X3))))))))).
% 2.71/1.54 thf(fact_125_ord__eq__less__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A]: (((A4 = B4) => ((((ord_less @ A) @ B4) @ C) => (((ord_less @ A) @ A4) @ C))))))))).
% 2.71/1.54 thf(fact_126_ord__less__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A]: (((((ord_less @ A) @ A4) @ B4) => ((B4 = C) => (((ord_less @ A) @ A4) @ C))))))))).
% 2.71/1.54 thf(fact_127_dual__order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A4:A]: (((((ord_less @ A) @ B4) @ A4) => (~ ((((ord_less @ A) @ A4) @ B4)))))))))).
% 2.71/1.54 thf(fact_128_less__imp__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (X3 != Y2)))))))).
% 2.71/1.54 thf(fact_129_less__not__sym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X3)))))))))).
% 2.71/1.54 thf(fact_130_less__induct, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A4:A]: (((! [X2:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X2) => (P @ Y5)))) => (P @ X2)))) => (P @ A4)))))))).
% 2.71/1.54 thf(fact_131_antisym__conv3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y2:A,X3:A]: (((~ ((((ord_less @ A) @ Y2) @ X3))) => ((~ ((((ord_less @ A) @ X3) @ Y2))) = (X3 = Y2))))))))).
% 2.71/1.54 thf(fact_132_less__imp__not__eq2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (Y2 != X3)))))))).
% 2.71/1.54 thf(fact_133_less__imp__triv, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A,P:$o]: (((((ord_less @ A) @ X3) @ Y2) => ((((ord_less @ A) @ Y2) @ X3) => P)))))))).
% 2.71/1.54 thf(fact_134_linorder__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((~ ((((ord_less @ A) @ X3) @ Y2))) => ((X3 != Y2) => (((ord_less @ A) @ Y2) @ X3))))))))).
% 2.71/1.54 thf(fact_135_dual__order_Oirrefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A]: ((~ ((((ord_less @ A) @ A4) @ A4))))))))).
% 2.71/1.54 thf(fact_136_order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A]: (((((ord_less @ A) @ A4) @ B4) => ((((ord_less @ A) @ B4) @ C) => (((ord_less @ A) @ A4) @ C))))))))).
% 2.71/1.54 thf(fact_137_less__imp__not__less, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X3)))))))))).
% 2.71/1.54 thf(fact_138_dual__order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A4:A,C:A]: (((((ord_less @ A) @ B4) @ A4) => ((((ord_less @ A) @ C) @ B4) => (((ord_less @ A) @ C) @ A4))))))))).
% 2.71/1.54 thf(fact_139_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((~ ((((ord_less @ A) @ X3) @ Y2))) = ((((ord_less @ A) @ Y2) @ X3) | (X3 = Y2))))))))).
% 2.71/1.54 thf(fact_140_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) => (A4 != B4)))))))).
% 2.71/1.54 thf(fact_141_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A4:A]: (((((ord_less @ A) @ B4) @ A4) => (A4 != B4)))))))).
% 2.71/1.54 thf(fact_142_Iic__subset__Iio__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ A4)) @ ((set_ord_lessThan @ A) @ B4)) = (((ord_less @ A) @ A4) @ B4)))))))).
% 2.71/1.54 thf(fact_143_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((A4 != B4) => ((((ord_less_eq @ A) @ A4) @ B4) => (((ord_less @ A) @ A4) @ B4))))))))).
% 2.71/1.54 thf(fact_144_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A4:A]: (((((ord_less @ A) @ B4) @ A4) => (((ord_less_eq @ A) @ B4) @ A4)))))))).
% 2.71/1.54 thf(fact_145_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B6:A,A6:A]: (((((ord_less_eq @ A) @ B6) @ A6) & (A6 != B6))))))))).
% 2.71/1.54 thf(fact_146_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B6:A,A6:A]: (((((ord_less @ A) @ B6) @ A6) | (A6 = B6))))))))).
% 2.71/1.54 thf(fact_147_order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) => (((ord_less_eq @ A) @ A4) @ B4)))))))).
% 2.71/1.54 thf(fact_148_dense__le__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A,Z:A]: (((((ord_less @ A) @ X3) @ Y2) => ((! [W:A]: (((((ord_less @ A) @ X3) @ W) => ((((ord_less @ A) @ W) @ Y2) => (((ord_less_eq @ A) @ W) @ Z))))) => (((ord_less_eq @ A) @ Y2) @ Z))))))))).
% 2.71/1.54 thf(fact_149_dense__ge__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z:A,X3:A,Y2:A]: (((((ord_less @ A) @ Z) @ X3) => ((! [W:A]: (((((ord_less @ A) @ Z) @ W) => ((((ord_less @ A) @ W) @ X3) => (((ord_less_eq @ A) @ Y2) @ W))))) => (((ord_less_eq @ A) @ Y2) @ Z))))))))).
% 2.71/1.54 thf(fact_150_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A4:A,C:A]: (((((ord_less @ A) @ B4) @ A4) => ((((ord_less_eq @ A) @ C) @ B4) => (((ord_less @ A) @ C) @ A4))))))))).
% 2.71/1.54 thf(fact_151_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A4:A,C:A]: (((((ord_less_eq @ A) @ B4) @ A4) => ((((ord_less @ A) @ C) @ B4) => (((ord_less @ A) @ C) @ A4))))))))).
% 2.71/1.54 thf(fact_152_order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [A6:A,B6:A]: (((((ord_less_eq @ A) @ A6) @ B6) & (A6 != B6))))))))).
% 2.71/1.54 thf(fact_153_order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A6:A,B6:A]: (((((ord_less @ A) @ A6) @ B6) | (A6 = B6))))))))).
% 2.71/1.54 thf(fact_154_order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A]: (((((ord_less @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ B4) @ C) => (((ord_less @ A) @ A4) @ C))))))))).
% 2.71/1.54 thf(fact_155_order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less @ A) @ B4) @ C) => (((ord_less @ A) @ A4) @ C))))))))).
% 2.71/1.54 thf(fact_156_not__le__imp__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y2:A,X3:A]: (((~ ((((ord_less_eq @ A) @ Y2) @ X3))) => (((ord_less @ A) @ X3) @ Y2)))))))).
% 2.71/1.54 thf(fact_157_less__le__not__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) & (~ ((((ord_less_eq @ A) @ Y) @ X))))))))))).
% 2.71/1.54 thf(fact_158_le__imp__less__or__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => ((((ord_less @ A) @ X3) @ Y2) | (X3 = Y2))))))))).
% 2.71/1.54 thf(fact_159_le__less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) | (((ord_less @ A) @ Y2) @ X3)))))))).
% 2.71/1.54 thf(fact_160_dense__le, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Y2:A,Z:A]: (((! [X2:A]: (((((ord_less @ A) @ X2) @ Y2) => (((ord_less_eq @ A) @ X2) @ Z)))) => (((ord_less_eq @ A) @ Y2) @ Z)))))))).
% 2.71/1.54 thf(fact_161_dense__ge, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z:A,Y2:A]: (((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (((ord_less_eq @ A) @ Y2) @ X2)))) => (((ord_less_eq @ A) @ Y2) @ Z)))))))).
% 2.71/1.54 thf(fact_162_less__le__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A,Z:A]: (((((ord_less @ A) @ X3) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z) => (((ord_less @ A) @ X3) @ Z))))))))).
% 2.71/1.54 thf(fact_163_le__less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A,Z:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => ((((ord_less @ A) @ Y2) @ Z) => (((ord_less @ A) @ X3) @ Z))))))))).
% 2.71/1.54 thf(fact_164_antisym__conv2, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => ((~ ((((ord_less @ A) @ X3) @ Y2))) = (X3 = Y2))))))))).
% 2.71/1.54 thf(fact_165_antisym__conv1, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((~ ((((ord_less @ A) @ X3) @ Y2))) => ((((ord_less_eq @ A) @ X3) @ Y2) = (X3 = Y2))))))))).
% 2.71/1.54 thf(fact_166_less__imp__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (((ord_less_eq @ A) @ X3) @ Y2)))))))).
% 2.71/1.54 thf(fact_167_le__neq__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((A4 != B4) => (((ord_less @ A) @ A4) @ B4))))))))).
% 2.71/1.54 thf(fact_168_not__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((~ ((((ord_less @ A) @ X3) @ Y2))) = (((ord_less_eq @ A) @ Y2) @ X3)))))))).
% 2.71/1.54 thf(fact_169_not__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y2))) = (((ord_less @ A) @ Y2) @ X3)))))))).
% 2.71/1.54 thf(fact_170_order__less__le__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,B4:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A4) @ B4) => ((((ord_less_eq @ C2) @ (F @ B4)) @ C) => ((! [X2:A,Y4:A]: (((((ord_less @ A) @ X2) @ Y4) => (((ord_less @ C2) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less @ C2) @ (F @ A4)) @ C)))))))))).
% 2.71/1.54 thf(fact_171_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B4:B,C:B]: (((((ord_less @ A) @ A4) @ (F @ B4)) => ((((ord_less_eq @ B) @ B4) @ C) => ((! [X2:B,Y4:B]: (((((ord_less_eq @ B) @ X2) @ Y4) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less @ A) @ A4) @ (F @ C))))))))))).
% 2.71/1.54 thf(fact_172_order__le__less__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,B4:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less @ C2) @ (F @ B4)) @ C) => ((! [X2:A,Y4:A]: (((((ord_less_eq @ A) @ X2) @ Y4) => (((ord_less_eq @ C2) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less @ C2) @ (F @ A4)) @ C)))))))))).
% 2.71/1.54 thf(fact_173_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B4:B,C:B]: (((((ord_less_eq @ A) @ A4) @ (F @ B4)) => ((((ord_less @ B) @ B4) @ C) => ((! [X2:B,Y4:B]: (((((ord_less @ B) @ X2) @ Y4) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less @ A) @ A4) @ (F @ C))))))))))).
% 2.71/1.54 thf(fact_174_less__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) & (X != Y))))))))).
% 2.71/1.54 thf(fact_175_le__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) | (X = Y))))))))).
% 2.71/1.54 thf(fact_176_leI, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((~ ((((ord_less @ A) @ X3) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X3)))))))).
% 2.71/1.54 thf(fact_177_leD, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y2:A,X3:A]: (((((ord_less_eq @ A) @ Y2) @ X3) => (~ ((((ord_less @ A) @ X3) @ Y2)))))))))).
% 2.71/1.54 thf(fact_178_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B3) | (A3 = B3)))))))).
% 2.71/1.54 thf(fact_179_subset__psubset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A2) @ C3)))))).
% 2.71/1.54 thf(fact_180_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) & (~ ((((ord_less_eq @ (set @ A)) @ B3) @ A3)))))))))).
% 2.71/1.54 thf(fact_181_psubset__subset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A2) @ C3)))))).
% 2.71/1.54 thf(fact_182_psubset__imp__subset, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.71/1.54 thf(fact_183_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) & (A3 != B3)))))))).
% 2.71/1.54 thf(fact_184_psubsetE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less @ (set @ A)) @ A2) @ B2) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A2)))))))).
% 2.71/1.54 thf(fact_185_not__Ici__eq__Icc, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [L4:A,L:A,H:A]: ((((set_ord_atLeast @ A) @ L4) != (((set_or331188842AtMost @ A) @ L) @ H)))))))).
% 2.71/1.54 thf(fact_186_less__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less @ (A > B)) = (^ [F2:(A > B),G:(A > B)]: (((((ord_less_eq @ (A > B)) @ F2) @ G) & (~ ((((ord_less_eq @ (A > B)) @ G) @ F2))))))))))).
% 2.71/1.54 thf(fact_187_not__Iic__eq__Ici, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [H:A,L4:A]: ((((set_ord_atMost @ A) @ H) != ((set_ord_atLeast @ A) @ L4)))))))).
% 2.71/1.54 thf(fact_188_strict__monoD, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X3:A,Y2:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less @ A) @ X3) @ Y2) => (((ord_less @ B) @ (F @ X3)) @ (F @ Y2)))))))))).
% 2.71/1.54 thf(fact_189_strict__monoI, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B)]: (((! [X2:A,Y4:A]: (((((ord_less @ A) @ X2) @ Y4) => (((ord_less @ B) @ (F @ X2)) @ (F @ Y4))))) => (((order_strict_mono @ A) @ B) @ F)))))))).
% 2.71/1.54 thf(fact_190_strict__mono__def, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (((order_strict_mono @ A) @ B) = (^ [F2:(A > B)]: ((! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (((ord_less @ B) @ (F2 @ X)) @ (F2 @ Y)))))))))))).
% 2.71/1.54 thf(fact_191_strict__mono__less, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X3:A,Y2:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less @ B) @ (F @ X3)) @ (F @ Y2)) = (((ord_less @ A) @ X3) @ Y2))))))))).
% 2.71/1.54 thf(fact_192_not__Ici__le__Icc, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [L:A,L4:A,H2:A]: ((~ ((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ L)) @ (((set_or331188842AtMost @ A) @ L4) @ H2)))))))))).
% 2.71/1.54 thf(fact_193_atLeastLessThan__subset__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A,D:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or1433837966ssThan @ A) @ A4) @ B4)) @ (((set_or1433837966ssThan @ A) @ C) @ D)) => ((((ord_less_eq @ A) @ B4) @ A4) | ((((ord_less_eq @ A) @ C) @ A4) & (((ord_less_eq @ A) @ B4) @ D)))))))))).
% 2.71/1.54 thf(fact_194_not__Ici__le__Iic, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [L:A,H2:A]: ((~ ((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ L)) @ ((set_ord_atMost @ A) @ H2)))))))))).
% 2.71/1.54 thf(fact_195_neg__less__iff__less, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [B4:A,A4:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ B4)) @ ((uminus_uminus @ A) @ A4)) = (((ord_less @ A) @ A4) @ B4)))))))).
% 2.71/1.54 thf(fact_196_neg__le__iff__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [B4:A,A4:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B4)) @ ((uminus_uminus @ A) @ A4)) = (((ord_less_eq @ A) @ A4) @ B4)))))))).
% 2.71/1.54 thf(fact_197_compl__le__compl__iff, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X3)) @ ((uminus_uminus @ A) @ Y2)) = (((ord_less_eq @ A) @ Y2) @ X3)))))))).
% 2.71/1.54 thf(fact_198_compl__eq__compl__iff, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((uminus_uminus @ A) @ X3) = ((uminus_uminus @ A) @ Y2)) = (X3 = Y2)))))))).
% 2.71/1.54 thf(fact_199_double__compl, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X3:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ X3)) = X3))))))).
% 2.71/1.54 thf(fact_200_neg__equal__iff__equal, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((uminus_uminus @ A) @ A4) = ((uminus_uminus @ A) @ B4)) = (A4 = B4)))))))).
% 2.71/1.54 thf(fact_201_uminus__apply, axiom, (! [B:$tType,A:$tType]: ((((uminus @ B) @ (type2 @ B)) => ((uminus_uminus @ (A > B)) = (^ [A3:(A > B),X:A]: (((uminus_uminus @ B) @ (A3 @ X))))))))).
% 2.71/1.54 thf(fact_202_add_Oinverse__inverse, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A4:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ A4)) = A4))))))).
% 2.71/1.54 thf(fact_203_psubsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C:A]: (((((ord_less @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C) @ A2) => (((member @ A) @ C) @ B2)))))).
% 2.71/1.54 thf(fact_204_psubset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A2) @ B2) => ((((ord_less @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A2) @ C3)))))).
% 2.71/1.54 thf(fact_205_fun__Compl__def, axiom, (! [B:$tType,A:$tType]: ((((uminus @ B) @ (type2 @ B)) => ((uminus_uminus @ (A > B)) = (^ [A3:(A > B),X:A]: (((uminus_uminus @ B) @ (A3 @ X))))))))).
% 2.71/1.54 thf(fact_206_equation__minus__iff, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((A4 = ((uminus_uminus @ A) @ B4)) = (B4 = ((uminus_uminus @ A) @ A4))))))))).
% 2.71/1.54 thf(fact_207_minus__equation__iff, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((uminus_uminus @ A) @ A4) = B4) = (((uminus_uminus @ A) @ B4) = A4)))))))).
% 2.71/1.54 thf(fact_208_le__imp__neg__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B4)) @ ((uminus_uminus @ A) @ A4))))))))).
% 2.71/1.54 thf(fact_209_minus__le__iff, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A4)) @ B4) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B4)) @ A4)))))))).
% 2.71/1.54 thf(fact_210_le__minus__iff, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ ((uminus_uminus @ A) @ B4)) = (((ord_less_eq @ A) @ B4) @ ((uminus_uminus @ A) @ A4))))))))).
% 2.71/1.54 thf(fact_211_compl__le__swap2, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [Y2:A,X3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y2)) @ X3) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X3)) @ Y2)))))))).
% 2.71/1.54 thf(fact_212_compl__le__swap1, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [Y2:A,X3:A]: (((((ord_less_eq @ A) @ Y2) @ ((uminus_uminus @ A) @ X3)) => (((ord_less_eq @ A) @ X3) @ ((uminus_uminus @ A) @ Y2))))))))).
% 2.71/1.54 thf(fact_213_compl__mono, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y2)) @ ((uminus_uminus @ A) @ X3))))))))).
% 2.71/1.54 thf(fact_214_compl__less__swap1, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [Y2:A,X3:A]: (((((ord_less @ A) @ Y2) @ ((uminus_uminus @ A) @ X3)) => (((ord_less @ A) @ X3) @ ((uminus_uminus @ A) @ Y2))))))))).
% 2.71/1.54 thf(fact_215_compl__less__swap2, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [Y2:A,X3:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ Y2)) @ X3) => (((ord_less @ A) @ ((uminus_uminus @ A) @ X3)) @ Y2)))))))).
% 2.71/1.54 thf(fact_216_less__minus__iff, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ ((uminus_uminus @ A) @ B4)) = (((ord_less @ A) @ B4) @ ((uminus_uminus @ A) @ A4))))))))).
% 2.71/1.54 thf(fact_217_minus__less__iff, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A4)) @ B4) = (((ord_less @ A) @ ((uminus_uminus @ A) @ B4)) @ A4)))))))).
% 2.71/1.54 thf(fact_218_compl__less__compl__iff, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ X3)) @ ((uminus_uminus @ A) @ Y2)) = (((ord_less @ A) @ Y2) @ X3)))))))).
% 2.71/1.54 thf(fact_219_complete__interval, axiom, (! [A:$tType]: ((((condit1037483654norder @ A) @ (type2 @ A)) => (! [A4:A,B4:A,P:(A > $o)]: (((((ord_less @ A) @ A4) @ B4) => ((P @ A4) => ((~ ((P @ B4))) => (? [C4:A]: (((((ord_less_eq @ A) @ A4) @ C4) & ((((ord_less_eq @ A) @ C4) @ B4) & ((! [X4:A]: ((((((ord_less_eq @ A) @ A4) @ X4) & (((ord_less @ A) @ X4) @ C4)) => (P @ X4)))) & (! [D2:A]: (((! [X2:A]: ((((((ord_less_eq @ A) @ A4) @ X2) & (((ord_less @ A) @ X2) @ D2)) => (P @ X2)))) => (((ord_less_eq @ A) @ D2) @ C4)))))))))))))))))).
% 2.71/1.54 thf(fact_220_pinf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T3:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ T3)))))))))))))).
% 2.71/1.54 thf(fact_221_ex__gt__or__lt, axiom, (! [A:$tType]: ((((condit1656338222tinuum @ A) @ (type2 @ A)) => (! [A4:A]: ((? [B5:A]: (((((ord_less @ A) @ A4) @ B5) | (((ord_less @ A) @ B5) @ A4)))))))))).
% 2.71/1.54 thf(fact_222_pinf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (((P @ X4) & (Q @ X4)) = ((P2 @ X4) & (Q2 @ X4)))))))))))))))).
% 2.71/1.54 thf(fact_223_pinf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (((P @ X4) | (Q @ X4)) = ((P2 @ X4) | (Q2 @ X4)))))))))))))))).
% 2.71/1.54 thf(fact_224_pinf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T3:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (X4 != T3)))))))))))).
% 2.71/1.54 thf(fact_225_pinf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T3:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (X4 != T3)))))))))))).
% 2.71/1.54 thf(fact_226_pinf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T3:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (~ ((((ord_less @ A) @ X4) @ T3)))))))))))))).
% 2.71/1.54 thf(fact_227_pinf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T3:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (((ord_less @ A) @ T3) @ X4)))))))))))).
% 2.71/1.54 thf(fact_228_pinf_I11_J, axiom, (! [C2:$tType,D3:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F3:D3]: ((? [Z3:C2]: ((! [X4:C2]: (((((ord_less @ C2) @ Z3) @ X4) => (F3 = F3)))))))))))).
% 2.71/1.54 thf(fact_229_minf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (((P @ X4) & (Q @ X4)) = ((P2 @ X4) & (Q2 @ X4)))))))))))))))).
% 2.71/1.54 thf(fact_230_minf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (((P @ X4) | (Q @ X4)) = ((P2 @ X4) | (Q2 @ X4)))))))))))))))).
% 2.71/1.54 thf(fact_231_minf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T3:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (X4 != T3)))))))))))).
% 2.71/1.54 thf(fact_232_minf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T3:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (X4 != T3)))))))))))).
% 2.71/1.54 thf(fact_233_minf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T3:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (((ord_less @ A) @ X4) @ T3)))))))))))).
% 2.71/1.54 thf(fact_234_minf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T3:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (~ ((((ord_less @ A) @ T3) @ X4)))))))))))))).
% 2.71/1.54 thf(fact_235_minf_I11_J, axiom, (! [C2:$tType,D3:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F3:D3]: ((? [Z3:C2]: ((! [X4:C2]: (((((ord_less @ C2) @ X4) @ Z3) => (F3 = F3)))))))))))).
% 2.71/1.54 thf(fact_236_minf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T3:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (~ ((((ord_less_eq @ A) @ T3) @ X4)))))))))))))).
% 2.71/1.54 thf(fact_237_minf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T3:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (((ord_less_eq @ A) @ X4) @ T3)))))))))))).
% 2.71/1.54 thf(fact_238_pinf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T3:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (((ord_less_eq @ A) @ T3) @ X4)))))))))))).
% 2.71/1.54 thf(fact_239_greaterThanLessThan__subseteq__atLeastLessThan__iff, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A,D:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or578182835ssThan @ A) @ A4) @ B4)) @ (((set_or1433837966ssThan @ A) @ C) @ D)) = ((((ord_less @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ C) @ A4) & (((ord_less_eq @ A) @ B4) @ D)))))))))).
% 2.71/1.54 thf(fact_240_greaterThanLessThan__subseteq__atLeastAtMost__iff, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A,D:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or578182835ssThan @ A) @ A4) @ B4)) @ (((set_or331188842AtMost @ A) @ C) @ D)) = ((((ord_less @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ C) @ A4) & (((ord_less_eq @ A) @ B4) @ D)))))))))).
% 2.71/1.54 thf(fact_241_greaterThanLessThan__iff, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [I:A,L:A,U:A]: (((((member @ A) @ I) @ (((set_or578182835ssThan @ A) @ L) @ U)) = ((((ord_less @ A) @ L) @ I) & (((ord_less @ A) @ I) @ U))))))))).
% 2.71/1.54 thf(fact_242_greaterThanLessThan__subseteq__greaterThanLessThan, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A,D:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or578182835ssThan @ A) @ A4) @ B4)) @ (((set_or578182835ssThan @ A) @ C) @ D)) = ((((ord_less @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ C) @ A4) & (((ord_less_eq @ A) @ B4) @ D)))))))))).
% 2.71/1.54 thf(fact_243_ivl__disj__un__two_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or331188842AtMost @ A) @ L) @ M)) @ (((set_or578182835ssThan @ A) @ M) @ U)) = (((set_or1433837966ssThan @ A) @ L) @ U)))))))))).
% 2.71/1.54 thf(fact_244_greaterThanLessThan__subseteq__greaterThanAtMost__iff, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C:A,D:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or578182835ssThan @ A) @ A4) @ B4)) @ (((set_or1361889807AtMost @ A) @ C) @ D)) = ((((ord_less @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ C) @ A4) & (((ord_less_eq @ A) @ B4) @ D)))))))))).
% 2.71/1.54 thf(fact_245_sup__apply, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ B) @ (type2 @ B)) => ((sup_sup @ (A > B)) = (^ [F2:(A > B),G:(A > B),X:A]: ((((sup_sup @ B) @ (F2 @ X)) @ (G @ X))))))))).
% 2.71/1.54 thf(fact_246_sup_Oidem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A4:A]: (((((sup_sup @ A) @ A4) @ A4) = A4))))))).
% 2.71/1.54 thf(fact_247_sup__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X3:A]: (((((sup_sup @ A) @ X3) @ X3) = X3))))))).
% 2.71/1.54 thf(fact_248_sup_Oleft__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((sup_sup @ A) @ A4) @ (((sup_sup @ A) @ A4) @ B4)) = (((sup_sup @ A) @ A4) @ B4)))))))).
% 2.71/1.54 thf(fact_249_sup__left__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X3:A,Y2:A]: (((((sup_sup @ A) @ X3) @ (((sup_sup @ A) @ X3) @ Y2)) = (((sup_sup @ A) @ X3) @ Y2)))))))).
% 2.71/1.54 thf(fact_250_sup_Oright__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A4) @ B4)) @ B4) = (((sup_sup @ A) @ A4) @ B4)))))))).
% 2.71/1.54 thf(fact_251_UnCI, axiom, (! [A:$tType,C:A,B2:(set @ A),A2:(set @ A)]: ((((~ ((((member @ A) @ C) @ B2))) => (((member @ A) @ C) @ A2)) => (((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A2) @ B2)))))).
% 2.71/1.54 thf(fact_252_Un__iff, axiom, (! [A:$tType,C:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) = ((((member @ A) @ C) @ A2) | (((member @ A) @ C) @ B2)))))).
% 2.71/1.54 thf(fact_253_sup_Obounded__iff, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B4:A,C:A,A4:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B4) @ C)) @ A4) = ((((ord_less_eq @ A) @ B4) @ A4) & (((ord_less_eq @ A) @ C) @ A4))))))))).
% 2.71/1.54 thf(fact_254_le__sup__iff, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X3:A,Y2:A,Z:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ X3) @ Y2)) @ Z) = ((((ord_less_eq @ A) @ X3) @ Z) & (((ord_less_eq @ A) @ Y2) @ Z))))))))).
% 2.71/1.54 thf(fact_255_Un__subset__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ C3) = ((((ord_less_eq @ (set @ A)) @ A2) @ C3) & (((ord_less_eq @ (set @ A)) @ B2) @ C3)))))).
% 2.71/1.54 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A7:$tType,A8:$tType]: ((((semilattice_sup @ A8) @ (type2 @ A8)) => ((semilattice_sup @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.71/1.54 thf(tcon_fun___Lattices_Oboolean__algebra, axiom, (! [A7:$tType,A8:$tType]: ((((boolean_algebra @ A8) @ (type2 @ A8)) => ((boolean_algebra @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.71/1.54 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: ((((preorder @ A8) @ (type2 @ A8)) => ((preorder @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.71/1.54 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: ((((order @ A8) @ (type2 @ A8)) => ((order @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.71/1.54 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: ((((ord @ A8) @ (type2 @ A8)) => ((ord @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.71/1.54 thf(tcon_fun___Groups_Ouminus, axiom, (! [A7:$tType,A8:$tType]: ((((uminus @ A8) @ (type2 @ A8)) => ((uminus @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.71/1.54 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1, axiom, (! [A7:$tType]: (((semilattice_sup @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.71/1.54 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_2, axiom, (! [A7:$tType]: (((boolean_algebra @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.71/1.54 thf(tcon_Set_Oset___Orderings_Opreorder_3, axiom, (! [A7:$tType]: (((preorder @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.71/1.54 thf(tcon_Set_Oset___Orderings_Oorder_4, axiom, (! [A7:$tType]: (((order @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.71/1.54 thf(tcon_Set_Oset___Orderings_Oord_5, axiom, (! [A7:$tType]: (((ord @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.71/1.54 thf(tcon_Set_Oset___Groups_Ouminus_6, axiom, (! [A7:$tType]: (((uminus @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.71/1.54 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_7, axiom, ((semilattice_sup @ $o) @ (type2 @ $o))).
% 2.71/1.54 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_8, axiom, ((boolean_algebra @ $o) @ (type2 @ $o))).
% 2.71/1.54 thf(tcon_HOL_Obool___Orderings_Opreorder_9, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.71/1.54 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.71/1.54 thf(tcon_HOL_Obool___Orderings_Oorder_10, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.71/1.54 thf(tcon_HOL_Obool___Orderings_Oord_11, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.71/1.54 thf(tcon_HOL_Obool___Groups_Ouminus_12, axiom, ((uminus @ $o) @ (type2 @ $o))).
% 2.71/1.54 thf(conj_0, hypothesis, ((((gram_L608943123e_subs @ a) @ l1) @ l2) & (((gram_L608943123e_subs @ a) @ l2) @ l1))).
% 2.71/1.54 thf(conj_1, conjecture, (((gram_L608943123e_subs @ a) @ l1) @ l2)).
% 2.71/1.54 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.LeYYwvo4TN/DTF2DTF12112.p
% 2.71/1.54 ------------------------
% 2.71/1.57 ---- Cleaned THF ---
% 2.71/1.57 thf(ty_t_Set_Oset,type,
% 2.71/1.57 set: $tType > $tType ).
% 2.71/1.57
% 2.71/1.57 thf(ty_t_itself,type,
% 2.71/1.57 itself: $tType > $tType ).
% 2.71/1.57
% 2.71/1.57 thf(ty_tf_a,type,
% 2.71/1.57 a: $tType ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_HOL_Otype,type,
% 2.71/1.57 type:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_Groups_Ouminus,type,
% 2.71/1.57 uminus:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_Orderings_Oord,type,
% 2.71/1.57 ord:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_Orderings_Oorder,type,
% 2.71/1.57 order:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_Groups_Ogroup__add,type,
% 2.71/1.57 group_add:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_Orderings_Ono__bot,type,
% 2.71/1.57 no_bot:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_Orderings_Ono__top,type,
% 2.71/1.57 no_top:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_Orderings_Olinorder,type,
% 2.71/1.57 linorder:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_Orderings_Opreorder,type,
% 2.71/1.57 preorder:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_Orderings_Owellorder,type,
% 2.71/1.57 wellorder:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_Orderings_Odense__order,type,
% 2.71/1.57 dense_order:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 2.71/1.57 boolean_algebra:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 2.71/1.57 semilattice_sup:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.71/1.57 dense_linorder:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.71/1.57 ordered_ab_group_add:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.71/1.57 condit1656338222tinuum:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.71/1.57 condit1037483654norder:
% 2.71/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Oleqv,type,
% 2.71/1.57 gram_L1456083582e_leqv:
% 2.71/1.57 !>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ A ) ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Osubs,type,
% 2.71/1.57 gram_L608943123e_subs:
% 2.71/1.57 !>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ A ) ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 2.71/1.57 uminus_uminus:
% 2.71/1.57 !>[A: $tType] : ( A > A ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Lattices_Osup__class_Osup,type,
% 2.71/1.57 sup_sup:
% 2.71/1.57 !>[A: $tType] : ( A > A > A ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.71/1.57 ord_less:
% 2.71/1.57 !>[A: $tType] : ( A > A > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.71/1.57 ord_less_eq:
% 2.71/1.57 !>[A: $tType] : ( A > A > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 2.71/1.57 order_antimono:
% 2.71/1.57 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 2.71/1.57 order_strict_mono:
% 2.71/1.57 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Pure_Otype,type,
% 2.71/1.57 type2:
% 2.71/1.57 !>[A: $tType] : ( itself @ A ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Set_OCollect,type,
% 2.71/1.57 collect:
% 2.71/1.57 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Set_OPow,type,
% 2.71/1.57 pow:
% 2.71/1.57 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Set_Opairwise,type,
% 2.71/1.57 pairwise:
% 2.71/1.57 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
% 2.71/1.57 set_ord_atLeast:
% 2.71/1.57 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
% 2.71/1.57 set_or331188842AtMost:
% 2.71/1.57 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
% 2.71/1.57 set_or1433837966ssThan:
% 2.71/1.57 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Set__Interval_Oord__class_OatMost,type,
% 2.71/1.57 set_ord_atMost:
% 2.71/1.57 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
% 2.71/1.57 set_ord_greaterThan:
% 2.71/1.57 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
% 2.71/1.57 set_or1361889807AtMost:
% 2.71/1.57 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
% 2.71/1.57 set_or578182835ssThan:
% 2.71/1.57 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
% 2.71/1.57 set_ord_lessThan:
% 2.71/1.57 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_c_member,type,
% 2.71/1.57 member:
% 2.71/1.57 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_v_L1,type,
% 2.71/1.57 l1: set @ ( set @ a ) ).
% 2.71/1.57
% 2.71/1.57 thf(sy_v_L2,type,
% 2.71/1.57 l2: set @ ( set @ a ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_0_subs__refl,axiom,
% 2.71/1.57 ! [A: $tType,L1: set @ ( set @ A )] : ( gram_L608943123e_subs @ A @ L1 @ L1 ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_1_subs__trans,axiom,
% 2.71/1.57 ! [A: $tType,L1: set @ ( set @ A ),L2: set @ ( set @ A ),L3: set @ ( set @ A )] :
% 2.71/1.57 ( ( gram_L608943123e_subs @ A @ L1 @ L2 )
% 2.71/1.57 => ( ( gram_L608943123e_subs @ A @ L2 @ L3 )
% 2.71/1.57 => ( gram_L608943123e_subs @ A @ L1 @ L3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_2_leqv__def,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( gram_L1456083582e_leqv @ A )
% 2.71/1.57 = ( ^ [L12: set @ ( set @ A ),L22: set @ ( set @ A )] :
% 2.71/1.57 ( ( gram_L608943123e_subs @ A @ L12 @ L22 )
% 2.71/1.57 & ( gram_L608943123e_subs @ A @ L22 @ L12 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_3_incl__subs,axiom,
% 2.71/1.57 ! [A: $tType,L2: set @ ( set @ A ),L1: set @ ( set @ A )] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ L2 @ L1 )
% 2.71/1.57 => ( gram_L608943123e_subs @ A @ L1 @ L2 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_4_subs__def,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( gram_L608943123e_subs @ A )
% 2.71/1.57 = ( ^ [L12: set @ ( set @ A ),L22: set @ ( set @ A )] :
% 2.71/1.57 ! [X: set @ A] :
% 2.71/1.57 ( ( member @ ( set @ A ) @ X @ L22 )
% 2.71/1.57 => ? [Y: set @ A] :
% 2.71/1.57 ( ( member @ ( set @ A ) @ Y @ L12 )
% 2.71/1.57 & ( ord_less_eq @ ( set @ A ) @ Y @ X ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_5_subsetI,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ! [X2: A] :
% 2.71/1.57 ( ( member @ A @ X2 @ A2 )
% 2.71/1.57 => ( member @ A @ X2 @ B2 ) )
% 2.71/1.57 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_6_subset__antisym,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.71/1.57 => ( A2 = B2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_7_order__refl,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_8_set__mp,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( ( member @ A @ X3 @ A2 )
% 2.71/1.57 => ( member @ A @ X3 @ B2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_9_in__mono,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( ( member @ A @ X3 @ A2 )
% 2.71/1.57 => ( member @ A @ X3 @ B2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_10_subsetD,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( ( member @ A @ C @ A2 )
% 2.71/1.57 => ( member @ A @ C @ B2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_11_subsetCE,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( ( member @ A @ C @ A2 )
% 2.71/1.57 => ( member @ A @ C @ B2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_12_equalityE,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( A2 = B2 )
% 2.71/1.57 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_13_subset__eq,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 2.71/1.57 = ( ^ [A3: set @ A,B3: set @ A] :
% 2.71/1.57 ! [X: A] :
% 2.71/1.57 ( ( member @ A @ X @ A3 )
% 2.71/1.57 => ( member @ A @ X @ B3 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_14_equalityD1,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( A2 = B2 )
% 2.71/1.57 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_15_equalityD2,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( A2 = B2 )
% 2.71/1.57 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_16_dual__order_Oantisym,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [B4: A,A4: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.71/1.57 => ( A4 = B4 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_17_dual__order_Otrans,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [B4: A,A4: A,C: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ C @ B4 )
% 2.71/1.57 => ( ord_less_eq @ A @ C @ A4 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_18_linorder__wlog,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [P: A > A > $o,A4: A,B4: A] :
% 2.71/1.57 ( ! [A5: A,B5: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.71/1.57 => ( P @ A5 @ B5 ) )
% 2.71/1.57 => ( ! [A5: A,B5: A] :
% 2.71/1.57 ( ( P @ B5 @ A5 )
% 2.71/1.57 => ( P @ A5 @ B5 ) )
% 2.71/1.57 => ( P @ A4 @ B4 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_19_dual__order_Orefl,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_20_order__trans,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A,Z: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ Y2 @ Z )
% 2.71/1.57 => ( ord_less_eq @ A @ X3 @ Z ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_21_order__class_Oorder_Oantisym,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.71/1.57 => ( A4 = B4 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_22_ord__le__eq__trans,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ord @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( B4 = C )
% 2.71/1.57 => ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_23_ord__eq__le__trans,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ord @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A] :
% 2.71/1.57 ( ( A4 = B4 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ B4 @ C )
% 2.71/1.57 => ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_24_antisym__conv,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [Y2: A,X3: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ Y2 @ X3 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.71/1.57 = ( X3 = Y2 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_25_le__cases3,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A,Z: A] :
% 2.71/1.57 ( ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.71/1.57 => ~ ( ord_less_eq @ A @ Y2 @ Z ) )
% 2.71/1.57 => ( ( ( ord_less_eq @ A @ Y2 @ X3 )
% 2.71/1.57 => ~ ( ord_less_eq @ A @ X3 @ Z ) )
% 2.71/1.57 => ( ( ( ord_less_eq @ A @ X3 @ Z )
% 2.71/1.57 => ~ ( ord_less_eq @ A @ Z @ Y2 ) )
% 2.71/1.57 => ( ( ( ord_less_eq @ A @ Z @ Y2 )
% 2.71/1.57 => ~ ( ord_less_eq @ A @ Y2 @ X3 ) )
% 2.71/1.57 => ( ( ( ord_less_eq @ A @ Y2 @ Z )
% 2.71/1.57 => ~ ( ord_less_eq @ A @ Z @ X3 ) )
% 2.71/1.57 => ~ ( ( ord_less_eq @ A @ Z @ X3 )
% 2.71/1.57 => ~ ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_26_order_Otrans,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ B4 @ C )
% 2.71/1.57 => ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_27_le__cases,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ~ ( ord_less_eq @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_28_eq__refl,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( X3 = Y2 )
% 2.71/1.57 => ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_29_linear,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.71/1.57 | ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_30_antisym,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ Y2 @ X3 )
% 2.71/1.57 => ( X3 = Y2 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_31_eq__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ( ( ^ [Y3: A,Z2: A] : ( Y3 = Z2 ) )
% 2.71/1.57 = ( ^ [X: A,Y: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ X @ Y )
% 2.71/1.57 & ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_32_ord__le__eq__subst,axiom,
% 2.71/1.57 ! [A: $tType,B: $tType] :
% 2.71/1.57 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.71/1.57 & ( ord @ A @ ( type2 @ A ) ) )
% 2.71/1.57 => ! [A4: A,B4: A,F: A > B,C: B] :
% 2.71/1.57 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ( F @ B4 )
% 2.71/1.57 = C )
% 2.71/1.57 => ( ! [X2: A,Y4: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 2.71/1.57 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.71/1.57 => ( ord_less_eq @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_33_ord__eq__le__subst,axiom,
% 2.71/1.57 ! [A: $tType,B: $tType] :
% 2.71/1.57 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.71/1.57 & ( ord @ A @ ( type2 @ A ) ) )
% 2.71/1.57 => ! [A4: A,F: B > A,B4: B,C: B] :
% 2.71/1.57 ( ( A4
% 2.71/1.57 = ( F @ B4 ) )
% 2.71/1.57 => ( ( ord_less_eq @ B @ B4 @ C )
% 2.71/1.57 => ( ! [X2: B,Y4: B] :
% 2.71/1.57 ( ( ord_less_eq @ B @ X2 @ Y4 )
% 2.71/1.57 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.71/1.57 => ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_34_order__subst2,axiom,
% 2.71/1.57 ! [A: $tType,C2: $tType] :
% 2.71/1.57 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.71/1.57 & ( order @ A @ ( type2 @ A ) ) )
% 2.71/1.57 => ! [A4: A,B4: A,F: A > C2,C: C2] :
% 2.71/1.57 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less_eq @ C2 @ ( F @ B4 ) @ C )
% 2.71/1.57 => ( ! [X2: A,Y4: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 2.71/1.57 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.71/1.57 => ( ord_less_eq @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_35_order__subst1,axiom,
% 2.71/1.57 ! [A: $tType,B: $tType] :
% 2.71/1.57 ( ( ( order @ B @ ( type2 @ B ) )
% 2.71/1.57 & ( order @ A @ ( type2 @ A ) ) )
% 2.71/1.57 => ! [A4: A,F: B > A,B4: B,C: B] :
% 2.71/1.57 ( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
% 2.71/1.57 => ( ( ord_less_eq @ B @ B4 @ C )
% 2.71/1.57 => ( ! [X2: B,Y4: B] :
% 2.71/1.57 ( ( ord_less_eq @ B @ X2 @ Y4 )
% 2.71/1.57 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.71/1.57 => ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_36_le__fun__def,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( ord @ B @ ( type2 @ B ) )
% 2.71/1.57 => ( ( ord_less_eq @ ( A > B ) )
% 2.71/1.57 = ( ^ [F2: A > B,G: A > B] :
% 2.71/1.57 ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_37_le__funI,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( ord @ B @ ( type2 @ B ) )
% 2.71/1.57 => ! [F: A > B,G2: A > B] :
% 2.71/1.57 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G2 @ X2 ) )
% 2.71/1.57 => ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_38_le__funE,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( ord @ B @ ( type2 @ B ) )
% 2.71/1.57 => ! [F: A > B,G2: A > B,X3: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 2.71/1.57 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_39_le__funD,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( ord @ B @ ( type2 @ B ) )
% 2.71/1.57 => ! [F: A > B,G2: A > B,X3: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 2.71/1.57 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_40_Collect__mono__iff,axiom,
% 2.71/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.71/1.57 = ( ! [X: A] :
% 2.71/1.57 ( ( P @ X )
% 2.71/1.57 => ( Q @ X ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_41_contra__subsetD,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( ~ ( member @ A @ C @ B2 )
% 2.71/1.57 => ~ ( member @ A @ C @ A2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_42_set__eq__subset,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ^ [Y3: set @ A,Z2: set @ A] : ( Y3 = Z2 ) )
% 2.71/1.57 = ( ^ [A3: set @ A,B3: set @ A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.71/1.57 & ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_43_subset__trans,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 2.71/1.57 => ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_44_Collect__mono,axiom,
% 2.71/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.71/1.57 ( ! [X2: A] :
% 2.71/1.57 ( ( P @ X2 )
% 2.71/1.57 => ( Q @ X2 ) )
% 2.71/1.57 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_45_mem__Collect__eq,axiom,
% 2.71/1.57 ! [A: $tType,A4: A,P: A > $o] :
% 2.71/1.57 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 2.71/1.57 = ( P @ A4 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_46_Collect__mem__eq,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A] :
% 2.71/1.57 ( ( collect @ A
% 2.71/1.57 @ ^ [X: A] : ( member @ A @ X @ A2 ) )
% 2.71/1.57 = A2 ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_47_Collect__cong,axiom,
% 2.71/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.71/1.57 ( ! [X2: A] :
% 2.71/1.57 ( ( P @ X2 )
% 2.71/1.57 = ( Q @ X2 ) )
% 2.71/1.57 => ( ( collect @ A @ P )
% 2.71/1.57 = ( collect @ A @ Q ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_48_ext,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType,F: A > B,G2: A > B] :
% 2.71/1.57 ( ! [X2: A] :
% 2.71/1.57 ( ( F @ X2 )
% 2.71/1.57 = ( G2 @ X2 ) )
% 2.71/1.57 => ( F = G2 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_49_subset__refl,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_50_rev__subsetD,axiom,
% 2.71/1.57 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( member @ A @ C @ A2 )
% 2.71/1.57 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( member @ A @ C @ B2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_51_subset__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 2.71/1.57 = ( ^ [A3: set @ A,B3: set @ A] :
% 2.71/1.57 ! [T: A] :
% 2.71/1.57 ( ( member @ A @ T @ A3 )
% 2.71/1.57 => ( member @ A @ T @ B3 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_52_set__rev__mp,axiom,
% 2.71/1.57 ! [A: $tType,X3: A,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( member @ A @ X3 @ A2 )
% 2.71/1.57 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( member @ A @ X3 @ B2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_53_antimono__def,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 & ( order @ B @ ( type2 @ B ) ) )
% 2.71/1.57 => ( ( order_antimono @ A @ B )
% 2.71/1.57 = ( ^ [F2: A > B] :
% 2.71/1.57 ! [X: A,Y: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ X @ Y )
% 2.71/1.57 => ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_54_antimonoI,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 & ( order @ B @ ( type2 @ B ) ) )
% 2.71/1.57 => ! [F: A > B] :
% 2.71/1.57 ( ! [X2: A,Y4: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 2.71/1.57 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X2 ) ) )
% 2.71/1.57 => ( order_antimono @ A @ B @ F ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_55_antimonoE,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 & ( order @ B @ ( type2 @ B ) ) )
% 2.71/1.57 => ! [F: A > B,X3: A,Y2: A] :
% 2.71/1.57 ( ( order_antimono @ A @ B @ F )
% 2.71/1.57 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_56_antimonoD,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 & ( order @ B @ ( type2 @ B ) ) )
% 2.71/1.57 => ! [F: A > B,X3: A,Y2: A] :
% 2.71/1.57 ( ( order_antimono @ A @ B @ F )
% 2.71/1.57 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_57_Pow__mono,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A2 ) @ ( pow @ A @ B2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_58_strict__mono__less__eq,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 & ( order @ B @ ( type2 @ B ) ) )
% 2.71/1.57 => ! [F: A > B,X3: A,Y2: A] :
% 2.71/1.57 ( ( order_strict_mono @ A @ B @ F )
% 2.71/1.57 => ( ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) )
% 2.71/1.57 = ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_59_pairwise__subset,axiom,
% 2.71/1.57 ! [A: $tType,P: A > A > $o,S: set @ A,T2: set @ A] :
% 2.71/1.57 ( ( pairwise @ A @ P @ S )
% 2.71/1.57 => ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
% 2.71/1.57 => ( pairwise @ A @ P @ T2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_60_greaterThan__subset__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X3 ) @ ( set_ord_greaterThan @ A @ Y2 ) )
% 2.71/1.57 = ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_61_lessThan__subset__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X3 ) @ ( set_ord_lessThan @ A @ Y2 ) )
% 2.71/1.57 = ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_62_atLeast__subset__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X3 ) @ ( set_ord_atLeast @ A @ Y2 ) )
% 2.71/1.57 = ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_63_lessThan__eq__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ( set_ord_lessThan @ A @ X3 )
% 2.71/1.57 = ( set_ord_lessThan @ A @ Y2 ) )
% 2.71/1.57 = ( X3 = Y2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_64_atLeast__eq__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ( set_ord_atLeast @ A @ X3 )
% 2.71/1.57 = ( set_ord_atLeast @ A @ Y2 ) )
% 2.71/1.57 = ( X3 = Y2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_65_greaterThan__eq__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ( set_ord_greaterThan @ A @ X3 )
% 2.71/1.57 = ( set_ord_greaterThan @ A @ Y2 ) )
% 2.71/1.57 = ( X3 = Y2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_66_atLeast__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ord @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [I: A,K: A] :
% 2.71/1.57 ( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) )
% 2.71/1.57 = ( ord_less_eq @ A @ K @ I ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_67_PowI,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_68_Pow__iff,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 2.71/1.57 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_69_Ioi__le__Ico,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A4 ) @ ( set_ord_atLeast @ A @ A4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_70_pairwise__def,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( pairwise @ A )
% 2.71/1.57 = ( ^ [R: A > A > $o,S2: set @ A] :
% 2.71/1.57 ! [X: A] :
% 2.71/1.57 ( ( member @ A @ X @ S2 )
% 2.71/1.57 => ! [Y: A] :
% 2.71/1.57 ( ( member @ A @ Y @ S2 )
% 2.71/1.57 => ( ( X != Y )
% 2.71/1.57 => ( R @ X @ Y ) ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_71_Pow__top,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A] : ( member @ ( set @ A ) @ A2 @ ( pow @ A @ A2 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_72_strict__mono__eq,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 & ( order @ B @ ( type2 @ B ) ) )
% 2.71/1.57 => ! [F: A > B,X3: A,Y2: A] :
% 2.71/1.57 ( ( order_strict_mono @ A @ B @ F )
% 2.71/1.57 => ( ( ( F @ X3 )
% 2.71/1.57 = ( F @ Y2 ) )
% 2.71/1.57 = ( X3 = Y2 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_73_PowD,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 2.71/1.57 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_74_Icc__subset__Ici__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [L: A,H: A,L4: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ H ) @ ( set_ord_atLeast @ A @ L4 ) )
% 2.71/1.57 = ( ~ ( ord_less_eq @ A @ L @ H )
% 2.71/1.57 | ( ord_less_eq @ A @ L4 @ L ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_75_Ici__subset__Ioi__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A4 ) @ ( set_ord_greaterThan @ A @ B4 ) )
% 2.71/1.57 = ( ord_less @ A @ B4 @ A4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_76_Compl__atLeast,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [K: A] :
% 2.71/1.57 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atLeast @ A @ K ) )
% 2.71/1.57 = ( set_ord_lessThan @ A @ K ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_77_Compl__lessThan,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [K: A] :
% 2.71/1.57 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) )
% 2.71/1.57 = ( set_ord_atLeast @ A @ K ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_78_atMost__subset__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X3 ) @ ( set_ord_atMost @ A @ Y2 ) )
% 2.71/1.57 = ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_79_ivl__subset,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [I: A,J: A,M: A,N: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1433837966ssThan @ A @ I @ J ) @ ( set_or1433837966ssThan @ A @ M @ N ) )
% 2.71/1.57 = ( ( ord_less_eq @ A @ J @ I )
% 2.71/1.57 | ( ( ord_less_eq @ A @ M @ I )
% 2.71/1.57 & ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_80_not__Iic__le__Ici,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( no_bot @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [H: A,L4: A] :
% 2.71/1.57 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_ord_atLeast @ A @ L4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_81_psubsetI,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( ( A2 != B2 )
% 2.71/1.57 => ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_82_ComplI,axiom,
% 2.71/1.57 ! [A: $tType,C: A,A2: set @ A] :
% 2.71/1.57 ( ~ ( member @ A @ C @ A2 )
% 2.71/1.57 => ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_83_Compl__iff,axiom,
% 2.71/1.57 ! [A: $tType,C: A,A2: set @ A] :
% 2.71/1.57 ( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.71/1.57 = ( ~ ( member @ A @ C @ A2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_84_Compl__eq__Compl__iff,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( ( uminus_uminus @ ( set @ A ) @ A2 )
% 2.71/1.57 = ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 2.71/1.57 = ( A2 = B2 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_85_atMost__eq__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ( set_ord_atMost @ A @ X3 )
% 2.71/1.57 = ( set_ord_atMost @ A @ Y2 ) )
% 2.71/1.57 = ( X3 = Y2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_86_atLeastAtMost__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ord @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [I: A,L: A,U: A] :
% 2.71/1.57 ( ( member @ A @ I @ ( set_or331188842AtMost @ A @ L @ U ) )
% 2.71/1.57 = ( ( ord_less_eq @ A @ L @ I )
% 2.71/1.57 & ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_87_Icc__eq__Icc,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [L: A,H: A,L4: A,H2: A] :
% 2.71/1.57 ( ( ( set_or331188842AtMost @ A @ L @ H )
% 2.71/1.57 = ( set_or331188842AtMost @ A @ L4 @ H2 ) )
% 2.71/1.57 = ( ( ( L = L4 )
% 2.71/1.57 & ( H = H2 ) )
% 2.71/1.57 | ( ~ ( ord_less_eq @ A @ L @ H )
% 2.71/1.57 & ~ ( ord_less_eq @ A @ L4 @ H2 ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_88_lessThan__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ord @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [I: A,K: A] :
% 2.71/1.57 ( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
% 2.71/1.57 = ( ord_less @ A @ I @ K ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_89_atMost__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ord @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [I: A,K: A] :
% 2.71/1.57 ( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) )
% 2.71/1.57 = ( ord_less_eq @ A @ I @ K ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_90_Compl__subset__Compl__iff,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 2.71/1.57 = ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_91_Compl__anti__mono,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_92_greaterThan__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ord @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [I: A,K: A] :
% 2.71/1.57 ( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K ) )
% 2.71/1.57 = ( ord_less @ A @ K @ I ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_93_atLeastLessThan__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ord @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [I: A,L: A,U: A] :
% 2.71/1.57 ( ( member @ A @ I @ ( set_or1433837966ssThan @ A @ L @ U ) )
% 2.71/1.57 = ( ( ord_less_eq @ A @ L @ I )
% 2.71/1.57 & ( ord_less @ A @ I @ U ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_94_atLeastatMost__subset__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A,D: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A4 @ B4 ) @ ( set_or331188842AtMost @ A @ C @ D ) )
% 2.71/1.57 = ( ~ ( ord_less_eq @ A @ A4 @ B4 )
% 2.71/1.57 | ( ( ord_less_eq @ A @ C @ A4 )
% 2.71/1.57 & ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_95_Compl__atMost,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [K: A] :
% 2.71/1.57 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atMost @ A @ K ) )
% 2.71/1.57 = ( set_ord_greaterThan @ A @ K ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_96_Compl__greaterThan,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [K: A] :
% 2.71/1.57 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_greaterThan @ A @ K ) )
% 2.71/1.57 = ( set_ord_atMost @ A @ K ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_97_Icc__subset__Iic__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [L: A,H: A,H2: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ H ) @ ( set_ord_atMost @ A @ H2 ) )
% 2.71/1.57 = ( ~ ( ord_less_eq @ A @ L @ H )
% 2.71/1.57 | ( ord_less_eq @ A @ H @ H2 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_98_not__Iic__le__Icc,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( no_bot @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [H: A,L4: A,H2: A] :
% 2.71/1.57 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_or331188842AtMost @ A @ L4 @ H2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_99_atLeastatMost__psubset__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A,D: A] :
% 2.71/1.57 ( ( ord_less @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A4 @ B4 ) @ ( set_or331188842AtMost @ A @ C @ D ) )
% 2.71/1.57 = ( ( ~ ( ord_less_eq @ A @ A4 @ B4 )
% 2.71/1.57 | ( ( ord_less_eq @ A @ C @ A4 )
% 2.71/1.57 & ( ord_less_eq @ A @ B4 @ D )
% 2.71/1.57 & ( ( ord_less @ A @ C @ A4 )
% 2.71/1.57 | ( ord_less @ A @ B4 @ D ) ) ) )
% 2.71/1.57 & ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_100_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A,D: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A4 @ B4 ) @ ( set_or1433837966ssThan @ A @ C @ D ) )
% 2.71/1.57 = ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ C @ A4 )
% 2.71/1.57 & ( ord_less @ A @ B4 @ D ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_101_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A,D: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1433837966ssThan @ A @ A4 @ B4 ) @ ( set_or331188842AtMost @ A @ C @ D ) )
% 2.71/1.57 = ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ C @ A4 )
% 2.71/1.57 & ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_102_atLeastLessThan__inj_I2_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A,D: A] :
% 2.71/1.57 ( ( ( set_or1433837966ssThan @ A @ A4 @ B4 )
% 2.71/1.57 = ( set_or1433837966ssThan @ A @ C @ D ) )
% 2.71/1.57 => ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less @ A @ C @ D )
% 2.71/1.57 => ( B4 = D ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_103_atLeastLessThan__inj_I1_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A,D: A] :
% 2.71/1.57 ( ( ( set_or1433837966ssThan @ A @ A4 @ B4 )
% 2.71/1.57 = ( set_or1433837966ssThan @ A @ C @ D ) )
% 2.71/1.57 => ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less @ A @ C @ D )
% 2.71/1.57 => ( A4 = C ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_104_atLeastLessThan__eq__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A,D: A] :
% 2.71/1.57 ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less @ A @ C @ D )
% 2.71/1.57 => ( ( ( set_or1433837966ssThan @ A @ A4 @ B4 )
% 2.71/1.57 = ( set_or1433837966ssThan @ A @ C @ D ) )
% 2.71/1.57 = ( ( A4 = C )
% 2.71/1.57 & ( B4 = D ) ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_105_lessThan__strict__subset__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [M: A,N: A] :
% 2.71/1.57 ( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
% 2.71/1.57 = ( ord_less @ A @ M @ N ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_106_not__Iic__eq__Icc,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( no_bot @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [H2: A,L: A,H: A] :
% 2.71/1.57 ( ( set_ord_atMost @ A @ H2 )
% 2.71/1.57 != ( set_or331188842AtMost @ A @ L @ H ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_107_ComplD,axiom,
% 2.71/1.57 ! [A: $tType,C: A,A2: set @ A] :
% 2.71/1.57 ( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.71/1.57 => ~ ( member @ A @ C @ A2 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_108_double__complement,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A] :
% 2.71/1.57 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.71/1.57 = A2 ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_109_ord__eq__less__subst,axiom,
% 2.71/1.57 ! [A: $tType,B: $tType] :
% 2.71/1.57 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.71/1.57 & ( ord @ A @ ( type2 @ A ) ) )
% 2.71/1.57 => ! [A4: A,F: B > A,B4: B,C: B] :
% 2.71/1.57 ( ( A4
% 2.71/1.57 = ( F @ B4 ) )
% 2.71/1.57 => ( ( ord_less @ B @ B4 @ C )
% 2.71/1.57 => ( ! [X2: B,Y4: B] :
% 2.71/1.57 ( ( ord_less @ B @ X2 @ Y4 )
% 2.71/1.57 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.71/1.57 => ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_110_ord__less__eq__subst,axiom,
% 2.71/1.57 ! [A: $tType,B: $tType] :
% 2.71/1.57 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.71/1.57 & ( ord @ A @ ( type2 @ A ) ) )
% 2.71/1.57 => ! [A4: A,B4: A,F: A > B,C: B] :
% 2.71/1.57 ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ( F @ B4 )
% 2.71/1.57 = C )
% 2.71/1.57 => ( ! [X2: A,Y4: A] :
% 2.71/1.57 ( ( ord_less @ A @ X2 @ Y4 )
% 2.71/1.57 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.71/1.57 => ( ord_less @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_111_order__less__subst1,axiom,
% 2.71/1.57 ! [A: $tType,B: $tType] :
% 2.71/1.57 ( ( ( order @ B @ ( type2 @ B ) )
% 2.71/1.57 & ( order @ A @ ( type2 @ A ) ) )
% 2.71/1.57 => ! [A4: A,F: B > A,B4: B,C: B] :
% 2.71/1.57 ( ( ord_less @ A @ A4 @ ( F @ B4 ) )
% 2.71/1.57 => ( ( ord_less @ B @ B4 @ C )
% 2.71/1.57 => ( ! [X2: B,Y4: B] :
% 2.71/1.57 ( ( ord_less @ B @ X2 @ Y4 )
% 2.71/1.57 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.71/1.57 => ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_112_order__less__subst2,axiom,
% 2.71/1.57 ! [A: $tType,C2: $tType] :
% 2.71/1.57 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.71/1.57 & ( order @ A @ ( type2 @ A ) ) )
% 2.71/1.57 => ! [A4: A,B4: A,F: A > C2,C: C2] :
% 2.71/1.57 ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less @ C2 @ ( F @ B4 ) @ C )
% 2.71/1.57 => ( ! [X2: A,Y4: A] :
% 2.71/1.57 ( ( ord_less @ A @ X2 @ Y4 )
% 2.71/1.57 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.71/1.57 => ( ord_less @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_113_lt__ex,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( no_bot @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A] :
% 2.71/1.57 ? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_114_gt__ex,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( no_top @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A] :
% 2.71/1.57 ? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_115_neqE,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( X3 != Y2 )
% 2.71/1.57 => ( ~ ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_116_neq__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( X3 != Y2 )
% 2.71/1.57 = ( ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 | ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_117_order_Oasym,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_118_dense,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( dense_order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ? [Z3: A] :
% 2.71/1.57 ( ( ord_less @ A @ X3 @ Z3 )
% 2.71/1.57 & ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_119_less__imp__neq,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ( X3 != Y2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_120_less__asym,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_121_less__asym_H,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_122_less__trans,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A,Z: A] :
% 2.71/1.57 ( ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ( ord_less @ A @ Y2 @ Z )
% 2.71/1.57 => ( ord_less @ A @ X3 @ Z ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_123_less__linear,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 | ( X3 = Y2 )
% 2.71/1.57 | ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_124_less__irrefl,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A] :
% 2.71/1.57 ~ ( ord_less @ A @ X3 @ X3 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_125_ord__eq__less__trans,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ord @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A] :
% 2.71/1.57 ( ( A4 = B4 )
% 2.71/1.57 => ( ( ord_less @ A @ B4 @ C )
% 2.71/1.57 => ( ord_less @ A @ A4 @ C ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_126_ord__less__eq__trans,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ord @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A] :
% 2.71/1.57 ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( B4 = C )
% 2.71/1.57 => ( ord_less @ A @ A4 @ C ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_127_dual__order_Oasym,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [B4: A,A4: A] :
% 2.71/1.57 ( ( ord_less @ A @ B4 @ A4 )
% 2.71/1.57 => ~ ( ord_less @ A @ A4 @ B4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_128_less__imp__not__eq,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ( X3 != Y2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_129_less__not__sym,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_130_less__induct,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [P: A > $o,A4: A] :
% 2.71/1.57 ( ! [X2: A] :
% 2.71/1.57 ( ! [Y5: A] :
% 2.71/1.57 ( ( ord_less @ A @ Y5 @ X2 )
% 2.71/1.57 => ( P @ Y5 ) )
% 2.71/1.57 => ( P @ X2 ) )
% 2.71/1.57 => ( P @ A4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_131_antisym__conv3,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [Y2: A,X3: A] :
% 2.71/1.57 ( ~ ( ord_less @ A @ Y2 @ X3 )
% 2.71/1.57 => ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 2.71/1.57 = ( X3 = Y2 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_132_less__imp__not__eq2,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ( Y2 != X3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_133_less__imp__triv,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A,P: $o] :
% 2.71/1.57 ( ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ( ord_less @ A @ Y2 @ X3 )
% 2.71/1.57 => P ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_134_linorder__cases,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ( X3 != Y2 )
% 2.71/1.57 => ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_135_dual__order_Oirrefl,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A] :
% 2.71/1.57 ~ ( ord_less @ A @ A4 @ A4 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_136_order_Ostrict__trans,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A] :
% 2.71/1.57 ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less @ A @ B4 @ C )
% 2.71/1.57 => ( ord_less @ A @ A4 @ C ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_137_less__imp__not__less,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_138_dual__order_Ostrict__trans,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [B4: A,A4: A,C: A] :
% 2.71/1.57 ( ( ord_less @ A @ B4 @ A4 )
% 2.71/1.57 => ( ( ord_less @ A @ C @ B4 )
% 2.71/1.57 => ( ord_less @ A @ C @ A4 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_139_not__less__iff__gr__or__eq,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 2.71/1.57 = ( ( ord_less @ A @ Y2 @ X3 )
% 2.71/1.57 | ( X3 = Y2 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_140_order_Ostrict__implies__not__eq,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( A4 != B4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_141_dual__order_Ostrict__implies__not__eq,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [B4: A,A4: A] :
% 2.71/1.57 ( ( ord_less @ A @ B4 @ A4 )
% 2.71/1.57 => ( A4 != B4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_142_Iic__subset__Iio__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A4 ) @ ( set_ord_lessThan @ A @ B4 ) )
% 2.71/1.57 = ( ord_less @ A @ A4 @ B4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_143_order_Onot__eq__order__implies__strict,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( A4 != B4 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.71/1.57 => ( ord_less @ A @ A4 @ B4 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_144_dual__order_Ostrict__implies__order,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [B4: A,A4: A] :
% 2.71/1.57 ( ( ord_less @ A @ B4 @ A4 )
% 2.71/1.57 => ( ord_less_eq @ A @ B4 @ A4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_145_dual__order_Ostrict__iff__order,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ( ( ord_less @ A )
% 2.71/1.57 = ( ^ [B6: A,A6: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ B6 @ A6 )
% 2.71/1.57 & ( A6 != B6 ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_146_dual__order_Oorder__iff__strict,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ( ( ord_less_eq @ A )
% 2.71/1.57 = ( ^ [B6: A,A6: A] :
% 2.71/1.57 ( ( ord_less @ A @ B6 @ A6 )
% 2.71/1.57 | ( A6 = B6 ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_147_order_Ostrict__implies__order,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_148_dense__le__bounded,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A,Z: A] :
% 2.71/1.57 ( ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ! [W: A] :
% 2.71/1.57 ( ( ord_less @ A @ X3 @ W )
% 2.71/1.57 => ( ( ord_less @ A @ W @ Y2 )
% 2.71/1.57 => ( ord_less_eq @ A @ W @ Z ) ) )
% 2.71/1.57 => ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_149_dense__ge__bounded,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [Z: A,X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less @ A @ Z @ X3 )
% 2.71/1.57 => ( ! [W: A] :
% 2.71/1.57 ( ( ord_less @ A @ Z @ W )
% 2.71/1.57 => ( ( ord_less @ A @ W @ X3 )
% 2.71/1.57 => ( ord_less_eq @ A @ Y2 @ W ) ) )
% 2.71/1.57 => ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_150_dual__order_Ostrict__trans2,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [B4: A,A4: A,C: A] :
% 2.71/1.57 ( ( ord_less @ A @ B4 @ A4 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ C @ B4 )
% 2.71/1.57 => ( ord_less @ A @ C @ A4 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_151_dual__order_Ostrict__trans1,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [B4: A,A4: A,C: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.71/1.57 => ( ( ord_less @ A @ C @ B4 )
% 2.71/1.57 => ( ord_less @ A @ C @ A4 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_152_order_Ostrict__iff__order,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ( ( ord_less @ A )
% 2.71/1.57 = ( ^ [A6: A,B6: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ A6 @ B6 )
% 2.71/1.57 & ( A6 != B6 ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_153_order_Oorder__iff__strict,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ( ( ord_less_eq @ A )
% 2.71/1.57 = ( ^ [A6: A,B6: A] :
% 2.71/1.57 ( ( ord_less @ A @ A6 @ B6 )
% 2.71/1.57 | ( A6 = B6 ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_154_order_Ostrict__trans2,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A] :
% 2.71/1.57 ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ B4 @ C )
% 2.71/1.57 => ( ord_less @ A @ A4 @ C ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_155_order_Ostrict__trans1,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less @ A @ B4 @ C )
% 2.71/1.57 => ( ord_less @ A @ A4 @ C ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_156_not__le__imp__less,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [Y2: A,X3: A] :
% 2.71/1.57 ( ~ ( ord_less_eq @ A @ Y2 @ X3 )
% 2.71/1.57 => ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_157_less__le__not__le,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ( ( ord_less @ A )
% 2.71/1.57 = ( ^ [X: A,Y: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ X @ Y )
% 2.71/1.57 & ~ ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_158_le__imp__less__or__eq,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 | ( X3 = Y2 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_159_le__less__linear,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.71/1.57 | ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_160_dense__le,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [Y2: A,Z: A] :
% 2.71/1.57 ( ! [X2: A] :
% 2.71/1.57 ( ( ord_less @ A @ X2 @ Y2 )
% 2.71/1.57 => ( ord_less_eq @ A @ X2 @ Z ) )
% 2.71/1.57 => ( ord_less_eq @ A @ Y2 @ Z ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_161_dense__ge,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [Z: A,Y2: A] :
% 2.71/1.57 ( ! [X2: A] :
% 2.71/1.57 ( ( ord_less @ A @ Z @ X2 )
% 2.71/1.57 => ( ord_less_eq @ A @ Y2 @ X2 ) )
% 2.71/1.57 => ( ord_less_eq @ A @ Y2 @ Z ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_162_less__le__trans,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A,Z: A] :
% 2.71/1.57 ( ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ Y2 @ Z )
% 2.71/1.57 => ( ord_less @ A @ X3 @ Z ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_163_le__less__trans,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A,Z: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ( ord_less @ A @ Y2 @ Z )
% 2.71/1.57 => ( ord_less @ A @ X3 @ Z ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_164_antisym__conv2,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 2.71/1.57 = ( X3 = Y2 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_165_antisym__conv1,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.71/1.57 = ( X3 = Y2 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_166_less__imp__le,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( preorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_167_le__neq__trans,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( A4 != B4 )
% 2.71/1.57 => ( ord_less @ A @ A4 @ B4 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_168_not__less,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
% 2.71/1.57 = ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_169_not__le,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ~ ( ord_less_eq @ A @ X3 @ Y2 ) )
% 2.71/1.57 = ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_170_order__less__le__subst2,axiom,
% 2.71/1.57 ! [A: $tType,C2: $tType] :
% 2.71/1.57 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.71/1.57 & ( order @ A @ ( type2 @ A ) ) )
% 2.71/1.57 => ! [A4: A,B4: A,F: A > C2,C: C2] :
% 2.71/1.57 ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less_eq @ C2 @ ( F @ B4 ) @ C )
% 2.71/1.57 => ( ! [X2: A,Y4: A] :
% 2.71/1.57 ( ( ord_less @ A @ X2 @ Y4 )
% 2.71/1.57 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.71/1.57 => ( ord_less @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_171_order__less__le__subst1,axiom,
% 2.71/1.57 ! [A: $tType,B: $tType] :
% 2.71/1.57 ( ( ( order @ B @ ( type2 @ B ) )
% 2.71/1.57 & ( order @ A @ ( type2 @ A ) ) )
% 2.71/1.57 => ! [A4: A,F: B > A,B4: B,C: B] :
% 2.71/1.57 ( ( ord_less @ A @ A4 @ ( F @ B4 ) )
% 2.71/1.57 => ( ( ord_less_eq @ B @ B4 @ C )
% 2.71/1.57 => ( ! [X2: B,Y4: B] :
% 2.71/1.57 ( ( ord_less_eq @ B @ X2 @ Y4 )
% 2.71/1.57 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.71/1.57 => ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_172_order__le__less__subst2,axiom,
% 2.71/1.57 ! [A: $tType,C2: $tType] :
% 2.71/1.57 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.71/1.57 & ( order @ A @ ( type2 @ A ) ) )
% 2.71/1.57 => ! [A4: A,B4: A,F: A > C2,C: C2] :
% 2.71/1.57 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less @ C2 @ ( F @ B4 ) @ C )
% 2.71/1.57 => ( ! [X2: A,Y4: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 2.71/1.57 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.71/1.57 => ( ord_less @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_173_order__le__less__subst1,axiom,
% 2.71/1.57 ! [A: $tType,B: $tType] :
% 2.71/1.57 ( ( ( order @ B @ ( type2 @ B ) )
% 2.71/1.57 & ( order @ A @ ( type2 @ A ) ) )
% 2.71/1.57 => ! [A4: A,F: B > A,B4: B,C: B] :
% 2.71/1.57 ( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
% 2.71/1.57 => ( ( ord_less @ B @ B4 @ C )
% 2.71/1.57 => ( ! [X2: B,Y4: B] :
% 2.71/1.57 ( ( ord_less @ B @ X2 @ Y4 )
% 2.71/1.57 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.71/1.57 => ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_174_less__le,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ( ( ord_less @ A )
% 2.71/1.57 = ( ^ [X: A,Y: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ X @ Y )
% 2.71/1.57 & ( X != Y ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_175_le__less,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 => ( ( ord_less_eq @ A )
% 2.71/1.57 = ( ^ [X: A,Y: A] :
% 2.71/1.57 ( ( ord_less @ A @ X @ Y )
% 2.71/1.57 | ( X = Y ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_176_leI,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ~ ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_177_leD,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [Y2: A,X3: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ Y2 @ X3 )
% 2.71/1.57 => ~ ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_178_subset__iff__psubset__eq,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 2.71/1.57 = ( ^ [A3: set @ A,B3: set @ A] :
% 2.71/1.57 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 2.71/1.57 | ( A3 = B3 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_179_subset__psubset__trans,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 2.71/1.57 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_180_subset__not__subset__eq,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ord_less @ ( set @ A ) )
% 2.71/1.57 = ( ^ [A3: set @ A,B3: set @ A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.71/1.57 & ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_181_psubset__subset__trans,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 2.71/1.57 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 2.71/1.57 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_182_psubset__imp__subset,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_183_psubset__eq,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ord_less @ ( set @ A ) )
% 2.71/1.57 = ( ^ [A3: set @ A,B3: set @ A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.71/1.57 & ( A3 != B3 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_184_psubsetE,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_185_not__Ici__eq__Icc,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( no_top @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [L4: A,L: A,H: A] :
% 2.71/1.57 ( ( set_ord_atLeast @ A @ L4 )
% 2.71/1.57 != ( set_or331188842AtMost @ A @ L @ H ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_186_less__fun__def,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( ord @ B @ ( type2 @ B ) )
% 2.71/1.57 => ( ( ord_less @ ( A > B ) )
% 2.71/1.57 = ( ^ [F2: A > B,G: A > B] :
% 2.71/1.57 ( ( ord_less_eq @ ( A > B ) @ F2 @ G )
% 2.71/1.57 & ~ ( ord_less_eq @ ( A > B ) @ G @ F2 ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_187_not__Iic__eq__Ici,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( no_top @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [H: A,L4: A] :
% 2.71/1.57 ( ( set_ord_atMost @ A @ H )
% 2.71/1.57 != ( set_ord_atLeast @ A @ L4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_188_strict__monoD,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 & ( order @ B @ ( type2 @ B ) ) )
% 2.71/1.57 => ! [F: A > B,X3: A,Y2: A] :
% 2.71/1.57 ( ( order_strict_mono @ A @ B @ F )
% 2.71/1.57 => ( ( ord_less @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_189_strict__monoI,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 & ( order @ B @ ( type2 @ B ) ) )
% 2.71/1.57 => ! [F: A > B] :
% 2.71/1.57 ( ! [X2: A,Y4: A] :
% 2.71/1.57 ( ( ord_less @ A @ X2 @ Y4 )
% 2.71/1.57 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 2.71/1.57 => ( order_strict_mono @ A @ B @ F ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_190_strict__mono__def,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( ( order @ A @ ( type2 @ A ) )
% 2.71/1.57 & ( order @ B @ ( type2 @ B ) ) )
% 2.71/1.57 => ( ( order_strict_mono @ A @ B )
% 2.71/1.57 = ( ^ [F2: A > B] :
% 2.71/1.57 ! [X: A,Y: A] :
% 2.71/1.57 ( ( ord_less @ A @ X @ Y )
% 2.71/1.57 => ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_191_strict__mono__less,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 & ( order @ B @ ( type2 @ B ) ) )
% 2.71/1.57 => ! [F: A > B,X3: A,Y2: A] :
% 2.71/1.57 ( ( order_strict_mono @ A @ B @ F )
% 2.71/1.57 => ( ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) )
% 2.71/1.57 = ( ord_less @ A @ X3 @ Y2 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_192_not__Ici__le__Icc,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( no_top @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [L: A,L4: A,H2: A] :
% 2.71/1.57 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or331188842AtMost @ A @ L4 @ H2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_193_atLeastLessThan__subset__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A,D: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1433837966ssThan @ A @ A4 @ B4 ) @ ( set_or1433837966ssThan @ A @ C @ D ) )
% 2.71/1.57 => ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.71/1.57 | ( ( ord_less_eq @ A @ C @ A4 )
% 2.71/1.57 & ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_194_not__Ici__le__Iic,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( no_top @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [L: A,H2: A] :
% 2.71/1.57 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_ord_atMost @ A @ H2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_195_neg__less__iff__less,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [B4: A,A4: A] :
% 2.71/1.57 ( ( ord_less @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A4 ) )
% 2.71/1.57 = ( ord_less @ A @ A4 @ B4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_196_neg__le__iff__le,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [B4: A,A4: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A4 ) )
% 2.71/1.57 = ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_197_compl__le__compl__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y2 ) )
% 2.71/1.57 = ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_198_compl__eq__compl__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ( uminus_uminus @ A @ X3 )
% 2.71/1.57 = ( uminus_uminus @ A @ Y2 ) )
% 2.71/1.57 = ( X3 = Y2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_199_double__compl,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A] :
% 2.71/1.57 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X3 ) )
% 2.71/1.57 = X3 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_200_neg__equal__iff__equal,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( group_add @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( ( uminus_uminus @ A @ A4 )
% 2.71/1.57 = ( uminus_uminus @ A @ B4 ) )
% 2.71/1.57 = ( A4 = B4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_201_uminus__apply,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( uminus @ B @ ( type2 @ B ) )
% 2.71/1.57 => ( ( uminus_uminus @ ( A > B ) )
% 2.71/1.57 = ( ^ [A3: A > B,X: A] : ( uminus_uminus @ B @ ( A3 @ X ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_202_add_Oinverse__inverse,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( group_add @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A] :
% 2.71/1.57 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A4 ) )
% 2.71/1.57 = A4 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_203_psubsetD,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
% 2.71/1.57 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( ( member @ A @ C @ A2 )
% 2.71/1.57 => ( member @ A @ C @ B2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_204_psubset__trans,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 2.71/1.57 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 2.71/1.57 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 2.71/1.57 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_205_fun__Compl__def,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( uminus @ B @ ( type2 @ B ) )
% 2.71/1.57 => ( ( uminus_uminus @ ( A > B ) )
% 2.71/1.57 = ( ^ [A3: A > B,X: A] : ( uminus_uminus @ B @ ( A3 @ X ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_206_equation__minus__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( group_add @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( A4
% 2.71/1.57 = ( uminus_uminus @ A @ B4 ) )
% 2.71/1.57 = ( B4
% 2.71/1.57 = ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_207_minus__equation__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( group_add @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( ( uminus_uminus @ A @ A4 )
% 2.71/1.57 = B4 )
% 2.71/1.57 = ( ( uminus_uminus @ A @ B4 )
% 2.71/1.57 = A4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_208_le__imp__neg__le,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.71/1.57 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_209_minus__le__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B4 )
% 2.71/1.57 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ A4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_210_le__minus__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ B4 ) )
% 2.71/1.57 = ( ord_less_eq @ A @ B4 @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_211_compl__le__swap2,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [Y2: A,X3: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ X3 )
% 2.71/1.57 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_212_compl__le__swap1,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [Y2: A,X3: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ Y2 @ ( uminus_uminus @ A @ X3 ) )
% 2.71/1.57 => ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_213_compl__mono,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.71/1.57 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_214_compl__less__swap1,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [Y2: A,X3: A] :
% 2.71/1.57 ( ( ord_less @ A @ Y2 @ ( uminus_uminus @ A @ X3 ) )
% 2.71/1.57 => ( ord_less @ A @ X3 @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_215_compl__less__swap2,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [Y2: A,X3: A] :
% 2.71/1.57 ( ( ord_less @ A @ ( uminus_uminus @ A @ Y2 ) @ X3 )
% 2.71/1.57 => ( ord_less @ A @ ( uminus_uminus @ A @ X3 ) @ Y2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_216_less__minus__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( ord_less @ A @ A4 @ ( uminus_uminus @ A @ B4 ) )
% 2.71/1.57 = ( ord_less @ A @ B4 @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_217_minus__less__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( ord_less @ A @ ( uminus_uminus @ A @ A4 ) @ B4 )
% 2.71/1.57 = ( ord_less @ A @ ( uminus_uminus @ A @ B4 ) @ A4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_218_compl__less__compl__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( ord_less @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y2 ) )
% 2.71/1.57 = ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_219_complete__interval,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,P: A > $o] :
% 2.71/1.57 ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( P @ A4 )
% 2.71/1.57 => ( ~ ( P @ B4 )
% 2.71/1.57 => ? [C4: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ A4 @ C4 )
% 2.71/1.57 & ( ord_less_eq @ A @ C4 @ B4 )
% 2.71/1.57 & ! [X4: A] :
% 2.71/1.57 ( ( ( ord_less_eq @ A @ A4 @ X4 )
% 2.71/1.57 & ( ord_less @ A @ X4 @ C4 ) )
% 2.71/1.57 => ( P @ X4 ) )
% 2.71/1.57 & ! [D2: A] :
% 2.71/1.57 ( ! [X2: A] :
% 2.71/1.57 ( ( ( ord_less_eq @ A @ A4 @ X2 )
% 2.71/1.57 & ( ord_less @ A @ X2 @ D2 ) )
% 2.71/1.57 => ( P @ X2 ) )
% 2.71/1.57 => ( ord_less_eq @ A @ D2 @ C4 ) ) ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_220_pinf_I6_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [T3: A] :
% 2.71/1.57 ? [Z3: A] :
% 2.71/1.57 ! [X4: A] :
% 2.71/1.57 ( ( ord_less @ A @ Z3 @ X4 )
% 2.71/1.57 => ~ ( ord_less_eq @ A @ X4 @ T3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_221_ex__gt__or__lt,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A] :
% 2.71/1.57 ? [B5: A] :
% 2.71/1.57 ( ( ord_less @ A @ A4 @ B5 )
% 2.71/1.57 | ( ord_less @ A @ B5 @ A4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_222_pinf_I1_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.71/1.57 ( ? [Z4: A] :
% 2.71/1.57 ! [X2: A] :
% 2.71/1.57 ( ( ord_less @ A @ Z4 @ X2 )
% 2.71/1.57 => ( ( P @ X2 )
% 2.71/1.57 = ( P2 @ X2 ) ) )
% 2.71/1.57 => ( ? [Z4: A] :
% 2.71/1.57 ! [X2: A] :
% 2.71/1.57 ( ( ord_less @ A @ Z4 @ X2 )
% 2.71/1.57 => ( ( Q @ X2 )
% 2.71/1.57 = ( Q2 @ X2 ) ) )
% 2.71/1.57 => ? [Z3: A] :
% 2.71/1.57 ! [X4: A] :
% 2.71/1.57 ( ( ord_less @ A @ Z3 @ X4 )
% 2.71/1.57 => ( ( ( P @ X4 )
% 2.71/1.57 & ( Q @ X4 ) )
% 2.71/1.57 = ( ( P2 @ X4 )
% 2.71/1.57 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_223_pinf_I2_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.71/1.57 ( ? [Z4: A] :
% 2.71/1.57 ! [X2: A] :
% 2.71/1.57 ( ( ord_less @ A @ Z4 @ X2 )
% 2.71/1.57 => ( ( P @ X2 )
% 2.71/1.57 = ( P2 @ X2 ) ) )
% 2.71/1.57 => ( ? [Z4: A] :
% 2.71/1.57 ! [X2: A] :
% 2.71/1.57 ( ( ord_less @ A @ Z4 @ X2 )
% 2.71/1.57 => ( ( Q @ X2 )
% 2.71/1.57 = ( Q2 @ X2 ) ) )
% 2.71/1.57 => ? [Z3: A] :
% 2.71/1.57 ! [X4: A] :
% 2.71/1.57 ( ( ord_less @ A @ Z3 @ X4 )
% 2.71/1.57 => ( ( ( P @ X4 )
% 2.71/1.57 | ( Q @ X4 ) )
% 2.71/1.57 = ( ( P2 @ X4 )
% 2.71/1.57 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_224_pinf_I3_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [T3: A] :
% 2.71/1.57 ? [Z3: A] :
% 2.71/1.57 ! [X4: A] :
% 2.71/1.57 ( ( ord_less @ A @ Z3 @ X4 )
% 2.71/1.57 => ( X4 != T3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_225_pinf_I4_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [T3: A] :
% 2.71/1.57 ? [Z3: A] :
% 2.71/1.57 ! [X4: A] :
% 2.71/1.57 ( ( ord_less @ A @ Z3 @ X4 )
% 2.71/1.57 => ( X4 != T3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_226_pinf_I5_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [T3: A] :
% 2.71/1.57 ? [Z3: A] :
% 2.71/1.57 ! [X4: A] :
% 2.71/1.57 ( ( ord_less @ A @ Z3 @ X4 )
% 2.71/1.57 => ~ ( ord_less @ A @ X4 @ T3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_227_pinf_I7_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [T3: A] :
% 2.71/1.57 ? [Z3: A] :
% 2.71/1.57 ! [X4: A] :
% 2.71/1.57 ( ( ord_less @ A @ Z3 @ X4 )
% 2.71/1.57 => ( ord_less @ A @ T3 @ X4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_228_pinf_I11_J,axiom,
% 2.71/1.57 ! [C2: $tType,D3: $tType] :
% 2.71/1.57 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 2.71/1.57 => ! [F3: D3] :
% 2.71/1.57 ? [Z3: C2] :
% 2.71/1.57 ! [X4: C2] :
% 2.71/1.57 ( ( ord_less @ C2 @ Z3 @ X4 )
% 2.71/1.57 => ( F3 = F3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_229_minf_I1_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.71/1.57 ( ? [Z4: A] :
% 2.71/1.57 ! [X2: A] :
% 2.71/1.57 ( ( ord_less @ A @ X2 @ Z4 )
% 2.71/1.57 => ( ( P @ X2 )
% 2.71/1.57 = ( P2 @ X2 ) ) )
% 2.71/1.57 => ( ? [Z4: A] :
% 2.71/1.57 ! [X2: A] :
% 2.71/1.57 ( ( ord_less @ A @ X2 @ Z4 )
% 2.71/1.57 => ( ( Q @ X2 )
% 2.71/1.57 = ( Q2 @ X2 ) ) )
% 2.71/1.57 => ? [Z3: A] :
% 2.71/1.57 ! [X4: A] :
% 2.71/1.57 ( ( ord_less @ A @ X4 @ Z3 )
% 2.71/1.57 => ( ( ( P @ X4 )
% 2.71/1.57 & ( Q @ X4 ) )
% 2.71/1.57 = ( ( P2 @ X4 )
% 2.71/1.57 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_230_minf_I2_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.71/1.57 ( ? [Z4: A] :
% 2.71/1.57 ! [X2: A] :
% 2.71/1.57 ( ( ord_less @ A @ X2 @ Z4 )
% 2.71/1.57 => ( ( P @ X2 )
% 2.71/1.57 = ( P2 @ X2 ) ) )
% 2.71/1.57 => ( ? [Z4: A] :
% 2.71/1.57 ! [X2: A] :
% 2.71/1.57 ( ( ord_less @ A @ X2 @ Z4 )
% 2.71/1.57 => ( ( Q @ X2 )
% 2.71/1.57 = ( Q2 @ X2 ) ) )
% 2.71/1.57 => ? [Z3: A] :
% 2.71/1.57 ! [X4: A] :
% 2.71/1.57 ( ( ord_less @ A @ X4 @ Z3 )
% 2.71/1.57 => ( ( ( P @ X4 )
% 2.71/1.57 | ( Q @ X4 ) )
% 2.71/1.57 = ( ( P2 @ X4 )
% 2.71/1.57 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_231_minf_I3_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [T3: A] :
% 2.71/1.57 ? [Z3: A] :
% 2.71/1.57 ! [X4: A] :
% 2.71/1.57 ( ( ord_less @ A @ X4 @ Z3 )
% 2.71/1.57 => ( X4 != T3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_232_minf_I4_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [T3: A] :
% 2.71/1.57 ? [Z3: A] :
% 2.71/1.57 ! [X4: A] :
% 2.71/1.57 ( ( ord_less @ A @ X4 @ Z3 )
% 2.71/1.57 => ( X4 != T3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_233_minf_I5_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [T3: A] :
% 2.71/1.57 ? [Z3: A] :
% 2.71/1.57 ! [X4: A] :
% 2.71/1.57 ( ( ord_less @ A @ X4 @ Z3 )
% 2.71/1.57 => ( ord_less @ A @ X4 @ T3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_234_minf_I7_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [T3: A] :
% 2.71/1.57 ? [Z3: A] :
% 2.71/1.57 ! [X4: A] :
% 2.71/1.57 ( ( ord_less @ A @ X4 @ Z3 )
% 2.71/1.57 => ~ ( ord_less @ A @ T3 @ X4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_235_minf_I11_J,axiom,
% 2.71/1.57 ! [C2: $tType,D3: $tType] :
% 2.71/1.57 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 2.71/1.57 => ! [F3: D3] :
% 2.71/1.57 ? [Z3: C2] :
% 2.71/1.57 ! [X4: C2] :
% 2.71/1.57 ( ( ord_less @ C2 @ X4 @ Z3 )
% 2.71/1.57 => ( F3 = F3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_236_minf_I8_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [T3: A] :
% 2.71/1.57 ? [Z3: A] :
% 2.71/1.57 ! [X4: A] :
% 2.71/1.57 ( ( ord_less @ A @ X4 @ Z3 )
% 2.71/1.57 => ~ ( ord_less_eq @ A @ T3 @ X4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_237_minf_I6_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [T3: A] :
% 2.71/1.57 ? [Z3: A] :
% 2.71/1.57 ! [X4: A] :
% 2.71/1.57 ( ( ord_less @ A @ X4 @ Z3 )
% 2.71/1.57 => ( ord_less_eq @ A @ X4 @ T3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_238_pinf_I8_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [T3: A] :
% 2.71/1.57 ? [Z3: A] :
% 2.71/1.57 ! [X4: A] :
% 2.71/1.57 ( ( ord_less @ A @ Z3 @ X4 )
% 2.71/1.57 => ( ord_less_eq @ A @ T3 @ X4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_239_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A,D: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_or578182835ssThan @ A @ A4 @ B4 ) @ ( set_or1433837966ssThan @ A @ C @ D ) )
% 2.71/1.57 = ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ C @ A4 )
% 2.71/1.57 & ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_240_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A,D: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_or578182835ssThan @ A @ A4 @ B4 ) @ ( set_or331188842AtMost @ A @ C @ D ) )
% 2.71/1.57 = ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ C @ A4 )
% 2.71/1.57 & ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_241_greaterThanLessThan__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( ord @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [I: A,L: A,U: A] :
% 2.71/1.57 ( ( member @ A @ I @ ( set_or578182835ssThan @ A @ L @ U ) )
% 2.71/1.57 = ( ( ord_less @ A @ L @ I )
% 2.71/1.57 & ( ord_less @ A @ I @ U ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_242_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A,D: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_or578182835ssThan @ A @ A4 @ B4 ) @ ( set_or578182835ssThan @ A @ C @ D ) )
% 2.71/1.57 = ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ C @ A4 )
% 2.71/1.57 & ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_243_ivl__disj__un__two_I4_J,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [L: A,M: A,U: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ L @ M )
% 2.71/1.57 => ( ( ord_less @ A @ M @ U )
% 2.71/1.57 => ( ( sup_sup @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ M ) @ ( set_or578182835ssThan @ A @ M @ U ) )
% 2.71/1.57 = ( set_or1433837966ssThan @ A @ L @ U ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_244_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A,C: A,D: A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_or578182835ssThan @ A @ A4 @ B4 ) @ ( set_or1361889807AtMost @ A @ C @ D ) )
% 2.71/1.57 = ( ( ord_less @ A @ A4 @ B4 )
% 2.71/1.57 => ( ( ord_less_eq @ A @ C @ A4 )
% 2.71/1.57 & ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_245_sup__apply,axiom,
% 2.71/1.57 ! [B: $tType,A: $tType] :
% 2.71/1.57 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 2.71/1.57 => ( ( sup_sup @ ( A > B ) )
% 2.71/1.57 = ( ^ [F2: A > B,G: A > B,X: A] : ( sup_sup @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_246_sup_Oidem,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A] :
% 2.71/1.57 ( ( sup_sup @ A @ A4 @ A4 )
% 2.71/1.57 = A4 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_247_sup__idem,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A] :
% 2.71/1.57 ( ( sup_sup @ A @ X3 @ X3 )
% 2.71/1.57 = X3 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_248_sup_Oleft__idem,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( sup_sup @ A @ A4 @ ( sup_sup @ A @ A4 @ B4 ) )
% 2.71/1.57 = ( sup_sup @ A @ A4 @ B4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_249_sup__left__idem,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A] :
% 2.71/1.57 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
% 2.71/1.57 = ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_250_sup_Oright__idem,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [A4: A,B4: A] :
% 2.71/1.57 ( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B4 ) @ B4 )
% 2.71/1.57 = ( sup_sup @ A @ A4 @ B4 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_251_UnCI,axiom,
% 2.71/1.57 ! [A: $tType,C: A,B2: set @ A,A2: set @ A] :
% 2.71/1.57 ( ( ~ ( member @ A @ C @ B2 )
% 2.71/1.57 => ( member @ A @ C @ A2 ) )
% 2.71/1.57 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_252_Un__iff,axiom,
% 2.71/1.57 ! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
% 2.71/1.57 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.71/1.57 = ( ( member @ A @ C @ A2 )
% 2.71/1.57 | ( member @ A @ C @ B2 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_253_sup_Obounded__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [B4: A,C: A,A4: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B4 @ C ) @ A4 )
% 2.71/1.57 = ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.71/1.57 & ( ord_less_eq @ A @ C @ A4 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_254_le__sup__iff,axiom,
% 2.71/1.57 ! [A: $tType] :
% 2.71/1.57 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.71/1.57 => ! [X3: A,Y2: A,Z: A] :
% 2.71/1.57 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z )
% 2.71/1.57 = ( ( ord_less_eq @ A @ X3 @ Z )
% 2.71/1.57 & ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(fact_255_Un__subset__iff,axiom,
% 2.71/1.57 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 2.71/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 )
% 2.71/1.57 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 2.71/1.57 & ( ord_less_eq @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 2.71/1.57 ! [A7: $tType,A8: $tType] :
% 2.71/1.57 ( ( semilattice_sup @ A8 @ ( type2 @ A8 ) )
% 2.71/1.57 => ( semilattice_sup @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 2.71/1.57 ! [A7: $tType,A8: $tType] :
% 2.71/1.57 ( ( boolean_algebra @ A8 @ ( type2 @ A8 ) )
% 2.71/1.57 => ( boolean_algebra @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.71/1.57 ! [A7: $tType,A8: $tType] :
% 2.71/1.57 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 2.71/1.57 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.71/1.57 ! [A7: $tType,A8: $tType] :
% 2.71/1.57 ( ( order @ A8 @ ( type2 @ A8 ) )
% 2.71/1.57 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_fun___Orderings_Oord,axiom,
% 2.71/1.57 ! [A7: $tType,A8: $tType] :
% 2.71/1.57 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 2.71/1.57 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_fun___Groups_Ouminus,axiom,
% 2.71/1.57 ! [A7: $tType,A8: $tType] :
% 2.71/1.57 ( ( uminus @ A8 @ ( type2 @ A8 ) )
% 2.71/1.57 => ( uminus @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
% 2.71/1.57 ! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_2,axiom,
% 2.71/1.57 ! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
% 2.71/1.57 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_Set_Oset___Orderings_Oorder_4,axiom,
% 2.71/1.57 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
% 2.71/1.57 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_Set_Oset___Groups_Ouminus_6,axiom,
% 2.71/1.57 ! [A7: $tType] : ( uminus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_7,axiom,
% 2.71/1.57 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_8,axiom,
% 2.71/1.57 boolean_algebra @ $o @ ( type2 @ $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_HOL_Obool___Orderings_Opreorder_9,axiom,
% 2.71/1.57 preorder @ $o @ ( type2 @ $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.71/1.57 linorder @ $o @ ( type2 @ $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_HOL_Obool___Orderings_Oorder_10,axiom,
% 2.71/1.57 order @ $o @ ( type2 @ $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_HOL_Obool___Orderings_Oord_11,axiom,
% 2.71/1.57 ord @ $o @ ( type2 @ $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(tcon_HOL_Obool___Groups_Ouminus_12,axiom,
% 2.71/1.57 uminus @ $o @ ( type2 @ $o ) ).
% 2.71/1.57
% 2.71/1.57 thf(conj_0,hypothesis,
% 2.71/1.57 ( ( gram_L608943123e_subs @ a @ l1 @ l2 )
% 2.71/1.57 & ( gram_L608943123e_subs @ a @ l2 @ l1 ) ) ).
% 2.71/1.57
% 2.71/1.57 thf(conj_1,conjecture,
% 2.71/1.57 gram_L608943123e_subs @ a @ l1 @ l2 ).
% 2.71/1.57 ------------------------
% 2.82/1.60 % (12246)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_12112 for (2999ds/183Mi)
% 2.82/1.60 % (12251)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_12112 for (2999ds/275Mi)
% 2.82/1.60 % (12248)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_12112 for (2999ds/27Mi)
% 2.82/1.61 % (12247)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_12112 for (2999ds/4Mi)
% 2.82/1.61 % (12249)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_12112 for (2999ds/2Mi)
% 2.82/1.61 % (12252)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_12112 for (2999ds/18Mi)
% 2.82/1.61 % (12250)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_12112 for (2999ds/2Mi)
% 2.82/1.61 % (12249)Instruction limit reached!
% 2.82/1.61 % (12249)------------------------------
% 2.82/1.61 % (12249)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.61 % (12249)Termination reason: Unknown
% 2.82/1.61 % (12249)Termination phase: shuffling
% 2.82/1.61 % (12250)Instruction limit reached!
% 2.82/1.61 % (12250)------------------------------
% 2.82/1.61 % (12250)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.61 % (12250)Termination reason: Unknown
% 2.82/1.61 % (12250)Termination phase: shuffling
% 2.82/1.61
% 2.82/1.61 % (12250)Memory used [KB]: 1407
% 2.82/1.61 % (12250)Time elapsed: 0.003 s
% 2.82/1.61 % (12250)Instructions burned: 2 (million)
% 2.82/1.61 % (12250)------------------------------
% 2.82/1.61 % (12250)------------------------------
% 2.82/1.61
% 2.82/1.61 % (12249)Memory used [KB]: 1407
% 2.82/1.61 % (12249)Time elapsed: 0.004 s
% 2.82/1.61 % (12249)Instructions burned: 3 (million)
% 2.82/1.61 % (12249)------------------------------
% 2.82/1.61 % (12249)------------------------------
% 2.82/1.61 % (12247)Instruction limit reached!
% 2.82/1.61 % (12247)------------------------------
% 2.82/1.61 % (12247)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.61 % (12247)Termination reason: Unknown
% 2.82/1.61 % (12247)Termination phase: shuffling
% 2.82/1.61
% 2.82/1.61 % (12247)Memory used [KB]: 1407
% 2.82/1.61 % (12247)Time elapsed: 0.005 s
% 2.82/1.61 % (12247)Instructions burned: 5 (million)
% 2.82/1.61 % (12247)------------------------------
% 2.82/1.61 % (12247)------------------------------
% 2.82/1.62 % (12248)Instruction limit reached!
% 2.82/1.62 % (12248)------------------------------
% 2.82/1.62 % (12248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.62 % (12248)Termination reason: Unknown
% 2.82/1.62 % (12248)Termination phase: Property scanning
% 2.82/1.62
% 2.82/1.62 % (12248)Memory used [KB]: 1791
% 2.82/1.62 % (12248)Time elapsed: 0.013 s
% 2.82/1.62 % (12248)Instructions burned: 27 (million)
% 2.82/1.62 % (12248)------------------------------
% 2.82/1.62 % (12248)------------------------------
% 2.82/1.62 % (12251)First to succeed.
% 2.82/1.62 % (12251)Refutation found. Thanks to Tanya!
% 2.82/1.62 % SZS status Theorem for DTF2THF_12112
% 2.82/1.62 % SZS output start Proof for DTF2THF_12112
% 2.82/1.62 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.82/1.62 thf(type_def_7, type, set: ($tType) > $tType).
% 2.82/1.62 thf(type_def_8, type, a: $tType).
% 2.82/1.62 thf(func_def_0, type, set: ($tType) > $tType).
% 2.82/1.62 thf(func_def_1, type, itself: ($tType) > $tType).
% 2.82/1.62 thf(func_def_2, type, a: $tType).
% 2.82/1.62 thf(func_def_3, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_4, type, uminus: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_5, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_6, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_7, type, group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_8, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_9, type, no_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_10, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_11, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_12, type, wellorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_13, type, dense_order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_14, type, boolean_algebra: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_15, type, semilattice_sup: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_16, type, dense_linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_17, type, ordered_ab_group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_18, type, condit1656338222tinuum: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_19, type, condit1037483654norder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.82/1.62 thf(func_def_20, type, gram_L1456083582e_leqv: !>[X0: $tType]:(set @ set @ X0 > set @ set @ X0 > $o)).
% 2.82/1.62 thf(func_def_21, type, gram_L608943123e_subs: !>[X0: $tType]:(set @ set @ X0 > set @ set @ X0 > $o)).
% 2.82/1.62 thf(func_def_22, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 2.82/1.62 thf(func_def_23, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.82/1.62 thf(func_def_24, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.82/1.62 thf(func_def_25, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.82/1.62 thf(func_def_26, type, order_antimono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 2.82/1.62 thf(func_def_27, type, order_strict_mono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 2.82/1.62 thf(func_def_28, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.82/1.62 thf(func_def_29, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.82/1.62 thf(func_def_30, type, pow: !>[X0: $tType]:(set @ X0 > set @ set @ X0)).
% 2.82/1.62 thf(func_def_31, type, pairwise: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > $o)).
% 2.82/1.62 thf(func_def_32, type, set_ord_atLeast: !>[X0: $tType]:(X0 > set @ X0)).
% 2.82/1.62 thf(func_def_33, type, set_or331188842AtMost: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 2.82/1.62 thf(func_def_34, type, set_or1433837966ssThan: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 2.82/1.62 thf(func_def_35, type, set_ord_atMost: !>[X0: $tType]:(X0 > set @ X0)).
% 2.82/1.62 thf(func_def_36, type, set_ord_greaterThan: !>[X0: $tType]:(X0 > set @ X0)).
% 2.82/1.62 thf(func_def_37, type, set_or1361889807AtMost: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 2.82/1.62 thf(func_def_38, type, set_or578182835ssThan: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 2.82/1.62 thf(func_def_39, type, set_ord_lessThan: !>[X0: $tType]:(X0 > set @ X0)).
% 2.82/1.62 thf(func_def_40, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.82/1.62 thf(func_def_41, type, l1: set @ set @ a).
% 2.82/1.62 thf(func_def_42, type, l2: set @ set @ a).
% 2.82/1.62 thf(func_def_58, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.82/1.62 thf(f873,plain,(
% 2.82/1.62 $false),
% 2.82/1.62 inference(subsumption_resolution,[status(thm)],[f852,f861])).
% 2.82/1.62 thf(f861,plain,(
% 2.82/1.62 ((gram_L608943123e_subs @ a @ l1 @ l2) = $true)),
% 2.82/1.62 inference(cnf_transformation,[status(thm)],[f301])).
% 2.82/1.62 thf(f301,plain,(
% 2.82/1.62 ((gram_L608943123e_subs @ a @ l2 @ l1) = $true) & ((gram_L608943123e_subs @ a @ l1 @ l2) = $true)),
% 2.82/1.62 inference(fool_elimination,[status(thm)],[f300])).
% 2.82/1.62 thf(f300,plain,(
% 2.82/1.62 (gram_L608943123e_subs @ a @ l2 @ l1) & (gram_L608943123e_subs @ a @ l1 @ l2)),
% 2.82/1.62 inference(rectify,[status(thm)],[f276])).
% 2.82/1.62 thf(f276,axiom,(
% 2.82/1.62 (gram_L608943123e_subs @ a @ l2 @ l1) & (gram_L608943123e_subs @ a @ l1 @ l2)),
% 2.82/1.62 file('/export/starexec/sandbox/tmp/tmp.LeYYwvo4TN/DTF2THF_12112.p',conj_0)).
% 2.82/1.62 thf(f852,plain,(
% 2.82/1.62 ((gram_L608943123e_subs @ a @ l1 @ l2) != $true)),
% 2.82/1.62 inference(cnf_transformation,[status(thm)],[f822])).
% 2.82/1.62 thf(f822,plain,(
% 2.82/1.62 ((gram_L608943123e_subs @ a @ l1 @ l2) != $true)),
% 2.82/1.62 inference(flattening,[status(thm)],[f325])).
% 2.82/1.62 thf(f325,plain,(
% 2.82/1.62 ~((gram_L608943123e_subs @ a @ l1 @ l2) = $true)),
% 2.82/1.62 inference(fool_elimination,[status(thm)],[f324])).
% 2.82/1.62 thf(f324,plain,(
% 2.82/1.62 ~(gram_L608943123e_subs @ a @ l1 @ l2)),
% 2.82/1.62 inference(rectify,[status(thm)],[f278])).
% 2.82/1.62 thf(f278,negated_conjecture,(
% 2.82/1.62 ~(gram_L608943123e_subs @ a @ l1 @ l2)),
% 2.82/1.62 inference(negated_conjecture,[status(cth)],[f277])).
% 2.82/1.62 thf(f277,conjecture,(
% 2.82/1.62 (gram_L608943123e_subs @ a @ l1 @ l2)),
% 2.82/1.62 file('/export/starexec/sandbox/tmp/tmp.LeYYwvo4TN/DTF2THF_12112.p',conj_1)).
% 2.82/1.62 % SZS output end Proof for DTF2THF_12112
% 2.82/1.62 % (12251)------------------------------
% 2.82/1.62 % (12251)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.62 % (12251)Termination reason: Refutation
% 2.82/1.62
% 2.82/1.62 % (12251)Memory used [KB]: 6396
% 2.82/1.62 % (12251)Time elapsed: 0.014 s
% 2.82/1.62 % (12251)Instructions burned: 38 (million)
% 2.82/1.62 % (12251)------------------------------
% 2.82/1.62 % (12251)------------------------------
% 2.82/1.62 % (12245)Success in time 0.024 s
%------------------------------------------------------------------------------