%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM196^1 : 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 : n011.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 2.85s 1.60s
% Output : Refutation 2.88s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : COM196^1 : TPTP v9.2.1. Released v7.0.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.34 % Computer : n011.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 04:17:02 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.46 ---- Original DTF file ---
% 0.20/0.46 thf(spec,logic,$$dhol).
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % File : COM196^1 : TPTP v9.2.1. Released v7.0.0.
% 0.20/0.46 % Domain : Computing Theory
% 0.20/0.46 % Problem : Grammars and languages 1357
% 0.20/0.46 % Version : [Bla16] axioms : Especial.
% 0.20/0.46 % English :
% 0.20/0.46
% 0.20/0.46 % Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% 0.20/0.46 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.20/0.46 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.20/0.46 % Source : [Bla16]
% 0.20/0.46 % Names : gram_lang__1357.p [Bla16]
% 0.20/0.46
% 0.20/0.46 % Status : Theorem
% 0.20/0.46 % Rating : 0.33 v8.1.0, 0.25 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% 0.20/0.46 % Syntax : Number of formulae : 326 ( 79 unt; 41 typ; 0 def)
% 0.20/0.46 % Number of atoms : 913 ( 176 equ; 0 cnn)
% 0.20/0.46 % Maximal formula atoms : 13 ( 3 avg)
% 0.20/0.46 % Number of connectives : 3609 ( 87 ~; 16 |; 48 &;2971 @)
% 0.20/0.46 % ( 0 <=>; 487 =>; 0 <=; 0 <~>)
% 0.20/0.46 % Maximal formula depth : 21 ( 8 avg)
% 0.20/0.46 % Number of types : 3 ( 2 usr)
% 0.20/0.46 % Number of type conns : 166 ( 166 >; 0 *; 0 +; 0 <<)
% 0.20/0.46 % Number of symbols : 40 ( 39 usr; 1 con; 0-4 aty)
% 0.20/0.46 % Number of variables : 999 ( 50 ^; 872 !; 43 ?; 999 :)
% 0.20/0.46 % ( 34 !>; 0 ?*; 0 @-; 0 @+)
% 0.20/0.46 % SPC : TH1_THM_EQU_NAR
% 0.20/0.46
% 0.20/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.20/0.46 % 2016-07-13 14:48:39.061
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 %----Could-be-implicit typings (4)
% 0.20/0.46 thf(ty_t_Set_Oset,type,
% 0.20/0.46 set: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_DTree_OT,type,
% 0.20/0.46 t: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_DTree_ON,type,
% 0.20/0.46 n: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_itself,type,
% 0.20/0.46 itself: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 %----Explicit typings (37)
% 0.20/0.46 thf(sy_cl_HOL_Otype,type,
% 0.20/0.46 type:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Obot,type,
% 0.20/0.46 bot:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.20/0.46 ord:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Otop,type,
% 0.20/0.46 top:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.20/0.46 order:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Ono__bot,type,
% 0.20/0.46 no_bot:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Ono__top,type,
% 0.20/0.46 no_top:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.20/0.46 linorder:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.20/0.46 preorder:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.20/0.46 order_bot:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Oorder__top,type,
% 0.20/0.46 order_top:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Owellorder,type,
% 0.20/0.46 wellorder:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Odense__order,type,
% 0.20/0.46 dense_order:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.20/0.46 dense_linorder:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.20/0.46 comple187826305attice:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
% 0.20/0.46 comple1035589618norder:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.20/0.46 condit1656338222tinuum:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
% 0.20/0.46 condit378418413attice:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.20/0.46 condit1037483654norder:
% 0.20/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
% 0.20/0.46 complete_Sup_Sup:
% 0.20/0.46 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OL,type,
% 0.20/0.46 gram_L1451583628elle_L: ( set @ n ) > n > ( set @ ( set @ t ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OLL,type,
% 0.20/0.46 gram_L861977280lle_LL: ( set @ n ) > n > ( set @ ( set @ t ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OLr,type,
% 0.20/0.46 gram_L861977318lle_Lr: ( set @ n ) > n > ( set @ ( set @ t ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Oleqv,type,
% 0.20/0.46 gram_L1456083582e_leqv:
% 0.20/0.46 !>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ A ) ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Osubs,type,
% 0.20/0.46 gram_L608943123e_subs:
% 0.20/0.46 !>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ A ) ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.20/0.46 bot_bot:
% 0.20/0.46 !>[A: $tType] : A ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.20/0.46 ord_less:
% 0.20/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.20/0.46 ord_less_eq:
% 0.20/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 0.20/0.46 order_antimono:
% 0.20/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Oordering__top,type,
% 0.20/0.46 ordering_top:
% 0.20/0.46 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.20/0.46 top_top:
% 0.20/0.46 !>[A: $tType] : A ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Pure_Otype,type,
% 0.20/0.46 type2:
% 0.20/0.46 !>[A: $tType] : ( itself @ A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_OCollect,type,
% 0.20/0.46 collect:
% 0.20/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_OPow,type,
% 0.20/0.46 pow:
% 0.20/0.46 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_Ois__empty,type,
% 0.20/0.46 is_empty:
% 0.20/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_member,type,
% 0.20/0.46 member:
% 0.20/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_ts,type,
% 0.20/0.46 ts: n ).
% 0.20/0.46
% 0.20/0.46 %----Relevant facts (256)
% 0.20/0.46 thf(fact_0_Lr__leqv__L,axiom,
% 0.20/0.46 ! [Ts: n] : ( gram_L1456083582e_leqv @ t @ ( gram_L861977318lle_Lr @ ( top_top @ ( set @ n ) ) @ Ts ) @ ( gram_L1451583628elle_L @ ( top_top @ ( set @ n ) ) @ Ts ) ) ).
% 0.20/0.46
% 0.20/0.46 % Lr_leqv_L
% 0.20/0.46 thf(fact_1_LL__leqv__L,axiom,
% 0.20/0.46 ! [Ts: n] : ( gram_L1456083582e_leqv @ t @ ( gram_L861977280lle_LL @ ( top_top @ ( set @ n ) ) @ Ts ) @ ( gram_L1451583628elle_L @ ( top_top @ ( set @ n ) ) @ Ts ) ) ).
% 0.20/0.46
% 0.20/0.46 % LL_leqv_L
% 0.20/0.46 thf(fact_2_leqv__Sym,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( gram_L1456083582e_leqv @ A )
% 0.20/0.46 = ( ^ [L1: set @ ( set @ A ),L2: set @ ( set @ A )] : ( gram_L1456083582e_leqv @ A @ L2 @ L1 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % leqv_Sym
% 0.20/0.46 thf(fact_3_leqv__sym,axiom,
% 0.20/0.46 ! [A: $tType,L12: set @ ( set @ A ),L22: set @ ( set @ A )] :
% 0.20/0.46 ( ( gram_L1456083582e_leqv @ A @ L12 @ L22 )
% 0.20/0.46 => ( gram_L1456083582e_leqv @ A @ L22 @ L12 ) ) ).
% 0.20/0.46
% 0.20/0.46 % leqv_sym
% 0.20/0.46 thf(fact_4_leqv__refl,axiom,
% 0.20/0.46 ! [A: $tType,L12: set @ ( set @ A )] : ( gram_L1456083582e_leqv @ A @ L12 @ L12 ) ).
% 0.20/0.46
% 0.20/0.46 % leqv_refl
% 0.20/0.46 thf(fact_5_leqv__trans,axiom,
% 0.20/0.46 ! [A: $tType,L12: set @ ( set @ A ),L22: set @ ( set @ A ),L3: set @ ( set @ A )] :
% 0.20/0.46 ( ( gram_L1456083582e_leqv @ A @ L12 @ L22 )
% 0.20/0.46 => ( ( gram_L1456083582e_leqv @ A @ L22 @ L3 )
% 0.20/0.46 => ( gram_L1456083582e_leqv @ A @ L12 @ L3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % leqv_trans
% 0.20/0.46 thf(fact_6_UNIV__I,axiom,
% 0.20/0.46 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % UNIV_I
% 0.20/0.46 thf(fact_7_iso__tuple__UNIV__I,axiom,
% 0.20/0.46 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % iso_tuple_UNIV_I
% 0.20/0.46 thf(fact_8_top__apply,axiom,
% 0.20/0.46 ! [C: $tType,D: $tType] :
% 0.20/0.46 ( ( top @ C @ ( type2 @ C ) )
% 0.20/0.46 => ( ( top_top @ ( D > C ) )
% 0.20/0.46 = ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % top_apply
% 0.20/0.46 thf(fact_9_Lr__subs__L,axiom,
% 0.20/0.46 ! [Ts: n] : ( gram_L608943123e_subs @ t @ ( gram_L861977318lle_Lr @ ( top_top @ ( set @ n ) ) @ Ts ) @ ( gram_L1451583628elle_L @ ( top_top @ ( set @ n ) ) @ Ts ) ) ).
% 0.20/0.46
% 0.20/0.46 % Lr_subs_L
% 0.20/0.46 thf(fact_10_UNIV__eq__I,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A] :
% 0.20/0.46 ( ! [X3: A] : ( member @ A @ X3 @ A2 )
% 0.20/0.46 => ( ( top_top @ ( set @ A ) )
% 0.20/0.46 = A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % UNIV_eq_I
% 0.20/0.46 thf(fact_11_UNIV__witness,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % UNIV_witness
% 0.20/0.46 thf(fact_12_Lr__LL,axiom,
% 0.20/0.46 ! [Ns: set @ n,N: n] : ( ord_less_eq @ ( set @ ( set @ t ) ) @ ( gram_L861977318lle_Lr @ Ns @ N ) @ ( gram_L861977280lle_LL @ Ns @ N ) ) ).
% 0.20/0.46
% 0.20/0.46 % Lr_LL
% 0.20/0.46 thf(fact_13_leqv__def,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( gram_L1456083582e_leqv @ A )
% 0.20/0.46 = ( ^ [L1: set @ ( set @ A ),L2: set @ ( set @ A )] :
% 0.20/0.46 ( ( gram_L608943123e_subs @ A @ L1 @ L2 )
% 0.20/0.46 & ( gram_L608943123e_subs @ A @ L2 @ L1 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % leqv_def
% 0.20/0.46 thf(fact_14_subs__leqv,axiom,
% 0.20/0.46 ! [A: $tType,L12: set @ ( set @ A ),L22: set @ ( set @ A )] :
% 0.20/0.46 ( ( gram_L1456083582e_leqv @ A @ L12 @ L22 )
% 0.20/0.46 => ( gram_L608943123e_subs @ A @ L12 @ L22 ) ) ).
% 0.20/0.46
% 0.20/0.46 % subs_leqv
% 0.20/0.46 thf(fact_15_subs__leqv__sym,axiom,
% 0.20/0.46 ! [A: $tType,L12: set @ ( set @ A ),L22: set @ ( set @ A )] :
% 0.20/0.46 ( ( gram_L1456083582e_leqv @ A @ L12 @ L22 )
% 0.20/0.46 => ( gram_L608943123e_subs @ A @ L22 @ L12 ) ) ).
% 0.20/0.46
% 0.20/0.46 % subs_leqv_sym
% 0.20/0.46 thf(fact_16_Sup__UNIV,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.20/0.46 => ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
% 0.20/0.46 = ( top_top @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Sup_UNIV
% 0.20/0.46 thf(fact_17_LL__L,axiom,
% 0.20/0.46 ! [Ns: set @ n,N: n] : ( ord_less_eq @ ( set @ ( set @ t ) ) @ ( gram_L861977280lle_LL @ Ns @ N ) @ ( gram_L1451583628elle_L @ Ns @ N ) ) ).
% 0.20/0.46
% 0.20/0.46 % LL_L
% 0.20/0.46 thf(fact_18_order__refl,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.20/0.46
% 0.20/0.46 % order_refl
% 0.20/0.46 thf(fact_19_subsetI,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.20/0.46 ( ! [X3: A] :
% 0.20/0.46 ( ( member @ A @ X3 @ A2 )
% 0.20/0.46 => ( member @ A @ X3 @ B2 ) )
% 0.20/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % subsetI
% 0.20/0.46 thf(fact_20_subset__antisym,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.20/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.20/0.46 => ( A2 = B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % subset_antisym
% 0.20/0.46 thf(fact_21_set__mp,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.20/0.46 => ( ( member @ A @ X @ A2 )
% 0.20/0.46 => ( member @ A @ X @ B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % set_mp
% 0.20/0.46 thf(fact_22_in__mono,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.20/0.46 => ( ( member @ A @ X @ A2 )
% 0.20/0.46 => ( member @ A @ X @ B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % in_mono
% 0.20/0.46 thf(fact_23_subsetD,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.20/0.46 => ( ( member @ A @ C2 @ A2 )
% 0.20/0.46 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % subsetD
% 0.20/0.46 thf(fact_24_subsetCE,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.20/0.46 => ( ( member @ A @ C2 @ A2 )
% 0.20/0.46 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % subsetCE
% 0.20/0.46 thf(fact_25_equalityE,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.20/0.46 ( ( A2 = B2 )
% 0.20/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.20/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % equalityE
% 0.20/0.46 thf(fact_26_subset__eq,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.20/0.46 = ( ^ [A3: set @ A,B3: set @ A] :
% 0.20/0.46 ! [X2: A] :
% 0.20/0.46 ( ( member @ A @ X2 @ A3 )
% 0.20/0.46 => ( member @ A @ X2 @ B3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % subset_eq
% 0.20/0.46 thf(fact_27_equalityD1,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.20/0.46 ( ( A2 = B2 )
% 0.20/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % equalityD1
% 0.20/0.46 thf(fact_28_equalityD2,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.20/0.46 ( ( A2 = B2 )
% 0.20/0.46 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % equalityD2
% 0.20/0.46 thf(fact_29_set__rev__mp,axiom,
% 0.20/0.46 ! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
% 0.20/0.46 ( ( member @ A @ X @ A2 )
% 0.20/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.20/0.46 => ( member @ A @ X @ B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % set_rev_mp
% 0.20/0.46 thf(fact_30_subset__iff,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.20/0.46 = ( ^ [A3: set @ A,B3: set @ A] :
% 0.20/0.46 ! [T: A] :
% 0.20/0.46 ( ( member @ A @ T @ A3 )
% 0.20/0.46 => ( member @ A @ T @ B3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % subset_iff
% 0.20/0.46 thf(fact_31_rev__subsetD,axiom,
% 0.20/0.46 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 0.20/0.46 ( ( member @ A @ C2 @ A2 )
% 0.20/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.20/0.46 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % rev_subsetD
% 0.20/0.46 thf(fact_32_subset__refl,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.20/0.46
% 0.20/0.46 % subset_refl
% 0.20/0.46 thf(fact_33_Collect__mono,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.20/0.46 ( ! [X3: A] :
% 0.20/0.46 ( ( P @ X3 )
% 0.20/0.46 => ( Q @ X3 ) )
% 0.20/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_mono
% 0.20/0.46 thf(fact_34_subset__trans,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.20/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.20/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % subset_trans
% 0.20/0.46 thf(fact_35_le__funD,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType] :
% 0.20/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.20/0.46 => ! [F: A > B,G: A > B,X: A] :
% 0.20/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.20/0.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_funD
% 0.20/0.46 thf(fact_36_le__funE,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType] :
% 0.20/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.20/0.46 => ! [F: A > B,G: A > B,X: A] :
% 0.20/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.20/0.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_funE
% 0.20/0.46 thf(fact_37_le__funI,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType] :
% 0.20/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.20/0.46 => ! [F: A > B,G: A > B] :
% 0.20/0.46 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.20/0.46 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_funI
% 0.20/0.46 thf(fact_38_set__eq__subset,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( ^ [Y: set @ A,Z: set @ A] : ( Y = Z ) )
% 0.20/0.46 = ( ^ [A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.20/0.46 & ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % set_eq_subset
% 0.20/0.46 thf(fact_39_contra__subsetD,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.20/0.46 => ( ~ ( member @ A @ C2 @ B2 )
% 0.20/0.46 => ~ ( member @ A @ C2 @ A2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % contra_subsetD
% 0.20/0.46 thf(fact_40_le__fun__def,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType] :
% 0.20/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.20/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.20/0.46 = ( ^ [F2: A > B,G2: A > B] :
% 0.20/0.46 ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_fun_def
% 0.20/0.46 thf(fact_41_Collect__mono__iff,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.20/0.46 = ( ! [X2: A] :
% 0.20/0.46 ( ( P @ X2 )
% 0.20/0.46 => ( Q @ X2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_mono_iff
% 0.20/0.46 thf(fact_42_order__subst1,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType] :
% 0.20/0.46 ( ( ( order @ B @ ( type2 @ B ) )
% 0.20/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.20/0.46 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 0.20/0.46 ( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
% 0.20/0.46 => ( ( ord_less_eq @ B @ B4 @ C2 )
% 0.20/0.46 => ( ! [X3: B,Y2: B] :
% 0.20/0.46 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 0.20/0.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.20/0.46 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % order_subst1
% 0.20/0.46 thf(fact_43_order__subst2,axiom,
% 0.20/0.46 ! [A: $tType,C: $tType] :
% 0.20/0.46 ( ( ( order @ C @ ( type2 @ C ) )
% 0.20/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.20/0.46 => ! [A4: A,B4: A,F: A > C,C2: C] :
% 0.20/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.20/0.46 => ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
% 0.20/0.46 => ( ! [X3: A,Y2: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.20/0.46 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.20/0.46 => ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % order_subst2
% 0.20/0.46 thf(fact_44_ord__eq__le__subst,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType] :
% 0.20/0.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.20/0.46 & ( ord @ A @ ( type2 @ A ) ) )
% 0.20/0.46 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 0.20/0.46 ( ( A4
% 0.20/0.46 = ( F @ B4 ) )
% 0.20/0.46 => ( ( ord_less_eq @ B @ B4 @ C2 )
% 0.20/0.46 => ( ! [X3: B,Y2: B] :
% 0.20/0.46 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 0.20/0.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.20/0.46 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ord_eq_le_subst
% 0.20/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.20/0.46 ! [A: $tType,A4: A,P: A > $o] :
% 0.20/0.46 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 0.20/0.46 = ( P @ A4 ) ) ).
% 0.20/0.46
% 0.20/0.46 % mem_Collect_eq
% 0.20/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A] :
% 0.20/0.46 ( ( collect @ A
% 0.20/0.46 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 0.20/0.46 = A2 ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_mem_eq
% 0.20/0.46 thf(fact_47_Collect__cong,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.20/0.46 ( ! [X3: A] :
% 0.20/0.46 ( ( P @ X3 )
% 0.20/0.46 = ( Q @ X3 ) )
% 0.20/0.46 => ( ( collect @ A @ P )
% 0.20/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_cong
% 0.20/0.46 thf(fact_48_ext,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.20/0.46 ( ! [X3: A] :
% 0.20/0.46 ( ( F @ X3 )
% 0.20/0.46 = ( G @ X3 ) )
% 0.20/0.46 => ( F = G ) ) ).
% 0.20/0.46
% 0.20/0.46 % ext
% 0.20/0.46 thf(fact_49_ord__le__eq__subst,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType] :
% 0.20/0.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.20/0.46 & ( ord @ A @ ( type2 @ A ) ) )
% 0.20/0.46 => ! [A4: A,B4: A,F: A > B,C2: B] :
% 0.20/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.20/0.46 => ( ( ( F @ B4 )
% 0.20/0.46 = C2 )
% 0.20/0.46 => ( ! [X3: A,Y2: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.20/0.46 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.20/0.46 => ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ord_le_eq_subst
% 0.20/0.46 thf(fact_50_Union__mono,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ ( set @ A ),B2: set @ ( set @ A )] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ B2 )
% 0.20/0.46 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Union_mono
% 0.20/0.46 thf(fact_51_eq__iff,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.20/0.46 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.20/0.46 = ( ^ [X2: A,Y3: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.20/0.46 & ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % eq_iff
% 0.20/0.46 thf(fact_52_Union__least,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ ( set @ A ),C3: set @ A] :
% 0.20/0.46 ( ! [X4: set @ A] :
% 0.20/0.46 ( ( member @ ( set @ A ) @ X4 @ A2 )
% 0.20/0.46 => ( ord_less_eq @ ( set @ A ) @ X4 @ C3 ) )
% 0.20/0.46 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ C3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % Union_least
% 0.20/0.46 thf(fact_53_Union__upper,axiom,
% 0.20/0.46 ! [A: $tType,B2: set @ A,A2: set @ ( set @ A )] :
% 0.20/0.46 ( ( member @ ( set @ A ) @ B2 @ A2 )
% 0.20/0.46 => ( ord_less_eq @ ( set @ A ) @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Union_upper
% 0.20/0.46 thf(fact_54_antisym,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [X: A,Y4: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.20/0.46 => ( ( ord_less_eq @ A @ Y4 @ X )
% 0.20/0.46 => ( X = Y4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % antisym
% 0.20/0.46 thf(fact_55_linear,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [X: A,Y4: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.20/0.46 | ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % linear
% 0.20/0.46 thf(fact_56_eq__refl,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [X: A,Y4: A] :
% 0.20/0.46 ( ( X = Y4 )
% 0.20/0.46 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % eq_refl
% 0.20/0.46 thf(fact_57_le__cases,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [X: A,Y4: A] :
% 0.20/0.46 ( ~ ( ord_less_eq @ A @ X @ Y4 )
% 0.20/0.46 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_cases
% 0.20/0.46 thf(fact_58_order_Otrans,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A4: A,B4: A,C2: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.20/0.46 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 0.20/0.46 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % order.trans
% 0.20/0.46 thf(fact_59_le__cases3,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [X: A,Y4: A,Z2: A] :
% 0.20/0.46 ( ( ( ord_less_eq @ A @ X @ Y4 )
% 0.20/0.46 => ~ ( ord_less_eq @ A @ Y4 @ Z2 ) )
% 0.20/0.46 => ( ( ( ord_less_eq @ A @ Y4 @ X )
% 0.20/0.46 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 0.20/0.46 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 0.20/0.46 => ~ ( ord_less_eq @ A @ Z2 @ Y4 ) )
% 0.20/0.46 => ( ( ( ord_less_eq @ A @ Z2 @ Y4 )
% 0.20/0.46 => ~ ( ord_less_eq @ A @ Y4 @ X ) )
% 0.20/0.46 => ( ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 0.20/0.46 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 0.20/0.46 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 0.20/0.46 => ~ ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % le_cases3
% 0.20/0.46 thf(fact_60_antisym__conv,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [Y4: A,X: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ Y4 @ X )
% 0.20/0.46 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.20/0.46 = ( X = Y4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % antisym_conv
% 0.20/0.46 thf(fact_61_ord__eq__le__trans,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A4: A,B4: A,C2: A] :
% 0.20/0.46 ( ( A4 = B4 )
% 0.20/0.46 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 0.20/0.46 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ord_eq_le_trans
% 0.20/0.46 thf(fact_62_ord__le__eq__trans,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A4: A,B4: A,C2: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.20/0.46 => ( ( B4 = C2 )
% 0.20/0.46 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ord_le_eq_trans
% 0.20/0.46 thf(fact_63_order__class_Oorder_Oantisym,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A4: A,B4: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.20/0.46 => ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.20/0.46 => ( A4 = B4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % order_class.order.antisym
% 0.20/0.46 thf(fact_64_order__trans,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [X: A,Y4: A,Z2: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.20/0.46 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 0.20/0.46 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % order_trans
% 0.20/0.46 thf(fact_65_dual__order_Orefl,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 0.20/0.46
% 0.20/0.46 % dual_order.refl
% 0.20/0.46 thf(fact_66_linorder__wlog,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [P: A > A > $o,A4: A,B4: A] :
% 0.20/0.46 ( ! [A5: A,B5: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.20/0.46 => ( P @ A5 @ B5 ) )
% 0.20/0.46 => ( ! [A5: A,B5: A] :
% 0.20/0.46 ( ( P @ B5 @ A5 )
% 0.20/0.46 => ( P @ A5 @ B5 ) )
% 0.20/0.46 => ( P @ A4 @ B4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % linorder_wlog
% 0.20/0.46 thf(fact_67_dual__order_Otrans,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [B4: A,A4: A,C2: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.20/0.46 => ( ( ord_less_eq @ A @ C2 @ B4 )
% 0.20/0.46 => ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % dual_order.trans
% 0.20/0.46 thf(fact_68_subs__def,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( gram_L608943123e_subs @ A )
% 0.20/0.46 = ( ^ [L1: set @ ( set @ A ),L2: set @ ( set @ A )] :
% 0.20/0.46 ! [X2: set @ A] :
% 0.20/0.46 ( ( member @ ( set @ A ) @ X2 @ L2 )
% 0.20/0.46 => ? [Y3: set @ A] :
% 0.20/0.46 ( ( member @ ( set @ A ) @ Y3 @ L1 )
% 0.20/0.46 & ( ord_less_eq @ ( set @ A ) @ Y3 @ X2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % subs_def
% 0.20/0.46 thf(fact_69_incl__subs,axiom,
% 0.20/0.46 ! [A: $tType,L22: set @ ( set @ A ),L12: set @ ( set @ A )] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ L22 @ L12 )
% 0.20/0.46 => ( gram_L608943123e_subs @ A @ L12 @ L22 ) ) ).
% 0.20/0.46
% 0.20/0.46 % incl_subs
% 0.20/0.46 thf(fact_70_dual__order_Oantisym,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [B4: A,A4: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.20/0.46 => ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.20/0.46 => ( A4 = B4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % dual_order.antisym
% 0.20/0.46 thf(fact_71_Sup__eqI,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A2: set @ A,X: A] :
% 0.20/0.46 ( ! [Y2: A] :
% 0.20/0.46 ( ( member @ A @ Y2 @ A2 )
% 0.20/0.46 => ( ord_less_eq @ A @ Y2 @ X ) )
% 0.20/0.46 => ( ! [Y2: A] :
% 0.20/0.46 ( ! [Z3: A] :
% 0.20/0.46 ( ( member @ A @ Z3 @ A2 )
% 0.20/0.46 => ( ord_less_eq @ A @ Z3 @ Y2 ) )
% 0.20/0.46 => ( ord_less_eq @ A @ X @ Y2 ) )
% 0.20/0.46 => ( ( complete_Sup_Sup @ A @ A2 )
% 0.20/0.46 = X ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Sup_eqI
% 0.20/0.46 thf(fact_72_Sup__mono,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A2: set @ A,B2: set @ A] :
% 0.20/0.46 ( ! [A5: A] :
% 0.20/0.46 ( ( member @ A @ A5 @ A2 )
% 0.20/0.46 => ? [X5: A] :
% 0.20/0.46 ( ( member @ A @ X5 @ B2 )
% 0.20/0.46 & ( ord_less_eq @ A @ A5 @ X5 ) ) )
% 0.20/0.46 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Sup_mono
% 0.20/0.46 thf(fact_73_Sup__least,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A2: set @ A,Z2: A] :
% 0.20/0.46 ( ! [X3: A] :
% 0.20/0.46 ( ( member @ A @ X3 @ A2 )
% 0.20/0.46 => ( ord_less_eq @ A @ X3 @ Z2 ) )
% 0.20/0.46 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ Z2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Sup_least
% 0.20/0.46 thf(fact_74_Sup__upper,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [X: A,A2: set @ A] :
% 0.20/0.46 ( ( member @ A @ X @ A2 )
% 0.20/0.46 => ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Sup_upper
% 0.20/0.46 thf(fact_75_Sup__le__iff,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A2: set @ A,B4: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ B4 )
% 0.20/0.46 = ( ! [X2: A] :
% 0.20/0.46 ( ( member @ A @ X2 @ A2 )
% 0.20/0.46 => ( ord_less_eq @ A @ X2 @ B4 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Sup_le_iff
% 0.20/0.46 thf(fact_76_Sup__upper2,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [U: A,A2: set @ A,V: A] :
% 0.20/0.46 ( ( member @ A @ U @ A2 )
% 0.20/0.46 => ( ( ord_less_eq @ A @ V @ U )
% 0.20/0.46 => ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Sup_upper2
% 0.20/0.46 thf(fact_77_Sup__subset__mono,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A2: set @ A,B2: set @ A] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.20/0.46 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Sup_subset_mono
% 0.20/0.46 thf(fact_78_subs__trans,axiom,
% 0.20/0.46 ! [A: $tType,L12: set @ ( set @ A ),L22: set @ ( set @ A ),L3: set @ ( set @ A )] :
% 0.20/0.46 ( ( gram_L608943123e_subs @ A @ L12 @ L22 )
% 0.20/0.46 => ( ( gram_L608943123e_subs @ A @ L22 @ L3 )
% 0.20/0.46 => ( gram_L608943123e_subs @ A @ L12 @ L3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % subs_trans
% 0.20/0.46 thf(fact_79_subs__refl,axiom,
% 0.20/0.46 ! [A: $tType,L12: set @ ( set @ A )] : ( gram_L608943123e_subs @ A @ L12 @ L12 ) ).
% 0.20/0.46
% 0.20/0.46 % subs_refl
% 0.20/0.46 thf(fact_80_top__set__def,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( top_top @ ( set @ A ) )
% 0.20/0.46 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % top_set_def
% 0.20/0.46 thf(fact_81_Union__UNIV,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
% 0.20/0.46 = ( top_top @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Union_UNIV
% 0.20/0.46 thf(fact_82_top_Oextremum__uniqueI,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( order_top @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A4: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
% 0.20/0.46 => ( A4
% 0.20/0.46 = ( top_top @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % top.extremum_uniqueI
% 0.20/0.46 thf(fact_83_top_Oextremum__unique,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( order_top @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A4: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
% 0.20/0.46 = ( A4
% 0.20/0.46 = ( top_top @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % top.extremum_unique
% 0.20/0.46 thf(fact_84_top__greatest,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( order_top @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( top_top @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % top_greatest
% 0.20/0.46 thf(fact_85_subset__UNIV,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % subset_UNIV
% 0.20/0.46 thf(fact_86_Lr__incl__L,axiom,
% 0.20/0.46 ! [Ns: set @ n,Ts: n] : ( ord_less_eq @ ( set @ ( set @ t ) ) @ ( gram_L861977318lle_Lr @ Ns @ Ts ) @ ( gram_L1451583628elle_L @ Ns @ Ts ) ) ).
% 0.20/0.46
% 0.20/0.46 % Lr_incl_L
% 0.20/0.46 thf(fact_87_cSup__eq,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.20/0.46 & ( no_bot @ A @ ( type2 @ A ) ) )
% 0.20/0.46 => ! [X6: set @ A,A4: A] :
% 0.20/0.46 ( ! [X3: A] :
% 0.20/0.46 ( ( member @ A @ X3 @ X6 )
% 0.20/0.46 => ( ord_less_eq @ A @ X3 @ A4 ) )
% 0.20/0.46 => ( ! [Y2: A] :
% 0.20/0.46 ( ! [X5: A] :
% 0.20/0.46 ( ( member @ A @ X5 @ X6 )
% 0.20/0.46 => ( ord_less_eq @ A @ X5 @ Y2 ) )
% 0.20/0.46 => ( ord_less_eq @ A @ A4 @ Y2 ) )
% 0.20/0.46 => ( ( complete_Sup_Sup @ A @ X6 )
% 0.20/0.46 = A4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % cSup_eq
% 0.20/0.46 thf(fact_88_cSup__eq__maximum,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [Z2: A,X6: set @ A] :
% 0.20/0.46 ( ( member @ A @ Z2 @ X6 )
% 0.20/0.46 => ( ! [X3: A] :
% 0.20/0.46 ( ( member @ A @ X3 @ X6 )
% 0.20/0.46 => ( ord_less_eq @ A @ X3 @ Z2 ) )
% 0.20/0.46 => ( ( complete_Sup_Sup @ A @ X6 )
% 0.20/0.46 = Z2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % cSup_eq_maximum
% 0.20/0.46 thf(fact_89_top__empty__eq,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( top_top @ ( A > $o ) )
% 0.20/0.46 = ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % top_empty_eq
% 0.20/0.46 thf(fact_90_Sup__eq__top__iff,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( comple1035589618norder @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A2: set @ A] :
% 0.20/0.46 ( ( ( complete_Sup_Sup @ A @ A2 )
% 0.20/0.46 = ( top_top @ A ) )
% 0.20/0.46 = ( ! [X2: A] :
% 0.20/0.46 ( ( ord_less @ A @ X2 @ ( top_top @ A ) )
% 0.20/0.46 => ? [Y3: A] :
% 0.20/0.46 ( ( member @ A @ Y3 @ A2 )
% 0.20/0.46 & ( ord_less @ A @ X2 @ Y3 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Sup_eq_top_iff
% 0.20/0.46 thf(fact_91_antimonoD,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType] :
% 0.20/0.46 ( ( ( order @ A @ ( type2 @ A ) )
% 0.20/0.46 & ( order @ B @ ( type2 @ B ) ) )
% 0.20/0.46 => ! [F: A > B,X: A,Y4: A] :
% 0.20/0.46 ( ( order_antimono @ A @ B @ F )
% 0.20/0.46 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.20/0.46 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % antimonoD
% 0.20/0.46 thf(fact_92_antimonoE,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType] :
% 0.20/0.46 ( ( ( order @ A @ ( type2 @ A ) )
% 0.20/0.46 & ( order @ B @ ( type2 @ B ) ) )
% 0.20/0.46 => ! [F: A > B,X: A,Y4: A] :
% 0.20/0.46 ( ( order_antimono @ A @ B @ F )
% 0.20/0.46 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.20/0.46 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % antimonoE
% 0.20/0.46 thf(fact_93_antimonoI,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType] :
% 0.20/0.46 ( ( ( order @ A @ ( type2 @ A ) )
% 0.20/0.46 & ( order @ B @ ( type2 @ B ) ) )
% 0.20/0.46 => ! [F: A > B] :
% 0.20/0.46 ( ! [X3: A,Y2: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.20/0.46 => ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X3 ) ) )
% 0.20/0.46 => ( order_antimono @ A @ B @ F ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % antimonoI
% 0.20/0.46 thf(fact_94_antimono__def,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType] :
% 0.20/0.46 ( ( ( order @ A @ ( type2 @ A ) )
% 0.20/0.46 & ( order @ B @ ( type2 @ B ) ) )
% 0.20/0.46 => ( ( order_antimono @ A @ B )
% 0.20/0.46 = ( ^ [F2: A > B] :
% 0.20/0.46 ! [X2: A,Y3: A] :
% 0.20/0.46 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.20/0.46 => ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % antimono_def
% 0.20/0.46 thf(fact_95_Pow__mono,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.20/0.46 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A2 ) @ ( pow @ A @ B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Pow_mono
% 0.20/0.46 thf(fact_96_less__eq__Sup,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A2: set @ A,U: A] :
% 0.20/0.46 ( ! [V2: A] :
% 0.20/0.46 ( ( member @ A @ V2 @ A2 )
% 0.20/0.46 => ( ord_less_eq @ A @ U @ V2 ) )
% 0.20/0.46 => ( ( A2
% 0.20/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_eq_Sup
% 0.20/0.46 thf(fact_97_bot__apply,axiom,
% 0.20/0.46 ! [C: $tType,D: $tType] :
% 0.20/0.46 ( ( bot @ C @ ( type2 @ C ) )
% 0.20/0.46 => ( ( bot_bot @ ( D > C ) )
% 0.20/0.46 = ( ^ [X2: D] : ( bot_bot @ C ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % bot_apply
% 0.20/0.46 thf(fact_98_empty__iff,axiom,
% 0.20/0.46 ! [A: $tType,C2: A] :
% 0.20/0.46 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % empty_iff
% 0.20/0.46 thf(fact_99_all__not__in__conv,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A] :
% 0.20/0.46 ( ( ! [X2: A] :
% 0.20/0.46 ~ ( member @ A @ X2 @ A2 ) )
% 0.20/0.46 = ( A2
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % all_not_in_conv
% 0.20/0.46 thf(fact_100_Collect__empty__eq,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o] :
% 0.20/0.46 ( ( ( collect @ A @ P )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 = ( ! [X2: A] :
% 0.20/0.46 ~ ( P @ X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_empty_eq
% 0.20/0.46 thf(fact_101_empty__Collect__eq,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o] :
% 0.20/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.20/0.46 = ( collect @ A @ P ) )
% 0.20/0.46 = ( ! [X2: A] :
% 0.20/0.46 ~ ( P @ X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % empty_Collect_eq
% 0.20/0.46 thf(fact_102_psubsetI,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.20/0.46 => ( ( A2 != B2 )
% 0.20/0.46 => ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % psubsetI
% 0.20/0.46 thf(fact_103_UN__ball__bex__simps_I3_J,axiom,
% 0.20/0.46 ! [D: $tType,A2: set @ ( set @ D ),P: D > $o] :
% 0.20/0.46 ( ( ? [X2: D] :
% 0.20/0.46 ( ( member @ D @ X2 @ ( complete_Sup_Sup @ ( set @ D ) @ A2 ) )
% 0.20/0.46 & ( P @ X2 ) ) )
% 0.20/0.46 = ( ? [X2: set @ D] :
% 0.20/0.46 ( ( member @ ( set @ D ) @ X2 @ A2 )
% 0.20/0.46 & ? [Y3: D] :
% 0.20/0.46 ( ( member @ D @ Y3 @ X2 )
% 0.20/0.46 & ( P @ Y3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % UN_ball_bex_simps(3)
% 0.20/0.46 thf(fact_104_UN__ball__bex__simps_I1_J,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ ( set @ A ),P: A > $o] :
% 0.20/0.46 ( ( ! [X2: A] :
% 0.20/0.46 ( ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
% 0.20/0.46 => ( P @ X2 ) ) )
% 0.20/0.46 = ( ! [X2: set @ A] :
% 0.20/0.46 ( ( member @ ( set @ A ) @ X2 @ A2 )
% 0.20/0.46 => ! [Y3: A] :
% 0.20/0.46 ( ( member @ A @ Y3 @ X2 )
% 0.20/0.46 => ( P @ Y3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % UN_ball_bex_simps(1)
% 0.20/0.46 thf(fact_105_UnionI,axiom,
% 0.20/0.46 ! [A: $tType,X6: set @ A,C3: set @ ( set @ A ),A2: A] :
% 0.20/0.46 ( ( member @ ( set @ A ) @ X6 @ C3 )
% 0.20/0.46 => ( ( member @ A @ A2 @ X6 )
% 0.20/0.46 => ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % UnionI
% 0.20/0.46 thf(fact_106_Union__iff,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,C3: set @ ( set @ A )] :
% 0.20/0.46 ( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
% 0.20/0.46 = ( ? [X2: set @ A] :
% 0.20/0.46 ( ( member @ ( set @ A ) @ X2 @ C3 )
% 0.20/0.46 & ( member @ A @ A2 @ X2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Union_iff
% 0.20/0.46 thf(fact_107_top1I,axiom,
% 0.20/0.46 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 0.20/0.46
% 0.20/0.46 % top1I
% 0.20/0.46 thf(fact_108_Sup__bot__conv_I1_J,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A2: set @ A] :
% 0.20/0.46 ( ( ( complete_Sup_Sup @ A @ A2 )
% 0.20/0.46 = ( bot_bot @ A ) )
% 0.20/0.46 = ( ! [X2: A] :
% 0.20/0.46 ( ( member @ A @ X2 @ A2 )
% 0.20/0.46 => ( X2
% 0.20/0.46 = ( bot_bot @ A ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Sup_bot_conv(1)
% 0.20/0.46 thf(fact_109_Sup__bot__conv_I2_J,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.20/0.46 => ! [A2: set @ A] :
% 0.20/0.46 ( ( ( bot_bot @ A )
% 0.20/0.46 = ( complete_Sup_Sup @ A @ A2 ) )
% 0.20/0.46 = ( ! [X2: A] :
% 0.20/0.46 ( ( member @ A @ X2 @ A2 )
% 0.20/0.46 => ( X2
% 0.20/0.46 = ( bot_bot @ A ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Sup_bot_conv(2)
% 0.20/0.46 thf(fact_110_empty__subsetI,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% 0.20/0.46
% 0.20/0.46 % empty_subsetI
% 0.20/0.46 thf(fact_111_subset__empty,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 = ( A2
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % subset_empty
% 0.20/0.46 thf(fact_112_Pow__iff,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.20/0.46 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 0.20/0.46 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % Pow_iff
% 0.20/0.46 thf(fact_113_PowI,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.20/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.20/0.46 => ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % PowI
% 0.20/0.46 thf(fact_114_Union__Pow__eq,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A] :
% 0.20/0.46 ( ( complete_Sup_Sup @ ( set @ A ) @ ( pow @ A @ A2 ) )
% 0.20/0.46 = A2 ) ).
% 0.20/0.46
% 0.20/0.46 % Union_Pow_eq
% 0.20/0.46 thf(fact_115_Sup__empty,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.20/0.46 => ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 = ( bot_bot @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Sup_empty
% 0.20/0.46 thf(fact_116_Pow__UNIV,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( pow @ A @ ( top_top @ ( set @ A ) ) )
% 0.20/0.46 = ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Pow_UNIV
% 0.20/0.46 thf(fact_117_emptyE,axiom,
% 0.20/0.46 ! [A: $tType,A4: A] :
% 0.20/0.46 ~ ( member @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % emptyE
% 0.20/0.46 thf(fact_118_Pow__top,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A] : ( member @ ( set @ A ) @ A2 @ ( pow @ A @ A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % Pow_top
% 0.20/0.46 thf(fact_119_equals0D,axiom,
% 0.20/0.46 ! [A: $tType,A2: set @ A,A4: A] :
% 0.20/0.46 ( ( A2
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 => ~ ( member @ A @ A4 @ A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % equals0D
% 0.31/0.46 thf(fact_120_equals0I,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] :
% 0.31/0.46 ( ! [Y2: A] :
% 0.31/0.46 ~ ( member @ A @ Y2 @ A2 )
% 0.31/0.46 => ( A2
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % equals0I
% 0.31/0.46 thf(fact_121_Pow__bottom,axiom,
% 0.31/0.46 ! [A: $tType,B2: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow @ A @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pow_bottom
% 0.31/0.46 thf(fact_122_ex__in__conv,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] :
% 0.31/0.46 ( ( ? [X2: A] : ( member @ A @ X2 @ A2 ) )
% 0.31/0.46 = ( A2
% 0.31/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ex_in_conv
% 0.31/0.46 thf(fact_123_Pow__not__empty,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] :
% 0.31/0.46 ( ( pow @ A @ A2 )
% 0.31/0.46 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pow_not_empty
% 0.31/0.46 thf(fact_124_bot__fun__def,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( bot @ B @ ( type2 @ B ) )
% 0.31/0.46 => ( ( bot_bot @ ( A > B ) )
% 0.31/0.46 = ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot_fun_def
% 0.31/0.46 thf(fact_125_not__psubset__empty,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] :
% 0.31/0.46 ~ ( ord_less @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_psubset_empty
% 0.31/0.46 thf(fact_126_less__fun__def,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.31/0.46 => ( ( ord_less @ ( A > B ) )
% 0.31/0.46 = ( ^ [F2: A > B,G2: A > B] :
% 0.31/0.46 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.31/0.46 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_fun_def
% 0.31/0.46 thf(fact_127_UnionE,axiom,
% 0.31/0.46 ! [A: $tType,A2: A,C3: set @ ( set @ A )] :
% 0.31/0.46 ( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
% 0.31/0.46 => ~ ! [X4: set @ A] :
% 0.31/0.46 ( ( member @ A @ A2 @ X4 )
% 0.31/0.46 => ~ ( member @ ( set @ A ) @ X4 @ C3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % UnionE
% 0.31/0.46 thf(fact_128_ord__eq__less__subst,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.31/0.46 & ( ord @ A @ ( type2 @ A ) ) )
% 0.31/0.46 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 0.31/0.46 ( ( A4
% 0.31/0.46 = ( F @ B4 ) )
% 0.31/0.46 => ( ( ord_less @ B @ B4 @ C2 )
% 0.31/0.46 => ( ! [X3: B,Y2: B] :
% 0.31/0.46 ( ( ord_less @ B @ X3 @ Y2 )
% 0.31/0.46 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.46 => ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_less_subst
% 0.31/0.46 thf(fact_129_ord__less__eq__subst,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.31/0.46 & ( ord @ A @ ( type2 @ A ) ) )
% 0.31/0.46 => ! [A4: A,B4: A,F: A > B,C2: B] :
% 0.31/0.46 ( ( ord_less @ A @ A4 @ B4 )
% 0.31/0.46 => ( ( ( F @ B4 )
% 0.31/0.46 = C2 )
% 0.31/0.46 => ( ! [X3: A,Y2: A] :
% 0.31/0.46 ( ( ord_less @ A @ X3 @ Y2 )
% 0.31/0.46 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.46 => ( ord_less @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_less_eq_subst
% 0.31/0.46 thf(fact_130_order__less__subst1,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( order @ B @ ( type2 @ B ) )
% 0.31/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.31/0.46 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 0.31/0.46 ( ( ord_less @ A @ A4 @ ( F @ B4 ) )
% 0.31/0.46 => ( ( ord_less @ B @ B4 @ C2 )
% 0.31/0.46 => ( ! [X3: B,Y2: B] :
% 0.31/0.46 ( ( ord_less @ B @ X3 @ Y2 )
% 0.31/0.46 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.46 => ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_subst1
% 0.31/0.46 thf(fact_131_order__less__subst2,axiom,
% 0.31/0.46 ! [A: $tType,C: $tType] :
% 0.31/0.46 ( ( ( order @ C @ ( type2 @ C ) )
% 0.31/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.31/0.46 => ! [A4: A,B4: A,F: A > C,C2: C] :
% 0.31/0.46 ( ( ord_less @ A @ A4 @ B4 )
% 0.31/0.46 => ( ( ord_less @ C @ ( F @ B4 ) @ C2 )
% 0.31/0.46 => ( ! [X3: A,Y2: A] :
% 0.31/0.46 ( ( ord_less @ A @ X3 @ Y2 )
% 0.31/0.46 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.46 => ( ord_less @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_subst2
% 0.31/0.46 thf(fact_132_lt__ex,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A] :
% 0.31/0.46 ? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % lt_ex
% 0.31/0.46 thf(fact_133_gt__ex,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( no_top @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A] :
% 0.31/0.46 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 0.31/0.46
% 0.31/0.46 % gt_ex
% 0.31/0.46 thf(fact_134_Union__empty,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Union_empty
% 0.31/0.46 thf(fact_135_neqE,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( X != Y4 )
% 0.31/0.46 => ( ~ ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % neqE
% 0.31/0.46 thf(fact_136_neq__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( X != Y4 )
% 0.31/0.46 = ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 | ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % neq_iff
% 0.31/0.46 thf(fact_137_order_Oasym,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A,B4: A] :
% 0.31/0.46 ( ( ord_less @ A @ A4 @ B4 )
% 0.31/0.46 => ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.asym
% 0.31/0.46 thf(fact_138_dense,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 => ? [Z4: A] :
% 0.31/0.46 ( ( ord_less @ A @ X @ Z4 )
% 0.31/0.46 & ( ord_less @ A @ Z4 @ Y4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dense
% 0.31/0.46 thf(fact_139_Union__empty__conv,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ ( set @ A )] :
% 0.31/0.46 ( ( ( complete_Sup_Sup @ ( set @ A ) @ A2 )
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.31/0.46 = ( ! [X2: set @ A] :
% 0.31/0.46 ( ( member @ ( set @ A ) @ X2 @ A2 )
% 0.31/0.46 => ( X2
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Union_empty_conv
% 0.31/0.46 thf(fact_140_empty__Union__conv,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ ( set @ A )] :
% 0.31/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.31/0.46 = ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
% 0.31/0.46 = ( ! [X2: set @ A] :
% 0.31/0.46 ( ( member @ ( set @ A ) @ X2 @ A2 )
% 0.31/0.46 => ( X2
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % empty_Union_conv
% 0.31/0.46 thf(fact_141_less__imp__neq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 => ( X != Y4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_neq
% 0.31/0.46 thf(fact_142_less__asym,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_asym
% 0.31/0.46 thf(fact_143_less__asym_H,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A,B4: A] :
% 0.31/0.46 ( ( ord_less @ A @ A4 @ B4 )
% 0.31/0.46 => ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_asym'
% 0.31/0.46 thf(fact_144_less__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A,Z2: A] :
% 0.31/0.46 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 => ( ( ord_less @ A @ Y4 @ Z2 )
% 0.31/0.46 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_trans
% 0.31/0.46 thf(fact_145_less__linear,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 | ( X = Y4 )
% 0.31/0.46 | ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_linear
% 0.31/0.46 thf(fact_146_less__irrefl,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A] :
% 0.31/0.46 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_irrefl
% 0.31/0.46 thf(fact_147_ord__eq__less__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A,B4: A,C2: A] :
% 0.31/0.46 ( ( A4 = B4 )
% 0.31/0.46 => ( ( ord_less @ A @ B4 @ C2 )
% 0.31/0.46 => ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_less_trans
% 0.31/0.46 thf(fact_148_ord__less__eq__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A,B4: A,C2: A] :
% 0.31/0.46 ( ( ord_less @ A @ A4 @ B4 )
% 0.31/0.46 => ( ( B4 = C2 )
% 0.31/0.46 => ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_less_eq_trans
% 0.31/0.46 thf(fact_149_dual__order_Oasym,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [B4: A,A4: A] :
% 0.31/0.46 ( ( ord_less @ A @ B4 @ A4 )
% 0.31/0.46 => ~ ( ord_less @ A @ A4 @ B4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.asym
% 0.31/0.46 thf(fact_150_less__imp__not__eq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 => ( X != Y4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_not_eq
% 0.31/0.46 thf(fact_151_less__not__sym,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_not_sym
% 0.31/0.46 thf(fact_152_less__induct,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [P: A > $o,A4: A] :
% 0.31/0.46 ( ! [X3: A] :
% 0.31/0.46 ( ! [Y5: A] :
% 0.31/0.46 ( ( ord_less @ A @ Y5 @ X3 )
% 0.31/0.46 => ( P @ Y5 ) )
% 0.31/0.46 => ( P @ X3 ) )
% 0.31/0.46 => ( P @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_induct
% 0.31/0.46 thf(fact_153_antisym__conv3,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [Y4: A,X: A] :
% 0.31/0.46 ( ~ ( ord_less @ A @ Y4 @ X )
% 0.31/0.46 => ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.31/0.46 = ( X = Y4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym_conv3
% 0.31/0.46 thf(fact_154_less__imp__not__eq2,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 => ( Y4 != X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_not_eq2
% 0.31/0.46 thf(fact_155_less__imp__triv,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A,P: $o] :
% 0.31/0.46 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 => ( ( ord_less @ A @ Y4 @ X )
% 0.31/0.46 => P ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_triv
% 0.31/0.46 thf(fact_156_linorder__cases,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ~ ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 => ( ( X != Y4 )
% 0.31/0.46 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_cases
% 0.31/0.46 thf(fact_157_dual__order_Oirrefl,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A] :
% 0.31/0.46 ~ ( ord_less @ A @ A4 @ A4 ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.irrefl
% 0.31/0.46 thf(fact_158_order_Ostrict__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A,B4: A,C2: A] :
% 0.31/0.46 ( ( ord_less @ A @ A4 @ B4 )
% 0.31/0.46 => ( ( ord_less @ A @ B4 @ C2 )
% 0.31/0.46 => ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_trans
% 0.31/0.46 thf(fact_159_less__imp__not__less,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_not_less
% 0.31/0.46 thf(fact_160_bot_Oextremum__strict,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A] :
% 0.31/0.46 ~ ( ord_less @ A @ A4 @ ( bot_bot @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot.extremum_strict
% 0.31/0.46 thf(fact_161_bot_Onot__eq__extremum,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A] :
% 0.31/0.46 ( ( A4
% 0.31/0.46 != ( bot_bot @ A ) )
% 0.31/0.46 = ( ord_less @ A @ ( bot_bot @ A ) @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot.not_eq_extremum
% 0.31/0.46 thf(fact_162_dual__order_Ostrict__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [B4: A,A4: A,C2: A] :
% 0.31/0.46 ( ( ord_less @ A @ B4 @ A4 )
% 0.31/0.46 => ( ( ord_less @ A @ C2 @ B4 )
% 0.31/0.46 => ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_trans
% 0.31/0.46 thf(fact_163_not__less__iff__gr__or__eq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.31/0.46 = ( ( ord_less @ A @ Y4 @ X )
% 0.31/0.46 | ( X = Y4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_less_iff_gr_or_eq
% 0.31/0.46 thf(fact_164_order_Ostrict__implies__not__eq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A,B4: A] :
% 0.31/0.46 ( ( ord_less @ A @ A4 @ B4 )
% 0.31/0.46 => ( A4 != B4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_implies_not_eq
% 0.31/0.46 thf(fact_165_dual__order_Ostrict__implies__not__eq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [B4: A,A4: A] :
% 0.31/0.46 ( ( ord_less @ A @ B4 @ A4 )
% 0.31/0.46 => ( A4 != B4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_implies_not_eq
% 0.31/0.46 thf(fact_166_less__cSupD,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X6: set @ A,Z2: A] :
% 0.31/0.46 ( ( X6
% 0.31/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.31/0.46 => ( ( ord_less @ A @ Z2 @ ( complete_Sup_Sup @ A @ X6 ) )
% 0.31/0.46 => ? [X3: A] :
% 0.31/0.46 ( ( member @ A @ X3 @ X6 )
% 0.31/0.46 & ( ord_less @ A @ Z2 @ X3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_cSupD
% 0.31/0.46 thf(fact_167_less__cSupE,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [Y4: A,X6: set @ A] :
% 0.31/0.46 ( ( ord_less @ A @ Y4 @ ( complete_Sup_Sup @ A @ X6 ) )
% 0.31/0.46 => ( ( X6
% 0.31/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.31/0.46 => ~ ! [X3: A] :
% 0.31/0.46 ( ( member @ A @ X3 @ X6 )
% 0.31/0.46 => ~ ( ord_less @ A @ Y4 @ X3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_cSupE
% 0.31/0.46 thf(fact_168_ex__gt__or__lt,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A] :
% 0.31/0.46 ? [B5: A] :
% 0.31/0.46 ( ( ord_less @ A @ A4 @ B5 )
% 0.31/0.46 | ( ord_less @ A @ B5 @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ex_gt_or_lt
% 0.31/0.46 thf(fact_169_complete__interval,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A,B4: A,P: A > $o] :
% 0.31/0.46 ( ( ord_less @ A @ A4 @ B4 )
% 0.31/0.46 => ( ( P @ A4 )
% 0.31/0.46 => ( ~ ( P @ B4 )
% 0.31/0.46 => ? [C4: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A4 @ C4 )
% 0.31/0.46 & ( ord_less_eq @ A @ C4 @ B4 )
% 0.31/0.46 & ! [X5: A] :
% 0.31/0.46 ( ( ( ord_less_eq @ A @ A4 @ X5 )
% 0.31/0.46 & ( ord_less @ A @ X5 @ C4 ) )
% 0.31/0.46 => ( P @ X5 ) )
% 0.31/0.46 & ! [D2: A] :
% 0.31/0.46 ( ! [X3: A] :
% 0.31/0.46 ( ( ( ord_less_eq @ A @ A4 @ X3 )
% 0.31/0.46 & ( ord_less @ A @ X3 @ D2 ) )
% 0.31/0.46 => ( P @ X3 ) )
% 0.31/0.46 => ( ord_less_eq @ A @ D2 @ C4 ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % complete_interval
% 0.31/0.46 thf(fact_170_order_Onot__eq__order__implies__strict,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A,B4: A] :
% 0.31/0.46 ( ( A4 != B4 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.31/0.46 => ( ord_less @ A @ A4 @ B4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.not_eq_order_implies_strict
% 0.31/0.46 thf(fact_171_dual__order_Ostrict__implies__order,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [B4: A,A4: A] :
% 0.31/0.46 ( ( ord_less @ A @ B4 @ A4 )
% 0.31/0.46 => ( ord_less_eq @ A @ B4 @ A4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_implies_order
% 0.31/0.46 thf(fact_172_dual__order_Ostrict__iff__order,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ( ( ord_less @ A )
% 0.31/0.46 = ( ^ [B6: A,A6: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ B6 @ A6 )
% 0.31/0.46 & ( A6 != B6 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_iff_order
% 0.31/0.46 thf(fact_173_dual__order_Oorder__iff__strict,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ( ( ord_less_eq @ A )
% 0.31/0.46 = ( ^ [B6: A,A6: A] :
% 0.31/0.46 ( ( ord_less @ A @ B6 @ A6 )
% 0.31/0.46 | ( A6 = B6 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.order_iff_strict
% 0.31/0.46 thf(fact_174_order_Ostrict__implies__order,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A,B4: A] :
% 0.31/0.46 ( ( ord_less @ A @ A4 @ B4 )
% 0.31/0.46 => ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_implies_order
% 0.31/0.46 thf(fact_175_dense__le__bounded,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A,Z2: A] :
% 0.31/0.46 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 => ( ! [W: A] :
% 0.31/0.46 ( ( ord_less @ A @ X @ W )
% 0.31/0.46 => ( ( ord_less @ A @ W @ Y4 )
% 0.31/0.46 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.31/0.46 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dense_le_bounded
% 0.31/0.46 thf(fact_176_dense__ge__bounded,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [Z2: A,X: A,Y4: A] :
% 0.31/0.46 ( ( ord_less @ A @ Z2 @ X )
% 0.31/0.46 => ( ! [W: A] :
% 0.31/0.46 ( ( ord_less @ A @ Z2 @ W )
% 0.31/0.46 => ( ( ord_less @ A @ W @ X )
% 0.31/0.46 => ( ord_less_eq @ A @ Y4 @ W ) ) )
% 0.31/0.46 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dense_ge_bounded
% 0.31/0.46 thf(fact_177_dual__order_Ostrict__trans2,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [B4: A,A4: A,C2: A] :
% 0.31/0.46 ( ( ord_less @ A @ B4 @ A4 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ C2 @ B4 )
% 0.31/0.46 => ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_trans2
% 0.31/0.46 thf(fact_178_dual__order_Ostrict__trans1,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [B4: A,A4: A,C2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.31/0.46 => ( ( ord_less @ A @ C2 @ B4 )
% 0.31/0.46 => ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.strict_trans1
% 0.31/0.46 thf(fact_179_order_Ostrict__iff__order,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ( ( ord_less @ A )
% 0.31/0.46 = ( ^ [A6: A,B6: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A6 @ B6 )
% 0.31/0.46 & ( A6 != B6 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_iff_order
% 0.31/0.46 thf(fact_180_order_Oorder__iff__strict,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ( ( ord_less_eq @ A )
% 0.31/0.46 = ( ^ [A6: A,B6: A] :
% 0.31/0.46 ( ( ord_less @ A @ A6 @ B6 )
% 0.31/0.46 | ( A6 = B6 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.order_iff_strict
% 0.31/0.46 thf(fact_181_order_Ostrict__trans2,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A,B4: A,C2: A] :
% 0.31/0.46 ( ( ord_less @ A @ A4 @ B4 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 0.31/0.46 => ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_trans2
% 0.31/0.46 thf(fact_182_order_Ostrict__trans1,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A,B4: A,C2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.31/0.46 => ( ( ord_less @ A @ B4 @ C2 )
% 0.31/0.46 => ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.strict_trans1
% 0.31/0.46 thf(fact_183_not__le__imp__less,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [Y4: A,X: A] :
% 0.31/0.46 ( ~ ( ord_less_eq @ A @ Y4 @ X )
% 0.31/0.46 => ( ord_less @ A @ X @ Y4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_le_imp_less
% 0.31/0.46 thf(fact_184_less__le__not__le,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ( ( ord_less @ A )
% 0.31/0.46 = ( ^ [X2: A,Y3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.31/0.46 & ~ ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_le_not_le
% 0.31/0.46 thf(fact_185_le__imp__less__or__eq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.46 => ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 | ( X = Y4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_imp_less_or_eq
% 0.31/0.46 thf(fact_186_le__less__linear,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.46 | ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_less_linear
% 0.31/0.46 thf(fact_187_dense__le,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [Y4: A,Z2: A] :
% 0.31/0.46 ( ! [X3: A] :
% 0.31/0.46 ( ( ord_less @ A @ X3 @ Y4 )
% 0.31/0.46 => ( ord_less_eq @ A @ X3 @ Z2 ) )
% 0.31/0.46 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dense_le
% 0.31/0.46 thf(fact_188_dense__ge,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [Z2: A,Y4: A] :
% 0.31/0.46 ( ! [X3: A] :
% 0.31/0.46 ( ( ord_less @ A @ Z2 @ X3 )
% 0.31/0.46 => ( ord_less_eq @ A @ Y4 @ X3 ) )
% 0.31/0.46 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dense_ge
% 0.31/0.46 thf(fact_189_less__le__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A,Z2: A] :
% 0.31/0.46 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 0.31/0.46 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_le_trans
% 0.31/0.46 thf(fact_190_le__less__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A,Z2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.46 => ( ( ord_less @ A @ Y4 @ Z2 )
% 0.31/0.46 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_less_trans
% 0.31/0.46 thf(fact_191_antisym__conv2,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.46 => ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.31/0.46 = ( X = Y4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym_conv2
% 0.31/0.46 thf(fact_192_antisym__conv1,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ~ ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.46 = ( X = Y4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym_conv1
% 0.31/0.46 thf(fact_193_less__imp__le,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_imp_le
% 0.31/0.46 thf(fact_194_le__neq__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A,B4: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.31/0.46 => ( ( A4 != B4 )
% 0.31/0.46 => ( ord_less @ A @ A4 @ B4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_neq_trans
% 0.31/0.46 thf(fact_195_not__less,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.31/0.46 = ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_less
% 0.31/0.46 thf(fact_196_not__le,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ( ~ ( ord_less_eq @ A @ X @ Y4 ) )
% 0.31/0.46 = ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_le
% 0.31/0.46 thf(fact_197_order__less__le__subst2,axiom,
% 0.31/0.46 ! [A: $tType,C: $tType] :
% 0.31/0.46 ( ( ( order @ C @ ( type2 @ C ) )
% 0.31/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.31/0.46 => ! [A4: A,B4: A,F: A > C,C2: C] :
% 0.31/0.46 ( ( ord_less @ A @ A4 @ B4 )
% 0.31/0.46 => ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
% 0.31/0.46 => ( ! [X3: A,Y2: A] :
% 0.31/0.46 ( ( ord_less @ A @ X3 @ Y2 )
% 0.31/0.46 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.46 => ( ord_less @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_le_subst2
% 0.31/0.46 thf(fact_198_order__less__le__subst1,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( order @ B @ ( type2 @ B ) )
% 0.31/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.31/0.46 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 0.31/0.46 ( ( ord_less @ A @ A4 @ ( F @ B4 ) )
% 0.31/0.46 => ( ( ord_less_eq @ B @ B4 @ C2 )
% 0.31/0.46 => ( ! [X3: B,Y2: B] :
% 0.31/0.46 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.46 => ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_less_le_subst1
% 0.31/0.46 thf(fact_199_order__le__less__subst2,axiom,
% 0.31/0.46 ! [A: $tType,C: $tType] :
% 0.31/0.46 ( ( ( order @ C @ ( type2 @ C ) )
% 0.31/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.31/0.46 => ! [A4: A,B4: A,F: A > C,C2: C] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.31/0.46 => ( ( ord_less @ C @ ( F @ B4 ) @ C2 )
% 0.31/0.46 => ( ! [X3: A,Y2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.31/0.46 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.46 => ( ord_less @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_le_less_subst2
% 0.31/0.46 thf(fact_200_order__le__less__subst1,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( order @ B @ ( type2 @ B ) )
% 0.31/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.31/0.46 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
% 0.31/0.46 => ( ( ord_less @ B @ B4 @ C2 )
% 0.31/0.46 => ( ! [X3: B,Y2: B] :
% 0.31/0.46 ( ( ord_less @ B @ X3 @ Y2 )
% 0.31/0.46 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.31/0.46 => ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_le_less_subst1
% 0.31/0.46 thf(fact_201_less__le,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ( ( ord_less @ A )
% 0.31/0.46 = ( ^ [X2: A,Y3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.31/0.46 & ( X2 != Y3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_le
% 0.31/0.46 thf(fact_202_le__less,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.46 => ( ( ord_less_eq @ A )
% 0.31/0.46 = ( ^ [X2: A,Y3: A] :
% 0.31/0.46 ( ( ord_less @ A @ X2 @ Y3 )
% 0.31/0.46 | ( X2 = Y3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_less
% 0.31/0.46 thf(fact_203_leI,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X: A,Y4: A] :
% 0.31/0.46 ( ~ ( ord_less @ A @ X @ Y4 )
% 0.31/0.46 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % leI
% 0.31/0.46 thf(fact_204_leD,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [Y4: A,X: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ Y4 @ X )
% 0.31/0.46 => ~ ( ord_less @ A @ X @ Y4 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % leD
% 0.31/0.46 thf(fact_205_bot_Oextremum__uniqueI,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
% 0.31/0.46 => ( A4
% 0.31/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot.extremum_uniqueI
% 0.31/0.46 thf(fact_206_bot_Oextremum__unique,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
% 0.31/0.46 = ( A4
% 0.31/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot.extremum_unique
% 0.31/0.46 thf(fact_207_bot_Oextremum,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A4 ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot.extremum
% 0.31/0.46 thf(fact_208_top_Oextremum__strict,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_top @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A] :
% 0.31/0.46 ~ ( ord_less @ A @ ( top_top @ A ) @ A4 ) ) ).
% 0.31/0.46
% 0.31/0.46 % top.extremum_strict
% 0.31/0.46 thf(fact_209_top_Onot__eq__extremum,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_top @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A] :
% 0.31/0.46 ( ( A4
% 0.31/0.46 != ( top_top @ A ) )
% 0.31/0.46 = ( ord_less @ A @ A4 @ ( top_top @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % top.not_eq_extremum
% 0.31/0.46 thf(fact_210_subset__iff__psubset__eq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.31/0.46 = ( ^ [A3: set @ A,B3: set @ A] :
% 0.31/0.46 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.31/0.46 | ( A3 = B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_iff_psubset_eq
% 0.31/0.46 thf(fact_211_subset__psubset__trans,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 0.31/0.46 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_psubset_trans
% 0.31/0.46 thf(fact_212_subset__not__subset__eq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord_less @ ( set @ A ) )
% 0.31/0.46 = ( ^ [A3: set @ A,B3: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.31/0.46 & ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_not_subset_eq
% 0.31/0.46 thf(fact_213_psubset__subset__trans,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.31/0.46 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.31/0.46 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % psubset_subset_trans
% 0.31/0.46 thf(fact_214_psubset__imp__subset,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % psubset_imp_subset
% 0.31/0.46 thf(fact_215_psubset__eq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord_less @ ( set @ A ) )
% 0.31/0.46 = ( ^ [A3: set @ A,B3: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.31/0.46 & ( A3 != B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % psubset_eq
% 0.31/0.46 thf(fact_216_psubsetE,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % psubsetE
% 0.31/0.46 thf(fact_217_less__Sup__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( comple1035589618norder @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [A4: A,S: set @ A] :
% 0.31/0.46 ( ( ord_less @ A @ A4 @ ( complete_Sup_Sup @ A @ S ) )
% 0.31/0.46 = ( ? [X2: A] :
% 0.31/0.46 ( ( member @ A @ X2 @ S )
% 0.31/0.46 & ( ord_less @ A @ A4 @ X2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_Sup_iff
% 0.31/0.46 thf(fact_218_PowD,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % PowD
% 0.31/0.46 thf(fact_219_empty__not__UNIV,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( bot_bot @ ( set @ A ) )
% 0.31/0.46 != ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % empty_not_UNIV
% 0.31/0.46 thf(fact_220_cSup__eq__non__empty,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X6: set @ A,A4: A] :
% 0.31/0.46 ( ( X6
% 0.31/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.31/0.46 => ( ! [X3: A] :
% 0.31/0.46 ( ( member @ A @ X3 @ X6 )
% 0.31/0.46 => ( ord_less_eq @ A @ X3 @ A4 ) )
% 0.31/0.46 => ( ! [Y2: A] :
% 0.31/0.46 ( ! [X5: A] :
% 0.31/0.46 ( ( member @ A @ X5 @ X6 )
% 0.31/0.46 => ( ord_less_eq @ A @ X5 @ Y2 ) )
% 0.31/0.46 => ( ord_less_eq @ A @ A4 @ Y2 ) )
% 0.31/0.46 => ( ( complete_Sup_Sup @ A @ X6 )
% 0.31/0.46 = A4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % cSup_eq_non_empty
% 0.31/0.46 thf(fact_221_cSup__least,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 0.31/0.46 => ! [X6: set @ A,Z2: A] :
% 0.31/0.47 ( ( X6
% 0.31/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.31/0.47 => ( ! [X3: A] :
% 0.31/0.47 ( ( member @ A @ X3 @ X6 )
% 0.31/0.47 => ( ord_less_eq @ A @ X3 @ Z2 ) )
% 0.31/0.47 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X6 ) @ Z2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % cSup_least
% 0.31/0.47 thf(fact_222_subset__Pow__Union,axiom,
% 0.31/0.47 ! [A: $tType,A2: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_Pow_Union
% 0.31/0.47 thf(fact_223_le__Sup__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( comple1035589618norder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,A2: set @ A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A2 ) )
% 0.31/0.47 = ( ! [Y3: A] :
% 0.31/0.47 ( ( ord_less @ A @ Y3 @ X )
% 0.31/0.47 => ? [X2: A] :
% 0.31/0.47 ( ( member @ A @ X2 @ A2 )
% 0.31/0.47 & ( ord_less @ A @ Y3 @ X2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_Sup_iff
% 0.31/0.47 thf(fact_224_subset__emptyI,axiom,
% 0.31/0.47 ! [A: $tType,A2: set @ A] :
% 0.31/0.47 ( ! [X3: A] :
% 0.31/0.47 ~ ( member @ A @ X3 @ A2 )
% 0.31/0.47 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_emptyI
% 0.31/0.47 thf(fact_225_pinf_I6_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z4: A] :
% 0.31/0.47 ! [X5: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z4 @ X5 )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(6)
% 0.31/0.47 thf(fact_226_pinf_I8_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z4: A] :
% 0.31/0.47 ! [X5: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z4 @ X5 )
% 0.31/0.47 => ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(8)
% 0.31/0.47 thf(fact_227_minf_I6_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z4: A] :
% 0.31/0.47 ! [X5: A] :
% 0.31/0.47 ( ( ord_less @ A @ X5 @ Z4 )
% 0.31/0.47 => ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(6)
% 0.31/0.47 thf(fact_228_minf_I8_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z4: A] :
% 0.31/0.47 ! [X5: A] :
% 0.31/0.47 ( ( ord_less @ A @ X5 @ Z4 )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(8)
% 0.31/0.47 thf(fact_229_psubsetD,axiom,
% 0.31/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 0.31/0.47 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.47 => ( ( member @ A @ C2 @ A2 )
% 0.31/0.47 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % psubsetD
% 0.31/0.47 thf(fact_230_bot__set__def,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( bot_bot @ ( set @ A ) )
% 0.31/0.47 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % bot_set_def
% 0.31/0.47 thf(fact_231_psubset__trans,axiom,
% 0.31/0.47 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.31/0.47 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.47 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 0.31/0.47 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % psubset_trans
% 0.31/0.47 thf(fact_232_bot__empty__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( bot_bot @ ( A > $o ) )
% 0.31/0.47 = ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % bot_empty_eq
% 0.31/0.47 thf(fact_233_minf_I11_J,axiom,
% 0.31/0.47 ! [C: $tType,D: $tType] :
% 0.31/0.47 ( ( ord @ C @ ( type2 @ C ) )
% 0.31/0.47 => ! [F3: D] :
% 0.31/0.47 ? [Z4: C] :
% 0.31/0.47 ! [X5: C] :
% 0.31/0.47 ( ( ord_less @ C @ X5 @ Z4 )
% 0.31/0.47 => ( F3 = F3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(11)
% 0.31/0.47 thf(fact_234_minf_I7_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z4: A] :
% 0.31/0.47 ! [X5: A] :
% 0.31/0.47 ( ( ord_less @ A @ X5 @ Z4 )
% 0.31/0.47 => ~ ( ord_less @ A @ T2 @ X5 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(7)
% 0.31/0.47 thf(fact_235_minf_I5_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z4: A] :
% 0.31/0.47 ! [X5: A] :
% 0.31/0.47 ( ( ord_less @ A @ X5 @ Z4 )
% 0.31/0.47 => ( ord_less @ A @ X5 @ T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(5)
% 0.31/0.47 thf(fact_236_minf_I4_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z4: A] :
% 0.31/0.47 ! [X5: A] :
% 0.31/0.47 ( ( ord_less @ A @ X5 @ Z4 )
% 0.31/0.47 => ( X5 != T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(4)
% 0.31/0.47 thf(fact_237_minf_I3_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z4: A] :
% 0.31/0.47 ! [X5: A] :
% 0.31/0.47 ( ( ord_less @ A @ X5 @ Z4 )
% 0.31/0.47 => ( X5 != T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(3)
% 0.31/0.47 thf(fact_238_minf_I2_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.31/0.47 ( ? [Z3: A] :
% 0.31/0.47 ! [X3: A] :
% 0.31/0.47 ( ( ord_less @ A @ X3 @ Z3 )
% 0.31/0.47 => ( ( P @ X3 )
% 0.31/0.47 = ( P2 @ X3 ) ) )
% 0.31/0.47 => ( ? [Z3: A] :
% 0.31/0.47 ! [X3: A] :
% 0.31/0.47 ( ( ord_less @ A @ X3 @ Z3 )
% 0.31/0.47 => ( ( Q @ X3 )
% 0.31/0.47 = ( Q2 @ X3 ) ) )
% 0.31/0.47 => ? [Z4: A] :
% 0.31/0.47 ! [X5: A] :
% 0.31/0.47 ( ( ord_less @ A @ X5 @ Z4 )
% 0.31/0.47 => ( ( ( P @ X5 )
% 0.31/0.47 | ( Q @ X5 ) )
% 0.31/0.47 = ( ( P2 @ X5 )
% 0.31/0.47 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(2)
% 0.31/0.47 thf(fact_239_minf_I1_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.31/0.47 ( ? [Z3: A] :
% 0.31/0.47 ! [X3: A] :
% 0.31/0.47 ( ( ord_less @ A @ X3 @ Z3 )
% 0.31/0.47 => ( ( P @ X3 )
% 0.31/0.47 = ( P2 @ X3 ) ) )
% 0.31/0.47 => ( ? [Z3: A] :
% 0.31/0.47 ! [X3: A] :
% 0.31/0.47 ( ( ord_less @ A @ X3 @ Z3 )
% 0.31/0.47 => ( ( Q @ X3 )
% 0.31/0.47 = ( Q2 @ X3 ) ) )
% 0.31/0.47 => ? [Z4: A] :
% 0.31/0.47 ! [X5: A] :
% 0.31/0.47 ( ( ord_less @ A @ X5 @ Z4 )
% 0.31/0.47 => ( ( ( P @ X5 )
% 0.31/0.47 & ( Q @ X5 ) )
% 0.31/0.47 = ( ( P2 @ X5 )
% 0.31/0.47 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(1)
% 0.31/0.47 thf(fact_240_pinf_I11_J,axiom,
% 0.31/0.47 ! [C: $tType,D: $tType] :
% 0.31/0.47 ( ( ord @ C @ ( type2 @ C ) )
% 0.31/0.47 => ! [F3: D] :
% 0.31/0.47 ? [Z4: C] :
% 0.31/0.47 ! [X5: C] :
% 0.31/0.47 ( ( ord_less @ C @ Z4 @ X5 )
% 0.31/0.47 => ( F3 = F3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(11)
% 0.31/0.47 thf(fact_241_pinf_I7_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z4: A] :
% 0.31/0.47 ! [X5: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z4 @ X5 )
% 0.31/0.47 => ( ord_less @ A @ T2 @ X5 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(7)
% 0.31/0.47 thf(fact_242_pinf_I5_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z4: A] :
% 0.31/0.47 ! [X5: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z4 @ X5 )
% 0.31/0.47 => ~ ( ord_less @ A @ X5 @ T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(5)
% 0.31/0.47 thf(fact_243_pinf_I4_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z4: A] :
% 0.31/0.47 ! [X5: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z4 @ X5 )
% 0.31/0.47 => ( X5 != T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(4)
% 0.31/0.47 thf(fact_244_pinf_I3_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z4: A] :
% 0.31/0.47 ! [X5: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z4 @ X5 )
% 0.31/0.47 => ( X5 != T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(3)
% 0.31/0.47 thf(fact_245_pinf_I2_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.31/0.47 ( ? [Z3: A] :
% 0.31/0.47 ! [X3: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z3 @ X3 )
% 0.31/0.47 => ( ( P @ X3 )
% 0.31/0.47 = ( P2 @ X3 ) ) )
% 0.31/0.47 => ( ? [Z3: A] :
% 0.31/0.47 ! [X3: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z3 @ X3 )
% 0.31/0.47 => ( ( Q @ X3 )
% 0.31/0.47 = ( Q2 @ X3 ) ) )
% 0.31/0.47 => ? [Z4: A] :
% 0.31/0.47 ! [X5: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z4 @ X5 )
% 0.31/0.47 => ( ( ( P @ X5 )
% 0.31/0.47 | ( Q @ X5 ) )
% 0.31/0.47 = ( ( P2 @ X5 )
% 0.31/0.47 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(2)
% 0.31/0.47 thf(fact_246_pinf_I1_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.31/0.47 ( ? [Z3: A] :
% 0.31/0.47 ! [X3: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z3 @ X3 )
% 0.31/0.47 => ( ( P @ X3 )
% 0.31/0.47 = ( P2 @ X3 ) ) )
% 0.31/0.47 => ( ? [Z3: A] :
% 0.31/0.47 ! [X3: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z3 @ X3 )
% 0.31/0.47 => ( ( Q @ X3 )
% 0.31/0.47 = ( Q2 @ X3 ) ) )
% 0.31/0.47 => ? [Z4: A] :
% 0.31/0.47 ! [X5: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z4 @ X5 )
% 0.31/0.47 => ( ( ( P @ X5 )
% 0.31/0.47 & ( Q @ X5 ) )
% 0.31/0.47 = ( ( P2 @ X5 )
% 0.31/0.47 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(1)
% 0.31/0.47 thf(fact_247_top_Oordering__top__axioms,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order_top @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ordering_top @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % top.ordering_top_axioms
% 0.31/0.47 thf(fact_248_Set_Ois__empty__def,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( is_empty @ A )
% 0.31/0.47 = ( ^ [A3: set @ A] :
% 0.31/0.47 ( A3
% 0.31/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Set.is_empty_def
% 0.31/0.47 thf(fact_249_ordering__top_Oextremum,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
% 0.31/0.47 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.31/0.47 => ( Less_eq @ A4 @ Top ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering_top.extremum
% 0.31/0.47 thf(fact_250_ordering__top_Oextremum__strict,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
% 0.31/0.47 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.31/0.47 => ~ ( Less @ Top @ A4 ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering_top.extremum_strict
% 0.31/0.47 thf(fact_251_ordering__top_Oextremum__unique,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
% 0.31/0.47 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.31/0.47 => ( ( Less_eq @ Top @ A4 )
% 0.31/0.47 = ( A4 = Top ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering_top.extremum_unique
% 0.31/0.47 thf(fact_252_ordering__top_Onot__eq__extremum,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
% 0.31/0.47 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.31/0.47 => ( ( A4 != Top )
% 0.31/0.47 = ( Less @ A4 @ Top ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering_top.not_eq_extremum
% 0.31/0.47 thf(fact_253_ordering__top_Oextremum__uniqueI,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
% 0.31/0.47 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 0.31/0.47 => ( ( Less_eq @ Top @ A4 )
% 0.31/0.47 => ( A4 = Top ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering_top.extremum_uniqueI
% 0.31/0.47 thf(fact_254_Collect__empty__eq__bot,axiom,
% 0.31/0.47 ! [A: $tType,P: A > $o] :
% 0.31/0.47 ( ( ( collect @ A @ P )
% 0.31/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.31/0.47 = ( P
% 0.31/0.47 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_empty_eq_bot
% 0.31/0.47 thf(fact_255_top__conj_I2_J,axiom,
% 0.31/0.47 ! [A: $tType,P: $o,X: A] :
% 0.31/0.47 ( ( P
% 0.31/0.47 & ( top_top @ ( A > $o ) @ X ) )
% 0.31/0.47 = P ) ).
% 0.31/0.47
% 0.31/0.47 % top_conj(2)
% 0.31/0.47
% 0.31/0.47 %----Type constructors (28)
% 0.31/0.47 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( comple187826305attice @ A8 @ ( type2 @ A8 ) )
% 0.31/0.47 => ( condit378418413attice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( comple187826305attice @ A8 @ ( type2 @ A8 ) )
% 0.31/0.47 => ( comple187826305attice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( order_top @ A8 @ ( type2 @ A8 ) )
% 0.31/0.47 => ( order_top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( order_bot @ A8 @ ( type2 @ A8 ) )
% 0.31/0.47 => ( order_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 0.31/0.47 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( order @ A8 @ ( type2 @ A8 ) )
% 0.31/0.47 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Otop,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( top @ A8 @ ( type2 @ A8 ) )
% 0.31/0.47 => ( top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 0.31/0.47 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Obot,axiom,
% 0.31/0.47 ! [A7: $tType,A8: $tType] :
% 0.31/0.47 ( ( bot @ A8 @ ( type2 @ A8 ) )
% 0.31/0.47 => ( bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1,axiom,
% 0.31/0.47 ! [A7: $tType] : ( condit378418413attice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_2,axiom,
% 0.31/0.47 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Oorder__top_3,axiom,
% 0.31/0.47 ! [A7: $tType] : ( order_top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Oorder__bot_4,axiom,
% 0.31/0.47 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
% 0.31/0.47 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 0.31/0.47 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
% 0.31/0.47 ! [A7: $tType] : ( top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 0.31/0.47 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Obot_9,axiom,
% 0.31/0.47 ! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_10,axiom,
% 0.31/0.47 condit378418413attice @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_11,axiom,
% 0.31/0.47 comple187826305attice @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Oorder__top_12,axiom,
% 0.31/0.47 order_top @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Oorder__bot_13,axiom,
% 0.31/0.47 order_bot @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_14,axiom,
% 0.31/0.47 preorder @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.31/0.47 linorder @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_15,axiom,
% 0.31/0.47 order @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Otop_16,axiom,
% 0.31/0.47 top @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Oord_17,axiom,
% 0.31/0.47 ord @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Obot_18,axiom,
% 0.31/0.47 bot @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 %----Conjectures (1)
% 0.31/0.47 thf(conj_0,conjecture,
% 0.31/0.47 gram_L1456083582e_leqv @ t @ ( gram_L861977280lle_LL @ ( top_top @ ( set @ n ) ) @ ts ) @ ( gram_L861977318lle_Lr @ ( top_top @ ( set @ n ) ) @ ts ) ).
% 0.31/0.47
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 ------------------------
% 2.61/1.48 ---- Embedded in THF ---
% 2.61/1.48 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.61/1.48 %%% Generated on Tue Feb 24 04:17:03 EST 2026
% 2.61/1.48 %%% using '$$dhol' embedding, version 1.3.0.
% 2.61/1.48 %%% Logic specification used:
% 2.61/1.48 %%% thf(spec, logic, $$dhol).
% 2.61/1.48
% 2.61/1.48 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.LejkCnALQX/DTF2DTF24078.p
% 2.61/1.48 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.61/1.48 thf(ty_t_DTree_OT, type, t: $tType).
% 2.61/1.48 thf(ty_t_DTree_ON, type, n: $tType).
% 2.61/1.48 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.61/1.48 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple187826305attice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Complete__Lattices_Ocomplete__linorder, type, comple1035589618norder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice, type, condit378418413attice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.61/1.48 thf(sy_c_Complete__Lattices_OSup__class_OSup, type, complete_Sup_Sup: (!> [A:$tType]: (((set @ A) > A)))).
% 2.61/1.48 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OL, type, gram_L1451583628elle_L: ((set @ n) > (n > (set @ (set @ t))))).
% 2.61/1.48 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OLL, type, gram_L861977280lle_LL: ((set @ n) > (n > (set @ (set @ t))))).
% 2.61/1.48 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OLr, type, gram_L861977318lle_Lr: ((set @ n) > (n > (set @ (set @ t))))).
% 2.61/1.48 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Oleqv, type, gram_L1456083582e_leqv: (!> [A:$tType]: (((set @ (set @ A)) > ((set @ (set @ A)) > $o))))).
% 2.61/1.48 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Osubs, type, gram_L608943123e_subs: (!> [A:$tType]: (((set @ (set @ A)) > ((set @ (set @ A)) > $o))))).
% 2.61/1.48 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.61/1.48 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.61/1.48 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.61/1.48 thf(sy_c_Orderings_Oorder__class_Oantimono, type, order_antimono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.61/1.48 thf(sy_c_Orderings_Oordering__top, type, ordering_top: (!> [A:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > (A > $o)))))).
% 2.61/1.48 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.61/1.48 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.61/1.48 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.61/1.48 thf(sy_c_Set_OPow, type, pow: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))).
% 2.61/1.48 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.61/1.48 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.61/1.48 thf(sy_v_ts, type, ts: n).
% 2.61/1.48 thf(fact_0_Lr__leqv__L, axiom, (! [Ts:n]: ((((gram_L1456083582e_leqv @ t) @ ((gram_L861977318lle_Lr @ (top_top @ (set @ n))) @ Ts)) @ ((gram_L1451583628elle_L @ (top_top @ (set @ n))) @ Ts))))).
% 2.61/1.48 thf(fact_1_LL__leqv__L, axiom, (! [Ts:n]: ((((gram_L1456083582e_leqv @ t) @ ((gram_L861977280lle_LL @ (top_top @ (set @ n))) @ Ts)) @ ((gram_L1451583628elle_L @ (top_top @ (set @ n))) @ Ts))))).
% 2.61/1.48 thf(fact_2_leqv__Sym, axiom, (! [A:$tType]: (((gram_L1456083582e_leqv @ A) = (^ [L1:(set @ (set @ A)),L2:(set @ (set @ A))]: ((((gram_L1456083582e_leqv @ A) @ L2) @ L1))))))).
% 2.61/1.48 thf(fact_3_leqv__sym, axiom, (! [A:$tType,L12:(set @ (set @ A)),L22:(set @ (set @ A))]: (((((gram_L1456083582e_leqv @ A) @ L12) @ L22) => (((gram_L1456083582e_leqv @ A) @ L22) @ L12))))).
% 2.61/1.48 thf(fact_4_leqv__refl, axiom, (! [A:$tType,L12:(set @ (set @ A))]: ((((gram_L1456083582e_leqv @ A) @ L12) @ L12)))).
% 2.61/1.48 thf(fact_5_leqv__trans, axiom, (! [A:$tType,L12:(set @ (set @ A)),L22:(set @ (set @ A)),L3:(set @ (set @ A))]: (((((gram_L1456083582e_leqv @ A) @ L12) @ L22) => ((((gram_L1456083582e_leqv @ A) @ L22) @ L3) => (((gram_L1456083582e_leqv @ A) @ L12) @ L3)))))).
% 2.61/1.48 thf(fact_6_UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.61/1.48 thf(fact_7_iso__tuple__UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.61/1.48 thf(fact_8_top__apply, axiom, (! [C:$tType,D:$tType]: ((((top @ C) @ (type2 @ C)) => ((top_top @ (D > C)) = (^ [X2:D]: ((top_top @ C)))))))).
% 2.61/1.48 thf(fact_9_Lr__subs__L, axiom, (! [Ts:n]: ((((gram_L608943123e_subs @ t) @ ((gram_L861977318lle_Lr @ (top_top @ (set @ n))) @ Ts)) @ ((gram_L1451583628elle_L @ (top_top @ (set @ n))) @ Ts))))).
% 2.61/1.48 thf(fact_10_UNIV__eq__I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X3:A]: ((((member @ A) @ X3) @ A2))) => ((top_top @ (set @ A)) = A2))))).
% 2.61/1.48 thf(fact_11_UNIV__witness, axiom, (! [A:$tType]: ((? [X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))))).
% 2.61/1.48 thf(fact_12_Lr__LL, axiom, (! [Ns:(set @ n),N:n]: ((((ord_less_eq @ (set @ (set @ t))) @ ((gram_L861977318lle_Lr @ Ns) @ N)) @ ((gram_L861977280lle_LL @ Ns) @ N))))).
% 2.61/1.48 thf(fact_13_leqv__def, axiom, (! [A:$tType]: (((gram_L1456083582e_leqv @ A) = (^ [L1:(set @ (set @ A)),L2:(set @ (set @ A))]: (((((gram_L608943123e_subs @ A) @ L1) @ L2) & (((gram_L608943123e_subs @ A) @ L2) @ L1)))))))).
% 2.61/1.48 thf(fact_14_subs__leqv, axiom, (! [A:$tType,L12:(set @ (set @ A)),L22:(set @ (set @ A))]: (((((gram_L1456083582e_leqv @ A) @ L12) @ L22) => (((gram_L608943123e_subs @ A) @ L12) @ L22))))).
% 2.61/1.48 thf(fact_15_subs__leqv__sym, axiom, (! [A:$tType,L12:(set @ (set @ A)),L22:(set @ (set @ A))]: (((((gram_L1456083582e_leqv @ A) @ L12) @ L22) => (((gram_L608943123e_subs @ A) @ L22) @ L12))))).
% 2.61/1.48 thf(fact_16_Sup__UNIV, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (((complete_Sup_Sup @ A) @ (top_top @ (set @ A))) = (top_top @ A)))))).
% 2.61/1.48 thf(fact_17_LL__L, axiom, (! [Ns:(set @ n),N:n]: ((((ord_less_eq @ (set @ (set @ t))) @ ((gram_L861977280lle_LL @ Ns) @ N)) @ ((gram_L1451583628elle_L @ Ns) @ N))))).
% 2.61/1.48 thf(fact_18_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.61/1.48 thf(fact_19_subsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A2) => (((member @ A) @ X3) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.61/1.48 thf(fact_20_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => (A2 = B2)))))).
% 2.61/1.48 thf(fact_21_set__mp, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X) @ A2) => (((member @ A) @ X) @ B2)))))).
% 2.61/1.48 thf(fact_22_in__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X) @ A2) => (((member @ A) @ X) @ B2)))))).
% 2.61/1.48 thf(fact_23_subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ B2)))))).
% 2.61/1.48 thf(fact_24_subsetCE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ B2)))))).
% 2.61/1.48 thf(fact_25_equalityE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A2)))))))))).
% 2.61/1.48 thf(fact_26_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B3:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ A3) => (((member @ A) @ X2) @ B3)))))))))).
% 2.61/1.48 thf(fact_27_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.61/1.48 thf(fact_28_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 2.61/1.48 thf(fact_29_set__rev__mp, axiom, (! [A:$tType,X:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ X) @ A2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((member @ A) @ X) @ B2)))))).
% 2.61/1.48 thf(fact_30_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B3:(set @ A)]: ((! [T:A]: (((((member @ A) @ T) @ A3) => (((member @ A) @ T) @ B3)))))))))).
% 2.61/1.48 thf(fact_31_rev__subsetD, axiom, (! [A:$tType,C2:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C2) @ A2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((member @ A) @ C2) @ B2)))))).
% 2.61/1.48 thf(fact_32_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 2.61/1.48 thf(fact_33_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.61/1.48 thf(fact_34_subset__trans, axiom, (! [A:$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)))))).
% 2.61/1.48 thf(fact_35_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.61/1.48 thf(fact_36_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.61/1.48 thf(fact_37_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.61/1.48 thf(fact_38_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y:(set @ A),Z:(set @ A)]: ((Y = Z))) = (^ [A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) & (((ord_less_eq @ (set @ A)) @ B3) @ A3)))))))).
% 2.61/1.48 thf(fact_39_contra__subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((~ ((((member @ A) @ C2) @ B2))) => (~ ((((member @ A) @ C2) @ A2)))))))).
% 2.61/1.48 thf(fact_40_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X2:A]: ((((ord_less_eq @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))))).
% 2.61/1.48 thf(fact_41_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X2:A]: (((P @ X2) => (Q @ X2)))))))).
% 2.61/1.48 thf(fact_42_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B4:B,C2:B]: (((((ord_less_eq @ A) @ A4) @ (F @ B4)) => ((((ord_less_eq @ B) @ B4) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less_eq @ B) @ X3) @ Y2) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A4) @ (F @ C2))))))))))).
% 2.61/1.48 thf(fact_43_order__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,B4:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ C) @ (F @ B4)) @ C2) => ((! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ C) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ C) @ (F @ A4)) @ C2)))))))))).
% 2.61/1.48 thf(fact_44_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B4:B,C2:B]: (((A4 = (F @ B4)) => ((((ord_less_eq @ B) @ B4) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less_eq @ B) @ X3) @ Y2) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A4) @ (F @ C2))))))))))).
% 2.61/1.48 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A4:A,P:(A > $o)]: (((((member @ A) @ A4) @ ((collect @ A) @ P)) = (P @ A4))))).
% 2.61/1.48 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A2)))) = A2)))).
% 2.61/1.48 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.61/1.48 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.61/1.48 thf(fact_49_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A4:A,B4:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A4) @ B4) => (((F @ B4) = C2) => ((! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ B) @ (F @ A4)) @ C2)))))))))).
% 2.61/1.48 thf(fact_50_Union__mono, axiom, (! [A:$tType,A2:(set @ (set @ A)),B2:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ A2) @ B2) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ A2)) @ ((complete_Sup_Sup @ (set @ A)) @ B2)))))).
% 2.61/1.48 thf(fact_51_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X2))))))))).
% 2.61/1.48 thf(fact_52_Union__least, axiom, (! [A:$tType,A2:(set @ (set @ A)),C3:(set @ A)]: (((! [X4:(set @ A)]: (((((member @ (set @ A)) @ X4) @ A2) => (((ord_less_eq @ (set @ A)) @ X4) @ C3)))) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ A2)) @ C3))))).
% 2.61/1.48 thf(fact_53_Union__upper, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ (set @ A))]: (((((member @ (set @ A)) @ B2) @ A2) => (((ord_less_eq @ (set @ A)) @ B2) @ ((complete_Sup_Sup @ (set @ A)) @ A2)))))).
% 2.61/1.48 thf(fact_54_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ X) => (X = Y4))))))))).
% 2.61/1.48 thf(fact_55_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) | (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 2.61/1.48 thf(fact_56_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((X = Y4) => (((ord_less_eq @ A) @ X) @ Y4)))))))).
% 2.61/1.48 thf(fact_57_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y4))) => (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 2.61/1.48 thf(fact_58_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ B4) @ C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 2.61/1.48 thf(fact_59_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: ((((((ord_less_eq @ A) @ X) @ Y4) => (~ ((((ord_less_eq @ A) @ Y4) @ Z2)))) => (((((ord_less_eq @ A) @ Y4) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z2)))) => (((((ord_less_eq @ A) @ X) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y4)))) => (((((ord_less_eq @ A) @ Z2) @ Y4) => (~ ((((ord_less_eq @ A) @ Y4) @ X)))) => (((((ord_less_eq @ A) @ Y4) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y4))))))))))))))))).
% 2.61/1.48 thf(fact_60_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y4:A,X:A]: (((((ord_less_eq @ A) @ Y4) @ X) => ((((ord_less_eq @ A) @ X) @ Y4) = (X = Y4))))))))).
% 2.61/1.48 thf(fact_61_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C2:A]: (((A4 = B4) => ((((ord_less_eq @ A) @ B4) @ C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 2.61/1.48 thf(fact_62_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((B4 = C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 2.61/1.48 thf(fact_63_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ B4) @ A4) => (A4 = B4))))))))).
% 2.61/1.48 thf(fact_64_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ Z2) => (((ord_less_eq @ A) @ X) @ Z2))))))))).
% 2.61/1.48 thf(fact_65_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A]: ((((ord_less_eq @ A) @ A4) @ A4))))))).
% 2.61/1.48 thf(fact_66_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A4:A,B4:A]: (((! [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((P @ A5) @ B5)))) => ((! [A5:A,B5:A]: ((((P @ B5) @ A5) => ((P @ A5) @ B5)))) => ((P @ A4) @ B4))))))))).
% 2.61/1.48 thf(fact_67_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A4:A,C2:A]: (((((ord_less_eq @ A) @ B4) @ A4) => ((((ord_less_eq @ A) @ C2) @ B4) => (((ord_less_eq @ A) @ C2) @ A4))))))))).
% 2.61/1.48 thf(fact_68_subs__def, axiom, (! [A:$tType]: (((gram_L608943123e_subs @ A) = (^ [L1:(set @ (set @ A)),L2:(set @ (set @ A))]: ((! [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ L2) => (? [Y3:(set @ A)]: (((((member @ (set @ A)) @ Y3) @ L1) & (((ord_less_eq @ (set @ A)) @ Y3) @ X2))))))))))))).
% 2.61/1.48 thf(fact_69_incl__subs, axiom, (! [A:$tType,L22:(set @ (set @ A)),L12:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ L22) @ L12) => (((gram_L608943123e_subs @ A) @ L12) @ L22))))).
% 2.61/1.48 thf(fact_70_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A4:A]: (((((ord_less_eq @ A) @ B4) @ A4) => ((((ord_less_eq @ A) @ A4) @ B4) => (A4 = B4))))))))).
% 2.61/1.48 thf(fact_71_Sup__eqI, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ A),X:A]: (((! [Y2:A]: (((((member @ A) @ Y2) @ A2) => (((ord_less_eq @ A) @ Y2) @ X)))) => ((! [Y2:A]: (((! [Z3:A]: (((((member @ A) @ Z3) @ A2) => (((ord_less_eq @ A) @ Z3) @ Y2)))) => (((ord_less_eq @ A) @ X) @ Y2)))) => (((complete_Sup_Sup @ A) @ A2) = X))))))))).
% 2.61/1.48 thf(fact_72_Sup__mono, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ A),B2:(set @ A)]: (((! [A5:A]: (((((member @ A) @ A5) @ A2) => (? [X5:A]: (((((member @ A) @ X5) @ B2) & (((ord_less_eq @ A) @ A5) @ X5))))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ ((complete_Sup_Sup @ A) @ B2))))))))).
% 2.61/1.48 thf(fact_73_Sup__least, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ A),Z2:A]: (((! [X3:A]: (((((member @ A) @ X3) @ A2) => (((ord_less_eq @ A) @ X3) @ Z2)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ Z2)))))))).
% 2.61/1.48 thf(fact_74_Sup__upper, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X:A,A2:(set @ A)]: (((((member @ A) @ X) @ A2) => (((ord_less_eq @ A) @ X) @ ((complete_Sup_Sup @ A) @ A2))))))))).
% 2.61/1.48 thf(fact_75_Sup__le__iff, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ A),B4:A]: (((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ B4) = (! [X2:A]: (((((member @ A) @ X2) @ A2) => (((ord_less_eq @ A) @ X2) @ B4))))))))))).
% 2.61/1.48 thf(fact_76_Sup__upper2, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [U:A,A2:(set @ A),V:A]: (((((member @ A) @ U) @ A2) => ((((ord_less_eq @ A) @ V) @ U) => (((ord_less_eq @ A) @ V) @ ((complete_Sup_Sup @ A) @ A2)))))))))).
% 2.61/1.48 thf(fact_77_Sup__subset__mono, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A2)) @ ((complete_Sup_Sup @ A) @ B2))))))))).
% 2.61/1.48 thf(fact_78_subs__trans, axiom, (! [A:$tType,L12:(set @ (set @ A)),L22:(set @ (set @ A)),L3:(set @ (set @ A))]: (((((gram_L608943123e_subs @ A) @ L12) @ L22) => ((((gram_L608943123e_subs @ A) @ L22) @ L3) => (((gram_L608943123e_subs @ A) @ L12) @ L3)))))).
% 2.61/1.48 thf(fact_79_subs__refl, axiom, (! [A:$tType,L12:(set @ (set @ A))]: ((((gram_L608943123e_subs @ A) @ L12) @ L12)))).
% 2.61/1.48 thf(fact_80_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.61/1.48 thf(fact_81_Union__UNIV, axiom, (! [A:$tType]: ((((complete_Sup_Sup @ (set @ A)) @ (top_top @ (set @ (set @ A)))) = (top_top @ (set @ A)))))).
% 2.61/1.48 thf(fact_82_top_Oextremum__uniqueI, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A4:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A4) => (A4 = (top_top @ A))))))))).
% 2.61/1.48 thf(fact_83_top_Oextremum__unique, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A4:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A4) = (A4 = (top_top @ A))))))))).
% 2.61/1.48 thf(fact_84_top__greatest, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A4:A]: ((((ord_less_eq @ A) @ A4) @ (top_top @ A)))))))).
% 2.61/1.48 thf(fact_85_subset__UNIV, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ (top_top @ (set @ A)))))).
% 2.61/1.48 thf(fact_86_Lr__incl__L, axiom, (! [Ns:(set @ n),Ts:n]: ((((ord_less_eq @ (set @ (set @ t))) @ ((gram_L861977318lle_Lr @ Ns) @ Ts)) @ ((gram_L1451583628elle_L @ Ns) @ Ts))))).
% 2.61/1.48 thf(fact_87_cSup__eq, axiom, (! [A:$tType]: (((((condit378418413attice @ A) @ (type2 @ A)) & ((no_bot @ A) @ (type2 @ A))) => (! [X6:(set @ A),A4:A]: (((! [X3:A]: (((((member @ A) @ X3) @ X6) => (((ord_less_eq @ A) @ X3) @ A4)))) => ((! [Y2:A]: (((! [X5:A]: (((((member @ A) @ X5) @ X6) => (((ord_less_eq @ A) @ X5) @ Y2)))) => (((ord_less_eq @ A) @ A4) @ Y2)))) => (((complete_Sup_Sup @ A) @ X6) = A4))))))))).
% 2.61/1.48 thf(fact_88_cSup__eq__maximum, axiom, (! [A:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [Z2:A,X6:(set @ A)]: (((((member @ A) @ Z2) @ X6) => ((! [X3:A]: (((((member @ A) @ X3) @ X6) => (((ord_less_eq @ A) @ X3) @ Z2)))) => (((complete_Sup_Sup @ A) @ X6) = Z2))))))))).
% 2.61/1.48 thf(fact_89_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A))))))))).
% 2.61/1.48 thf(fact_90_Sup__eq__top__iff, axiom, (! [A:$tType]: ((((comple1035589618norder @ A) @ (type2 @ A)) => (! [A2:(set @ A)]: (((((complete_Sup_Sup @ A) @ A2) = (top_top @ A)) = (! [X2:A]: (((((ord_less @ A) @ X2) @ (top_top @ A)) => (? [Y3:A]: (((((member @ A) @ Y3) @ A2) & (((ord_less @ A) @ X2) @ Y3)))))))))))))).
% 2.61/1.48 thf(fact_91_antimonoD, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y4:A]: (((((order_antimono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X) @ Y4) => (((ord_less_eq @ B) @ (F @ Y4)) @ (F @ X)))))))))).
% 2.61/1.48 thf(fact_92_antimonoE, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y4:A]: (((((order_antimono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X) @ Y4) => (((ord_less_eq @ B) @ (F @ Y4)) @ (F @ X)))))))))).
% 2.61/1.48 thf(fact_93_antimonoI, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B)]: (((! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ B) @ (F @ Y2)) @ (F @ X3))))) => (((order_antimono @ A) @ B) @ F)))))))).
% 2.61/1.48 thf(fact_94_antimono__def, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (((order_antimono @ A) @ B) = (^ [F2:(A > B)]: ((! [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) => (((ord_less_eq @ B) @ (F2 @ Y3)) @ (F2 @ X2)))))))))))).
% 2.61/1.48 thf(fact_95_Pow__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ (set @ A))) @ ((pow @ A) @ A2)) @ ((pow @ A) @ B2)))))).
% 2.61/1.48 thf(fact_96_less__eq__Sup, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ A),U:A]: (((! [V2:A]: (((((member @ A) @ V2) @ A2) => (((ord_less_eq @ A) @ U) @ V2)))) => ((A2 != (bot_bot @ (set @ A))) => (((ord_less_eq @ A) @ U) @ ((complete_Sup_Sup @ A) @ A2)))))))))).
% 2.61/1.48 thf(fact_97_bot__apply, axiom, (! [C:$tType,D:$tType]: ((((bot @ C) @ (type2 @ C)) => ((bot_bot @ (D > C)) = (^ [X2:D]: ((bot_bot @ C)))))))).
% 2.61/1.48 thf(fact_98_empty__iff, axiom, (! [A:$tType,C2:A]: ((~ ((((member @ A) @ C2) @ (bot_bot @ (set @ A)))))))).
% 2.61/1.48 thf(fact_99_all__not__in__conv, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X2:A]: ((~ ((((member @ A) @ X2) @ A2))))) = (A2 = (bot_bot @ (set @ A))))))).
% 2.61/1.48 thf(fact_100_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X2:A]: ((~ ((P @ X2))))))))).
% 2.61/1.48 thf(fact_101_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X2:A]: ((~ ((P @ X2))))))))).
% 2.61/1.48 thf(fact_102_psubsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((A2 != B2) => (((ord_less @ (set @ A)) @ A2) @ B2)))))).
% 2.61/1.48 thf(fact_103_UN__ball__bex__simps_I3_J, axiom, (! [D:$tType,A2:(set @ (set @ D)),P:(D > $o)]: (((? [X2:D]: (((((member @ D) @ X2) @ ((complete_Sup_Sup @ (set @ D)) @ A2)) & (P @ X2)))) = (? [X2:(set @ D)]: (((((member @ (set @ D)) @ X2) @ A2) & (? [Y3:D]: (((((member @ D) @ Y3) @ X2) & (P @ Y3))))))))))).
% 2.61/1.48 thf(fact_104_UN__ball__bex__simps_I1_J, axiom, (! [A:$tType,A2:(set @ (set @ A)),P:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ ((complete_Sup_Sup @ (set @ A)) @ A2)) => (P @ X2)))) = (! [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ A2) => (! [Y3:A]: (((((member @ A) @ Y3) @ X2) => (P @ Y3))))))))))).
% 2.61/1.48 thf(fact_105_UnionI, axiom, (! [A:$tType,X6:(set @ A),C3:(set @ (set @ A)),A2:A]: (((((member @ (set @ A)) @ X6) @ C3) => ((((member @ A) @ A2) @ X6) => (((member @ A) @ A2) @ ((complete_Sup_Sup @ (set @ A)) @ C3))))))).
% 2.61/1.48 thf(fact_106_Union__iff, axiom, (! [A:$tType,A2:A,C3:(set @ (set @ A))]: (((((member @ A) @ A2) @ ((complete_Sup_Sup @ (set @ A)) @ C3)) = (? [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ C3) & (((member @ A) @ A2) @ X2)))))))).
% 2.61/1.48 thf(fact_107_top1I, axiom, (! [A:$tType,X:A]: (((top_top @ (A > $o)) @ X)))).
% 2.61/1.48 thf(fact_108_Sup__bot__conv_I1_J, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ A)]: (((((complete_Sup_Sup @ A) @ A2) = (bot_bot @ A)) = (! [X2:A]: (((((member @ A) @ X2) @ A2) => (X2 = (bot_bot @ A)))))))))))).
% 2.61/1.48 thf(fact_109_Sup__bot__conv_I2_J, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [A2:(set @ A)]: ((((bot_bot @ A) = ((complete_Sup_Sup @ A) @ A2)) = (! [X2:A]: (((((member @ A) @ X2) @ A2) => (X2 = (bot_bot @ A)))))))))))).
% 2.61/1.48 thf(fact_110_empty__subsetI, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A2)))).
% 2.61/1.48 thf(fact_111_subset__empty, axiom, (! [A:$tType,A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))) = (A2 = (bot_bot @ (set @ A))))))).
% 2.61/1.48 thf(fact_112_Pow__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((member @ (set @ A)) @ A2) @ ((pow @ A) @ B2)) = (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.61/1.48 thf(fact_113_PowI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((member @ (set @ A)) @ A2) @ ((pow @ A) @ B2)))))).
% 2.61/1.48 thf(fact_114_Union__Pow__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((complete_Sup_Sup @ (set @ A)) @ ((pow @ A) @ A2)) = A2)))).
% 2.61/1.48 thf(fact_115_Sup__empty, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (((complete_Sup_Sup @ A) @ (bot_bot @ (set @ A))) = (bot_bot @ A)))))).
% 2.61/1.48 thf(fact_116_Pow__UNIV, axiom, (! [A:$tType]: ((((pow @ A) @ (top_top @ (set @ A))) = (top_top @ (set @ (set @ A))))))).
% 2.61/1.48 thf(fact_117_emptyE, axiom, (! [A:$tType,A4:A]: ((~ ((((member @ A) @ A4) @ (bot_bot @ (set @ A)))))))).
% 2.61/1.48 thf(fact_118_Pow__top, axiom, (! [A:$tType,A2:(set @ A)]: ((((member @ (set @ A)) @ A2) @ ((pow @ A) @ A2))))).
% 2.61/1.48 thf(fact_119_equals0D, axiom, (! [A:$tType,A2:(set @ A),A4:A]: (((A2 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A4) @ A2))))))).
% 2.61/1.48 thf(fact_120_equals0I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [Y2:A]: ((~ ((((member @ A) @ Y2) @ A2))))) => (A2 = (bot_bot @ (set @ A))))))).
% 2.61/1.48 thf(fact_121_Pow__bottom, axiom, (! [A:$tType,B2:(set @ A)]: ((((member @ (set @ A)) @ (bot_bot @ (set @ A))) @ ((pow @ A) @ B2))))).
% 2.61/1.48 thf(fact_122_ex__in__conv, axiom, (! [A:$tType,A2:(set @ A)]: (((? [X2:A]: ((((member @ A) @ X2) @ A2))) = (A2 != (bot_bot @ (set @ A))))))).
% 2.61/1.48 thf(fact_123_Pow__not__empty, axiom, (! [A:$tType,A2:(set @ A)]: ((((pow @ A) @ A2) != (bot_bot @ (set @ (set @ A))))))).
% 2.61/1.48 thf(fact_124_bot__fun__def, axiom, (! [B:$tType,A:$tType]: ((((bot @ B) @ (type2 @ B)) => ((bot_bot @ (A > B)) = (^ [X2:A]: ((bot_bot @ B)))))))).
% 2.61/1.48 thf(fact_125_not__psubset__empty, axiom, (! [A:$tType,A2:(set @ A)]: ((~ ((((ord_less @ (set @ A)) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.61/1.48 thf(fact_126_less__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: (((((ord_less_eq @ (A > B)) @ F2) @ G2) & (~ ((((ord_less_eq @ (A > B)) @ G2) @ F2))))))))))).
% 2.61/1.48 thf(fact_127_UnionE, axiom, (! [A:$tType,A2:A,C3:(set @ (set @ A))]: (((((member @ A) @ A2) @ ((complete_Sup_Sup @ (set @ A)) @ C3)) => (~ ((! [X4:(set @ A)]: (((((member @ A) @ A2) @ X4) => (~ ((((member @ (set @ A)) @ X4) @ C3)))))))))))).
% 2.61/1.48 thf(fact_128_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B4:B,C2:B]: (((A4 = (F @ B4)) => ((((ord_less @ B) @ B4) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less @ B) @ X3) @ Y2) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ A) @ A4) @ (F @ C2))))))))))).
% 2.61/1.48 thf(fact_129_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A4:A,B4:A,F:(A > B),C2:B]: (((((ord_less @ A) @ A4) @ B4) => (((F @ B4) = C2) => ((! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (((ord_less @ B) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ B) @ (F @ A4)) @ C2)))))))))).
% 2.61/1.48 thf(fact_130_order__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B4:B,C2:B]: (((((ord_less @ A) @ A4) @ (F @ B4)) => ((((ord_less @ B) @ B4) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less @ B) @ X3) @ Y2) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ A) @ A4) @ (F @ C2))))))))))).
% 2.61/1.48 thf(fact_131_order__less__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,B4:A,F:(A > C),C2:C]: (((((ord_less @ A) @ A4) @ B4) => ((((ord_less @ C) @ (F @ B4)) @ C2) => ((! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (((ord_less @ C) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ C) @ (F @ A4)) @ C2)))))))))).
% 2.61/1.48 thf(fact_132_lt__ex, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [X:A]: ((? [Y2:A]: ((((ord_less @ A) @ Y2) @ X))))))))).
% 2.61/1.48 thf(fact_133_gt__ex, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [X:A]: ((? [X1:A]: ((((ord_less @ A) @ X) @ X1))))))))).
% 2.61/1.48 thf(fact_134_Union__empty, axiom, (! [A:$tType]: ((((complete_Sup_Sup @ (set @ A)) @ (bot_bot @ (set @ (set @ A)))) = (bot_bot @ (set @ A)))))).
% 2.61/1.48 thf(fact_135_neqE, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((X != Y4) => ((~ ((((ord_less @ A) @ X) @ Y4))) => (((ord_less @ A) @ Y4) @ X))))))))).
% 2.61/1.48 thf(fact_136_neq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((X != Y4) = ((((ord_less @ A) @ X) @ Y4) | (((ord_less @ A) @ Y4) @ X))))))))).
% 2.61/1.48 thf(fact_137_order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) => (~ ((((ord_less @ A) @ B4) @ A4)))))))))).
% 2.61/1.48 thf(fact_138_dense, axiom, (! [A:$tType]: ((((dense_order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (? [Z4:A]: (((((ord_less @ A) @ X) @ Z4) & (((ord_less @ A) @ Z4) @ Y4))))))))))).
% 2.61/1.48 thf(fact_139_Union__empty__conv, axiom, (! [A:$tType,A2:(set @ (set @ A))]: (((((complete_Sup_Sup @ (set @ A)) @ A2) = (bot_bot @ (set @ A))) = (! [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ A2) => (X2 = (bot_bot @ (set @ A)))))))))).
% 2.61/1.48 thf(fact_140_empty__Union__conv, axiom, (! [A:$tType,A2:(set @ (set @ A))]: ((((bot_bot @ (set @ A)) = ((complete_Sup_Sup @ (set @ A)) @ A2)) = (! [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ A2) => (X2 = (bot_bot @ (set @ A)))))))))).
% 2.61/1.48 thf(fact_141_less__imp__neq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (X != Y4)))))))).
% 2.61/1.48 thf(fact_142_less__asym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (~ ((((ord_less @ A) @ Y4) @ X)))))))))).
% 2.61/1.48 thf(fact_143_less__asym_H, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) => (~ ((((ord_less @ A) @ B4) @ A4)))))))))).
% 2.61/1.48 thf(fact_144_less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((ord_less @ A) @ X) @ Y4) => ((((ord_less @ A) @ Y4) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 2.61/1.48 thf(fact_145_less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) | ((X = Y4) | (((ord_less @ A) @ Y4) @ X))))))))).
% 2.61/1.48 thf(fact_146_less__irrefl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 2.61/1.48 thf(fact_147_ord__eq__less__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C2:A]: (((A4 = B4) => ((((ord_less @ A) @ B4) @ C2) => (((ord_less @ A) @ A4) @ C2))))))))).
% 2.61/1.48 thf(fact_148_ord__less__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C2:A]: (((((ord_less @ A) @ A4) @ B4) => ((B4 = C2) => (((ord_less @ A) @ A4) @ C2))))))))).
% 2.61/1.48 thf(fact_149_dual__order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A4:A]: (((((ord_less @ A) @ B4) @ A4) => (~ ((((ord_less @ A) @ A4) @ B4)))))))))).
% 2.61/1.48 thf(fact_150_less__imp__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (X != Y4)))))))).
% 2.61/1.48 thf(fact_151_less__not__sym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (~ ((((ord_less @ A) @ Y4) @ X)))))))))).
% 2.61/1.48 thf(fact_152_less__induct, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A4:A]: (((! [X3:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X3) => (P @ Y5)))) => (P @ X3)))) => (P @ A4)))))))).
% 2.61/1.48 thf(fact_153_antisym__conv3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y4:A,X:A]: (((~ ((((ord_less @ A) @ Y4) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y4))) = (X = Y4))))))))).
% 2.61/1.48 thf(fact_154_less__imp__not__eq2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (Y4 != X)))))))).
% 2.61/1.48 thf(fact_155_less__imp__triv, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,P:$o]: (((((ord_less @ A) @ X) @ Y4) => ((((ord_less @ A) @ Y4) @ X) => P)))))))).
% 2.61/1.48 thf(fact_156_linorder__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) => ((X != Y4) => (((ord_less @ A) @ Y4) @ X))))))))).
% 2.61/1.48 thf(fact_157_dual__order_Oirrefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A]: ((~ ((((ord_less @ A) @ A4) @ A4))))))))).
% 2.61/1.48 thf(fact_158_order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C2:A]: (((((ord_less @ A) @ A4) @ B4) => ((((ord_less @ A) @ B4) @ C2) => (((ord_less @ A) @ A4) @ C2))))))))).
% 2.61/1.48 thf(fact_159_less__imp__not__less, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (~ ((((ord_less @ A) @ Y4) @ X)))))))))).
% 2.61/1.48 thf(fact_160_bot_Oextremum__strict, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A4:A]: ((~ ((((ord_less @ A) @ A4) @ (bot_bot @ A)))))))))).
% 2.61/1.48 thf(fact_161_bot_Onot__eq__extremum, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A4:A]: (((A4 != (bot_bot @ A)) = (((ord_less @ A) @ (bot_bot @ A)) @ A4)))))))).
% 2.61/1.48 thf(fact_162_dual__order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A4:A,C2:A]: (((((ord_less @ A) @ B4) @ A4) => ((((ord_less @ A) @ C2) @ B4) => (((ord_less @ A) @ C2) @ A4))))))))).
% 2.61/1.48 thf(fact_163_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) = ((((ord_less @ A) @ Y4) @ X) | (X = Y4))))))))).
% 2.61/1.48 thf(fact_164_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) => (A4 != B4)))))))).
% 2.61/1.48 thf(fact_165_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A4:A]: (((((ord_less @ A) @ B4) @ A4) => (A4 != B4)))))))).
% 2.61/1.48 thf(fact_166_less__cSupD, axiom, (! [A:$tType]: ((((condit1037483654norder @ A) @ (type2 @ A)) => (! [X6:(set @ A),Z2:A]: (((X6 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ Z2) @ ((complete_Sup_Sup @ A) @ X6)) => (? [X3:A]: (((((member @ A) @ X3) @ X6) & (((ord_less @ A) @ Z2) @ X3)))))))))))).
% 2.61/1.48 thf(fact_167_less__cSupE, axiom, (! [A:$tType]: ((((condit1037483654norder @ A) @ (type2 @ A)) => (! [Y4:A,X6:(set @ A)]: (((((ord_less @ A) @ Y4) @ ((complete_Sup_Sup @ A) @ X6)) => ((X6 != (bot_bot @ (set @ A))) => (~ ((! [X3:A]: (((((member @ A) @ X3) @ X6) => (~ ((((ord_less @ A) @ Y4) @ X3)))))))))))))))).
% 2.61/1.48 thf(fact_168_ex__gt__or__lt, axiom, (! [A:$tType]: ((((condit1656338222tinuum @ A) @ (type2 @ A)) => (! [A4:A]: ((? [B5:A]: (((((ord_less @ A) @ A4) @ B5) | (((ord_less @ A) @ B5) @ A4)))))))))).
% 2.61/1.48 thf(fact_169_complete__interval, axiom, (! [A:$tType]: ((((condit1037483654norder @ A) @ (type2 @ A)) => (! [A4:A,B4:A,P:(A > $o)]: (((((ord_less @ A) @ A4) @ B4) => ((P @ A4) => ((~ ((P @ B4))) => (? [C4:A]: (((((ord_less_eq @ A) @ A4) @ C4) & ((((ord_less_eq @ A) @ C4) @ B4) & ((! [X5:A]: ((((((ord_less_eq @ A) @ A4) @ X5) & (((ord_less @ A) @ X5) @ C4)) => (P @ X5)))) & (! [D2:A]: (((! [X3:A]: ((((((ord_less_eq @ A) @ A4) @ X3) & (((ord_less @ A) @ X3) @ D2)) => (P @ X3)))) => (((ord_less_eq @ A) @ D2) @ C4)))))))))))))))))).
% 2.61/1.48 thf(fact_170_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((A4 != B4) => ((((ord_less_eq @ A) @ A4) @ B4) => (((ord_less @ A) @ A4) @ B4))))))))).
% 2.61/1.48 thf(fact_171_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A4:A]: (((((ord_less @ A) @ B4) @ A4) => (((ord_less_eq @ A) @ B4) @ A4)))))))).
% 2.61/1.48 thf(fact_172_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B6:A,A6:A]: (((((ord_less_eq @ A) @ B6) @ A6) & (A6 != B6))))))))).
% 2.61/1.48 thf(fact_173_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B6:A,A6:A]: (((((ord_less @ A) @ B6) @ A6) | (A6 = B6))))))))).
% 2.61/1.48 thf(fact_174_order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) => (((ord_less_eq @ A) @ A4) @ B4)))))))).
% 2.61/1.48 thf(fact_175_dense__le__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((ord_less @ A) @ X) @ Y4) => ((! [W:A]: (((((ord_less @ A) @ X) @ W) => ((((ord_less @ A) @ W) @ Y4) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y4) @ Z2))))))))).
% 2.61/1.48 thf(fact_176_dense__ge__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z2:A,X:A,Y4:A]: (((((ord_less @ A) @ Z2) @ X) => ((! [W:A]: (((((ord_less @ A) @ Z2) @ W) => ((((ord_less @ A) @ W) @ X) => (((ord_less_eq @ A) @ Y4) @ W))))) => (((ord_less_eq @ A) @ Y4) @ Z2))))))))).
% 2.61/1.48 thf(fact_177_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A4:A,C2:A]: (((((ord_less @ A) @ B4) @ A4) => ((((ord_less_eq @ A) @ C2) @ B4) => (((ord_less @ A) @ C2) @ A4))))))))).
% 2.61/1.48 thf(fact_178_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A4:A,C2:A]: (((((ord_less_eq @ A) @ B4) @ A4) => ((((ord_less @ A) @ C2) @ B4) => (((ord_less @ A) @ C2) @ A4))))))))).
% 2.61/1.48 thf(fact_179_order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [A6:A,B6:A]: (((((ord_less_eq @ A) @ A6) @ B6) & (A6 != B6))))))))).
% 2.61/1.48 thf(fact_180_order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A6:A,B6:A]: (((((ord_less @ A) @ A6) @ B6) | (A6 = B6))))))))).
% 2.61/1.48 thf(fact_181_order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C2:A]: (((((ord_less @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ B4) @ C2) => (((ord_less @ A) @ A4) @ C2))))))))).
% 2.61/1.48 thf(fact_182_order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less @ A) @ B4) @ C2) => (((ord_less @ A) @ A4) @ C2))))))))).
% 2.61/1.48 thf(fact_183_not__le__imp__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y4:A,X:A]: (((~ ((((ord_less_eq @ A) @ Y4) @ X))) => (((ord_less @ A) @ X) @ Y4)))))))).
% 2.61/1.48 thf(fact_184_less__le__not__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) & (~ ((((ord_less_eq @ A) @ Y3) @ X2))))))))))).
% 2.61/1.48 thf(fact_185_le__imp__less__or__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less @ A) @ X) @ Y4) | (X = Y4))))))))).
% 2.61/1.48 thf(fact_186_le__less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) | (((ord_less @ A) @ Y4) @ X)))))))).
% 2.61/1.48 thf(fact_187_dense__le, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Y4:A,Z2:A]: (((! [X3:A]: (((((ord_less @ A) @ X3) @ Y4) => (((ord_less_eq @ A) @ X3) @ Z2)))) => (((ord_less_eq @ A) @ Y4) @ Z2)))))))).
% 2.61/1.48 thf(fact_188_dense__ge, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z2:A,Y4:A]: (((! [X3:A]: (((((ord_less @ A) @ Z2) @ X3) => (((ord_less_eq @ A) @ Y4) @ X3)))) => (((ord_less_eq @ A) @ Y4) @ Z2)))))))).
% 2.61/1.48 thf(fact_189_less__le__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((ord_less @ A) @ X) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 2.61/1.48 thf(fact_190_le__less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less @ A) @ Y4) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 2.61/1.48 thf(fact_191_antisym__conv2, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((~ ((((ord_less @ A) @ X) @ Y4))) = (X = Y4))))))))).
% 2.61/1.48 thf(fact_192_antisym__conv1, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) => ((((ord_less_eq @ A) @ X) @ Y4) = (X = Y4))))))))).
% 2.61/1.48 thf(fact_193_less__imp__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (((ord_less_eq @ A) @ X) @ Y4)))))))).
% 2.61/1.48 thf(fact_194_le__neq__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((A4 != B4) => (((ord_less @ A) @ A4) @ B4))))))))).
% 2.61/1.48 thf(fact_195_not__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) = (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 2.61/1.48 thf(fact_196_not__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y4))) = (((ord_less @ A) @ Y4) @ X)))))))).
% 2.61/1.48 thf(fact_197_order__less__le__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,B4:A,F:(A > C),C2:C]: (((((ord_less @ A) @ A4) @ B4) => ((((ord_less_eq @ C) @ (F @ B4)) @ C2) => ((! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (((ord_less @ C) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ C) @ (F @ A4)) @ C2)))))))))).
% 2.61/1.48 thf(fact_198_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B4:B,C2:B]: (((((ord_less @ A) @ A4) @ (F @ B4)) => ((((ord_less_eq @ B) @ B4) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less_eq @ B) @ X3) @ Y2) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ A) @ A4) @ (F @ C2))))))))))).
% 2.61/1.48 thf(fact_199_order__le__less__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,B4:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less @ C) @ (F @ B4)) @ C2) => ((! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ C) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ C) @ (F @ A4)) @ C2)))))))))).
% 2.61/1.48 thf(fact_200_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B4:B,C2:B]: (((((ord_less_eq @ A) @ A4) @ (F @ B4)) => ((((ord_less @ B) @ B4) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less @ B) @ X3) @ Y2) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ A) @ A4) @ (F @ C2))))))))))).
% 2.61/1.48 thf(fact_201_less__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) & (X2 != Y3))))))))).
% 2.61/1.48 thf(fact_202_le__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X2:A,Y3:A]: (((((ord_less @ A) @ X2) @ Y3) | (X2 = Y3))))))))).
% 2.61/1.48 thf(fact_203_leI, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) => (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 2.61/1.48 thf(fact_204_leD, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y4:A,X:A]: (((((ord_less_eq @ A) @ Y4) @ X) => (~ ((((ord_less @ A) @ X) @ Y4)))))))))).
% 2.61/1.48 thf(fact_205_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A4:A]: (((((ord_less_eq @ A) @ A4) @ (bot_bot @ A)) => (A4 = (bot_bot @ A))))))))).
% 2.61/1.48 thf(fact_206_bot_Oextremum__unique, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A4:A]: (((((ord_less_eq @ A) @ A4) @ (bot_bot @ A)) = (A4 = (bot_bot @ A))))))))).
% 2.61/1.48 thf(fact_207_bot_Oextremum, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A4:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A4))))))).
% 2.61/1.48 thf(fact_208_top_Oextremum__strict, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A4:A]: ((~ ((((ord_less @ A) @ (top_top @ A)) @ A4))))))))).
% 2.61/1.48 thf(fact_209_top_Onot__eq__extremum, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A4:A]: (((A4 != (top_top @ A)) = (((ord_less @ A) @ A4) @ (top_top @ A))))))))).
% 2.61/1.48 thf(fact_210_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A3:(set @ A),B3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B3) | (A3 = B3)))))))).
% 2.61/1.48 thf(fact_211_subset__psubset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A2) @ C3)))))).
% 2.61/1.48 thf(fact_212_subset__not__subset__eq, axiom, (! [A:$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)))))))))).
% 2.61/1.48 thf(fact_213_psubset__subset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A2) @ C3)))))).
% 2.61/1.48 thf(fact_214_psubset__imp__subset, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.61/1.48 thf(fact_215_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) & (A3 != B3)))))))).
% 2.61/1.48 thf(fact_216_psubsetE, axiom, (! [A:$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)))))))).
% 2.61/1.48 thf(fact_217_less__Sup__iff, axiom, (! [A:$tType]: ((((comple1035589618norder @ A) @ (type2 @ A)) => (! [A4:A,S:(set @ A)]: (((((ord_less @ A) @ A4) @ ((complete_Sup_Sup @ A) @ S)) = (? [X2:A]: (((((member @ A) @ X2) @ S) & (((ord_less @ A) @ A4) @ X2))))))))))).
% 2.61/1.48 thf(fact_218_PowD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((member @ (set @ A)) @ A2) @ ((pow @ A) @ B2)) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.61/1.48 thf(fact_219_empty__not__UNIV, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) != (top_top @ (set @ A)))))).
% 2.61/1.48 thf(fact_220_cSup__eq__non__empty, axiom, (! [A:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [X6:(set @ A),A4:A]: (((X6 != (bot_bot @ (set @ A))) => ((! [X3:A]: (((((member @ A) @ X3) @ X6) => (((ord_less_eq @ A) @ X3) @ A4)))) => ((! [Y2:A]: (((! [X5:A]: (((((member @ A) @ X5) @ X6) => (((ord_less_eq @ A) @ X5) @ Y2)))) => (((ord_less_eq @ A) @ A4) @ Y2)))) => (((complete_Sup_Sup @ A) @ X6) = A4)))))))))).
% 2.61/1.48 thf(fact_221_cSup__least, axiom, (! [A:$tType]: ((((condit378418413attice @ A) @ (type2 @ A)) => (! [X6:(set @ A),Z2:A]: (((X6 != (bot_bot @ (set @ A))) => ((! [X3:A]: (((((member @ A) @ X3) @ X6) => (((ord_less_eq @ A) @ X3) @ Z2)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ X6)) @ Z2))))))))).
% 2.61/1.48 thf(fact_222_subset__Pow__Union, axiom, (! [A:$tType,A2:(set @ (set @ A))]: ((((ord_less_eq @ (set @ (set @ A))) @ A2) @ ((pow @ A) @ ((complete_Sup_Sup @ (set @ A)) @ A2)))))).
% 2.61/1.48 thf(fact_223_le__Sup__iff, axiom, (! [A:$tType]: ((((comple1035589618norder @ A) @ (type2 @ A)) => (! [X:A,A2:(set @ A)]: (((((ord_less_eq @ A) @ X) @ ((complete_Sup_Sup @ A) @ A2)) = (! [Y3:A]: (((((ord_less @ A) @ Y3) @ X) => (? [X2:A]: (((((member @ A) @ X2) @ A2) & (((ord_less @ A) @ Y3) @ X2)))))))))))))).
% 2.61/1.48 thf(fact_224_subset__emptyI, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X3:A]: ((~ ((((member @ A) @ X3) @ A2))))) => (((ord_less_eq @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.61/1.48 thf(fact_225_pinf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => (~ ((((ord_less_eq @ A) @ X5) @ T2)))))))))))))).
% 2.61/1.48 thf(fact_226_pinf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => (((ord_less_eq @ A) @ T2) @ X5)))))))))))).
% 2.61/1.48 thf(fact_227_minf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => (((ord_less_eq @ A) @ X5) @ T2)))))))))))).
% 2.61/1.48 thf(fact_228_minf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => (~ ((((ord_less_eq @ A) @ T2) @ X5)))))))))))))).
% 2.61/1.48 thf(fact_229_psubsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:A]: (((((ord_less @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ B2)))))).
% 2.61/1.48 thf(fact_230_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.61/1.48 thf(fact_231_psubset__trans, axiom, (! [A:$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)))))).
% 2.61/1.48 thf(fact_232_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X2:A]: ((((member @ A) @ X2) @ (bot_bot @ (set @ A))))))))).
% 2.61/1.48 thf(fact_233_minf_I11_J, axiom, (! [C:$tType,D:$tType]: ((((ord @ C) @ (type2 @ C)) => (! [F3:D]: ((? [Z4:C]: ((! [X5:C]: (((((ord_less @ C) @ X5) @ Z4) => (F3 = F3)))))))))))).
% 2.61/1.48 thf(fact_234_minf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => (~ ((((ord_less @ A) @ T2) @ X5)))))))))))))).
% 2.61/1.48 thf(fact_235_minf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => (((ord_less @ A) @ X5) @ T2)))))))))))).
% 2.61/1.48 thf(fact_236_minf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => (X5 != T2)))))))))))).
% 2.61/1.48 thf(fact_237_minf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => (X5 != T2)))))))))))).
% 2.61/1.48 thf(fact_238_minf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z3:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z3) => ((P @ X3) = (P2 @ X3))))))) => ((? [Z3:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z3) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => (((P @ X5) | (Q @ X5)) = ((P2 @ X5) | (Q2 @ X5)))))))))))))))).
% 2.61/1.48 thf(fact_239_minf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z3:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z3) => ((P @ X3) = (P2 @ X3))))))) => ((? [Z3:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z3) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => (((P @ X5) & (Q @ X5)) = ((P2 @ X5) & (Q2 @ X5)))))))))))))))).
% 2.61/1.48 thf(fact_240_pinf_I11_J, axiom, (! [C:$tType,D:$tType]: ((((ord @ C) @ (type2 @ C)) => (! [F3:D]: ((? [Z4:C]: ((! [X5:C]: (((((ord_less @ C) @ Z4) @ X5) => (F3 = F3)))))))))))).
% 2.61/1.48 thf(fact_241_pinf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => (((ord_less @ A) @ T2) @ X5)))))))))))).
% 2.61/1.48 thf(fact_242_pinf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => (~ ((((ord_less @ A) @ X5) @ T2)))))))))))))).
% 2.61/1.48 thf(fact_243_pinf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => (X5 != T2)))))))))))).
% 2.61/1.48 thf(fact_244_pinf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => (X5 != T2)))))))))))).
% 2.61/1.48 thf(fact_245_pinf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z3:A]: ((! [X3:A]: (((((ord_less @ A) @ Z3) @ X3) => ((P @ X3) = (P2 @ X3))))))) => ((? [Z3:A]: ((! [X3:A]: (((((ord_less @ A) @ Z3) @ X3) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => (((P @ X5) | (Q @ X5)) = ((P2 @ X5) | (Q2 @ X5)))))))))))))))).
% 2.61/1.48 thf(fact_246_pinf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z3:A]: ((! [X3:A]: (((((ord_less @ A) @ Z3) @ X3) => ((P @ X3) = (P2 @ X3))))))) => ((? [Z3:A]: ((! [X3:A]: (((((ord_less @ A) @ Z3) @ X3) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => (((P @ X5) & (Q @ X5)) = ((P2 @ X5) & (Q2 @ X5)))))))))))))))).
% 2.61/1.48 thf(fact_247_top_Oordering__top__axioms, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => ((((ordering_top @ A) @ (ord_less_eq @ A)) @ (ord_less @ A)) @ (top_top @ A)))))).
% 2.61/1.48 thf(fact_248_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A3:(set @ A)]: ((A3 = (bot_bot @ (set @ A))))))))).
% 2.61/1.48 thf(fact_249_ordering__top_Oextremum, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A4:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => ((Less_eq @ A4) @ Top))))).
% 2.61/1.48 thf(fact_250_ordering__top_Oextremum__strict, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A4:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => (~ (((Less @ Top) @ A4))))))).
% 2.61/1.48 thf(fact_251_ordering__top_Oextremum__unique, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A4:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => (((Less_eq @ Top) @ A4) = (A4 = Top)))))).
% 2.61/1.48 thf(fact_252_ordering__top_Onot__eq__extremum, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A4:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => ((A4 != Top) = ((Less @ A4) @ Top)))))).
% 2.61/1.48 thf(fact_253_ordering__top_Oextremum__uniqueI, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),Top:A,A4:A]: ((((((ordering_top @ A) @ Less_eq) @ Less) @ Top) => (((Less_eq @ Top) @ A4) => (A4 = Top)))))).
% 2.61/1.48 thf(fact_254_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 2.61/1.48 thf(fact_255_top__conj_I2_J, axiom, (! [A:$tType,P:$o,X:A]: (((P & ((top_top @ (A > $o)) @ X)) = P)))).
% 2.61/1.48 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice, axiom, (! [A7:$tType,A8:$tType]: ((((comple187826305attice @ A8) @ (type2 @ A8)) => ((condit378418413attice @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.61/1.48 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A7:$tType,A8:$tType]: ((((comple187826305attice @ A8) @ (type2 @ A8)) => ((comple187826305attice @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.61/1.48 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A7:$tType,A8:$tType]: ((((order_top @ A8) @ (type2 @ A8)) => ((order_top @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.61/1.48 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A7:$tType,A8:$tType]: ((((order_bot @ A8) @ (type2 @ A8)) => ((order_bot @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.61/1.48 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: ((((preorder @ A8) @ (type2 @ A8)) => ((preorder @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.61/1.48 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: ((((order @ A8) @ (type2 @ A8)) => ((order @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.61/1.48 thf(tcon_fun___Orderings_Otop, axiom, (! [A7:$tType,A8:$tType]: ((((top @ A8) @ (type2 @ A8)) => ((top @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.61/1.48 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: ((((ord @ A8) @ (type2 @ A8)) => ((ord @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.61/1.48 thf(tcon_fun___Orderings_Obot, axiom, (! [A7:$tType,A8:$tType]: ((((bot @ A8) @ (type2 @ A8)) => ((bot @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.61/1.48 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1, axiom, (! [A7:$tType]: (((condit378418413attice @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.61/1.48 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_2, axiom, (! [A7:$tType]: (((comple187826305attice @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.61/1.48 thf(tcon_Set_Oset___Orderings_Oorder__top_3, axiom, (! [A7:$tType]: (((order_top @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.61/1.48 thf(tcon_Set_Oset___Orderings_Oorder__bot_4, axiom, (! [A7:$tType]: (((order_bot @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.61/1.48 thf(tcon_Set_Oset___Orderings_Opreorder_5, axiom, (! [A7:$tType]: (((preorder @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.61/1.48 thf(tcon_Set_Oset___Orderings_Oorder_6, axiom, (! [A7:$tType]: (((order @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.61/1.48 thf(tcon_Set_Oset___Orderings_Otop_7, axiom, (! [A7:$tType]: (((top @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.61/1.48 thf(tcon_Set_Oset___Orderings_Oord_8, axiom, (! [A7:$tType]: (((ord @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.61/1.48 thf(tcon_Set_Oset___Orderings_Obot_9, axiom, (! [A7:$tType]: (((bot @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.61/1.48 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_10, axiom, ((condit378418413attice @ $o) @ (type2 @ $o))).
% 2.61/1.48 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_11, axiom, ((comple187826305attice @ $o) @ (type2 @ $o))).
% 2.61/1.48 thf(tcon_HOL_Obool___Orderings_Oorder__top_12, axiom, ((order_top @ $o) @ (type2 @ $o))).
% 2.61/1.48 thf(tcon_HOL_Obool___Orderings_Oorder__bot_13, axiom, ((order_bot @ $o) @ (type2 @ $o))).
% 2.61/1.48 thf(tcon_HOL_Obool___Orderings_Opreorder_14, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.61/1.48 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.61/1.48 thf(tcon_HOL_Obool___Orderings_Oorder_15, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.61/1.48 thf(tcon_HOL_Obool___Orderings_Otop_16, axiom, ((top @ $o) @ (type2 @ $o))).
% 2.61/1.48 thf(tcon_HOL_Obool___Orderings_Oord_17, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.61/1.48 thf(tcon_HOL_Obool___Orderings_Obot_18, axiom, ((bot @ $o) @ (type2 @ $o))).
% 2.61/1.48 thf(conj_0, conjecture, (((gram_L1456083582e_leqv @ t) @ ((gram_L861977280lle_LL @ (top_top @ (set @ n))) @ ts)) @ ((gram_L861977318lle_Lr @ (top_top @ (set @ n))) @ ts))).
% 2.61/1.48 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.LejkCnALQX/DTF2DTF24078.p
% 2.61/1.48 ------------------------
% 2.85/1.51 ---- Cleaned THF ---
% 2.85/1.51 thf(ty_t_Set_Oset,type,
% 2.85/1.51 set: $tType > $tType ).
% 2.85/1.51
% 2.85/1.51 thf(ty_t_DTree_OT,type,
% 2.85/1.51 t: $tType ).
% 2.85/1.51
% 2.85/1.51 thf(ty_t_DTree_ON,type,
% 2.85/1.51 n: $tType ).
% 2.85/1.51
% 2.85/1.51 thf(ty_t_itself,type,
% 2.85/1.51 itself: $tType > $tType ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_HOL_Otype,type,
% 2.85/1.51 type:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Orderings_Obot,type,
% 2.85/1.51 bot:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Orderings_Oord,type,
% 2.85/1.51 ord:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Orderings_Otop,type,
% 2.85/1.51 top:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Orderings_Oorder,type,
% 2.85/1.51 order:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Orderings_Ono__bot,type,
% 2.85/1.51 no_bot:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Orderings_Ono__top,type,
% 2.85/1.51 no_top:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Orderings_Olinorder,type,
% 2.85/1.51 linorder:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Orderings_Opreorder,type,
% 2.85/1.51 preorder:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.85/1.51 order_bot:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Orderings_Oorder__top,type,
% 2.85/1.51 order_top:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Orderings_Owellorder,type,
% 2.85/1.51 wellorder:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Orderings_Odense__order,type,
% 2.85/1.51 dense_order:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.85/1.51 dense_linorder:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 2.85/1.51 comple187826305attice:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
% 2.85/1.51 comple1035589618norder:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.85/1.51 condit1656338222tinuum:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
% 2.85/1.51 condit378418413attice:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.85/1.51 condit1037483654norder:
% 2.85/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
% 2.85/1.51 complete_Sup_Sup:
% 2.85/1.51 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OL,type,
% 2.85/1.51 gram_L1451583628elle_L: ( set @ n ) > n > ( set @ ( set @ t ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OLL,type,
% 2.85/1.51 gram_L861977280lle_LL: ( set @ n ) > n > ( set @ ( set @ t ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OLr,type,
% 2.85/1.51 gram_L861977318lle_Lr: ( set @ n ) > n > ( set @ ( set @ t ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Oleqv,type,
% 2.85/1.51 gram_L1456083582e_leqv:
% 2.85/1.51 !>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ A ) ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Osubs,type,
% 2.85/1.51 gram_L608943123e_subs:
% 2.85/1.51 !>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ A ) ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.85/1.51 bot_bot:
% 2.85/1.51 !>[A: $tType] : A ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.85/1.51 ord_less:
% 2.85/1.51 !>[A: $tType] : ( A > A > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.85/1.51 ord_less_eq:
% 2.85/1.51 !>[A: $tType] : ( A > A > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 2.85/1.51 order_antimono:
% 2.85/1.51 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_Orderings_Oordering__top,type,
% 2.85/1.51 ordering_top:
% 2.85/1.51 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.85/1.51 top_top:
% 2.85/1.51 !>[A: $tType] : A ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_Pure_Otype,type,
% 2.85/1.51 type2:
% 2.85/1.51 !>[A: $tType] : ( itself @ A ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_Set_OCollect,type,
% 2.85/1.51 collect:
% 2.85/1.51 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_Set_OPow,type,
% 2.85/1.51 pow:
% 2.85/1.51 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_Set_Ois__empty,type,
% 2.85/1.51 is_empty:
% 2.85/1.51 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_c_member,type,
% 2.85/1.51 member:
% 2.85/1.51 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(sy_v_ts,type,
% 2.85/1.51 ts: n ).
% 2.85/1.51
% 2.85/1.51 thf(fact_0_Lr__leqv__L,axiom,
% 2.85/1.51 ! [Ts: n] : ( gram_L1456083582e_leqv @ t @ ( gram_L861977318lle_Lr @ ( top_top @ ( set @ n ) ) @ Ts ) @ ( gram_L1451583628elle_L @ ( top_top @ ( set @ n ) ) @ Ts ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_1_LL__leqv__L,axiom,
% 2.85/1.51 ! [Ts: n] : ( gram_L1456083582e_leqv @ t @ ( gram_L861977280lle_LL @ ( top_top @ ( set @ n ) ) @ Ts ) @ ( gram_L1451583628elle_L @ ( top_top @ ( set @ n ) ) @ Ts ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_2_leqv__Sym,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( gram_L1456083582e_leqv @ A )
% 2.85/1.51 = ( ^ [L1: set @ ( set @ A ),L2: set @ ( set @ A )] : ( gram_L1456083582e_leqv @ A @ L2 @ L1 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_3_leqv__sym,axiom,
% 2.85/1.51 ! [A: $tType,L12: set @ ( set @ A ),L22: set @ ( set @ A )] :
% 2.85/1.51 ( ( gram_L1456083582e_leqv @ A @ L12 @ L22 )
% 2.85/1.51 => ( gram_L1456083582e_leqv @ A @ L22 @ L12 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_4_leqv__refl,axiom,
% 2.85/1.51 ! [A: $tType,L12: set @ ( set @ A )] : ( gram_L1456083582e_leqv @ A @ L12 @ L12 ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_5_leqv__trans,axiom,
% 2.85/1.51 ! [A: $tType,L12: set @ ( set @ A ),L22: set @ ( set @ A ),L3: set @ ( set @ A )] :
% 2.85/1.51 ( ( gram_L1456083582e_leqv @ A @ L12 @ L22 )
% 2.85/1.51 => ( ( gram_L1456083582e_leqv @ A @ L22 @ L3 )
% 2.85/1.51 => ( gram_L1456083582e_leqv @ A @ L12 @ L3 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_6_UNIV__I,axiom,
% 2.85/1.51 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_7_iso__tuple__UNIV__I,axiom,
% 2.85/1.51 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_8_top__apply,axiom,
% 2.85/1.51 ! [C: $tType,D: $tType] :
% 2.85/1.51 ( ( top @ C @ ( type2 @ C ) )
% 2.85/1.51 => ( ( top_top @ ( D > C ) )
% 2.85/1.51 = ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_9_Lr__subs__L,axiom,
% 2.85/1.51 ! [Ts: n] : ( gram_L608943123e_subs @ t @ ( gram_L861977318lle_Lr @ ( top_top @ ( set @ n ) ) @ Ts ) @ ( gram_L1451583628elle_L @ ( top_top @ ( set @ n ) ) @ Ts ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_10_UNIV__eq__I,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A] :
% 2.85/1.51 ( ! [X3: A] : ( member @ A @ X3 @ A2 )
% 2.85/1.51 => ( ( top_top @ ( set @ A ) )
% 2.85/1.51 = A2 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_11_UNIV__witness,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_12_Lr__LL,axiom,
% 2.85/1.51 ! [Ns: set @ n,N: n] : ( ord_less_eq @ ( set @ ( set @ t ) ) @ ( gram_L861977318lle_Lr @ Ns @ N ) @ ( gram_L861977280lle_LL @ Ns @ N ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_13_leqv__def,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( gram_L1456083582e_leqv @ A )
% 2.85/1.51 = ( ^ [L1: set @ ( set @ A ),L2: set @ ( set @ A )] :
% 2.85/1.51 ( ( gram_L608943123e_subs @ A @ L1 @ L2 )
% 2.85/1.51 & ( gram_L608943123e_subs @ A @ L2 @ L1 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_14_subs__leqv,axiom,
% 2.85/1.51 ! [A: $tType,L12: set @ ( set @ A ),L22: set @ ( set @ A )] :
% 2.85/1.51 ( ( gram_L1456083582e_leqv @ A @ L12 @ L22 )
% 2.85/1.51 => ( gram_L608943123e_subs @ A @ L12 @ L22 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_15_subs__leqv__sym,axiom,
% 2.85/1.51 ! [A: $tType,L12: set @ ( set @ A ),L22: set @ ( set @ A )] :
% 2.85/1.51 ( ( gram_L1456083582e_leqv @ A @ L12 @ L22 )
% 2.85/1.51 => ( gram_L608943123e_subs @ A @ L22 @ L12 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_16_Sup__UNIV,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.85/1.51 => ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
% 2.85/1.51 = ( top_top @ A ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_17_LL__L,axiom,
% 2.85/1.51 ! [Ns: set @ n,N: n] : ( ord_less_eq @ ( set @ ( set @ t ) ) @ ( gram_L861977280lle_LL @ Ns @ N ) @ ( gram_L1451583628elle_L @ Ns @ N ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_18_order__refl,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_19_subsetI,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.85/1.51 ( ! [X3: A] :
% 2.85/1.51 ( ( member @ A @ X3 @ A2 )
% 2.85/1.51 => ( member @ A @ X3 @ B2 ) )
% 2.85/1.51 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_20_subset__antisym,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.85/1.51 => ( A2 = B2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_21_set__mp,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( ( member @ A @ X @ A2 )
% 2.85/1.51 => ( member @ A @ X @ B2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_22_in__mono,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( ( member @ A @ X @ A2 )
% 2.85/1.51 => ( member @ A @ X @ B2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_23_subsetD,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( ( member @ A @ C2 @ A2 )
% 2.85/1.51 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_24_subsetCE,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( ( member @ A @ C2 @ A2 )
% 2.85/1.51 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_25_equalityE,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.85/1.51 ( ( A2 = B2 )
% 2.85/1.51 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_26_subset__eq,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) )
% 2.85/1.51 = ( ^ [A3: set @ A,B3: set @ A] :
% 2.85/1.51 ! [X2: A] :
% 2.85/1.51 ( ( member @ A @ X2 @ A3 )
% 2.85/1.51 => ( member @ A @ X2 @ B3 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_27_equalityD1,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.85/1.51 ( ( A2 = B2 )
% 2.85/1.51 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_28_equalityD2,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.85/1.51 ( ( A2 = B2 )
% 2.85/1.51 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_29_set__rev__mp,axiom,
% 2.85/1.51 ! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
% 2.85/1.51 ( ( member @ A @ X @ A2 )
% 2.85/1.51 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( member @ A @ X @ B2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_30_subset__iff,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) )
% 2.85/1.51 = ( ^ [A3: set @ A,B3: set @ A] :
% 2.85/1.51 ! [T: A] :
% 2.85/1.51 ( ( member @ A @ T @ A3 )
% 2.85/1.51 => ( member @ A @ T @ B3 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_31_rev__subsetD,axiom,
% 2.85/1.51 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 2.85/1.51 ( ( member @ A @ C2 @ A2 )
% 2.85/1.51 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_32_subset__refl,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_33_Collect__mono,axiom,
% 2.85/1.51 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.85/1.51 ( ! [X3: A] :
% 2.85/1.51 ( ( P @ X3 )
% 2.85/1.51 => ( Q @ X3 ) )
% 2.85/1.51 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_34_subset__trans,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 2.85/1.51 => ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_35_le__funD,axiom,
% 2.85/1.51 ! [B: $tType,A: $tType] :
% 2.85/1.51 ( ( ord @ B @ ( type2 @ B ) )
% 2.85/1.51 => ! [F: A > B,G: A > B,X: A] :
% 2.85/1.51 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.85/1.51 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_36_le__funE,axiom,
% 2.85/1.51 ! [B: $tType,A: $tType] :
% 2.85/1.51 ( ( ord @ B @ ( type2 @ B ) )
% 2.85/1.51 => ! [F: A > B,G: A > B,X: A] :
% 2.85/1.51 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.85/1.51 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_37_le__funI,axiom,
% 2.85/1.51 ! [B: $tType,A: $tType] :
% 2.85/1.51 ( ( ord @ B @ ( type2 @ B ) )
% 2.85/1.51 => ! [F: A > B,G: A > B] :
% 2.85/1.51 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 2.85/1.51 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_38_set__eq__subset,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( ^ [Y: set @ A,Z: set @ A] : ( Y = Z ) )
% 2.85/1.51 = ( ^ [A3: set @ A,B3: set @ A] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.85/1.51 & ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_39_contra__subsetD,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( ~ ( member @ A @ C2 @ B2 )
% 2.85/1.51 => ~ ( member @ A @ C2 @ A2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_40_le__fun__def,axiom,
% 2.85/1.51 ! [B: $tType,A: $tType] :
% 2.85/1.51 ( ( ord @ B @ ( type2 @ B ) )
% 2.85/1.51 => ( ( ord_less_eq @ ( A > B ) )
% 2.85/1.51 = ( ^ [F2: A > B,G2: A > B] :
% 2.85/1.51 ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_41_Collect__mono__iff,axiom,
% 2.85/1.51 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.85/1.51 = ( ! [X2: A] :
% 2.85/1.51 ( ( P @ X2 )
% 2.85/1.51 => ( Q @ X2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_42_order__subst1,axiom,
% 2.85/1.51 ! [A: $tType,B: $tType] :
% 2.85/1.51 ( ( ( order @ B @ ( type2 @ B ) )
% 2.85/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.85/1.51 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 2.85/1.51 ( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
% 2.85/1.51 => ( ( ord_less_eq @ B @ B4 @ C2 )
% 2.85/1.51 => ( ! [X3: B,Y2: B] :
% 2.85/1.51 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 2.85/1.51 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.85/1.51 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_43_order__subst2,axiom,
% 2.85/1.51 ! [A: $tType,C: $tType] :
% 2.85/1.51 ( ( ( order @ C @ ( type2 @ C ) )
% 2.85/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.85/1.51 => ! [A4: A,B4: A,F: A > C,C2: C] :
% 2.85/1.51 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.85/1.51 => ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
% 2.85/1.51 => ( ! [X3: A,Y2: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.85/1.51 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.85/1.51 => ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_44_ord__eq__le__subst,axiom,
% 2.85/1.51 ! [A: $tType,B: $tType] :
% 2.85/1.51 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.85/1.51 & ( ord @ A @ ( type2 @ A ) ) )
% 2.85/1.51 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 2.85/1.51 ( ( A4
% 2.85/1.51 = ( F @ B4 ) )
% 2.85/1.51 => ( ( ord_less_eq @ B @ B4 @ C2 )
% 2.85/1.51 => ( ! [X3: B,Y2: B] :
% 2.85/1.51 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 2.85/1.51 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.85/1.51 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_45_mem__Collect__eq,axiom,
% 2.85/1.51 ! [A: $tType,A4: A,P: A > $o] :
% 2.85/1.51 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 2.85/1.51 = ( P @ A4 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_46_Collect__mem__eq,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A] :
% 2.85/1.51 ( ( collect @ A
% 2.85/1.51 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 2.85/1.51 = A2 ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_47_Collect__cong,axiom,
% 2.85/1.51 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.85/1.51 ( ! [X3: A] :
% 2.85/1.51 ( ( P @ X3 )
% 2.85/1.51 = ( Q @ X3 ) )
% 2.85/1.51 => ( ( collect @ A @ P )
% 2.85/1.51 = ( collect @ A @ Q ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_48_ext,axiom,
% 2.85/1.51 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.85/1.51 ( ! [X3: A] :
% 2.85/1.51 ( ( F @ X3 )
% 2.85/1.51 = ( G @ X3 ) )
% 2.85/1.51 => ( F = G ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_49_ord__le__eq__subst,axiom,
% 2.85/1.51 ! [A: $tType,B: $tType] :
% 2.85/1.51 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.85/1.51 & ( ord @ A @ ( type2 @ A ) ) )
% 2.85/1.51 => ! [A4: A,B4: A,F: A > B,C2: B] :
% 2.85/1.51 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.85/1.51 => ( ( ( F @ B4 )
% 2.85/1.51 = C2 )
% 2.85/1.51 => ( ! [X3: A,Y2: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.85/1.51 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.85/1.51 => ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_50_Union__mono,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ ( set @ A ),B2: set @ ( set @ A )] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ B2 )
% 2.85/1.51 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_51_eq__iff,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 2.85/1.51 = ( ^ [X2: A,Y3: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.85/1.51 & ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_52_Union__least,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ ( set @ A ),C3: set @ A] :
% 2.85/1.51 ( ! [X4: set @ A] :
% 2.85/1.51 ( ( member @ ( set @ A ) @ X4 @ A2 )
% 2.85/1.51 => ( ord_less_eq @ ( set @ A ) @ X4 @ C3 ) )
% 2.85/1.51 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ C3 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_53_Union__upper,axiom,
% 2.85/1.51 ! [A: $tType,B2: set @ A,A2: set @ ( set @ A )] :
% 2.85/1.51 ( ( member @ ( set @ A ) @ B2 @ A2 )
% 2.85/1.51 => ( ord_less_eq @ ( set @ A ) @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_54_antisym,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.85/1.51 => ( ( ord_less_eq @ A @ Y4 @ X )
% 2.85/1.51 => ( X = Y4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_55_linear,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.85/1.51 | ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_56_eq__refl,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( X = Y4 )
% 2.85/1.51 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_57_le__cases,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ~ ( ord_less_eq @ A @ X @ Y4 )
% 2.85/1.51 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_58_order_Otrans,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,B4: A,C2: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.85/1.51 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 2.85/1.51 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_59_le__cases3,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A,Z2: A] :
% 2.85/1.51 ( ( ( ord_less_eq @ A @ X @ Y4 )
% 2.85/1.51 => ~ ( ord_less_eq @ A @ Y4 @ Z2 ) )
% 2.85/1.51 => ( ( ( ord_less_eq @ A @ Y4 @ X )
% 2.85/1.51 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 2.85/1.51 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 2.85/1.51 => ~ ( ord_less_eq @ A @ Z2 @ Y4 ) )
% 2.85/1.51 => ( ( ( ord_less_eq @ A @ Z2 @ Y4 )
% 2.85/1.51 => ~ ( ord_less_eq @ A @ Y4 @ X ) )
% 2.85/1.51 => ( ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 2.85/1.51 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 2.85/1.51 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 2.85/1.51 => ~ ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_60_antisym__conv,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [Y4: A,X: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ Y4 @ X )
% 2.85/1.51 => ( ( ord_less_eq @ A @ X @ Y4 )
% 2.85/1.51 = ( X = Y4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_61_ord__eq__le__trans,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( ord @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,B4: A,C2: A] :
% 2.85/1.51 ( ( A4 = B4 )
% 2.85/1.51 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 2.85/1.51 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_62_ord__le__eq__trans,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( ord @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,B4: A,C2: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.85/1.51 => ( ( B4 = C2 )
% 2.85/1.51 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_63_order__class_Oorder_Oantisym,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,B4: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.85/1.51 => ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.85/1.51 => ( A4 = B4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_64_order__trans,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A,Z2: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.85/1.51 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 2.85/1.51 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_65_dual__order_Orefl,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_66_linorder__wlog,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [P: A > A > $o,A4: A,B4: A] :
% 2.85/1.51 ( ! [A5: A,B5: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.85/1.51 => ( P @ A5 @ B5 ) )
% 2.85/1.51 => ( ! [A5: A,B5: A] :
% 2.85/1.51 ( ( P @ B5 @ A5 )
% 2.85/1.51 => ( P @ A5 @ B5 ) )
% 2.85/1.51 => ( P @ A4 @ B4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_67_dual__order_Otrans,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [B4: A,A4: A,C2: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.85/1.51 => ( ( ord_less_eq @ A @ C2 @ B4 )
% 2.85/1.51 => ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_68_subs__def,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( gram_L608943123e_subs @ A )
% 2.85/1.51 = ( ^ [L1: set @ ( set @ A ),L2: set @ ( set @ A )] :
% 2.85/1.51 ! [X2: set @ A] :
% 2.85/1.51 ( ( member @ ( set @ A ) @ X2 @ L2 )
% 2.85/1.51 => ? [Y3: set @ A] :
% 2.85/1.51 ( ( member @ ( set @ A ) @ Y3 @ L1 )
% 2.85/1.51 & ( ord_less_eq @ ( set @ A ) @ Y3 @ X2 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_69_incl__subs,axiom,
% 2.85/1.51 ! [A: $tType,L22: set @ ( set @ A ),L12: set @ ( set @ A )] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ L22 @ L12 )
% 2.85/1.51 => ( gram_L608943123e_subs @ A @ L12 @ L22 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_70_dual__order_Oantisym,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [B4: A,A4: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.85/1.51 => ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.85/1.51 => ( A4 = B4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_71_Sup__eqI,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A2: set @ A,X: A] :
% 2.85/1.51 ( ! [Y2: A] :
% 2.85/1.51 ( ( member @ A @ Y2 @ A2 )
% 2.85/1.51 => ( ord_less_eq @ A @ Y2 @ X ) )
% 2.85/1.51 => ( ! [Y2: A] :
% 2.85/1.51 ( ! [Z3: A] :
% 2.85/1.51 ( ( member @ A @ Z3 @ A2 )
% 2.85/1.51 => ( ord_less_eq @ A @ Z3 @ Y2 ) )
% 2.85/1.51 => ( ord_less_eq @ A @ X @ Y2 ) )
% 2.85/1.51 => ( ( complete_Sup_Sup @ A @ A2 )
% 2.85/1.51 = X ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_72_Sup__mono,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A2: set @ A,B2: set @ A] :
% 2.85/1.51 ( ! [A5: A] :
% 2.85/1.51 ( ( member @ A @ A5 @ A2 )
% 2.85/1.51 => ? [X5: A] :
% 2.85/1.51 ( ( member @ A @ X5 @ B2 )
% 2.85/1.51 & ( ord_less_eq @ A @ A5 @ X5 ) ) )
% 2.85/1.51 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_73_Sup__least,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A2: set @ A,Z2: A] :
% 2.85/1.51 ( ! [X3: A] :
% 2.85/1.51 ( ( member @ A @ X3 @ A2 )
% 2.85/1.51 => ( ord_less_eq @ A @ X3 @ Z2 ) )
% 2.85/1.51 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ Z2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_74_Sup__upper,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,A2: set @ A] :
% 2.85/1.51 ( ( member @ A @ X @ A2 )
% 2.85/1.51 => ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_75_Sup__le__iff,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A2: set @ A,B4: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ B4 )
% 2.85/1.51 = ( ! [X2: A] :
% 2.85/1.51 ( ( member @ A @ X2 @ A2 )
% 2.85/1.51 => ( ord_less_eq @ A @ X2 @ B4 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_76_Sup__upper2,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [U: A,A2: set @ A,V: A] :
% 2.85/1.51 ( ( member @ A @ U @ A2 )
% 2.85/1.51 => ( ( ord_less_eq @ A @ V @ U )
% 2.85/1.51 => ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_77_Sup__subset__mono,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A2: set @ A,B2: set @ A] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_78_subs__trans,axiom,
% 2.85/1.51 ! [A: $tType,L12: set @ ( set @ A ),L22: set @ ( set @ A ),L3: set @ ( set @ A )] :
% 2.85/1.51 ( ( gram_L608943123e_subs @ A @ L12 @ L22 )
% 2.85/1.51 => ( ( gram_L608943123e_subs @ A @ L22 @ L3 )
% 2.85/1.51 => ( gram_L608943123e_subs @ A @ L12 @ L3 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_79_subs__refl,axiom,
% 2.85/1.51 ! [A: $tType,L12: set @ ( set @ A )] : ( gram_L608943123e_subs @ A @ L12 @ L12 ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_80_top__set__def,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( top_top @ ( set @ A ) )
% 2.85/1.51 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_81_Union__UNIV,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
% 2.85/1.51 = ( top_top @ ( set @ A ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_82_top_Oextremum__uniqueI,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order_top @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
% 2.85/1.51 => ( A4
% 2.85/1.51 = ( top_top @ A ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_83_top_Oextremum__unique,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order_top @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
% 2.85/1.51 = ( A4
% 2.85/1.51 = ( top_top @ A ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_84_top__greatest,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order_top @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( top_top @ A ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_85_subset__UNIV,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_86_Lr__incl__L,axiom,
% 2.85/1.51 ! [Ns: set @ n,Ts: n] : ( ord_less_eq @ ( set @ ( set @ t ) ) @ ( gram_L861977318lle_Lr @ Ns @ Ts ) @ ( gram_L1451583628elle_L @ Ns @ Ts ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_87_cSup__eq,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 2.85/1.51 & ( no_bot @ A @ ( type2 @ A ) ) )
% 2.85/1.51 => ! [X6: set @ A,A4: A] :
% 2.85/1.51 ( ! [X3: A] :
% 2.85/1.51 ( ( member @ A @ X3 @ X6 )
% 2.85/1.51 => ( ord_less_eq @ A @ X3 @ A4 ) )
% 2.85/1.51 => ( ! [Y2: A] :
% 2.85/1.51 ( ! [X5: A] :
% 2.85/1.51 ( ( member @ A @ X5 @ X6 )
% 2.85/1.51 => ( ord_less_eq @ A @ X5 @ Y2 ) )
% 2.85/1.51 => ( ord_less_eq @ A @ A4 @ Y2 ) )
% 2.85/1.51 => ( ( complete_Sup_Sup @ A @ X6 )
% 2.85/1.51 = A4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_88_cSup__eq__maximum,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [Z2: A,X6: set @ A] :
% 2.85/1.51 ( ( member @ A @ Z2 @ X6 )
% 2.85/1.51 => ( ! [X3: A] :
% 2.85/1.51 ( ( member @ A @ X3 @ X6 )
% 2.85/1.51 => ( ord_less_eq @ A @ X3 @ Z2 ) )
% 2.85/1.51 => ( ( complete_Sup_Sup @ A @ X6 )
% 2.85/1.51 = Z2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_89_top__empty__eq,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( top_top @ ( A > $o ) )
% 2.85/1.51 = ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_90_Sup__eq__top__iff,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( comple1035589618norder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A2: set @ A] :
% 2.85/1.51 ( ( ( complete_Sup_Sup @ A @ A2 )
% 2.85/1.51 = ( top_top @ A ) )
% 2.85/1.51 = ( ! [X2: A] :
% 2.85/1.51 ( ( ord_less @ A @ X2 @ ( top_top @ A ) )
% 2.85/1.51 => ? [Y3: A] :
% 2.85/1.51 ( ( member @ A @ Y3 @ A2 )
% 2.85/1.51 & ( ord_less @ A @ X2 @ Y3 ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_91_antimonoD,axiom,
% 2.85/1.51 ! [B: $tType,A: $tType] :
% 2.85/1.51 ( ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 & ( order @ B @ ( type2 @ B ) ) )
% 2.85/1.51 => ! [F: A > B,X: A,Y4: A] :
% 2.85/1.51 ( ( order_antimono @ A @ B @ F )
% 2.85/1.51 => ( ( ord_less_eq @ A @ X @ Y4 )
% 2.85/1.51 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_92_antimonoE,axiom,
% 2.85/1.51 ! [B: $tType,A: $tType] :
% 2.85/1.51 ( ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 & ( order @ B @ ( type2 @ B ) ) )
% 2.85/1.51 => ! [F: A > B,X: A,Y4: A] :
% 2.85/1.51 ( ( order_antimono @ A @ B @ F )
% 2.85/1.51 => ( ( ord_less_eq @ A @ X @ Y4 )
% 2.85/1.51 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_93_antimonoI,axiom,
% 2.85/1.51 ! [B: $tType,A: $tType] :
% 2.85/1.51 ( ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 & ( order @ B @ ( type2 @ B ) ) )
% 2.85/1.51 => ! [F: A > B] :
% 2.85/1.51 ( ! [X3: A,Y2: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.85/1.51 => ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X3 ) ) )
% 2.85/1.51 => ( order_antimono @ A @ B @ F ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_94_antimono__def,axiom,
% 2.85/1.51 ! [B: $tType,A: $tType] :
% 2.85/1.51 ( ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 & ( order @ B @ ( type2 @ B ) ) )
% 2.85/1.51 => ( ( order_antimono @ A @ B )
% 2.85/1.51 = ( ^ [F2: A > B] :
% 2.85/1.51 ! [X2: A,Y3: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.85/1.51 => ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_95_Pow__mono,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A2 ) @ ( pow @ A @ B2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_96_less__eq__Sup,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A2: set @ A,U: A] :
% 2.85/1.51 ( ! [V2: A] :
% 2.85/1.51 ( ( member @ A @ V2 @ A2 )
% 2.85/1.51 => ( ord_less_eq @ A @ U @ V2 ) )
% 2.85/1.51 => ( ( A2
% 2.85/1.51 != ( bot_bot @ ( set @ A ) ) )
% 2.85/1.51 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_97_bot__apply,axiom,
% 2.85/1.51 ! [C: $tType,D: $tType] :
% 2.85/1.51 ( ( bot @ C @ ( type2 @ C ) )
% 2.85/1.51 => ( ( bot_bot @ ( D > C ) )
% 2.85/1.51 = ( ^ [X2: D] : ( bot_bot @ C ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_98_empty__iff,axiom,
% 2.85/1.51 ! [A: $tType,C2: A] :
% 2.85/1.51 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_99_all__not__in__conv,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A] :
% 2.85/1.51 ( ( ! [X2: A] :
% 2.85/1.51 ~ ( member @ A @ X2 @ A2 ) )
% 2.85/1.51 = ( A2
% 2.85/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_100_Collect__empty__eq,axiom,
% 2.85/1.51 ! [A: $tType,P: A > $o] :
% 2.85/1.51 ( ( ( collect @ A @ P )
% 2.85/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.85/1.51 = ( ! [X2: A] :
% 2.85/1.51 ~ ( P @ X2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_101_empty__Collect__eq,axiom,
% 2.85/1.51 ! [A: $tType,P: A > $o] :
% 2.85/1.51 ( ( ( bot_bot @ ( set @ A ) )
% 2.85/1.51 = ( collect @ A @ P ) )
% 2.85/1.51 = ( ! [X2: A] :
% 2.85/1.51 ~ ( P @ X2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_102_psubsetI,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( ( A2 != B2 )
% 2.85/1.51 => ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_103_UN__ball__bex__simps_I3_J,axiom,
% 2.85/1.51 ! [D: $tType,A2: set @ ( set @ D ),P: D > $o] :
% 2.85/1.51 ( ( ? [X2: D] :
% 2.85/1.51 ( ( member @ D @ X2 @ ( complete_Sup_Sup @ ( set @ D ) @ A2 ) )
% 2.85/1.51 & ( P @ X2 ) ) )
% 2.85/1.51 = ( ? [X2: set @ D] :
% 2.85/1.51 ( ( member @ ( set @ D ) @ X2 @ A2 )
% 2.85/1.51 & ? [Y3: D] :
% 2.85/1.51 ( ( member @ D @ Y3 @ X2 )
% 2.85/1.51 & ( P @ Y3 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_104_UN__ball__bex__simps_I1_J,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ ( set @ A ),P: A > $o] :
% 2.85/1.51 ( ( ! [X2: A] :
% 2.85/1.51 ( ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
% 2.85/1.51 => ( P @ X2 ) ) )
% 2.85/1.51 = ( ! [X2: set @ A] :
% 2.85/1.51 ( ( member @ ( set @ A ) @ X2 @ A2 )
% 2.85/1.51 => ! [Y3: A] :
% 2.85/1.51 ( ( member @ A @ Y3 @ X2 )
% 2.85/1.51 => ( P @ Y3 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_105_UnionI,axiom,
% 2.85/1.51 ! [A: $tType,X6: set @ A,C3: set @ ( set @ A ),A2: A] :
% 2.85/1.51 ( ( member @ ( set @ A ) @ X6 @ C3 )
% 2.85/1.51 => ( ( member @ A @ A2 @ X6 )
% 2.85/1.51 => ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_106_Union__iff,axiom,
% 2.85/1.51 ! [A: $tType,A2: A,C3: set @ ( set @ A )] :
% 2.85/1.51 ( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
% 2.85/1.51 = ( ? [X2: set @ A] :
% 2.85/1.51 ( ( member @ ( set @ A ) @ X2 @ C3 )
% 2.85/1.51 & ( member @ A @ A2 @ X2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_107_top1I,axiom,
% 2.85/1.51 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_108_Sup__bot__conv_I1_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A2: set @ A] :
% 2.85/1.51 ( ( ( complete_Sup_Sup @ A @ A2 )
% 2.85/1.51 = ( bot_bot @ A ) )
% 2.85/1.51 = ( ! [X2: A] :
% 2.85/1.51 ( ( member @ A @ X2 @ A2 )
% 2.85/1.51 => ( X2
% 2.85/1.51 = ( bot_bot @ A ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_109_Sup__bot__conv_I2_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A2: set @ A] :
% 2.85/1.51 ( ( ( bot_bot @ A )
% 2.85/1.51 = ( complete_Sup_Sup @ A @ A2 ) )
% 2.85/1.51 = ( ! [X2: A] :
% 2.85/1.51 ( ( member @ A @ X2 @ A2 )
% 2.85/1.51 => ( X2
% 2.85/1.51 = ( bot_bot @ A ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_110_empty__subsetI,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_111_subset__empty,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.85/1.51 = ( A2
% 2.85/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_112_Pow__iff,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.85/1.51 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 2.85/1.51 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_113_PowI,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_114_Union__Pow__eq,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A] :
% 2.85/1.51 ( ( complete_Sup_Sup @ ( set @ A ) @ ( pow @ A @ A2 ) )
% 2.85/1.51 = A2 ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_115_Sup__empty,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 2.85/1.51 => ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
% 2.85/1.51 = ( bot_bot @ A ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_116_Pow__UNIV,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( pow @ A @ ( top_top @ ( set @ A ) ) )
% 2.85/1.51 = ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_117_emptyE,axiom,
% 2.85/1.51 ! [A: $tType,A4: A] :
% 2.85/1.51 ~ ( member @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_118_Pow__top,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A] : ( member @ ( set @ A ) @ A2 @ ( pow @ A @ A2 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_119_equals0D,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,A4: A] :
% 2.85/1.51 ( ( A2
% 2.85/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.85/1.51 => ~ ( member @ A @ A4 @ A2 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_120_equals0I,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A] :
% 2.85/1.51 ( ! [Y2: A] :
% 2.85/1.51 ~ ( member @ A @ Y2 @ A2 )
% 2.85/1.51 => ( A2
% 2.85/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_121_Pow__bottom,axiom,
% 2.85/1.51 ! [A: $tType,B2: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow @ A @ B2 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_122_ex__in__conv,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A] :
% 2.85/1.51 ( ( ? [X2: A] : ( member @ A @ X2 @ A2 ) )
% 2.85/1.51 = ( A2
% 2.85/1.51 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_123_Pow__not__empty,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A] :
% 2.85/1.51 ( ( pow @ A @ A2 )
% 2.85/1.51 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_124_bot__fun__def,axiom,
% 2.85/1.51 ! [B: $tType,A: $tType] :
% 2.85/1.51 ( ( bot @ B @ ( type2 @ B ) )
% 2.85/1.51 => ( ( bot_bot @ ( A > B ) )
% 2.85/1.51 = ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_125_not__psubset__empty,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A] :
% 2.85/1.51 ~ ( ord_less @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_126_less__fun__def,axiom,
% 2.85/1.51 ! [B: $tType,A: $tType] :
% 2.85/1.51 ( ( ord @ B @ ( type2 @ B ) )
% 2.85/1.51 => ( ( ord_less @ ( A > B ) )
% 2.85/1.51 = ( ^ [F2: A > B,G2: A > B] :
% 2.85/1.51 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 2.85/1.51 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_127_UnionE,axiom,
% 2.85/1.51 ! [A: $tType,A2: A,C3: set @ ( set @ A )] :
% 2.85/1.51 ( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
% 2.85/1.51 => ~ ! [X4: set @ A] :
% 2.85/1.51 ( ( member @ A @ A2 @ X4 )
% 2.85/1.51 => ~ ( member @ ( set @ A ) @ X4 @ C3 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_128_ord__eq__less__subst,axiom,
% 2.85/1.51 ! [A: $tType,B: $tType] :
% 2.85/1.51 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.85/1.51 & ( ord @ A @ ( type2 @ A ) ) )
% 2.85/1.51 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 2.85/1.51 ( ( A4
% 2.85/1.51 = ( F @ B4 ) )
% 2.85/1.51 => ( ( ord_less @ B @ B4 @ C2 )
% 2.85/1.51 => ( ! [X3: B,Y2: B] :
% 2.85/1.51 ( ( ord_less @ B @ X3 @ Y2 )
% 2.85/1.51 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.85/1.51 => ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_129_ord__less__eq__subst,axiom,
% 2.85/1.51 ! [A: $tType,B: $tType] :
% 2.85/1.51 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.85/1.51 & ( ord @ A @ ( type2 @ A ) ) )
% 2.85/1.51 => ! [A4: A,B4: A,F: A > B,C2: B] :
% 2.85/1.51 ( ( ord_less @ A @ A4 @ B4 )
% 2.85/1.51 => ( ( ( F @ B4 )
% 2.85/1.51 = C2 )
% 2.85/1.51 => ( ! [X3: A,Y2: A] :
% 2.85/1.51 ( ( ord_less @ A @ X3 @ Y2 )
% 2.85/1.51 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.85/1.51 => ( ord_less @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_130_order__less__subst1,axiom,
% 2.85/1.51 ! [A: $tType,B: $tType] :
% 2.85/1.51 ( ( ( order @ B @ ( type2 @ B ) )
% 2.85/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.85/1.51 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 2.85/1.51 ( ( ord_less @ A @ A4 @ ( F @ B4 ) )
% 2.85/1.51 => ( ( ord_less @ B @ B4 @ C2 )
% 2.85/1.51 => ( ! [X3: B,Y2: B] :
% 2.85/1.51 ( ( ord_less @ B @ X3 @ Y2 )
% 2.85/1.51 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.85/1.51 => ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_131_order__less__subst2,axiom,
% 2.85/1.51 ! [A: $tType,C: $tType] :
% 2.85/1.51 ( ( ( order @ C @ ( type2 @ C ) )
% 2.85/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.85/1.51 => ! [A4: A,B4: A,F: A > C,C2: C] :
% 2.85/1.51 ( ( ord_less @ A @ A4 @ B4 )
% 2.85/1.51 => ( ( ord_less @ C @ ( F @ B4 ) @ C2 )
% 2.85/1.51 => ( ! [X3: A,Y2: A] :
% 2.85/1.51 ( ( ord_less @ A @ X3 @ Y2 )
% 2.85/1.51 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.85/1.51 => ( ord_less @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_132_lt__ex,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( no_bot @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A] :
% 2.85/1.51 ? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_133_gt__ex,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( no_top @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A] :
% 2.85/1.51 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_134_Union__empty,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
% 2.85/1.51 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_135_neqE,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( X != Y4 )
% 2.85/1.51 => ( ~ ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_136_neq__iff,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( X != Y4 )
% 2.85/1.51 = ( ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 | ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_137_order_Oasym,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,B4: A] :
% 2.85/1.51 ( ( ord_less @ A @ A4 @ B4 )
% 2.85/1.51 => ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_138_dense,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( dense_order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 => ? [Z4: A] :
% 2.85/1.51 ( ( ord_less @ A @ X @ Z4 )
% 2.85/1.51 & ( ord_less @ A @ Z4 @ Y4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_139_Union__empty__conv,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ ( set @ A )] :
% 2.85/1.51 ( ( ( complete_Sup_Sup @ ( set @ A ) @ A2 )
% 2.85/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.85/1.51 = ( ! [X2: set @ A] :
% 2.85/1.51 ( ( member @ ( set @ A ) @ X2 @ A2 )
% 2.85/1.51 => ( X2
% 2.85/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_140_empty__Union__conv,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ ( set @ A )] :
% 2.85/1.51 ( ( ( bot_bot @ ( set @ A ) )
% 2.85/1.51 = ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
% 2.85/1.51 = ( ! [X2: set @ A] :
% 2.85/1.51 ( ( member @ ( set @ A ) @ X2 @ A2 )
% 2.85/1.51 => ( X2
% 2.85/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_141_less__imp__neq,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 => ( X != Y4 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_142_less__asym,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_143_less__asym_H,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,B4: A] :
% 2.85/1.51 ( ( ord_less @ A @ A4 @ B4 )
% 2.85/1.51 => ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_144_less__trans,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A,Z2: A] :
% 2.85/1.51 ( ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 => ( ( ord_less @ A @ Y4 @ Z2 )
% 2.85/1.51 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_145_less__linear,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 | ( X = Y4 )
% 2.85/1.51 | ( ord_less @ A @ Y4 @ X ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_146_less__irrefl,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A] :
% 2.85/1.51 ~ ( ord_less @ A @ X @ X ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_147_ord__eq__less__trans,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( ord @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,B4: A,C2: A] :
% 2.85/1.51 ( ( A4 = B4 )
% 2.85/1.51 => ( ( ord_less @ A @ B4 @ C2 )
% 2.85/1.51 => ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_148_ord__less__eq__trans,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( ord @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,B4: A,C2: A] :
% 2.85/1.51 ( ( ord_less @ A @ A4 @ B4 )
% 2.85/1.51 => ( ( B4 = C2 )
% 2.85/1.51 => ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_149_dual__order_Oasym,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [B4: A,A4: A] :
% 2.85/1.51 ( ( ord_less @ A @ B4 @ A4 )
% 2.85/1.51 => ~ ( ord_less @ A @ A4 @ B4 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_150_less__imp__not__eq,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 => ( X != Y4 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_151_less__not__sym,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_152_less__induct,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [P: A > $o,A4: A] :
% 2.85/1.51 ( ! [X3: A] :
% 2.85/1.51 ( ! [Y5: A] :
% 2.85/1.51 ( ( ord_less @ A @ Y5 @ X3 )
% 2.85/1.51 => ( P @ Y5 ) )
% 2.85/1.51 => ( P @ X3 ) )
% 2.85/1.51 => ( P @ A4 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_153_antisym__conv3,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [Y4: A,X: A] :
% 2.85/1.51 ( ~ ( ord_less @ A @ Y4 @ X )
% 2.85/1.51 => ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 2.85/1.51 = ( X = Y4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_154_less__imp__not__eq2,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 => ( Y4 != X ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_155_less__imp__triv,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A,P: $o] :
% 2.85/1.51 ( ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 => ( ( ord_less @ A @ Y4 @ X )
% 2.85/1.51 => P ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_156_linorder__cases,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ~ ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 => ( ( X != Y4 )
% 2.85/1.51 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_157_dual__order_Oirrefl,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A] :
% 2.85/1.51 ~ ( ord_less @ A @ A4 @ A4 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_158_order_Ostrict__trans,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,B4: A,C2: A] :
% 2.85/1.51 ( ( ord_less @ A @ A4 @ B4 )
% 2.85/1.51 => ( ( ord_less @ A @ B4 @ C2 )
% 2.85/1.51 => ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_159_less__imp__not__less,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_160_bot_Oextremum__strict,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A] :
% 2.85/1.51 ~ ( ord_less @ A @ A4 @ ( bot_bot @ A ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_161_bot_Onot__eq__extremum,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A] :
% 2.85/1.51 ( ( A4
% 2.85/1.51 != ( bot_bot @ A ) )
% 2.85/1.51 = ( ord_less @ A @ ( bot_bot @ A ) @ A4 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_162_dual__order_Ostrict__trans,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [B4: A,A4: A,C2: A] :
% 2.85/1.51 ( ( ord_less @ A @ B4 @ A4 )
% 2.85/1.51 => ( ( ord_less @ A @ C2 @ B4 )
% 2.85/1.51 => ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_163_not__less__iff__gr__or__eq,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 2.85/1.51 = ( ( ord_less @ A @ Y4 @ X )
% 2.85/1.51 | ( X = Y4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_164_order_Ostrict__implies__not__eq,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,B4: A] :
% 2.85/1.51 ( ( ord_less @ A @ A4 @ B4 )
% 2.85/1.51 => ( A4 != B4 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_165_dual__order_Ostrict__implies__not__eq,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [B4: A,A4: A] :
% 2.85/1.51 ( ( ord_less @ A @ B4 @ A4 )
% 2.85/1.51 => ( A4 != B4 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_166_less__cSupD,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X6: set @ A,Z2: A] :
% 2.85/1.51 ( ( X6
% 2.85/1.51 != ( bot_bot @ ( set @ A ) ) )
% 2.85/1.51 => ( ( ord_less @ A @ Z2 @ ( complete_Sup_Sup @ A @ X6 ) )
% 2.85/1.51 => ? [X3: A] :
% 2.85/1.51 ( ( member @ A @ X3 @ X6 )
% 2.85/1.51 & ( ord_less @ A @ Z2 @ X3 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_167_less__cSupE,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [Y4: A,X6: set @ A] :
% 2.85/1.51 ( ( ord_less @ A @ Y4 @ ( complete_Sup_Sup @ A @ X6 ) )
% 2.85/1.51 => ( ( X6
% 2.85/1.51 != ( bot_bot @ ( set @ A ) ) )
% 2.85/1.51 => ~ ! [X3: A] :
% 2.85/1.51 ( ( member @ A @ X3 @ X6 )
% 2.85/1.51 => ~ ( ord_less @ A @ Y4 @ X3 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_168_ex__gt__or__lt,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A] :
% 2.85/1.51 ? [B5: A] :
% 2.85/1.51 ( ( ord_less @ A @ A4 @ B5 )
% 2.85/1.51 | ( ord_less @ A @ B5 @ A4 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_169_complete__interval,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,B4: A,P: A > $o] :
% 2.85/1.51 ( ( ord_less @ A @ A4 @ B4 )
% 2.85/1.51 => ( ( P @ A4 )
% 2.85/1.51 => ( ~ ( P @ B4 )
% 2.85/1.51 => ? [C4: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ A4 @ C4 )
% 2.85/1.51 & ( ord_less_eq @ A @ C4 @ B4 )
% 2.85/1.51 & ! [X5: A] :
% 2.85/1.51 ( ( ( ord_less_eq @ A @ A4 @ X5 )
% 2.85/1.51 & ( ord_less @ A @ X5 @ C4 ) )
% 2.85/1.51 => ( P @ X5 ) )
% 2.85/1.51 & ! [D2: A] :
% 2.85/1.51 ( ! [X3: A] :
% 2.85/1.51 ( ( ( ord_less_eq @ A @ A4 @ X3 )
% 2.85/1.51 & ( ord_less @ A @ X3 @ D2 ) )
% 2.85/1.51 => ( P @ X3 ) )
% 2.85/1.51 => ( ord_less_eq @ A @ D2 @ C4 ) ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_170_order_Onot__eq__order__implies__strict,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,B4: A] :
% 2.85/1.51 ( ( A4 != B4 )
% 2.85/1.51 => ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.85/1.51 => ( ord_less @ A @ A4 @ B4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_171_dual__order_Ostrict__implies__order,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [B4: A,A4: A] :
% 2.85/1.51 ( ( ord_less @ A @ B4 @ A4 )
% 2.85/1.51 => ( ord_less_eq @ A @ B4 @ A4 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_172_dual__order_Ostrict__iff__order,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ( ( ord_less @ A )
% 2.85/1.51 = ( ^ [B6: A,A6: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ B6 @ A6 )
% 2.85/1.51 & ( A6 != B6 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_173_dual__order_Oorder__iff__strict,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ( ( ord_less_eq @ A )
% 2.85/1.51 = ( ^ [B6: A,A6: A] :
% 2.85/1.51 ( ( ord_less @ A @ B6 @ A6 )
% 2.85/1.51 | ( A6 = B6 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_174_order_Ostrict__implies__order,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,B4: A] :
% 2.85/1.51 ( ( ord_less @ A @ A4 @ B4 )
% 2.85/1.51 => ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_175_dense__le__bounded,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A,Z2: A] :
% 2.85/1.51 ( ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 => ( ! [W: A] :
% 2.85/1.51 ( ( ord_less @ A @ X @ W )
% 2.85/1.51 => ( ( ord_less @ A @ W @ Y4 )
% 2.85/1.51 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 2.85/1.51 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_176_dense__ge__bounded,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [Z2: A,X: A,Y4: A] :
% 2.85/1.51 ( ( ord_less @ A @ Z2 @ X )
% 2.85/1.51 => ( ! [W: A] :
% 2.85/1.51 ( ( ord_less @ A @ Z2 @ W )
% 2.85/1.51 => ( ( ord_less @ A @ W @ X )
% 2.85/1.51 => ( ord_less_eq @ A @ Y4 @ W ) ) )
% 2.85/1.51 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_177_dual__order_Ostrict__trans2,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [B4: A,A4: A,C2: A] :
% 2.85/1.51 ( ( ord_less @ A @ B4 @ A4 )
% 2.85/1.51 => ( ( ord_less_eq @ A @ C2 @ B4 )
% 2.85/1.51 => ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_178_dual__order_Ostrict__trans1,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [B4: A,A4: A,C2: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.85/1.51 => ( ( ord_less @ A @ C2 @ B4 )
% 2.85/1.51 => ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_179_order_Ostrict__iff__order,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ( ( ord_less @ A )
% 2.85/1.51 = ( ^ [A6: A,B6: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ A6 @ B6 )
% 2.85/1.51 & ( A6 != B6 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_180_order_Oorder__iff__strict,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ( ( ord_less_eq @ A )
% 2.85/1.51 = ( ^ [A6: A,B6: A] :
% 2.85/1.51 ( ( ord_less @ A @ A6 @ B6 )
% 2.85/1.51 | ( A6 = B6 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_181_order_Ostrict__trans2,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,B4: A,C2: A] :
% 2.85/1.51 ( ( ord_less @ A @ A4 @ B4 )
% 2.85/1.51 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 2.85/1.51 => ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_182_order_Ostrict__trans1,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,B4: A,C2: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.85/1.51 => ( ( ord_less @ A @ B4 @ C2 )
% 2.85/1.51 => ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_183_not__le__imp__less,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [Y4: A,X: A] :
% 2.85/1.51 ( ~ ( ord_less_eq @ A @ Y4 @ X )
% 2.85/1.51 => ( ord_less @ A @ X @ Y4 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_184_less__le__not__le,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ( ( ord_less @ A )
% 2.85/1.51 = ( ^ [X2: A,Y3: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.85/1.51 & ~ ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_185_le__imp__less__or__eq,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.85/1.51 => ( ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 | ( X = Y4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_186_le__less__linear,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.85/1.51 | ( ord_less @ A @ Y4 @ X ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_187_dense__le,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [Y4: A,Z2: A] :
% 2.85/1.51 ( ! [X3: A] :
% 2.85/1.51 ( ( ord_less @ A @ X3 @ Y4 )
% 2.85/1.51 => ( ord_less_eq @ A @ X3 @ Z2 ) )
% 2.85/1.51 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_188_dense__ge,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [Z2: A,Y4: A] :
% 2.85/1.51 ( ! [X3: A] :
% 2.85/1.51 ( ( ord_less @ A @ Z2 @ X3 )
% 2.85/1.51 => ( ord_less_eq @ A @ Y4 @ X3 ) )
% 2.85/1.51 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_189_less__le__trans,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A,Z2: A] :
% 2.85/1.51 ( ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 2.85/1.51 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_190_le__less__trans,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A,Z2: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.85/1.51 => ( ( ord_less @ A @ Y4 @ Z2 )
% 2.85/1.51 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_191_antisym__conv2,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.85/1.51 => ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 2.85/1.51 = ( X = Y4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_192_antisym__conv1,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ~ ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 => ( ( ord_less_eq @ A @ X @ Y4 )
% 2.85/1.51 = ( X = Y4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_193_less__imp__le,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_194_le__neq__trans,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,B4: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.85/1.51 => ( ( A4 != B4 )
% 2.85/1.51 => ( ord_less @ A @ A4 @ B4 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_195_not__less,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 2.85/1.51 = ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_196_not__le,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ( ~ ( ord_less_eq @ A @ X @ Y4 ) )
% 2.85/1.51 = ( ord_less @ A @ Y4 @ X ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_197_order__less__le__subst2,axiom,
% 2.85/1.51 ! [A: $tType,C: $tType] :
% 2.85/1.51 ( ( ( order @ C @ ( type2 @ C ) )
% 2.85/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.85/1.51 => ! [A4: A,B4: A,F: A > C,C2: C] :
% 2.85/1.51 ( ( ord_less @ A @ A4 @ B4 )
% 2.85/1.51 => ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
% 2.85/1.51 => ( ! [X3: A,Y2: A] :
% 2.85/1.51 ( ( ord_less @ A @ X3 @ Y2 )
% 2.85/1.51 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.85/1.51 => ( ord_less @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_198_order__less__le__subst1,axiom,
% 2.85/1.51 ! [A: $tType,B: $tType] :
% 2.85/1.51 ( ( ( order @ B @ ( type2 @ B ) )
% 2.85/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.85/1.51 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 2.85/1.51 ( ( ord_less @ A @ A4 @ ( F @ B4 ) )
% 2.85/1.51 => ( ( ord_less_eq @ B @ B4 @ C2 )
% 2.85/1.51 => ( ! [X3: B,Y2: B] :
% 2.85/1.51 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 2.85/1.51 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.85/1.51 => ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_199_order__le__less__subst2,axiom,
% 2.85/1.51 ! [A: $tType,C: $tType] :
% 2.85/1.51 ( ( ( order @ C @ ( type2 @ C ) )
% 2.85/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.85/1.51 => ! [A4: A,B4: A,F: A > C,C2: C] :
% 2.85/1.51 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.85/1.51 => ( ( ord_less @ C @ ( F @ B4 ) @ C2 )
% 2.85/1.51 => ( ! [X3: A,Y2: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.85/1.51 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.85/1.51 => ( ord_less @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_200_order__le__less__subst1,axiom,
% 2.85/1.51 ! [A: $tType,B: $tType] :
% 2.85/1.51 ( ( ( order @ B @ ( type2 @ B ) )
% 2.85/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.85/1.51 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 2.85/1.51 ( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
% 2.85/1.51 => ( ( ord_less @ B @ B4 @ C2 )
% 2.85/1.51 => ( ! [X3: B,Y2: B] :
% 2.85/1.51 ( ( ord_less @ B @ X3 @ Y2 )
% 2.85/1.51 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.85/1.51 => ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_201_less__le,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ( ( ord_less @ A )
% 2.85/1.51 = ( ^ [X2: A,Y3: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.85/1.51 & ( X2 != Y3 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_202_le__less,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.85/1.51 => ( ( ord_less_eq @ A )
% 2.85/1.51 = ( ^ [X2: A,Y3: A] :
% 2.85/1.51 ( ( ord_less @ A @ X2 @ Y3 )
% 2.85/1.51 | ( X2 = Y3 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_203_leI,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,Y4: A] :
% 2.85/1.51 ( ~ ( ord_less @ A @ X @ Y4 )
% 2.85/1.51 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_204_leD,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [Y4: A,X: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ Y4 @ X )
% 2.85/1.51 => ~ ( ord_less @ A @ X @ Y4 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_205_bot_Oextremum__uniqueI,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
% 2.85/1.51 => ( A4
% 2.85/1.51 = ( bot_bot @ A ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_206_bot_Oextremum__unique,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
% 2.85/1.51 = ( A4
% 2.85/1.51 = ( bot_bot @ A ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_207_bot_Oextremum,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A4 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_208_top_Oextremum__strict,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order_top @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A] :
% 2.85/1.51 ~ ( ord_less @ A @ ( top_top @ A ) @ A4 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_209_top_Onot__eq__extremum,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order_top @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A] :
% 2.85/1.51 ( ( A4
% 2.85/1.51 != ( top_top @ A ) )
% 2.85/1.51 = ( ord_less @ A @ A4 @ ( top_top @ A ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_210_subset__iff__psubset__eq,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) )
% 2.85/1.51 = ( ^ [A3: set @ A,B3: set @ A] :
% 2.85/1.51 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 2.85/1.51 | ( A3 = B3 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_211_subset__psubset__trans,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 2.85/1.51 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_212_subset__not__subset__eq,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( ord_less @ ( set @ A ) )
% 2.85/1.51 = ( ^ [A3: set @ A,B3: set @ A] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.85/1.51 & ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_213_psubset__subset__trans,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 2.85/1.51 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 2.85/1.51 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_214_psubset__imp__subset,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.85/1.51 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_215_psubset__eq,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( ord_less @ ( set @ A ) )
% 2.85/1.51 = ( ^ [A3: set @ A,B3: set @ A] :
% 2.85/1.51 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.85/1.51 & ( A3 != B3 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_216_psubsetE,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.85/1.51 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_217_less__Sup__iff,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( comple1035589618norder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [A4: A,S: set @ A] :
% 2.85/1.51 ( ( ord_less @ A @ A4 @ ( complete_Sup_Sup @ A @ S ) )
% 2.85/1.51 = ( ? [X2: A] :
% 2.85/1.51 ( ( member @ A @ X2 @ S )
% 2.85/1.51 & ( ord_less @ A @ A4 @ X2 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_218_PowD,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.85/1.51 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 2.85/1.51 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_219_empty__not__UNIV,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( bot_bot @ ( set @ A ) )
% 2.85/1.51 != ( top_top @ ( set @ A ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_220_cSup__eq__non__empty,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X6: set @ A,A4: A] :
% 2.85/1.51 ( ( X6
% 2.85/1.51 != ( bot_bot @ ( set @ A ) ) )
% 2.85/1.51 => ( ! [X3: A] :
% 2.85/1.51 ( ( member @ A @ X3 @ X6 )
% 2.85/1.51 => ( ord_less_eq @ A @ X3 @ A4 ) )
% 2.85/1.51 => ( ! [Y2: A] :
% 2.85/1.51 ( ! [X5: A] :
% 2.85/1.51 ( ( member @ A @ X5 @ X6 )
% 2.85/1.51 => ( ord_less_eq @ A @ X5 @ Y2 ) )
% 2.85/1.51 => ( ord_less_eq @ A @ A4 @ Y2 ) )
% 2.85/1.51 => ( ( complete_Sup_Sup @ A @ X6 )
% 2.85/1.51 = A4 ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_221_cSup__least,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( condit378418413attice @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X6: set @ A,Z2: A] :
% 2.85/1.51 ( ( X6
% 2.85/1.51 != ( bot_bot @ ( set @ A ) ) )
% 2.85/1.51 => ( ! [X3: A] :
% 2.85/1.51 ( ( member @ A @ X3 @ X6 )
% 2.85/1.51 => ( ord_less_eq @ A @ X3 @ Z2 ) )
% 2.85/1.51 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X6 ) @ Z2 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_222_subset__Pow__Union,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_223_le__Sup__iff,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( comple1035589618norder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [X: A,A2: set @ A] :
% 2.85/1.51 ( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A2 ) )
% 2.85/1.51 = ( ! [Y3: A] :
% 2.85/1.51 ( ( ord_less @ A @ Y3 @ X )
% 2.85/1.51 => ? [X2: A] :
% 2.85/1.51 ( ( member @ A @ X2 @ A2 )
% 2.85/1.51 & ( ord_less @ A @ Y3 @ X2 ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_224_subset__emptyI,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A] :
% 2.85/1.51 ( ! [X3: A] :
% 2.85/1.51 ~ ( member @ A @ X3 @ A2 )
% 2.85/1.51 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_225_pinf_I6_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [T2: A] :
% 2.85/1.51 ? [Z4: A] :
% 2.85/1.51 ! [X5: A] :
% 2.85/1.51 ( ( ord_less @ A @ Z4 @ X5 )
% 2.85/1.51 => ~ ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_226_pinf_I8_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [T2: A] :
% 2.85/1.51 ? [Z4: A] :
% 2.85/1.51 ! [X5: A] :
% 2.85/1.51 ( ( ord_less @ A @ Z4 @ X5 )
% 2.85/1.51 => ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_227_minf_I6_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [T2: A] :
% 2.85/1.51 ? [Z4: A] :
% 2.85/1.51 ! [X5: A] :
% 2.85/1.51 ( ( ord_less @ A @ X5 @ Z4 )
% 2.85/1.51 => ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_228_minf_I8_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [T2: A] :
% 2.85/1.51 ? [Z4: A] :
% 2.85/1.51 ! [X5: A] :
% 2.85/1.51 ( ( ord_less @ A @ X5 @ Z4 )
% 2.85/1.51 => ~ ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_229_psubsetD,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 2.85/1.51 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( ( member @ A @ C2 @ A2 )
% 2.85/1.51 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_230_bot__set__def,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( bot_bot @ ( set @ A ) )
% 2.85/1.51 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_231_psubset__trans,axiom,
% 2.85/1.51 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 2.85/1.51 ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
% 2.85/1.51 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 2.85/1.51 => ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_232_bot__empty__eq,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( bot_bot @ ( A > $o ) )
% 2.85/1.51 = ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_233_minf_I11_J,axiom,
% 2.85/1.51 ! [C: $tType,D: $tType] :
% 2.85/1.51 ( ( ord @ C @ ( type2 @ C ) )
% 2.85/1.51 => ! [F3: D] :
% 2.85/1.51 ? [Z4: C] :
% 2.85/1.51 ! [X5: C] :
% 2.85/1.51 ( ( ord_less @ C @ X5 @ Z4 )
% 2.85/1.51 => ( F3 = F3 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_234_minf_I7_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [T2: A] :
% 2.85/1.51 ? [Z4: A] :
% 2.85/1.51 ! [X5: A] :
% 2.85/1.51 ( ( ord_less @ A @ X5 @ Z4 )
% 2.85/1.51 => ~ ( ord_less @ A @ T2 @ X5 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_235_minf_I5_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [T2: A] :
% 2.85/1.51 ? [Z4: A] :
% 2.85/1.51 ! [X5: A] :
% 2.85/1.51 ( ( ord_less @ A @ X5 @ Z4 )
% 2.85/1.51 => ( ord_less @ A @ X5 @ T2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_236_minf_I4_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [T2: A] :
% 2.85/1.51 ? [Z4: A] :
% 2.85/1.51 ! [X5: A] :
% 2.85/1.51 ( ( ord_less @ A @ X5 @ Z4 )
% 2.85/1.51 => ( X5 != T2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_237_minf_I3_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [T2: A] :
% 2.85/1.51 ? [Z4: A] :
% 2.85/1.51 ! [X5: A] :
% 2.85/1.51 ( ( ord_less @ A @ X5 @ Z4 )
% 2.85/1.51 => ( X5 != T2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_238_minf_I2_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.85/1.51 ( ? [Z3: A] :
% 2.85/1.51 ! [X3: A] :
% 2.85/1.51 ( ( ord_less @ A @ X3 @ Z3 )
% 2.85/1.51 => ( ( P @ X3 )
% 2.85/1.51 = ( P2 @ X3 ) ) )
% 2.85/1.51 => ( ? [Z3: A] :
% 2.85/1.51 ! [X3: A] :
% 2.85/1.51 ( ( ord_less @ A @ X3 @ Z3 )
% 2.85/1.51 => ( ( Q @ X3 )
% 2.85/1.51 = ( Q2 @ X3 ) ) )
% 2.85/1.51 => ? [Z4: A] :
% 2.85/1.51 ! [X5: A] :
% 2.85/1.51 ( ( ord_less @ A @ X5 @ Z4 )
% 2.85/1.51 => ( ( ( P @ X5 )
% 2.85/1.51 | ( Q @ X5 ) )
% 2.85/1.51 = ( ( P2 @ X5 )
% 2.85/1.51 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_239_minf_I1_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.85/1.51 ( ? [Z3: A] :
% 2.85/1.51 ! [X3: A] :
% 2.85/1.51 ( ( ord_less @ A @ X3 @ Z3 )
% 2.85/1.51 => ( ( P @ X3 )
% 2.85/1.51 = ( P2 @ X3 ) ) )
% 2.85/1.51 => ( ? [Z3: A] :
% 2.85/1.51 ! [X3: A] :
% 2.85/1.51 ( ( ord_less @ A @ X3 @ Z3 )
% 2.85/1.51 => ( ( Q @ X3 )
% 2.85/1.51 = ( Q2 @ X3 ) ) )
% 2.85/1.51 => ? [Z4: A] :
% 2.85/1.51 ! [X5: A] :
% 2.85/1.51 ( ( ord_less @ A @ X5 @ Z4 )
% 2.85/1.51 => ( ( ( P @ X5 )
% 2.85/1.51 & ( Q @ X5 ) )
% 2.85/1.51 = ( ( P2 @ X5 )
% 2.85/1.51 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_240_pinf_I11_J,axiom,
% 2.85/1.51 ! [C: $tType,D: $tType] :
% 2.85/1.51 ( ( ord @ C @ ( type2 @ C ) )
% 2.85/1.51 => ! [F3: D] :
% 2.85/1.51 ? [Z4: C] :
% 2.85/1.51 ! [X5: C] :
% 2.85/1.51 ( ( ord_less @ C @ Z4 @ X5 )
% 2.85/1.51 => ( F3 = F3 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_241_pinf_I7_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [T2: A] :
% 2.85/1.51 ? [Z4: A] :
% 2.85/1.51 ! [X5: A] :
% 2.85/1.51 ( ( ord_less @ A @ Z4 @ X5 )
% 2.85/1.51 => ( ord_less @ A @ T2 @ X5 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_242_pinf_I5_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [T2: A] :
% 2.85/1.51 ? [Z4: A] :
% 2.85/1.51 ! [X5: A] :
% 2.85/1.51 ( ( ord_less @ A @ Z4 @ X5 )
% 2.85/1.51 => ~ ( ord_less @ A @ X5 @ T2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_243_pinf_I4_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [T2: A] :
% 2.85/1.51 ? [Z4: A] :
% 2.85/1.51 ! [X5: A] :
% 2.85/1.51 ( ( ord_less @ A @ Z4 @ X5 )
% 2.85/1.51 => ( X5 != T2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_244_pinf_I3_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [T2: A] :
% 2.85/1.51 ? [Z4: A] :
% 2.85/1.51 ! [X5: A] :
% 2.85/1.51 ( ( ord_less @ A @ Z4 @ X5 )
% 2.85/1.51 => ( X5 != T2 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_245_pinf_I2_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.85/1.51 ( ? [Z3: A] :
% 2.85/1.51 ! [X3: A] :
% 2.85/1.51 ( ( ord_less @ A @ Z3 @ X3 )
% 2.85/1.51 => ( ( P @ X3 )
% 2.85/1.51 = ( P2 @ X3 ) ) )
% 2.85/1.51 => ( ? [Z3: A] :
% 2.85/1.51 ! [X3: A] :
% 2.85/1.51 ( ( ord_less @ A @ Z3 @ X3 )
% 2.85/1.51 => ( ( Q @ X3 )
% 2.85/1.51 = ( Q2 @ X3 ) ) )
% 2.85/1.51 => ? [Z4: A] :
% 2.85/1.51 ! [X5: A] :
% 2.85/1.51 ( ( ord_less @ A @ Z4 @ X5 )
% 2.85/1.51 => ( ( ( P @ X5 )
% 2.85/1.51 | ( Q @ X5 ) )
% 2.85/1.51 = ( ( P2 @ X5 )
% 2.85/1.51 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_246_pinf_I1_J,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.85/1.51 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.85/1.51 ( ? [Z3: A] :
% 2.85/1.51 ! [X3: A] :
% 2.85/1.51 ( ( ord_less @ A @ Z3 @ X3 )
% 2.85/1.51 => ( ( P @ X3 )
% 2.85/1.51 = ( P2 @ X3 ) ) )
% 2.85/1.51 => ( ? [Z3: A] :
% 2.85/1.51 ! [X3: A] :
% 2.85/1.51 ( ( ord_less @ A @ Z3 @ X3 )
% 2.85/1.51 => ( ( Q @ X3 )
% 2.85/1.51 = ( Q2 @ X3 ) ) )
% 2.85/1.51 => ? [Z4: A] :
% 2.85/1.51 ! [X5: A] :
% 2.85/1.51 ( ( ord_less @ A @ Z4 @ X5 )
% 2.85/1.51 => ( ( ( P @ X5 )
% 2.85/1.51 & ( Q @ X5 ) )
% 2.85/1.51 = ( ( P2 @ X5 )
% 2.85/1.51 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_247_top_Oordering__top__axioms,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( order_top @ A @ ( type2 @ A ) )
% 2.85/1.51 => ( ordering_top @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ A ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_248_Set_Ois__empty__def,axiom,
% 2.85/1.51 ! [A: $tType] :
% 2.85/1.51 ( ( is_empty @ A )
% 2.85/1.51 = ( ^ [A3: set @ A] :
% 2.85/1.51 ( A3
% 2.85/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_249_ordering__top_Oextremum,axiom,
% 2.85/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
% 2.85/1.51 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 2.85/1.51 => ( Less_eq @ A4 @ Top ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_250_ordering__top_Oextremum__strict,axiom,
% 2.85/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
% 2.85/1.51 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 2.85/1.51 => ~ ( Less @ Top @ A4 ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_251_ordering__top_Oextremum__unique,axiom,
% 2.85/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
% 2.85/1.51 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 2.85/1.51 => ( ( Less_eq @ Top @ A4 )
% 2.85/1.51 = ( A4 = Top ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_252_ordering__top_Onot__eq__extremum,axiom,
% 2.85/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
% 2.85/1.51 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 2.85/1.51 => ( ( A4 != Top )
% 2.85/1.51 = ( Less @ A4 @ Top ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_253_ordering__top_Oextremum__uniqueI,axiom,
% 2.85/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
% 2.85/1.51 ( ( ordering_top @ A @ Less_eq @ Less @ Top )
% 2.85/1.51 => ( ( Less_eq @ Top @ A4 )
% 2.85/1.51 => ( A4 = Top ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_254_Collect__empty__eq__bot,axiom,
% 2.85/1.51 ! [A: $tType,P: A > $o] :
% 2.85/1.51 ( ( ( collect @ A @ P )
% 2.85/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.85/1.51 = ( P
% 2.85/1.51 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(fact_255_top__conj_I2_J,axiom,
% 2.85/1.51 ! [A: $tType,P: $o,X: A] :
% 2.85/1.51 ( ( P
% 2.85/1.51 & ( top_top @ ( A > $o ) @ X ) )
% 2.85/1.51 = P ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
% 2.85/1.51 ! [A7: $tType,A8: $tType] :
% 2.85/1.51 ( ( comple187826305attice @ A8 @ ( type2 @ A8 ) )
% 2.85/1.51 => ( condit378418413attice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 2.85/1.51 ! [A7: $tType,A8: $tType] :
% 2.85/1.51 ( ( comple187826305attice @ A8 @ ( type2 @ A8 ) )
% 2.85/1.51 => ( comple187826305attice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 2.85/1.51 ! [A7: $tType,A8: $tType] :
% 2.85/1.51 ( ( order_top @ A8 @ ( type2 @ A8 ) )
% 2.85/1.51 => ( order_top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.85/1.51 ! [A7: $tType,A8: $tType] :
% 2.85/1.51 ( ( order_bot @ A8 @ ( type2 @ A8 ) )
% 2.85/1.51 => ( order_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.85/1.51 ! [A7: $tType,A8: $tType] :
% 2.85/1.51 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 2.85/1.51 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.85/1.51 ! [A7: $tType,A8: $tType] :
% 2.85/1.51 ( ( order @ A8 @ ( type2 @ A8 ) )
% 2.85/1.51 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_fun___Orderings_Otop,axiom,
% 2.85/1.51 ! [A7: $tType,A8: $tType] :
% 2.85/1.51 ( ( top @ A8 @ ( type2 @ A8 ) )
% 2.85/1.51 => ( top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_fun___Orderings_Oord,axiom,
% 2.85/1.51 ! [A7: $tType,A8: $tType] :
% 2.85/1.51 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 2.85/1.51 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_fun___Orderings_Obot,axiom,
% 2.85/1.51 ! [A7: $tType,A8: $tType] :
% 2.85/1.51 ( ( bot @ A8 @ ( type2 @ A8 ) )
% 2.85/1.51 => ( bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1,axiom,
% 2.85/1.51 ! [A7: $tType] : ( condit378418413attice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_2,axiom,
% 2.85/1.51 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_Set_Oset___Orderings_Oorder__top_3,axiom,
% 2.85/1.51 ! [A7: $tType] : ( order_top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_Set_Oset___Orderings_Oorder__bot_4,axiom,
% 2.85/1.51 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
% 2.85/1.51 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 2.85/1.51 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
% 2.85/1.51 ! [A7: $tType] : ( top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 2.85/1.51 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_Set_Oset___Orderings_Obot_9,axiom,
% 2.85/1.51 ! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_10,axiom,
% 2.85/1.51 condit378418413attice @ $o @ ( type2 @ $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_11,axiom,
% 2.85/1.51 comple187826305attice @ $o @ ( type2 @ $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_HOL_Obool___Orderings_Oorder__top_12,axiom,
% 2.85/1.51 order_top @ $o @ ( type2 @ $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_HOL_Obool___Orderings_Oorder__bot_13,axiom,
% 2.85/1.51 order_bot @ $o @ ( type2 @ $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_HOL_Obool___Orderings_Opreorder_14,axiom,
% 2.85/1.51 preorder @ $o @ ( type2 @ $o ) ).
% 2.85/1.51
% 2.85/1.51 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.85/1.51 linorder @ $o @ ( type2 @ $o ) ).
% 2.85/1.52
% 2.85/1.52 thf(tcon_HOL_Obool___Orderings_Oorder_15,axiom,
% 2.85/1.52 order @ $o @ ( type2 @ $o ) ).
% 2.85/1.52
% 2.85/1.52 thf(tcon_HOL_Obool___Orderings_Otop_16,axiom,
% 2.85/1.52 top @ $o @ ( type2 @ $o ) ).
% 2.85/1.52
% 2.85/1.52 thf(tcon_HOL_Obool___Orderings_Oord_17,axiom,
% 2.85/1.52 ord @ $o @ ( type2 @ $o ) ).
% 2.85/1.52
% 2.85/1.52 thf(tcon_HOL_Obool___Orderings_Obot_18,axiom,
% 2.85/1.52 bot @ $o @ ( type2 @ $o ) ).
% 2.85/1.52
% 2.85/1.52 thf(conj_0,conjecture,
% 2.85/1.52 gram_L1456083582e_leqv @ t @ ( gram_L861977280lle_LL @ ( top_top @ ( set @ n ) ) @ ts ) @ ( gram_L861977318lle_Lr @ ( top_top @ ( set @ n ) ) @ ts ) ).
% 2.85/1.52 ------------------------
% 2.85/1.54 % (24409)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_24078 for (2999ds/27Mi)
% 2.85/1.54 % (24408)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_24078 for (2999ds/4Mi)
% 2.85/1.54 % (24410)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_24078 for (2999ds/2Mi)
% 2.85/1.54 % (24408)Instruction limit reached!
% 2.85/1.54 % (24408)------------------------------
% 2.85/1.54 % (24408)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.54 % (24408)Termination reason: Unknown
% 2.85/1.54 % (24408)Termination phase: shuffling
% 2.85/1.54
% 2.85/1.54 % (24408)Memory used [KB]: 1407
% 2.85/1.54 % (24408)Time elapsed: 0.003 s
% 2.85/1.54 % (24408)Instructions burned: 4 (million)
% 2.85/1.54 % (24408)------------------------------
% 2.85/1.54 % (24408)------------------------------
% 2.85/1.54 % (24410)Instruction limit reached!
% 2.85/1.54 % (24410)------------------------------
% 2.85/1.54 % (24410)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.54 % (24410)Termination reason: Unknown
% 2.85/1.54 % (24410)Termination phase: shuffling
% 2.85/1.54
% 2.85/1.54 % (24410)Memory used [KB]: 1279
% 2.85/1.54 % (24410)Time elapsed: 0.002 s
% 2.85/1.54 % (24410)Instructions burned: 3 (million)
% 2.85/1.54 % (24410)------------------------------
% 2.85/1.54 % (24410)------------------------------
% 2.85/1.55 % (24412)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_24078 for (2999ds/275Mi)
% 2.85/1.55 % (24407)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_24078 for (2999ds/183Mi)
% 2.85/1.55 % (24411)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_24078 for (2999ds/2Mi)
% 2.85/1.55 % (24413)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_24078 for (2999ds/18Mi)
% 2.85/1.55 % (24409)Instruction limit reached!
% 2.85/1.55 % (24409)------------------------------
% 2.85/1.55 % (24409)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.55 % (24409)Termination reason: Unknown
% 2.85/1.55 % (24409)Termination phase: Property scanning
% 2.85/1.55
% 2.85/1.55 % (24409)Memory used [KB]: 1791
% 2.85/1.55 % (24409)Time elapsed: 0.011 s
% 2.85/1.55 % (24409)Instructions burned: 30 (million)
% 2.85/1.55 % (24409)------------------------------
% 2.85/1.55 % (24409)------------------------------
% 2.85/1.55 % (24411)Instruction limit reached!
% 2.85/1.55 % (24411)------------------------------
% 2.85/1.55 % (24411)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.55 % (24411)Termination reason: Unknown
% 2.85/1.55 % (24411)Termination phase: shuffling
% 2.85/1.55
% 2.85/1.55 % (24411)Memory used [KB]: 1279
% 2.85/1.55 % (24411)Time elapsed: 0.004 s
% 2.85/1.55 % (24411)Instructions burned: 3 (million)
% 2.85/1.55 % (24411)------------------------------
% 2.85/1.55 % (24411)------------------------------
% 2.85/1.55 % (24414)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24078 for (2999ds/3Mi)
% 2.85/1.55 % (24415)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_24078 for (2999ds/37Mi)
% 2.85/1.55 % (24414)Instruction limit reached!
% 2.85/1.55 % (24414)------------------------------
% 2.85/1.55 % (24414)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.55 % (24414)Termination reason: Unknown
% 2.85/1.55 % (24414)Termination phase: shuffling
% 2.85/1.55
% 2.85/1.55 % (24414)Memory used [KB]: 1407
% 2.85/1.55 % (24414)Time elapsed: 0.003 s
% 2.85/1.55 % (24414)Instructions burned: 5 (million)
% 2.85/1.55 % (24414)------------------------------
% 2.85/1.55 % (24414)------------------------------
% 2.85/1.56 % (24413)Instruction limit reached!
% 2.85/1.56 % (24413)------------------------------
% 2.85/1.56 % (24413)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.56 % (24413)Termination reason: Unknown
% 2.85/1.56 % (24413)Termination phase: shuffling
% 2.85/1.56
% 2.85/1.56 % (24413)Memory used [KB]: 1663
% 2.85/1.56 % (24413)Time elapsed: 0.014 s
% 2.85/1.56 % (24413)Instructions burned: 18 (million)
% 2.85/1.56 % (24413)------------------------------
% 2.85/1.56 % (24413)------------------------------
% 2.85/1.56 % (24418)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_24078 for (2999ds/1041Mi)
% 2.85/1.56 % (24415)Instruction limit reached!
% 2.85/1.56 % (24415)------------------------------
% 2.85/1.56 % (24415)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.56 % (24415)Termination reason: Unknown
% 2.85/1.56 % (24415)Termination phase: Property scanning
% 2.85/1.56
% 2.85/1.56 % (24415)Memory used [KB]: 1918
% 2.85/1.56 % (24415)Time elapsed: 0.013 s
% 2.85/1.56 % (24415)Instructions burned: 37 (million)
% 2.85/1.56 % (24415)------------------------------
% 2.85/1.56 % (24415)------------------------------
% 2.85/1.57 % (24416)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_24078 for (2999ds/15Mi)
% 2.85/1.57 % (24417)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24078 for (2999ds/3Mi)
% 2.85/1.57 % (24417)Instruction limit reached!
% 2.85/1.57 % (24417)------------------------------
% 2.85/1.57 % (24417)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.57 % (24417)Termination reason: Unknown
% 2.85/1.57 % (24417)Termination phase: shuffling
% 2.85/1.57
% 2.85/1.57 % (24417)Memory used [KB]: 1407
% 2.85/1.57 % (24417)Time elapsed: 0.004 s
% 2.85/1.57 % (24417)Instructions burned: 4 (million)
% 2.85/1.57 % (24417)------------------------------
% 2.85/1.57 % (24417)------------------------------
% 2.85/1.57 % (24416)Instruction limit reached!
% 2.85/1.57 % (24416)------------------------------
% 2.85/1.57 % (24416)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.57 % (24416)Termination reason: Unknown
% 2.85/1.57 % (24416)Termination phase: shuffling
% 2.85/1.57
% 2.85/1.57 % (24416)Memory used [KB]: 1663
% 2.85/1.57 % (24416)Time elapsed: 0.007 s
% 2.85/1.57 % (24416)Instructions burned: 15 (million)
% 2.85/1.57 % (24416)------------------------------
% 2.85/1.57 % (24416)------------------------------
% 2.85/1.57 % (24420)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_24078 for (2999ds/16Mi)
% 2.85/1.58 % (24419)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_24078 for (2999ds/7Mi)
% 2.85/1.58 % (24420)Instruction limit reached!
% 2.85/1.58 % (24420)------------------------------
% 2.85/1.58 % (24420)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.58 % (24420)Termination reason: Unknown
% 2.85/1.58 % (24420)Termination phase: shuffling
% 2.85/1.58
% 2.85/1.58 % (24420)Memory used [KB]: 1663
% 2.85/1.58 % (24420)Time elapsed: 0.008 s
% 2.85/1.58 % (24420)Instructions burned: 17 (million)
% 2.85/1.58 % (24420)------------------------------
% 2.85/1.58 % (24420)------------------------------
% 2.85/1.58 % (24422)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_24078 for (2999ds/3Mi)
% 2.85/1.58 % (24422)Instruction limit reached!
% 2.85/1.58 % (24422)------------------------------
% 2.85/1.58 % (24422)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.58 % (24422)Termination reason: Unknown
% 2.85/1.58 % (24422)Termination phase: shuffling
% 2.85/1.58
% 2.85/1.58 % (24422)Memory used [KB]: 1279
% 2.85/1.58 % (24422)Time elapsed: 0.003 s
% 2.85/1.58 % (24422)Instructions burned: 3 (million)
% 2.85/1.58 % (24422)------------------------------
% 2.85/1.58 % (24422)------------------------------
% 2.85/1.58 % (24419)Instruction limit reached!
% 2.85/1.58 % (24419)------------------------------
% 2.85/1.58 % (24419)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.58 % (24419)Termination reason: Unknown
% 2.85/1.58 % (24419)Termination phase: shuffling
% 2.85/1.58
% 2.85/1.58 % (24419)Memory used [KB]: 1407
% 2.85/1.58 % (24419)Time elapsed: 0.007 s
% 2.85/1.58 % (24419)Instructions burned: 8 (million)
% 2.85/1.58 % (24419)------------------------------
% 2.85/1.58 % (24419)------------------------------
% 2.85/1.58 % (24421)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_24078 for (2999ds/3Mi)
% 2.85/1.59 % (24421)Instruction limit reached!
% 2.85/1.59 % (24421)------------------------------
% 2.85/1.59 % (24421)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.59 % (24421)Termination reason: Unknown
% 2.85/1.59 % (24421)Termination phase: shuffling
% 2.85/1.59
% 2.85/1.59 % (24421)Memory used [KB]: 1407
% 2.85/1.59 % (24421)Time elapsed: 0.004 s
% 2.85/1.59 % (24421)Instructions burned: 5 (million)
% 2.85/1.59 % (24421)------------------------------
% 2.85/1.59 % (24421)------------------------------
% 2.85/1.59 % (24423)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_24078 for (2999ds/7Mi)
% 2.85/1.59 % (24412)First to succeed.
% 2.85/1.59 % (24424)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_24078 for (2999ds/3Mi)
% 2.85/1.59 % (24423)Instruction limit reached!
% 2.85/1.59 % (24423)------------------------------
% 2.85/1.59 % (24423)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.59 % (24423)Termination reason: Unknown
% 2.85/1.59 % (24423)Termination phase: shuffling
% 2.85/1.59
% 2.85/1.59 % (24423)Memory used [KB]: 1535
% 2.85/1.59 % (24423)Time elapsed: 0.005 s
% 2.85/1.59 % (24423)Instructions burned: 9 (million)
% 2.85/1.59 % (24423)------------------------------
% 2.85/1.59 % (24423)------------------------------
% 2.85/1.60 % (24424)Instruction limit reached!
% 2.85/1.60 % (24424)------------------------------
% 2.85/1.60 % (24424)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.60 % (24424)Termination reason: Unknown
% 2.85/1.60 % (24424)Termination phase: shuffling
% 2.85/1.60
% 2.85/1.60 % (24424)Memory used [KB]: 1407
% 2.85/1.60 % (24424)Time elapsed: 0.003 s
% 2.85/1.60 % (24424)Instructions burned: 6 (million)
% 2.85/1.60 % (24424)------------------------------
% 2.85/1.60 % (24424)------------------------------
% 2.85/1.60 % (24412)Refutation found. Thanks to Tanya!
% 2.85/1.60 % SZS status Theorem for DTF2THF_24078
% 2.85/1.60 % SZS output start Proof for DTF2THF_24078
% 2.85/1.60 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.85/1.60 thf(type_def_7, type, set: ($tType) > $tType).
% 2.85/1.60 thf(type_def_8, type, n: $tType).
% 2.85/1.60 thf(type_def_9, type, t: $tType).
% 2.85/1.60 thf(func_def_0, type, set: ($tType) > $tType).
% 2.85/1.60 thf(func_def_1, type, t: $tType).
% 2.85/1.60 thf(func_def_2, type, n: $tType).
% 2.85/1.60 thf(func_def_3, type, itself: ($tType) > $tType).
% 2.85/1.60 thf(func_def_4, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_5, type, bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_6, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_7, type, top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_8, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_9, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_10, type, no_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_11, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_12, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_13, type, order_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_14, type, order_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_15, type, wellorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_16, type, dense_order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_17, type, dense_linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_18, type, comple187826305attice: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_19, type, comple1035589618norder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_20, type, condit1656338222tinuum: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_21, type, condit378418413attice: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_22, type, condit1037483654norder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.85/1.60 thf(func_def_23, type, complete_Sup_Sup: !>[X0: $tType]:(set @ X0 > X0)).
% 2.85/1.60 thf(func_def_24, type, gram_L1451583628elle_L: set @ n > n > set @ set @ t).
% 2.85/1.60 thf(func_def_25, type, gram_L861977280lle_LL: set @ n > n > set @ set @ t).
% 2.85/1.60 thf(func_def_26, type, gram_L861977318lle_Lr: set @ n > n > set @ set @ t).
% 2.85/1.60 thf(func_def_27, type, gram_L1456083582e_leqv: !>[X0: $tType]:(set @ set @ X0 > set @ set @ X0 > $o)).
% 2.85/1.60 thf(func_def_28, type, gram_L608943123e_subs: !>[X0: $tType]:(set @ set @ X0 > set @ set @ X0 > $o)).
% 2.85/1.60 thf(func_def_29, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.85/1.60 thf(func_def_30, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.85/1.60 thf(func_def_31, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.85/1.60 thf(func_def_32, type, order_antimono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 2.85/1.60 thf(func_def_33, type, ordering_top: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > X0 > $o)).
% 2.85/1.60 thf(func_def_34, type, top_top: !>[X0: $tType]:(X0)).
% 2.85/1.60 thf(func_def_35, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.85/1.60 thf(func_def_36, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.85/1.60 thf(func_def_37, type, pow: !>[X0: $tType]:(set @ X0 > set @ set @ X0)).
% 2.85/1.60 thf(func_def_38, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 2.85/1.60 thf(func_def_39, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.85/1.60 thf(func_def_40, type, ts: n).
% 2.85/1.60 thf(func_def_56, type, sK0: !>[X0: $tType]:(set @ X0 > X0)).
% 2.85/1.60 thf(func_def_57, type, sK1: !>[X0: $tType]:(set @ set @ X0 > X0 > set @ X0)).
% 2.85/1.60 thf(func_def_58, type, sK2: !>[X0: $tType]:(set @ X0 > X0)).
% 2.85/1.60 thf(func_def_59, type, sK3: !>[X0: $tType]:(set @ X0 > set @ set @ X0 > set @ X0)).
% 2.85/1.60 thf(func_def_60, type, sK4: !>[X0: $tType]:(set @ set @ X0 > X0 > set @ X0)).
% 2.85/1.60 thf(func_def_61, type, sK5: !>[X0: $tType]:(set @ set @ X0 > set @ X0)).
% 2.85/1.60 thf(func_def_62, type, sK6: !>[X0: $tType]:(X0)).
% 2.85/1.60 thf(func_def_63, type, sK7: !>[X0: $tType]:(set @ set @ X0 > set @ X0)).
% 2.85/1.60 thf(func_def_64, type, sK8: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > X0)).
% 2.85/1.60 thf(func_def_65, type, sK9: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > set @ X0)).
% 2.85/1.60 thf(func_def_66, type, sK10: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > X0)).
% 2.85/1.60 thf(func_def_67, type, sK11: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > set @ X0)).
% 2.85/1.60 thf(func_def_68, type, sK12: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > X0)).
% 2.85/1.60 thf(func_def_69, type, sK13: !>[X0: $tType]:((X0 > $o) > set @ set @ X0 > X0)).
% 2.85/1.60 thf(func_def_70, type, sK14: !>[X0: $tType]:(set @ X0 > X0)).
% 2.85/1.60 thf(func_def_71, type, sK15: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.85/1.60 thf(func_def_72, type, sK16: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.85/1.60 thf(func_def_73, type, sK17: !>[X0: $tType]:(set @ X0 > X0)).
% 2.85/1.60 thf(func_def_74, type, sK18: !>[X0: $tType]:(set @ X0 > X0)).
% 2.85/1.60 thf(func_def_76, type, ph20: !>[X0: $tType]:(X0)).
% 2.85/1.60 thf(f1433,plain,(
% 2.85/1.60 $false),
% 2.85/1.60 inference(avatar_sat_refutation,[status(thm)],[f1070,f1200,f1225,f1259,f1287,f1432])).
% 2.85/1.60 thf(f1432,plain,(
% 2.85/1.60 ~spl19_1 | spl19_5 | spl19_6 | spl19_7),
% 2.85/1.60 inference(avatar_contradiction_clause,[status(thm)],[f1431])).
% 2.85/1.60 thf(f1431,plain,(
% 2.85/1.60 $false | (~spl19_1 | spl19_5 | spl19_6 | spl19_7)),
% 2.85/1.60 inference(subsumption_resolution,[status(thm)],[f1430,f1188])).
% 2.85/1.60 thf(f1188,plain,(
% 2.85/1.60 ($false != (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) | spl19_5),
% 2.85/1.60 inference(avatar_component_clause,[status(thm)],[f1187])).
% 2.85/1.60 thf(f1187,definition,(
% 2.85/1.60 spl19_5 <=> ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts)))),
% 2.85/1.60 introduced(definition,[new_symbols(naming,[spl19_5])],[avatar_definition])).
% 2.85/1.60 thf(f1430,plain,(
% 2.85/1.60 ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) | (~spl19_1 | spl19_6 | spl19_7)),
% 2.85/1.60 inference(subsumption_resolution,[status(thm)],[f1429,f1192])).
% 2.85/1.60 thf(f1192,plain,(
% 2.85/1.60 ($false != (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts))) | spl19_6),
% 2.85/1.60 inference(avatar_component_clause,[status(thm)],[f1191])).
% 2.85/1.60 thf(f1191,definition,(
% 2.85/1.60 spl19_6 <=> ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts)))),
% 2.85/1.60 introduced(definition,[new_symbols(naming,[spl19_6])],[avatar_definition])).
% 2.88/1.60 thf(f1429,plain,(
% 2.88/1.60 ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts))) | ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) | (~spl19_1 | spl19_7)),
% 2.88/1.60 inference(subsumption_resolution,[status(thm)],[f1387,f1196])).
% 2.88/1.60 thf(f1196,plain,(
% 2.88/1.60 ((gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts)) != $false) | spl19_7),
% 2.88/1.60 inference(avatar_component_clause,[status(thm)],[f1195])).
% 2.88/1.60 thf(f1195,definition,(
% 2.88/1.60 spl19_7 <=> ((gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts)) = $false)),
% 2.88/1.60 introduced(definition,[new_symbols(naming,[spl19_7])],[avatar_definition])).
% 2.88/1.60 thf(f1387,plain,(
% 2.88/1.60 ((gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts)) = $false) | ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) | ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts))) | ~spl19_1),
% 2.88/1.60 inference(trivial_inequality_removal,[status(thm)],[f1368])).
% 2.88/1.60 thf(f1368,plain,(
% 2.88/1.60 ($false = $true) | ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) | ((gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts)) = $false) | ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts))) | ~spl19_1),
% 2.88/1.60 inference(superposition,[status(thm)],[f1020,f1338])).
% 2.88/1.60 thf(f1338,plain,(
% 2.88/1.60 ( ! [X0 : set @ set @ t] : (($false = (gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts) @ X0)) | ($false = (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ X0)) | ($false = (gram_L608943123e_subs @ t @ X0 @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts))) | ($false = (gram_L608943123e_subs @ t @ X0 @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts)))) ) | ~spl19_1),
% 2.88/1.60 inference(binary_proxy_clausification,[status(thm)],[f1337])).
% 2.88/1.60 thf(f1337,plain,(
% 2.88/1.60 ( ! [X0 : set @ set @ t] : (($false = (gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts) @ X0)) | (((gram_L608943123e_subs @ t @ X0 @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts)) & (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ X0)) != $true) | ($false = (gram_L608943123e_subs @ t @ X0 @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts)))) ) | ~spl19_1),
% 2.88/1.60 inference(binary_proxy_clausification,[status(thm)],[f1336])).
% 2.88/1.60 thf(f1336,plain,(
% 2.88/1.60 ( ! [X0 : set @ set @ t] : ((((gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts) @ X0) & (gram_L608943123e_subs @ t @ X0 @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) != $true) | (((gram_L608943123e_subs @ t @ X0 @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts)) & (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ X0)) != $true)) ) | ~spl19_1),
% 2.88/1.60 inference(trivial_inequality_removal,[status(thm)],[f1335])).
% 2.88/1.60 thf(f1335,plain,(
% 2.88/1.60 ( ! [X0 : set @ set @ t] : (($false = $true) | (((gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts) @ X0) & (gram_L608943123e_subs @ t @ X0 @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) != $true) | (((gram_L608943123e_subs @ t @ X0 @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts)) & (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ X0)) != $true)) ) | ~spl19_1),
% 2.88/1.60 inference(boolean_simplification,[status(thm)],[f1312])).
% 2.88/1.60 thf(f1312,plain,(
% 2.88/1.60 ( ! [X0 : set @ set @ t] : ((((gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts) @ X0) & (gram_L608943123e_subs @ t @ X0 @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) != $true) | (((gram_L608943123e_subs @ t @ X0 @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts)) & (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ X0)) != $true) | (((gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts) @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts)) & $false) = $true)) ) | ~spl19_1),
% 2.88/1.60 inference(superposition,[status(thm)],[f1054,f1065])).
% 2.88/1.60 thf(f1065,plain,(
% 2.88/1.60 ($false = (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) | ~spl19_1),
% 2.88/1.60 inference(avatar_component_clause,[status(thm)],[f1063])).
% 2.88/1.60 thf(f1063,definition,(
% 2.88/1.60 spl19_1 <=> ($false = (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts)))),
% 2.88/1.60 introduced(definition,[new_symbols(naming,[spl19_1])],[avatar_definition])).
% 2.88/1.60 thf(f1054,plain,(
% 2.88/1.60 ( ! [X0 : $tType,X2 : set @ set @ X0,X3 : set @ set @ X0,X1 : set @ set @ X0] : ((((gram_L608943123e_subs @ X0 @ X3 @ X1) & (gram_L608943123e_subs @ X0 @ X1 @ X3)) = $true) | (((gram_L608943123e_subs @ X0 @ X3 @ X2) & (gram_L608943123e_subs @ X0 @ X2 @ X3)) != $true) | (((gram_L608943123e_subs @ X0 @ X2 @ X1) & (gram_L608943123e_subs @ X0 @ X1 @ X2)) != $true)) )),
% 2.88/1.60 inference(beta_eta_normalization,[status(thm)],[f1034])).
% 2.88/1.60 thf(f1034,plain,(
% 2.88/1.60 ( ! [X0 : $tType,X2 : set @ set @ X0,X3 : set @ set @ X0,X1 : set @ set @ X0] : ((((^[Y0 : set @ set @ X0]: ((^[Y1 : set @ set @ X0]: ((gram_L608943123e_subs @ X0 @ Y1 @ Y0) & (gram_L608943123e_subs @ X0 @ Y0 @ Y1))))) @ X1 @ X2) != $true) | (((^[Y0 : set @ set @ X0]: ((^[Y1 : set @ set @ X0]: ((gram_L608943123e_subs @ X0 @ Y1 @ Y0) & (gram_L608943123e_subs @ X0 @ Y0 @ Y1))))) @ X2 @ X3) != $true) | (((^[Y0 : set @ set @ X0]: ((^[Y1 : set @ set @ X0]: ((gram_L608943123e_subs @ X0 @ Y1 @ Y0) & (gram_L608943123e_subs @ X0 @ Y0 @ Y1))))) @ X1 @ X3) = $true)) )),
% 2.88/1.60 inference(definition_unfolding,[status(thm)],[f976,f948,f948,f948])).
% 2.88/1.60 thf(f948,plain,(
% 2.88/1.60 ( ! [X0 : $tType] : ((gram_L1456083582e_leqv @ X0 = (^[Y0 : set @ set @ X0]: ((^[Y1 : set @ set @ X0]: ((gram_L608943123e_subs @ X0 @ Y1 @ Y0) & (gram_L608943123e_subs @ X0 @ Y0 @ Y1))))))) )),
% 2.88/1.60 inference(cnf_transformation,[status(thm)],[f593])).
% 2.88/1.60 thf(f593,plain,(
% 2.88/1.60 ! [X0 : $tType] : (gram_L1456083582e_leqv @ X0 = (^[Y0 : set @ set @ X0]: ((^[Y1 : set @ set @ X0]: ((gram_L608943123e_subs @ X0 @ Y1 @ Y0) & (gram_L608943123e_subs @ X0 @ Y0 @ Y1))))))),
% 2.88/1.60 inference(fool_elimination,[status(thm)],[f592])).
% 2.88/1.60 thf(f592,plain,(
% 2.88/1.60 ! [X0 : $tType] : (gram_L1456083582e_leqv @ X0 = (^[X1 : set @ set @ X0, X2 : set @ set @ X0] : ((gram_L608943123e_subs @ X0 @ X1 @ X2) & (gram_L608943123e_subs @ X0 @ X2 @ X1))))),
% 2.88/1.60 inference(rectify,[status(thm)],[f14])).
% 2.88/1.60 thf(f14,axiom,(
% 2.88/1.60 ! [X0 : $tType] : (gram_L1456083582e_leqv @ X0 = (^[X3 : set @ set @ X0, X4 : set @ set @ X0] : ((gram_L608943123e_subs @ X0 @ X3 @ X4) & (gram_L608943123e_subs @ X0 @ X4 @ X3))))),
% 2.88/1.60 file('/export/starexec/sandbox/tmp/tmp.LejkCnALQX/DTF2THF_24078.p',fact_13_leqv__def)).
% 2.88/1.60 thf(f976,plain,(
% 2.88/1.60 ( ! [X0 : $tType,X2 : set @ set @ X0,X3 : set @ set @ X0,X1 : set @ set @ X0] : (((gram_L1456083582e_leqv @ X0 @ X1 @ X3) = $true) | ((gram_L1456083582e_leqv @ X0 @ X1 @ X2) != $true) | ($true != (gram_L1456083582e_leqv @ X0 @ X2 @ X3))) )),
% 2.88/1.60 inference(cnf_transformation,[status(thm)],[f884])).
% 2.88/1.60 thf(f884,plain,(
% 2.88/1.60 ! [X0 : $tType,X1 : set @ set @ X0,X2 : set @ set @ X0,X3 : set @ set @ X0] : (((gram_L1456083582e_leqv @ X0 @ X1 @ X3) = $true) | ((gram_L1456083582e_leqv @ X0 @ X1 @ X2) != $true) | ($true != (gram_L1456083582e_leqv @ X0 @ X2 @ X3)))),
% 2.88/1.60 inference(flattening,[status(thm)],[f883])).
% 2.88/1.60 thf(f883,plain,(
% 2.88/1.60 ! [X0 : $tType,X1 : set @ set @ X0,X2 : set @ set @ X0,X3 : set @ set @ X0] : ((((gram_L1456083582e_leqv @ X0 @ X1 @ X3) = $true) | ($true != (gram_L1456083582e_leqv @ X0 @ X2 @ X3))) | ((gram_L1456083582e_leqv @ X0 @ X1 @ X2) != $true))),
% 2.88/1.60 inference(ennf_transformation,[status(thm)],[f557])).
% 2.88/1.60 thf(f557,plain,(
% 2.88/1.60 ! [X0 : $tType,X1 : set @ set @ X0,X2 : set @ set @ X0,X3 : set @ set @ X0] : (((gram_L1456083582e_leqv @ X0 @ X1 @ X2) = $true) => (($true = (gram_L1456083582e_leqv @ X0 @ X2 @ X3)) => ((gram_L1456083582e_leqv @ X0 @ X1 @ X3) = $true)))),
% 2.88/1.60 inference(fool_elimination,[status(thm)],[f556])).
% 2.88/1.60 thf(f556,plain,(
% 2.88/1.60 ! [X0 : $tType,X1 : set @ set @ X0,X2 : set @ set @ X0,X3 : set @ set @ X0] : ((gram_L1456083582e_leqv @ X0 @ X1 @ X2) => ((gram_L1456083582e_leqv @ X0 @ X2 @ X3) => (gram_L1456083582e_leqv @ X0 @ X1 @ X3)))),
% 2.88/1.60 inference(rectify,[status(thm)],[f6])).
% 2.88/1.60 thf(f6,axiom,(
% 2.88/1.60 ! [X0 : $tType,X5 : set @ set @ X0,X6 : set @ set @ X0,X7 : set @ set @ X0] : ((gram_L1456083582e_leqv @ X0 @ X5 @ X6) => ((gram_L1456083582e_leqv @ X0 @ X6 @ X7) => (gram_L1456083582e_leqv @ X0 @ X5 @ X7)))),
% 2.88/1.60 file('/export/starexec/sandbox/tmp/tmp.LejkCnALQX/DTF2THF_24078.p',fact_5_leqv__trans)).
% 2.88/1.60 thf(f1020,plain,(
% 2.88/1.60 ( ! [X0 : n] : (((gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ X0) @ (gram_L1451583628elle_L @ top_top @ set @ n @ X0)) = $true)) )),
% 2.88/1.60 inference(cnf_transformation,[status(thm)],[f641])).
% 2.88/1.60 thf(f641,plain,(
% 2.88/1.60 ! [X0 : n] : ((gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ X0) @ (gram_L1451583628elle_L @ top_top @ set @ n @ X0)) = $true)),
% 2.88/1.60 inference(fool_elimination,[status(thm)],[f640])).
% 2.88/1.60 thf(f640,plain,(
% 2.88/1.60 ! [X0 : n] : (gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ X0) @ (gram_L1451583628elle_L @ top_top @ set @ n @ X0))),
% 2.88/1.60 inference(rectify,[status(thm)],[f10])).
% 2.88/1.60 thf(f10,axiom,(
% 2.88/1.60 ! [X2 : n] : (gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ X2) @ (gram_L1451583628elle_L @ top_top @ set @ n @ X2))),
% 2.88/1.60 file('/export/starexec/sandbox/tmp/tmp.LejkCnALQX/DTF2THF_24078.p',fact_9_Lr__subs__L)).
% 2.88/1.60 thf(f1287,plain,(
% 2.88/1.60 ~spl19_7),
% 2.88/1.60 inference(avatar_contradiction_clause,[status(thm)],[f1286])).
% 2.88/1.60 thf(f1286,plain,(
% 2.88/1.60 $false | ~spl19_7),
% 2.88/1.60 inference(trivial_inequality_removal,[status(thm)],[f1285])).
% 2.88/1.60 thf(f1285,plain,(
% 2.88/1.60 ($false = $true) | ~spl19_7),
% 2.88/1.60 inference(boolean_simplification,[status(thm)],[f1275])).
% 2.88/1.60 thf(f1275,plain,(
% 2.88/1.60 (((gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts)) & $false) = $true) | ~spl19_7),
% 2.88/1.60 inference(superposition,[status(thm)],[f1058,f1197])).
% 2.88/1.60 thf(f1197,plain,(
% 2.88/1.60 ((gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts)) = $false) | ~spl19_7),
% 2.88/1.60 inference(avatar_component_clause,[status(thm)],[f1195])).
% 2.88/1.60 thf(f1058,plain,(
% 2.88/1.60 ( ! [X0 : n] : ((((gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ X0) @ (gram_L861977280lle_LL @ top_top @ set @ n @ X0)) & (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ X0) @ (gram_L1451583628elle_L @ top_top @ set @ n @ X0))) = $true)) )),
% 2.88/1.60 inference(beta_eta_normalization,[status(thm)],[f1039])).
% 2.88/1.60 thf(f1039,plain,(
% 2.88/1.60 ( ! [X0 : n] : ((((^[Y0 : set @ set @ t]: ((^[Y1 : set @ set @ t]: ((gram_L608943123e_subs @ t @ Y1 @ Y0) & (gram_L608943123e_subs @ t @ Y0 @ Y1))))) @ (gram_L861977280lle_LL @ top_top @ set @ n @ X0) @ (gram_L1451583628elle_L @ top_top @ set @ n @ X0)) = $true)) )),
% 2.88/1.60 inference(definition_unfolding,[status(thm)],[f1029,f948])).
% 2.88/1.60 thf(f1029,plain,(
% 2.88/1.60 ( ! [X0 : n] : (((gram_L1456083582e_leqv @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ X0) @ (gram_L1451583628elle_L @ top_top @ set @ n @ X0)) = $true)) )),
% 2.88/1.60 inference(cnf_transformation,[status(thm)],[f491])).
% 2.88/1.60 thf(f491,plain,(
% 2.88/1.60 ! [X0 : n] : ((gram_L1456083582e_leqv @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ X0) @ (gram_L1451583628elle_L @ top_top @ set @ n @ X0)) = $true)),
% 2.88/1.60 inference(fool_elimination,[status(thm)],[f490])).
% 2.88/1.60 thf(f490,plain,(
% 2.88/1.60 ! [X0 : n] : (gram_L1456083582e_leqv @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ X0) @ (gram_L1451583628elle_L @ top_top @ set @ n @ X0))),
% 2.88/1.60 inference(rectify,[status(thm)],[f2])).
% 2.88/1.60 thf(f2,axiom,(
% 2.88/1.60 ! [X2 : n] : (gram_L1456083582e_leqv @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ X2) @ (gram_L1451583628elle_L @ top_top @ set @ n @ X2))),
% 2.88/1.60 file('/export/starexec/sandbox/tmp/tmp.LejkCnALQX/DTF2THF_24078.p',fact_1_LL__leqv__L)).
% 2.88/1.60 thf(f1259,plain,(
% 2.88/1.60 ~spl19_6),
% 2.88/1.60 inference(avatar_contradiction_clause,[status(thm)],[f1258])).
% 2.88/1.60 thf(f1258,plain,(
% 2.88/1.60 $false | ~spl19_6),
% 2.88/1.60 inference(trivial_inequality_removal,[status(thm)],[f1257])).
% 2.88/1.60 thf(f1257,plain,(
% 2.88/1.60 ($false = $true) | ~spl19_6),
% 2.88/1.60 inference(boolean_simplification,[status(thm)],[f1240])).
% 2.88/1.60 thf(f1240,plain,(
% 2.88/1.60 ($true = ($false & (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts)))) | ~spl19_6),
% 2.88/1.60 inference(superposition,[status(thm)],[f1058,f1193])).
% 2.88/1.60 thf(f1193,plain,(
% 2.88/1.60 ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts))) | ~spl19_6),
% 2.88/1.60 inference(avatar_component_clause,[status(thm)],[f1191])).
% 2.88/1.60 thf(f1225,plain,(
% 2.88/1.60 ~spl19_5),
% 2.88/1.60 inference(avatar_contradiction_clause,[status(thm)],[f1224])).
% 2.88/1.60 thf(f1224,plain,(
% 2.88/1.60 $false | ~spl19_5),
% 2.88/1.60 inference(trivial_inequality_removal,[status(thm)],[f1223])).
% 2.88/1.60 thf(f1223,plain,(
% 2.88/1.60 ($false = $true) | ~spl19_5),
% 2.88/1.60 inference(boolean_simplification,[status(thm)],[f1204])).
% 2.88/1.60 thf(f1204,plain,(
% 2.88/1.60 (($false & (gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts) @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts))) = $true) | ~spl19_5),
% 2.88/1.60 inference(superposition,[status(thm)],[f1055,f1189])).
% 2.88/1.60 thf(f1189,plain,(
% 2.88/1.60 ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) | ~spl19_5),
% 2.88/1.60 inference(avatar_component_clause,[status(thm)],[f1187])).
% 2.88/1.60 thf(f1055,plain,(
% 2.88/1.60 ( ! [X0 : n] : ((((gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ X0) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ X0)) & (gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ X0) @ (gram_L1451583628elle_L @ top_top @ set @ n @ X0))) = $true)) )),
% 2.88/1.60 inference(beta_eta_normalization,[status(thm)],[f1038])).
% 2.88/1.60 thf(f1038,plain,(
% 2.88/1.60 ( ! [X0 : n] : ((((^[Y0 : set @ set @ t]: ((^[Y1 : set @ set @ t]: ((gram_L608943123e_subs @ t @ Y1 @ Y0) & (gram_L608943123e_subs @ t @ Y0 @ Y1))))) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ X0) @ (gram_L1451583628elle_L @ top_top @ set @ n @ X0)) = $true)) )),
% 2.88/1.60 inference(definition_unfolding,[status(thm)],[f1019,f948])).
% 2.88/1.60 thf(f1019,plain,(
% 2.88/1.60 ( ! [X0 : n] : (((gram_L1456083582e_leqv @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ X0) @ (gram_L1451583628elle_L @ top_top @ set @ n @ X0)) = $true)) )),
% 2.88/1.60 inference(cnf_transformation,[status(thm)],[f581])).
% 2.88/1.60 thf(f581,plain,(
% 2.88/1.60 ! [X0 : n] : ((gram_L1456083582e_leqv @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ X0) @ (gram_L1451583628elle_L @ top_top @ set @ n @ X0)) = $true)),
% 2.88/1.60 inference(fool_elimination,[status(thm)],[f580])).
% 2.88/1.60 thf(f580,plain,(
% 2.88/1.60 ! [X0 : n] : (gram_L1456083582e_leqv @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ X0) @ (gram_L1451583628elle_L @ top_top @ set @ n @ X0))),
% 2.88/1.60 inference(rectify,[status(thm)],[f1])).
% 2.88/1.60 thf(f1,axiom,(
% 2.88/1.60 ! [X2 : n] : (gram_L1456083582e_leqv @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ X2) @ (gram_L1451583628elle_L @ top_top @ set @ n @ X2))),
% 2.88/1.60 file('/export/starexec/sandbox/tmp/tmp.LejkCnALQX/DTF2THF_24078.p',fact_0_Lr__leqv__L)).
% 2.88/1.60 thf(f1200,plain,(
% 2.88/1.60 spl19_6 | spl19_5 | spl19_7 | ~spl19_2),
% 2.88/1.60 inference(avatar_split_clause,[status(thm)],[f1159,f1067,f1195,f1187,f1191])).
% 2.88/1.60 thf(f1067,definition,(
% 2.88/1.60 spl19_2 <=> ($false = (gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts) @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts)))),
% 2.88/1.60 introduced(definition,[new_symbols(naming,[spl19_2])],[avatar_definition])).
% 2.88/1.60 thf(f1159,plain,(
% 2.88/1.60 ((gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts)) = $false) | ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts))) | ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) | ~spl19_2),
% 2.88/1.60 inference(trivial_inequality_removal,[status(thm)],[f1158])).
% 2.88/1.60 thf(f1158,plain,(
% 2.88/1.60 ((gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts)) = $false) | ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) | ($false = $true) | ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts))) | ~spl19_2),
% 2.88/1.60 inference(boolean_simplification,[status(thm)],[f1136])).
% 2.88/1.60 thf(f1136,plain,(
% 2.88/1.60 ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) | ($false = (gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts))) | (((gram_L608943123e_subs @ t @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts)) & $false) = $true) | ((gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L1451583628elle_L @ top_top @ set @ n @ ts)) = $false) | ~spl19_2),
% 2.88/1.60 inference(superposition,[status(thm)],[f1055,f1101])).
% 2.88/1.60 thf(f1101,plain,(
% 2.88/1.60 ( ! [X0 : set @ set @ t] : (($false = (gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts) @ X0)) | ($false = (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ X0)) | ($false = (gram_L608943123e_subs @ t @ X0 @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts))) | ($false = (gram_L608943123e_subs @ t @ X0 @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts)))) ) | ~spl19_2),
% 2.88/1.60 inference(binary_proxy_clausification,[status(thm)],[f1100])).
% 2.88/1.60 thf(f1100,plain,(
% 2.88/1.60 ( ! [X0 : set @ set @ t] : (($false = (gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts) @ X0)) | ($false = (gram_L608943123e_subs @ t @ X0 @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) | (((gram_L608943123e_subs @ t @ X0 @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts)) & (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ X0)) != $true)) ) | ~spl19_2),
% 2.88/1.60 inference(binary_proxy_clausification,[status(thm)],[f1099])).
% 2.88/1.60 thf(f1099,plain,(
% 2.88/1.60 ( ! [X0 : set @ set @ t] : ((((gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts) @ X0) & (gram_L608943123e_subs @ t @ X0 @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) != $true) | (((gram_L608943123e_subs @ t @ X0 @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts)) & (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ X0)) != $true)) ) | ~spl19_2),
% 2.88/1.60 inference(trivial_inequality_removal,[status(thm)],[f1098])).
% 2.88/1.60 thf(f1098,plain,(
% 2.88/1.60 ( ! [X0 : set @ set @ t] : (($false = $true) | (((gram_L608943123e_subs @ t @ X0 @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts)) & (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ X0)) != $true) | (((gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts) @ X0) & (gram_L608943123e_subs @ t @ X0 @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) != $true)) ) | ~spl19_2),
% 2.88/1.60 inference(boolean_simplification,[status(thm)],[f1073])).
% 2.88/1.60 thf(f1073,plain,(
% 2.88/1.60 ( ! [X0 : set @ set @ t] : (($true = ($false & (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts)))) | (((gram_L608943123e_subs @ t @ X0 @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts)) & (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ X0)) != $true) | (((gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts) @ X0) & (gram_L608943123e_subs @ t @ X0 @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))) != $true)) ) | ~spl19_2),
% 2.88/1.60 inference(superposition,[status(thm)],[f1054,f1069])).
% 2.88/1.60 thf(f1069,plain,(
% 2.88/1.60 ($false = (gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts) @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts))) | ~spl19_2),
% 2.88/1.60 inference(avatar_component_clause,[status(thm)],[f1067])).
% 2.88/1.60 thf(f1070,plain,(
% 2.88/1.60 spl19_1 | spl19_2),
% 2.88/1.60 inference(avatar_split_clause,[status(thm)],[f1061,f1067,f1063])).
% 2.88/1.60 thf(f1061,plain,(
% 2.88/1.60 ($false = (gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts) @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts))) | ($false = (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts)))),
% 2.88/1.60 inference(binary_proxy_clausification,[status(thm)],[f1060])).
% 2.88/1.60 thf(f1060,plain,(
% 2.88/1.60 ($true != ((gram_L608943123e_subs @ t @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts) @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts)) & (gram_L608943123e_subs @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))))),
% 2.88/1.60 inference(beta_eta_normalization,[status(thm)],[f1031])).
% 2.88/1.60 thf(f1031,plain,(
% 2.88/1.60 (((^[Y0 : set @ set @ t]: ((^[Y1 : set @ set @ t]: ((gram_L608943123e_subs @ t @ Y1 @ Y0) & (gram_L608943123e_subs @ t @ Y0 @ Y1))))) @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts)) != $true)),
% 2.88/1.60 inference(definition_unfolding,[status(thm)],[f949,f948])).
% 2.88/1.60 thf(f949,plain,(
% 2.88/1.60 ((gram_L1456083582e_leqv @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts)) != $true)),
% 2.88/1.60 inference(cnf_transformation,[status(thm)],[f834])).
% 2.88/1.60 thf(f834,plain,(
% 2.88/1.60 ((gram_L1456083582e_leqv @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts)) != $true)),
% 2.88/1.60 inference(flattening,[status(thm)],[f717])).
% 2.88/1.60 thf(f717,plain,(
% 2.88/1.60 ~((gram_L1456083582e_leqv @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts)) = $true)),
% 2.88/1.60 inference(fool_elimination,[status(thm)],[f716])).
% 2.88/1.60 thf(f716,plain,(
% 2.88/1.60 ~(gram_L1456083582e_leqv @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))),
% 2.88/1.60 inference(rectify,[status(thm)],[f286])).
% 2.88/1.60 thf(f286,negated_conjecture,(
% 2.88/1.60 ~(gram_L1456083582e_leqv @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))),
% 2.88/1.60 inference(negated_conjecture,[status(cth)],[f285])).
% 2.88/1.60 thf(f285,conjecture,(
% 2.88/1.60 (gram_L1456083582e_leqv @ t @ (gram_L861977280lle_LL @ top_top @ set @ n @ ts) @ (gram_L861977318lle_Lr @ top_top @ set @ n @ ts))),
% 2.88/1.60 file('/export/starexec/sandbox/tmp/tmp.LejkCnALQX/DTF2THF_24078.p',conj_0)).
% 2.88/1.60 % SZS output end Proof for DTF2THF_24078
% 2.88/1.60 % (24412)------------------------------
% 2.88/1.60 % (24412)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.60 % (24412)Termination reason: Refutation
% 2.88/1.60
% 2.88/1.60 % (24412)Memory used [KB]: 6652
% 2.88/1.60 % (24412)Time elapsed: 0.050 s
% 2.88/1.60 % (24412)Instructions burned: 112 (million)
% 2.88/1.60 % (24412)------------------------------
% 2.88/1.60 % (24412)------------------------------
% 2.88/1.60 % (24406)Success in time 0.071 s
%------------------------------------------------------------------------------