%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT244^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n007.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 08:45:16 AM UTC 2026
% Result : Theorem 2.90s 1.56s
% Output : Refutation 2.90s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : DAT244^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n007.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 06:11:23 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : DAT244^1 : TPTP v9.2.1. Released v7.0.0.
% 0.30/0.46 % Domain : Data Structures
% 0.30/0.46 % Problem : Infinite streams (sequences/lists) 29
% 0.30/0.46 % Version : [Bla16] axioms : Especial.
% 0.30/0.46 % English :
% 0.30/0.46
% 0.30/0.46 % Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% 0.30/0.46 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.30/0.46 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.30/0.46 % Source : [Bla16]
% 0.30/0.46 % Names : stream__29.p [Bla16]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.33 v9.1.0, 0.00 v7.2.0, 0.25 v7.1.0
% 0.30/0.46 % Syntax : Number of formulae : 332 ( 138 unt; 47 typ; 0 def)
% 0.30/0.46 % Number of atoms : 707 ( 301 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.30/0.46 % Number of connectives : 3749 ( 64 ~; 7 |; 37 &;3320 @)
% 0.30/0.46 % ( 0 <=>; 321 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 21 ( 8 avg)
% 0.30/0.46 % Number of types : 2 ( 1 usr)
% 0.30/0.46 % Number of type conns : 407 ( 407 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 49 ( 46 usr; 5 con; 0-7 aty)
% 0.30/0.46 % Number of variables : 1224 ( 62 ^;1102 !; 11 ?;1224 :)
% 0.30/0.46 % ( 49 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.46 % SPC : TH1_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.30/0.46 % 2016-07-13 14:40:28.127
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 %----Could-be-implicit typings (4)
% 0.30/0.46 thf(ty_t_Stream__Mirabelle__hbrgyiwlrc_Ostream,type,
% 0.30/0.46 stream170649215stream: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Set_Oset,type,
% 0.30/0.46 set: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_itself,type,
% 0.30/0.46 itself: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_tf_a,type,
% 0.30/0.46 a: $tType ).
% 0.30/0.46
% 0.30/0.46 %----Explicit typings (43)
% 0.30/0.46 thf(sy_cl_HOL_Otype,type,
% 0.30/0.46 type:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Obot,type,
% 0.30/0.46 bot:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.30/0.46 ord:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Otop,type,
% 0.30/0.46 top:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.30/0.46 order:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.30/0.46 linorder:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.30/0.46 preorder:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.30/0.46 order_bot:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oorder__top,type,
% 0.30/0.46 order_top:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 0.30/0.46 boolean_algebra:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.30/0.46 ordered_ab_group_add:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_BNF__Composition_Oid__bnf,type,
% 0.30/0.46 bNF_id_bnf:
% 0.30/0.46 !>[A: $tType] : ( A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Fun_Ocomp,type,
% 0.30/0.46 comp:
% 0.30/0.46 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Fun_Oinj__on,type,
% 0.30/0.46 inj_on:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Fun_Ooverride__on,type,
% 0.30/0.46 override_on:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.30/0.46 uminus_uminus:
% 0.30/0.46 !>[A: $tType] : ( A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_If,type,
% 0.30/0.46 if:
% 0.30/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.30/0.46 bot_bot:
% 0.30/0.46 !>[A: $tType] : A ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.46 ord_less_eq:
% 0.30/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.30/0.46 top_top:
% 0.30/0.46 !>[A: $tType] : A ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Pure_Otype,type,
% 0.30/0.46 type2:
% 0.30/0.46 !>[A: $tType] : ( itself @ A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_OBall,type,
% 0.30/0.46 ball:
% 0.30/0.46 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_OCollect,type,
% 0.30/0.46 collect:
% 0.30/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_Oimage,type,
% 0.30/0.46 image:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_Oinsert,type,
% 0.30/0.46 insert:
% 0.30/0.46 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_Ois__empty,type,
% 0.30/0.46 is_empty:
% 0.30/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_Ois__singleton,type,
% 0.30/0.46 is_singleton:
% 0.30/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_Opairwise,type,
% 0.30/0.46 pairwise:
% 0.30/0.46 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_Othe__elem,type,
% 0.30/0.46 the_elem:
% 0.30/0.46 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Osmember,type,
% 0.30/0.46 stream1586597341member:
% 0.30/0.46 !>[A: $tType] : ( A > ( stream170649215stream @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_OSCons,type,
% 0.30/0.46 stream641971652_SCons:
% 0.30/0.46 !>[A: $tType] : ( A > ( stream170649215stream @ A ) > ( stream170649215stream @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Ocase__stream,type,
% 0.30/0.46 stream1342653232stream:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > ( stream170649215stream @ A ) > B ) > ( stream170649215stream @ A ) > B ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Ocorec__stream,type,
% 0.30/0.46 stream660621732stream:
% 0.30/0.46 !>[C: $tType,A: $tType] : ( ( C > A ) > ( C > $o ) > ( C > ( stream170649215stream @ A ) ) > ( C > C ) > C > ( stream170649215stream @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Opred__stream,type,
% 0.30/0.46 stream1153105665stream:
% 0.30/0.46 !>[A: $tType] : ( ( A > $o ) > ( stream170649215stream @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Oshd,type,
% 0.30/0.46 stream_Mirabelle_shd:
% 0.30/0.46 !>[A: $tType] : ( ( stream170649215stream @ A ) > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Osmap,type,
% 0.30/0.46 stream2128578057e_smap:
% 0.30/0.46 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( stream170649215stream @ A ) > ( stream170649215stream @ Aa ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Osset,type,
% 0.30/0.46 stream30925839e_sset:
% 0.30/0.46 !>[A: $tType] : ( ( stream170649215stream @ A ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Ostl,type,
% 0.30/0.46 stream_Mirabelle_stl:
% 0.30/0.46 !>[A: $tType] : ( ( stream170649215stream @ A ) > ( stream170649215stream @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Ostream__all2,type,
% 0.30/0.46 stream686525009m_all2:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( stream170649215stream @ A ) > ( stream170649215stream @ B ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_member,type,
% 0.30/0.46 member:
% 0.30/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_s,type,
% 0.30/0.46 s: stream170649215stream @ a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_x,type,
% 0.30/0.46 x: a ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_y,type,
% 0.30/0.46 y: a ).
% 0.30/0.46
% 0.30/0.46 %----Relevant facts (256)
% 0.30/0.46 thf(fact_0_stream_Oinject,axiom,
% 0.30/0.46 ! [A: $tType,X1: A,X2: stream170649215stream @ A,Y1: A,Y2: stream170649215stream @ A] :
% 0.30/0.46 ( ( ( stream641971652_SCons @ A @ X1 @ X2 )
% 0.30/0.46 = ( stream641971652_SCons @ A @ Y1 @ Y2 ) )
% 0.30/0.46 = ( ( X1 = Y1 )
% 0.30/0.46 & ( X2 = Y2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.inject
% 0.30/0.46 thf(fact_1_stream_Oset__intros_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A2: stream170649215stream @ A,A1: A] :
% 0.30/0.46 ( ( member @ A @ X @ ( stream30925839e_sset @ A @ A2 ) )
% 0.30/0.46 => ( member @ A @ X @ ( stream30925839e_sset @ A @ ( stream641971652_SCons @ A @ A1 @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.set_intros(2)
% 0.30/0.46 thf(fact_2_stream_Oset__intros_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,A1: A,A2: stream170649215stream @ A] : ( member @ A @ A1 @ ( stream30925839e_sset @ A @ ( stream641971652_SCons @ A @ A1 @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.set_intros(1)
% 0.30/0.46 thf(fact_3_stream_Oexhaust,axiom,
% 0.30/0.46 ! [A: $tType,Y: stream170649215stream @ A] :
% 0.30/0.46 ~ ! [X12: A,X22: stream170649215stream @ A] :
% 0.30/0.46 ( Y
% 0.30/0.46 != ( stream641971652_SCons @ A @ X12 @ X22 ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.exhaust
% 0.30/0.46 thf(fact_4_stream_Oset__cases,axiom,
% 0.30/0.46 ! [A: $tType,E: A,A3: stream170649215stream @ A] :
% 0.30/0.46 ( ( member @ A @ E @ ( stream30925839e_sset @ A @ A3 ) )
% 0.30/0.46 => ( ! [Z2: stream170649215stream @ A] :
% 0.30/0.46 ( A3
% 0.30/0.46 != ( stream641971652_SCons @ A @ E @ Z2 ) )
% 0.30/0.46 => ~ ! [Z1: A,Z2: stream170649215stream @ A] :
% 0.30/0.46 ( ( A3
% 0.30/0.46 = ( stream641971652_SCons @ A @ Z1 @ Z2 ) )
% 0.30/0.46 => ~ ( member @ A @ E @ ( stream30925839e_sset @ A @ Z2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.set_cases
% 0.30/0.46 thf(fact_5_stream_Oset__induct,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A3: stream170649215stream @ A,P: A > ( stream170649215stream @ A ) > $o] :
% 0.30/0.46 ( ( member @ A @ X @ ( stream30925839e_sset @ A @ A3 ) )
% 0.30/0.46 => ( ! [Z1: A,Z2: stream170649215stream @ A] : ( P @ Z1 @ ( stream641971652_SCons @ A @ Z1 @ Z2 ) )
% 0.30/0.46 => ( ! [Z1: A,Z2: stream170649215stream @ A,Xa: A] :
% 0.30/0.46 ( ( member @ A @ Xa @ ( stream30925839e_sset @ A @ Z2 ) )
% 0.30/0.46 => ( ( P @ Xa @ Z2 )
% 0.30/0.46 => ( P @ Xa @ ( stream641971652_SCons @ A @ Z1 @ Z2 ) ) ) )
% 0.30/0.46 => ( P @ X @ A3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.set_induct
% 0.30/0.46 thf(fact_6_Stream__Mirabelle__hbrgyiwlrc_Osmember__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( stream1586597341member @ A )
% 0.30/0.46 = ( ^ [X3: A,S: stream170649215stream @ A] : ( member @ A @ X3 @ ( stream30925839e_sset @ A @ S ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Stream_Mirabelle_hbrgyiwlrc.smember_def
% 0.30/0.46 thf(fact_7_stream_Oset,axiom,
% 0.30/0.46 ! [A: $tType,X1: A,X2: stream170649215stream @ A] :
% 0.30/0.46 ( ( stream30925839e_sset @ A @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
% 0.30/0.46 = ( insert @ A @ X1 @ ( stream30925839e_sset @ A @ X2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.set
% 0.30/0.46 thf(fact_8_stream_Opred__inject,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,A3: A,Aa2: stream170649215stream @ A] :
% 0.30/0.46 ( ( stream1153105665stream @ A @ P @ ( stream641971652_SCons @ A @ A3 @ Aa2 ) )
% 0.30/0.46 = ( ( P @ A3 )
% 0.30/0.46 & ( stream1153105665stream @ A @ P @ Aa2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.pred_inject
% 0.30/0.46 thf(fact_9_stream_Ocase,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > ( stream170649215stream @ A ) > B,X1: A,X2: stream170649215stream @ A] :
% 0.30/0.46 ( ( stream1342653232stream @ A @ B @ F @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
% 0.30/0.46 = ( F @ X1 @ X2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.case
% 0.30/0.46 thf(fact_10_stream_Ocorec__code,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType] :
% 0.30/0.46 ( ( stream660621732stream @ C @ A )
% 0.30/0.46 = ( ^ [G1: C > A,Q2: C > $o,G21: C > ( stream170649215stream @ A ),G22: C > C,A4: C] : ( stream641971652_SCons @ A @ ( G1 @ A4 ) @ ( if @ ( stream170649215stream @ A ) @ ( Q2 @ A4 ) @ ( G21 @ A4 ) @ ( stream660621732stream @ C @ A @ G1 @ Q2 @ G21 @ G22 @ ( G22 @ A4 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.corec_code
% 0.30/0.46 thf(fact_11_stream_Opred__cong,axiom,
% 0.30/0.46 ! [A: $tType,X: stream170649215stream @ A,Ya: stream170649215stream @ A,P: A > $o,Pa: A > $o] :
% 0.30/0.46 ( ( X = Ya )
% 0.30/0.46 => ( ! [Z: A] :
% 0.30/0.46 ( ( member @ A @ Z @ ( stream30925839e_sset @ A @ Ya ) )
% 0.30/0.46 => ( ( P @ Z )
% 0.30/0.46 = ( Pa @ Z ) ) )
% 0.30/0.46 => ( ( stream1153105665stream @ A @ P @ X )
% 0.30/0.46 = ( stream1153105665stream @ A @ Pa @ Ya ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.pred_cong
% 0.30/0.46 thf(fact_12_stream_Opred__mono__strong,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,X: stream170649215stream @ A,Pa: A > $o] :
% 0.30/0.46 ( ( stream1153105665stream @ A @ P @ X )
% 0.30/0.46 => ( ! [Z: A] :
% 0.30/0.46 ( ( member @ A @ Z @ ( stream30925839e_sset @ A @ X ) )
% 0.30/0.46 => ( ( P @ Z )
% 0.30/0.46 => ( Pa @ Z ) ) )
% 0.30/0.46 => ( stream1153105665stream @ A @ Pa @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.pred_mono_strong
% 0.30/0.46 thf(fact_13_stream_Osel_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,X1: A,X2: stream170649215stream @ A] :
% 0.30/0.46 ( ( stream_Mirabelle_shd @ A @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
% 0.30/0.46 = X1 ) ).
% 0.30/0.46
% 0.30/0.46 % stream.sel(1)
% 0.30/0.46 thf(fact_14_stream_Oset__sel_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,A3: stream170649215stream @ A] : ( member @ A @ ( stream_Mirabelle_shd @ A @ A3 ) @ ( stream30925839e_sset @ A @ A3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.set_sel(1)
% 0.30/0.46 thf(fact_15_stream_Osel_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,X1: A,X2: stream170649215stream @ A] :
% 0.30/0.46 ( ( stream_Mirabelle_stl @ A @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
% 0.30/0.46 = X2 ) ).
% 0.30/0.46
% 0.30/0.46 % stream.sel(2)
% 0.30/0.46 thf(fact_16_stream_Oset__sel_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A3: stream170649215stream @ A] :
% 0.30/0.46 ( ( member @ A @ X @ ( stream30925839e_sset @ A @ ( stream_Mirabelle_stl @ A @ A3 ) ) )
% 0.30/0.46 => ( member @ A @ X @ ( stream30925839e_sset @ A @ A3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.set_sel(2)
% 0.30/0.46 thf(fact_17_stream_Omap,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B,X1: A,X2: stream170649215stream @ A] :
% 0.30/0.46 ( ( stream2128578057e_smap @ A @ B @ F @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
% 0.30/0.46 = ( stream641971652_SCons @ B @ ( F @ X1 ) @ ( stream2128578057e_smap @ A @ B @ F @ X2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.map
% 0.30/0.46 thf(fact_18_stream_Ocollapse,axiom,
% 0.30/0.46 ! [A: $tType,Stream: stream170649215stream @ A] :
% 0.30/0.46 ( ( stream641971652_SCons @ A @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) )
% 0.30/0.46 = Stream ) ).
% 0.30/0.46
% 0.30/0.46 % stream.collapse
% 0.30/0.46 thf(fact_19_stream_Omap__sel_I2_J,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B,A3: stream170649215stream @ A] :
% 0.30/0.46 ( ( stream_Mirabelle_stl @ B @ ( stream2128578057e_smap @ A @ B @ F @ A3 ) )
% 0.30/0.46 = ( stream2128578057e_smap @ A @ B @ F @ ( stream_Mirabelle_stl @ A @ A3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.map_sel(2)
% 0.30/0.46 thf(fact_20_stream_Omap__sel_I1_J,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B,A3: stream170649215stream @ A] :
% 0.30/0.46 ( ( stream_Mirabelle_shd @ B @ ( stream2128578057e_smap @ A @ B @ F @ A3 ) )
% 0.30/0.46 = ( F @ ( stream_Mirabelle_shd @ A @ A3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.map_sel(1)
% 0.30/0.46 thf(fact_21_stream_Ocorec__sel_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,Q22: C > $o,A3: C,G12: C > A,G212: C > ( stream170649215stream @ A ),G222: C > C] :
% 0.30/0.46 ( ( ( Q22 @ A3 )
% 0.30/0.46 => ( ( stream_Mirabelle_stl @ A @ ( stream660621732stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ A3 ) )
% 0.30/0.46 = ( G212 @ A3 ) ) )
% 0.30/0.46 & ( ~ ( Q22 @ A3 )
% 0.30/0.46 => ( ( stream_Mirabelle_stl @ A @ ( stream660621732stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ A3 ) )
% 0.30/0.46 = ( stream660621732stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ ( G222 @ A3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.corec_sel(2)
% 0.30/0.46 thf(fact_22_stream_Ocorec__sel_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,G12: C > A,Q22: C > $o,G212: C > ( stream170649215stream @ A ),G222: C > C,A3: C] :
% 0.30/0.46 ( ( stream_Mirabelle_shd @ A @ ( stream660621732stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ A3 ) )
% 0.30/0.46 = ( G12 @ A3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.corec_sel(1)
% 0.30/0.46 thf(fact_23_stream_Oexpand,axiom,
% 0.30/0.46 ! [A: $tType,Stream: stream170649215stream @ A,Stream2: stream170649215stream @ A] :
% 0.30/0.46 ( ( ( ( stream_Mirabelle_shd @ A @ Stream )
% 0.30/0.46 = ( stream_Mirabelle_shd @ A @ Stream2 ) )
% 0.30/0.46 & ( ( stream_Mirabelle_stl @ A @ Stream )
% 0.30/0.46 = ( stream_Mirabelle_stl @ A @ Stream2 ) ) )
% 0.30/0.46 => ( Stream = Stream2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.expand
% 0.30/0.46 thf(fact_24_stream_Ocoinduct,axiom,
% 0.30/0.46 ! [A: $tType,R: ( stream170649215stream @ A ) > ( stream170649215stream @ A ) > $o,Stream: stream170649215stream @ A,Stream2: stream170649215stream @ A] :
% 0.30/0.46 ( ( R @ Stream @ Stream2 )
% 0.30/0.46 => ( ! [Stream3: stream170649215stream @ A,Stream4: stream170649215stream @ A] :
% 0.30/0.46 ( ( R @ Stream3 @ Stream4 )
% 0.30/0.46 => ( ( ( stream_Mirabelle_shd @ A @ Stream3 )
% 0.30/0.46 = ( stream_Mirabelle_shd @ A @ Stream4 ) )
% 0.30/0.46 & ( R @ ( stream_Mirabelle_stl @ A @ Stream3 ) @ ( stream_Mirabelle_stl @ A @ Stream4 ) ) ) )
% 0.30/0.46 => ( Stream = Stream2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.coinduct
% 0.30/0.46 thf(fact_25_stream_Ocase__eq__if,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( stream1342653232stream @ A @ B )
% 0.30/0.46 = ( ^ [F2: A > ( stream170649215stream @ A ) > B,Stream5: stream170649215stream @ A] : ( F2 @ ( stream_Mirabelle_shd @ A @ Stream5 ) @ ( stream_Mirabelle_stl @ A @ Stream5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.case_eq_if
% 0.30/0.46 thf(fact_26_stream_Ocoinduct__strong,axiom,
% 0.30/0.46 ! [A: $tType,R: ( stream170649215stream @ A ) > ( stream170649215stream @ A ) > $o,Stream: stream170649215stream @ A,Stream2: stream170649215stream @ A] :
% 0.30/0.46 ( ( R @ Stream @ Stream2 )
% 0.30/0.46 => ( ! [Stream3: stream170649215stream @ A,Stream4: stream170649215stream @ A] :
% 0.30/0.46 ( ( R @ Stream3 @ Stream4 )
% 0.30/0.46 => ( ( ( stream_Mirabelle_shd @ A @ Stream3 )
% 0.30/0.46 = ( stream_Mirabelle_shd @ A @ Stream4 ) )
% 0.30/0.46 & ( ( R @ ( stream_Mirabelle_stl @ A @ Stream3 ) @ ( stream_Mirabelle_stl @ A @ Stream4 ) )
% 0.30/0.46 | ( ( stream_Mirabelle_stl @ A @ Stream3 )
% 0.30/0.46 = ( stream_Mirabelle_stl @ A @ Stream4 ) ) ) ) )
% 0.30/0.46 => ( Stream = Stream2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.coinduct_strong
% 0.30/0.46 thf(fact_27_stream_Ocorec__disc,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType] :
% 0.30/0.46 ( ( stream660621732stream @ C @ A )
% 0.30/0.46 = ( stream660621732stream @ C @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.corec_disc
% 0.30/0.46 thf(fact_28_stream_Osplit__sel__asm,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,P: B > $o,F: A > ( stream170649215stream @ A ) > B,Stream: stream170649215stream @ A] :
% 0.30/0.46 ( ( P @ ( stream1342653232stream @ A @ B @ F @ Stream ) )
% 0.30/0.46 = ( ~ ( ( Stream
% 0.30/0.46 = ( stream641971652_SCons @ A @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) )
% 0.30/0.46 & ~ ( P @ ( F @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.split_sel_asm
% 0.30/0.46 thf(fact_29_stream_Osplit__sel,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,P: B > $o,F: A > ( stream170649215stream @ A ) > B,Stream: stream170649215stream @ A] :
% 0.30/0.46 ( ( P @ ( stream1342653232stream @ A @ B @ F @ Stream ) )
% 0.30/0.46 = ( ( Stream
% 0.30/0.46 = ( stream641971652_SCons @ A @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) )
% 0.30/0.46 => ( P @ ( F @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.split_sel
% 0.30/0.46 thf(fact_30_stream_Oexhaust__sel,axiom,
% 0.30/0.46 ! [A: $tType,Stream: stream170649215stream @ A] :
% 0.30/0.46 ( Stream
% 0.30/0.46 = ( stream641971652_SCons @ A @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.exhaust_sel
% 0.30/0.46 thf(fact_31_stream_Oinj__map__strong,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X: stream170649215stream @ A,Xa2: stream170649215stream @ A,F: A > B,Fa: A > B] :
% 0.30/0.46 ( ! [Z: A,Za: A] :
% 0.30/0.46 ( ( member @ A @ Z @ ( stream30925839e_sset @ A @ X ) )
% 0.30/0.46 => ( ( member @ A @ Za @ ( stream30925839e_sset @ A @ Xa2 ) )
% 0.30/0.46 => ( ( ( F @ Z )
% 0.30/0.46 = ( Fa @ Za ) )
% 0.30/0.46 => ( Z = Za ) ) ) )
% 0.30/0.46 => ( ( ( stream2128578057e_smap @ A @ B @ F @ X )
% 0.30/0.46 = ( stream2128578057e_smap @ A @ B @ Fa @ Xa2 ) )
% 0.30/0.46 => ( X = Xa2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.inj_map_strong
% 0.30/0.46 thf(fact_32_stream_Omap__cong0,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X: stream170649215stream @ A,F: A > B,G: A > B] :
% 0.30/0.46 ( ! [Z: A] :
% 0.30/0.46 ( ( member @ A @ Z @ ( stream30925839e_sset @ A @ X ) )
% 0.30/0.46 => ( ( F @ Z )
% 0.30/0.46 = ( G @ Z ) ) )
% 0.30/0.46 => ( ( stream2128578057e_smap @ A @ B @ F @ X )
% 0.30/0.46 = ( stream2128578057e_smap @ A @ B @ G @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.map_cong0
% 0.30/0.46 thf(fact_33_stream_Omap__cong,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X: stream170649215stream @ A,Ya: stream170649215stream @ A,F: A > B,G: A > B] :
% 0.30/0.46 ( ( X = Ya )
% 0.30/0.46 => ( ! [Z: A] :
% 0.30/0.46 ( ( member @ A @ Z @ ( stream30925839e_sset @ A @ Ya ) )
% 0.30/0.46 => ( ( F @ Z )
% 0.30/0.46 = ( G @ Z ) ) )
% 0.30/0.46 => ( ( stream2128578057e_smap @ A @ B @ F @ X )
% 0.30/0.46 = ( stream2128578057e_smap @ A @ B @ G @ Ya ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.map_cong
% 0.30/0.46 thf(fact_34_insertCI,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 0.30/0.46 ( ( ~ ( member @ A @ A3 @ B2 )
% 0.30/0.46 => ( A3 = B3 ) )
% 0.30/0.46 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insertCI
% 0.30/0.46 thf(fact_35_insert__iff,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,B3: A,A5: set @ A] :
% 0.30/0.46 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A5 ) )
% 0.30/0.46 = ( ( A3 = B3 )
% 0.30/0.46 | ( member @ A @ A3 @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_iff
% 0.30/0.46 thf(fact_36_insert__absorb2,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A5: set @ A] :
% 0.30/0.46 ( ( insert @ A @ X @ ( insert @ A @ X @ A5 ) )
% 0.30/0.46 = ( insert @ A @ X @ A5 ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_absorb2
% 0.30/0.46 thf(fact_37_insertE,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,B3: A,A5: set @ A] :
% 0.30/0.46 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A5 ) )
% 0.30/0.46 => ( ( A3 != B3 )
% 0.30/0.46 => ( member @ A @ A3 @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insertE
% 0.30/0.46 thf(fact_38_insertI1,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,B2: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % insertI1
% 0.30/0.46 thf(fact_39_insertI2,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 0.30/0.46 ( ( member @ A @ A3 @ B2 )
% 0.30/0.46 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insertI2
% 0.30/0.46 thf(fact_40_Set_Oset__insert,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A5: set @ A] :
% 0.30/0.46 ( ( member @ A @ X @ A5 )
% 0.30/0.46 => ~ ! [B4: set @ A] :
% 0.30/0.46 ( ( A5
% 0.30/0.46 = ( insert @ A @ X @ B4 ) )
% 0.30/0.46 => ( member @ A @ X @ B4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Set.set_insert
% 0.30/0.46 thf(fact_41_insert__ident,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ~ ( member @ A @ X @ A5 )
% 0.30/0.46 => ( ~ ( member @ A @ X @ B2 )
% 0.30/0.46 => ( ( ( insert @ A @ X @ A5 )
% 0.30/0.46 = ( insert @ A @ X @ B2 ) )
% 0.30/0.46 = ( A5 = B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_ident
% 0.30/0.46 thf(fact_42_insert__absorb,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,A5: set @ A] :
% 0.30/0.46 ( ( member @ A @ A3 @ A5 )
% 0.30/0.46 => ( ( insert @ A @ A3 @ A5 )
% 0.30/0.46 = A5 ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_absorb
% 0.30/0.46 thf(fact_43_insert__eq__iff,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,A5: set @ A,B3: A,B2: set @ A] :
% 0.30/0.46 ( ~ ( member @ A @ A3 @ A5 )
% 0.30/0.46 => ( ~ ( member @ A @ B3 @ B2 )
% 0.30/0.46 => ( ( ( insert @ A @ A3 @ A5 )
% 0.30/0.46 = ( insert @ A @ B3 @ B2 ) )
% 0.30/0.46 = ( ( ( A3 = B3 )
% 0.30/0.46 => ( A5 = B2 ) )
% 0.30/0.46 & ( ( A3 != B3 )
% 0.30/0.46 => ? [C2: set @ A] :
% 0.30/0.46 ( ( A5
% 0.30/0.46 = ( insert @ A @ B3 @ C2 ) )
% 0.30/0.46 & ~ ( member @ A @ B3 @ C2 )
% 0.30/0.46 & ( B2
% 0.30/0.46 = ( insert @ A @ A3 @ C2 ) )
% 0.30/0.46 & ~ ( member @ A @ A3 @ C2 ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_eq_iff
% 0.30/0.46 thf(fact_44_mk__disjoint__insert,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,A5: set @ A] :
% 0.30/0.46 ( ( member @ A @ A3 @ A5 )
% 0.30/0.46 => ? [B4: set @ A] :
% 0.30/0.46 ( ( A5
% 0.30/0.46 = ( insert @ A @ A3 @ B4 ) )
% 0.30/0.46 & ~ ( member @ A @ A3 @ B4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mk_disjoint_insert
% 0.30/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,P: A > $o] :
% 0.30/0.46 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.30/0.46 = ( P @ A3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Collect_eq
% 0.30/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] :
% 0.30/0.46 ( ( collect @ A
% 0.30/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ A5 ) )
% 0.30/0.46 = A5 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mem_eq
% 0.30/0.46 thf(fact_47_Collect__cong,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X4: A] :
% 0.30/0.46 ( ( P @ X4 )
% 0.30/0.46 = ( Q @ X4 ) )
% 0.30/0.46 => ( ( collect @ A @ P )
% 0.30/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_cong
% 0.30/0.46 thf(fact_48_ext,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/0.46 ( ! [X4: A] :
% 0.30/0.46 ( ( F @ X4 )
% 0.30/0.46 = ( G @ X4 ) )
% 0.30/0.46 => ( F = G ) ) ).
% 0.30/0.46
% 0.30/0.46 % ext
% 0.30/0.46 thf(fact_49_insert__commute,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Y: A,A5: set @ A] :
% 0.30/0.46 ( ( insert @ A @ X @ ( insert @ A @ Y @ A5 ) )
% 0.30/0.46 = ( insert @ A @ Y @ ( insert @ A @ X @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_commute
% 0.30/0.46 thf(fact_50_stream_Orel__coinduct,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,P: ( stream170649215stream @ A ) > ( stream170649215stream @ B ) > $o,X: stream170649215stream @ A,Y: stream170649215stream @ B,R: A > B > $o] :
% 0.30/0.46 ( ( P @ X @ Y )
% 0.30/0.46 => ( ! [Stream3: stream170649215stream @ A,Stream4: stream170649215stream @ B] :
% 0.30/0.46 ( ( P @ Stream3 @ Stream4 )
% 0.30/0.46 => ( ( R @ ( stream_Mirabelle_shd @ A @ Stream3 ) @ ( stream_Mirabelle_shd @ B @ Stream4 ) )
% 0.30/0.46 & ( P @ ( stream_Mirabelle_stl @ A @ Stream3 ) @ ( stream_Mirabelle_stl @ B @ Stream4 ) ) ) )
% 0.30/0.46 => ( stream686525009m_all2 @ A @ B @ R @ X @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.rel_coinduct
% 0.30/0.46 thf(fact_51_stream_Orel__sel,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( stream686525009m_all2 @ A @ B )
% 0.30/0.46 = ( ^ [R2: A > B > $o,A4: stream170649215stream @ A,B5: stream170649215stream @ B] :
% 0.30/0.46 ( ( R2 @ ( stream_Mirabelle_shd @ A @ A4 ) @ ( stream_Mirabelle_shd @ B @ B5 ) )
% 0.30/0.46 & ( stream686525009m_all2 @ A @ B @ R2 @ ( stream_Mirabelle_stl @ A @ A4 ) @ ( stream_Mirabelle_stl @ B @ B5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.rel_sel
% 0.30/0.46 thf(fact_52_stream_Opred__set,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( stream1153105665stream @ A )
% 0.30/0.46 = ( ^ [P2: A > $o,X3: stream170649215stream @ A] :
% 0.30/0.46 ! [Y3: A] :
% 0.30/0.46 ( ( member @ A @ Y3 @ ( stream30925839e_sset @ A @ X3 ) )
% 0.30/0.46 => ( P2 @ Y3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.pred_set
% 0.30/0.46 thf(fact_53_stream_Opred__map,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: stream170649215stream @ A] :
% 0.30/0.46 ( ( stream1153105665stream @ B @ Q @ ( stream2128578057e_smap @ A @ B @ F @ X ) )
% 0.30/0.46 = ( stream1153105665stream @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.pred_map
% 0.30/0.46 thf(fact_54_stream_Oset__map,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B,V: stream170649215stream @ A] :
% 0.30/0.46 ( ( stream30925839e_sset @ B @ ( stream2128578057e_smap @ A @ B @ F @ V ) )
% 0.30/0.46 = ( image @ A @ B @ F @ ( stream30925839e_sset @ A @ V ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.set_map
% 0.30/0.46 thf(fact_55_pairwise__insert,axiom,
% 0.30/0.46 ! [A: $tType,R3: A > A > $o,X: A,S2: set @ A] :
% 0.30/0.46 ( ( pairwise @ A @ R3 @ ( insert @ A @ X @ S2 ) )
% 0.30/0.46 = ( ! [Y3: A] :
% 0.30/0.46 ( ( ( member @ A @ Y3 @ S2 )
% 0.30/0.46 & ( Y3 != X ) )
% 0.30/0.46 => ( ( R3 @ X @ Y3 )
% 0.30/0.46 & ( R3 @ Y3 @ X ) ) )
% 0.30/0.46 & ( pairwise @ A @ R3 @ S2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pairwise_insert
% 0.30/0.46 thf(fact_56_stream_Orel__inject,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: A > B > $o,X1: A,X2: stream170649215stream @ A,Y1: B,Y2: stream170649215stream @ B] :
% 0.30/0.46 ( ( stream686525009m_all2 @ A @ B @ R @ ( stream641971652_SCons @ A @ X1 @ X2 ) @ ( stream641971652_SCons @ B @ Y1 @ Y2 ) )
% 0.30/0.46 = ( ( R @ X1 @ Y1 )
% 0.30/0.46 & ( stream686525009m_all2 @ A @ B @ R @ X2 @ Y2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.rel_inject
% 0.30/0.46 thf(fact_57_singletonI,axiom,
% 0.30/0.46 ! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % singletonI
% 0.30/0.46 thf(fact_58_image__eqI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B3: A,F: B > A,X: B,A5: set @ B] :
% 0.30/0.46 ( ( B3
% 0.30/0.46 = ( F @ X ) )
% 0.30/0.46 => ( ( member @ B @ X @ A5 )
% 0.30/0.46 => ( member @ A @ B3 @ ( image @ B @ A @ F @ A5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_eqI
% 0.30/0.46 thf(fact_59_empty__Collect__eq,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o] :
% 0.30/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/0.46 = ( collect @ A @ P ) )
% 0.30/0.46 = ( ! [X3: A] :
% 0.30/0.46 ~ ( P @ X3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % empty_Collect_eq
% 0.30/0.46 thf(fact_60_Collect__empty__eq,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o] :
% 0.30/0.46 ( ( ( collect @ A @ P )
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = ( ! [X3: A] :
% 0.30/0.46 ~ ( P @ X3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_empty_eq
% 0.30/0.46 thf(fact_61_all__not__in__conv,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] :
% 0.30/0.46 ( ( ! [X3: A] :
% 0.30/0.46 ~ ( member @ A @ X3 @ A5 ) )
% 0.30/0.46 = ( A5
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % all_not_in_conv
% 0.30/0.46 thf(fact_62_empty__iff,axiom,
% 0.30/0.46 ! [A: $tType,C3: A] :
% 0.30/0.46 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % empty_iff
% 0.30/0.46 thf(fact_63_image__is__empty,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
% 0.30/0.46 ( ( ( image @ B @ A @ F @ A5 )
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = ( A5
% 0.30/0.46 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_is_empty
% 0.30/0.46 thf(fact_64_empty__is__image,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
% 0.30/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/0.46 = ( image @ B @ A @ F @ A5 ) )
% 0.30/0.46 = ( A5
% 0.30/0.46 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % empty_is_image
% 0.30/0.46 thf(fact_65_image__empty,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: B > A] :
% 0.30/0.46 ( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_empty
% 0.30/0.46 thf(fact_66_image__insert,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,A3: B,B2: set @ B] :
% 0.30/0.46 ( ( image @ B @ A @ F @ ( insert @ B @ A3 @ B2 ) )
% 0.30/0.46 = ( insert @ A @ ( F @ A3 ) @ ( image @ B @ A @ F @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_insert
% 0.30/0.46 thf(fact_67_insert__image,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X: A,A5: set @ A,F: A > B] :
% 0.30/0.46 ( ( member @ A @ X @ A5 )
% 0.30/0.46 => ( ( insert @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A5 ) )
% 0.30/0.46 = ( image @ A @ B @ F @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_image
% 0.30/0.46 thf(fact_68_ball__empty,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,X5: A] :
% 0.30/0.46 ( ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( P @ X5 ) ) ).
% 0.30/0.46
% 0.30/0.46 % ball_empty
% 0.30/0.46 thf(fact_69_pairwise__empty,axiom,
% 0.30/0.46 ! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pairwise_empty
% 0.30/0.46 thf(fact_70_rev__image__eqI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X: A,A5: set @ A,B3: B,F: A > B] :
% 0.30/0.46 ( ( member @ A @ X @ A5 )
% 0.30/0.46 => ( ( B3
% 0.30/0.46 = ( F @ X ) )
% 0.30/0.46 => ( member @ B @ B3 @ ( image @ A @ B @ F @ A5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rev_image_eqI
% 0.30/0.46 thf(fact_71_pairwise__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( pairwise @ A )
% 0.30/0.46 = ( ^ [R2: A > A > $o,S3: set @ A] :
% 0.30/0.46 ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ S3 )
% 0.30/0.46 => ! [Y3: A] :
% 0.30/0.46 ( ( member @ A @ Y3 @ S3 )
% 0.30/0.46 => ( ( X3 != Y3 )
% 0.30/0.46 => ( R2 @ X3 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pairwise_def
% 0.30/0.46 thf(fact_72_ball__imageD,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: A > $o] :
% 0.30/0.46 ( ! [X4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ ( image @ B @ A @ F @ A5 ) )
% 0.30/0.46 => ( P @ X4 ) )
% 0.30/0.46 => ! [X5: B] :
% 0.30/0.46 ( ( member @ B @ X5 @ A5 )
% 0.30/0.46 => ( P @ ( F @ X5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ball_imageD
% 0.30/0.46 thf(fact_73_image__cong,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G: A > B] :
% 0.30/0.46 ( ( M = N )
% 0.30/0.46 => ( ! [X4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ N )
% 0.30/0.46 => ( ( F @ X4 )
% 0.30/0.46 = ( G @ X4 ) ) )
% 0.30/0.46 => ( ( image @ A @ B @ F @ M )
% 0.30/0.46 = ( image @ A @ B @ G @ N ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_cong
% 0.30/0.46 thf(fact_74_ex__in__conv,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] :
% 0.30/0.46 ( ( ? [X3: A] : ( member @ A @ X3 @ A5 ) )
% 0.30/0.46 = ( A5
% 0.30/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ex_in_conv
% 0.30/0.46 thf(fact_75_bex__imageD,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: A > $o] :
% 0.30/0.46 ( ? [X5: A] :
% 0.30/0.46 ( ( member @ A @ X5 @ ( image @ B @ A @ F @ A5 ) )
% 0.30/0.46 & ( P @ X5 ) )
% 0.30/0.46 => ? [X4: B] :
% 0.30/0.46 ( ( member @ B @ X4 @ A5 )
% 0.30/0.46 & ( P @ ( F @ X4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bex_imageD
% 0.30/0.46 thf(fact_76_image__iff,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,Z3: A,F: B > A,A5: set @ B] :
% 0.30/0.46 ( ( member @ A @ Z3 @ ( image @ B @ A @ F @ A5 ) )
% 0.30/0.46 = ( ? [X3: B] :
% 0.30/0.46 ( ( member @ B @ X3 @ A5 )
% 0.30/0.46 & ( Z3
% 0.30/0.46 = ( F @ X3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_iff
% 0.30/0.46 thf(fact_77_equals0I,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] :
% 0.30/0.46 ( ! [Y4: A] :
% 0.30/0.46 ~ ( member @ A @ Y4 @ A5 )
% 0.30/0.46 => ( A5
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % equals0I
% 0.30/0.46 thf(fact_78_equals0D,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,A3: A] :
% 0.30/0.46 ( ( A5
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ~ ( member @ A @ A3 @ A5 ) ) ).
% 0.30/0.46
% 0.30/0.46 % equals0D
% 0.30/0.46 thf(fact_79_Ball__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ball @ A )
% 0.30/0.46 = ( ^ [A6: set @ A,P2: A > $o] :
% 0.30/0.46 ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ A6 )
% 0.30/0.46 => ( P2 @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ball_def
% 0.30/0.46 thf(fact_80_imageI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X: A,A5: set @ A,F: A > B] :
% 0.30/0.46 ( ( member @ A @ X @ A5 )
% 0.30/0.46 => ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % imageI
% 0.30/0.46 thf(fact_81_emptyE,axiom,
% 0.30/0.46 ! [A: $tType,A3: A] :
% 0.30/0.46 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % emptyE
% 0.30/0.46 thf(fact_82_stream_Orel__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( stream686525009m_all2 @ A @ A
% 0.30/0.46 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.30/0.46 = ( ^ [Y5: stream170649215stream @ A,Z4: stream170649215stream @ A] : ( Y5 = Z4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.rel_eq
% 0.30/0.46 thf(fact_83_stream_Orel__refl,axiom,
% 0.30/0.46 ! [B: $tType,Ra: B > B > $o,X: stream170649215stream @ B] :
% 0.30/0.46 ( ! [X4: B] : ( Ra @ X4 @ X4 )
% 0.30/0.46 => ( stream686525009m_all2 @ B @ B @ Ra @ X @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.rel_refl
% 0.30/0.46 thf(fact_84_pairwise__singleton,axiom,
% 0.30/0.46 ! [A: $tType,P: A > A > $o,A5: A] : ( pairwise @ A @ P @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pairwise_singleton
% 0.30/0.46 thf(fact_85_stream_Omap__o__corec,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > A,Ga: C > $o,Gb: C > ( stream170649215stream @ A ),Gc: C > C] :
% 0.30/0.46 ( ( comp @ ( stream170649215stream @ A ) @ ( stream170649215stream @ B ) @ C @ ( stream2128578057e_smap @ A @ B @ F ) @ ( stream660621732stream @ C @ A @ G @ Ga @ Gb @ Gc ) )
% 0.30/0.46 = ( stream660621732stream @ C @ B @ ( comp @ A @ B @ C @ F @ G ) @ Ga @ ( comp @ ( stream170649215stream @ A ) @ ( stream170649215stream @ B ) @ C @ ( stream2128578057e_smap @ A @ B @ F ) @ Gb ) @ Gc ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.map_o_corec
% 0.30/0.46 thf(fact_86_stream_Omap__comp,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: stream170649215stream @ A] :
% 0.30/0.46 ( ( stream2128578057e_smap @ B @ C @ G @ ( stream2128578057e_smap @ A @ B @ F @ V ) )
% 0.30/0.46 = ( stream2128578057e_smap @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.map_comp
% 0.30/0.46 thf(fact_87_singletonD,axiom,
% 0.30/0.46 ! [A: $tType,B3: A,A3: A] :
% 0.30/0.46 ( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 => ( B3 = A3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % singletonD
% 0.30/0.46 thf(fact_88_singleton__iff,axiom,
% 0.30/0.46 ! [A: $tType,B3: A,A3: A] :
% 0.30/0.46 ( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 = ( B3 = A3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % singleton_iff
% 0.30/0.46 thf(fact_89_doubleton__eq__iff,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,B3: A,C3: A,D: A] :
% 0.30/0.46 ( ( ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 = ( insert @ A @ C3 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.46 = ( ( ( A3 = C3 )
% 0.30/0.46 & ( B3 = D ) )
% 0.30/0.46 | ( ( A3 = D )
% 0.30/0.46 & ( B3 = C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % doubleton_eq_iff
% 0.30/0.46 thf(fact_90_insert__not__empty,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,A5: set @ A] :
% 0.30/0.46 ( ( insert @ A @ A3 @ A5 )
% 0.30/0.46 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_not_empty
% 0.30/0.46 thf(fact_91_singleton__inject,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,B3: A] :
% 0.30/0.46 ( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 => ( A3 = B3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % singleton_inject
% 0.30/0.46 thf(fact_92_stream_Orel__refl__strong,axiom,
% 0.30/0.46 ! [A: $tType,X: stream170649215stream @ A,Ra: A > A > $o] :
% 0.30/0.46 ( ! [Z: A] :
% 0.30/0.46 ( ( member @ A @ Z @ ( stream30925839e_sset @ A @ X ) )
% 0.30/0.46 => ( Ra @ Z @ Z ) )
% 0.30/0.46 => ( stream686525009m_all2 @ A @ A @ Ra @ X @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.rel_refl_strong
% 0.30/0.46 thf(fact_93_stream_Orel__mono__strong,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: A > B > $o,X: stream170649215stream @ A,Y: stream170649215stream @ B,Ra: A > B > $o] :
% 0.30/0.46 ( ( stream686525009m_all2 @ A @ B @ R @ X @ Y )
% 0.30/0.46 => ( ! [Z: A,Yb: B] :
% 0.30/0.46 ( ( member @ A @ Z @ ( stream30925839e_sset @ A @ X ) )
% 0.30/0.46 => ( ( member @ B @ Yb @ ( stream30925839e_sset @ B @ Y ) )
% 0.30/0.46 => ( ( R @ Z @ Yb )
% 0.30/0.46 => ( Ra @ Z @ Yb ) ) ) )
% 0.30/0.46 => ( stream686525009m_all2 @ A @ B @ Ra @ X @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.rel_mono_strong
% 0.30/0.46 thf(fact_94_stream_Orel__cong,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X: stream170649215stream @ A,Ya: stream170649215stream @ A,Y: stream170649215stream @ B,Xa2: stream170649215stream @ B,R: A > B > $o,Ra: A > B > $o] :
% 0.30/0.46 ( ( X = Ya )
% 0.30/0.46 => ( ( Y = Xa2 )
% 0.30/0.46 => ( ! [Z: A,Yb: B] :
% 0.30/0.46 ( ( member @ A @ Z @ ( stream30925839e_sset @ A @ Ya ) )
% 0.30/0.46 => ( ( member @ B @ Yb @ ( stream30925839e_sset @ B @ Xa2 ) )
% 0.30/0.46 => ( ( R @ Z @ Yb )
% 0.30/0.46 = ( Ra @ Z @ Yb ) ) ) )
% 0.30/0.46 => ( ( stream686525009m_all2 @ A @ B @ R @ X @ Y )
% 0.30/0.46 = ( stream686525009m_all2 @ A @ B @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.rel_cong
% 0.30/0.46 thf(fact_95_stream_Orel__intros,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: A > B > $o,X1: A,Y1: B,X2: stream170649215stream @ A,Y2: stream170649215stream @ B] :
% 0.30/0.46 ( ( R @ X1 @ Y1 )
% 0.30/0.46 => ( ( stream686525009m_all2 @ A @ B @ R @ X2 @ Y2 )
% 0.30/0.46 => ( stream686525009m_all2 @ A @ B @ R @ ( stream641971652_SCons @ A @ X1 @ X2 ) @ ( stream641971652_SCons @ B @ Y1 @ Y2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.rel_intros
% 0.30/0.46 thf(fact_96_stream_Orel__cases,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: A > B > $o,A3: stream170649215stream @ A,B3: stream170649215stream @ B] :
% 0.30/0.46 ( ( stream686525009m_all2 @ A @ B @ R @ A3 @ B3 )
% 0.30/0.46 => ~ ! [X1a: A,X2a: stream170649215stream @ A] :
% 0.30/0.46 ( ( A3
% 0.30/0.46 = ( stream641971652_SCons @ A @ X1a @ X2a ) )
% 0.30/0.46 => ! [Y1a: B,Y2a: stream170649215stream @ B] :
% 0.30/0.46 ( ( B3
% 0.30/0.46 = ( stream641971652_SCons @ B @ Y1a @ Y2a ) )
% 0.30/0.46 => ( ( R @ X1a @ Y1a )
% 0.30/0.46 => ~ ( stream686525009m_all2 @ A @ B @ R @ X2a @ Y2a ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.rel_cases
% 0.30/0.46 thf(fact_97_Ball__image__comp,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,A5: set @ B,G: A > $o] :
% 0.30/0.46 ( ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ ( image @ B @ A @ F @ A5 ) )
% 0.30/0.46 => ( G @ X3 ) ) )
% 0.30/0.46 = ( ! [X3: B] :
% 0.30/0.46 ( ( member @ B @ X3 @ A5 )
% 0.30/0.46 => ( comp @ A @ $o @ B @ G @ F @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ball_image_comp
% 0.30/0.46 thf(fact_98_not__False__in__image__Ball,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,A5: set @ A] :
% 0.30/0.46 ( ( ~ ( member @ $o @ $false @ ( image @ A @ $o @ P @ A5 ) ) )
% 0.30/0.46 = ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ A5 )
% 0.30/0.46 => ( P @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % not_False_in_image_Ball
% 0.30/0.46 thf(fact_99_comp__apply,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType] :
% 0.30/0.46 ( ( comp @ B @ A @ C )
% 0.30/0.46 = ( ^ [F2: B > A,G2: C > B,X3: C] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_apply
% 0.30/0.46 thf(fact_100_the__elem__eq,axiom,
% 0.30/0.46 ! [A: $tType,X: A] :
% 0.30/0.46 ( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 = X ) ).
% 0.30/0.46
% 0.30/0.46 % the_elem_eq
% 0.30/0.46 thf(fact_101_bot__apply,axiom,
% 0.30/0.46 ! [C: $tType,D2: $tType] :
% 0.30/0.46 ( ( bot @ C @ ( type2 @ C ) )
% 0.30/0.46 => ( ( bot_bot @ ( D2 > C ) )
% 0.30/0.46 = ( ^ [X3: D2] : ( bot_bot @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot_apply
% 0.30/0.46 thf(fact_102_is__singletonI,axiom,
% 0.30/0.46 ! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % is_singletonI
% 0.30/0.46 thf(fact_103_image__comp,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,R3: set @ C] :
% 0.30/0.46 ( ( image @ B @ A @ F @ ( image @ C @ B @ G @ R3 ) )
% 0.30/0.46 = ( image @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ R3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_comp
% 0.30/0.46 thf(fact_104_bot__set__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( bot_bot @ ( set @ A ) )
% 0.30/0.46 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot_set_def
% 0.30/0.46 thf(fact_105_is__singleton__the__elem,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( is_singleton @ A )
% 0.30/0.46 = ( ^ [A6: set @ A] :
% 0.30/0.46 ( A6
% 0.30/0.46 = ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % is_singleton_the_elem
% 0.30/0.46 thf(fact_106_is__singletonI_H,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] :
% 0.30/0.46 ( ( A5
% 0.30/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ! [X4: A,Y4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ A5 )
% 0.30/0.46 => ( ( member @ A @ Y4 @ A5 )
% 0.30/0.46 => ( X4 = Y4 ) ) )
% 0.30/0.46 => ( is_singleton @ A @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % is_singletonI'
% 0.30/0.46 thf(fact_107_Sup_OSUP__cong,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A5: set @ B,B2: set @ B,C4: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 0.30/0.46 ( ( A5 = B2 )
% 0.30/0.46 => ( ! [X4: B] :
% 0.30/0.46 ( ( member @ B @ X4 @ B2 )
% 0.30/0.46 => ( ( C4 @ X4 )
% 0.30/0.46 = ( D3 @ X4 ) ) )
% 0.30/0.46 => ( ( Sup @ ( image @ B @ A @ C4 @ A5 ) )
% 0.30/0.46 = ( Sup @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup.SUP_cong
% 0.30/0.46 thf(fact_108_Inf_OINF__cong,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A5: set @ B,B2: set @ B,C4: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 0.30/0.46 ( ( A5 = B2 )
% 0.30/0.46 => ( ! [X4: B] :
% 0.30/0.46 ( ( member @ B @ X4 @ B2 )
% 0.30/0.46 => ( ( C4 @ X4 )
% 0.30/0.46 = ( D3 @ X4 ) ) )
% 0.30/0.46 => ( ( Inf @ ( image @ B @ A @ C4 @ A5 ) )
% 0.30/0.46 = ( Inf @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Inf.INF_cong
% 0.30/0.46 thf(fact_109_bot__fun__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( bot @ B @ ( type2 @ B ) )
% 0.30/0.46 => ( ( bot_bot @ ( A > B ) )
% 0.30/0.46 = ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot_fun_def
% 0.30/0.46 thf(fact_110_comp__apply__eq,axiom,
% 0.30/0.46 ! [B: $tType,D2: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H: D2 > A,K: C > D2] :
% 0.30/0.46 ( ( ( F @ ( G @ X ) )
% 0.30/0.46 = ( H @ ( K @ X ) ) )
% 0.30/0.46 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 0.30/0.46 = ( comp @ D2 @ A @ C @ H @ K @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_apply_eq
% 0.30/0.46 thf(fact_111_comp__eq__dest__lhs,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,A3: C > B,B3: A > C,C3: A > B,V: A] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 0.30/0.46 = C3 )
% 0.30/0.46 => ( ( A3 @ ( B3 @ V ) )
% 0.30/0.46 = ( C3 @ V ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_eq_dest_lhs
% 0.30/0.46 thf(fact_112_comp__eq__elim,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,D2: $tType,A: $tType,A3: C > B,B3: A > C,C3: D2 > B,D: A > D2] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 0.30/0.46 = ( comp @ D2 @ B @ A @ C3 @ D ) )
% 0.30/0.46 => ! [V2: A] :
% 0.30/0.46 ( ( A3 @ ( B3 @ V2 ) )
% 0.30/0.46 = ( C3 @ ( D @ V2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_eq_elim
% 0.30/0.46 thf(fact_113_comp__eq__dest,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,D2: $tType,A: $tType,A3: C > B,B3: A > C,C3: D2 > B,D: A > D2,V: A] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 0.30/0.46 = ( comp @ D2 @ B @ A @ C3 @ D ) )
% 0.30/0.46 => ( ( A3 @ ( B3 @ V ) )
% 0.30/0.46 = ( C3 @ ( D @ V ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_eq_dest
% 0.30/0.46 thf(fact_114_comp__assoc,axiom,
% 0.30/0.46 ! [B: $tType,D2: $tType,C: $tType,A: $tType,F: D2 > B,G: C > D2,H: A > C] :
% 0.30/0.46 ( ( comp @ C @ B @ A @ ( comp @ D2 @ B @ C @ F @ G ) @ H )
% 0.30/0.46 = ( comp @ D2 @ B @ A @ F @ ( comp @ C @ D2 @ A @ G @ H ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_assoc
% 0.30/0.46 thf(fact_115_comp__def,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.46 ( ( comp @ B @ C @ A )
% 0.30/0.46 = ( ^ [F2: B > C,G2: A > B,X3: A] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_def
% 0.30/0.46 thf(fact_116_is__singletonE,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] :
% 0.30/0.46 ( ( is_singleton @ A @ A5 )
% 0.30/0.46 => ~ ! [X4: A] :
% 0.30/0.46 ( A5
% 0.30/0.46 != ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % is_singletonE
% 0.30/0.46 thf(fact_117_is__singleton__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( is_singleton @ A )
% 0.30/0.46 = ( ^ [A6: set @ A] :
% 0.30/0.46 ? [X3: A] :
% 0.30/0.46 ( A6
% 0.30/0.46 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % is_singleton_def
% 0.30/0.46 thf(fact_118_the__elem__image__unique,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A5: set @ A,F: A > B,X: A] :
% 0.30/0.46 ( ( A5
% 0.30/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ! [Y4: A] :
% 0.30/0.46 ( ( member @ A @ Y4 @ A5 )
% 0.30/0.46 => ( ( F @ Y4 )
% 0.30/0.46 = ( F @ X ) ) )
% 0.30/0.46 => ( ( the_elem @ B @ ( image @ A @ B @ F @ A5 ) )
% 0.30/0.46 = ( F @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % the_elem_image_unique
% 0.30/0.46 thf(fact_119_Sup_OSUP__image,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A5: set @ C] :
% 0.30/0.46 ( ( Sup @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A5 ) ) )
% 0.30/0.46 = ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup.SUP_image
% 0.30/0.46 thf(fact_120_Inf_OINF__image,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A5: set @ C] :
% 0.30/0.46 ( ( Inf @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A5 ) ) )
% 0.30/0.46 = ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Inf.INF_image
% 0.30/0.46 thf(fact_121_image__eq__imp__comp,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,D2: $tType,C: $tType,F: B > A,A5: set @ B,G: C > A,B2: set @ C,H: A > D2] :
% 0.30/0.46 ( ( ( image @ B @ A @ F @ A5 )
% 0.30/0.46 = ( image @ C @ A @ G @ B2 ) )
% 0.30/0.46 => ( ( image @ B @ D2 @ ( comp @ A @ D2 @ B @ H @ F ) @ A5 )
% 0.30/0.46 = ( image @ C @ D2 @ ( comp @ A @ D2 @ C @ H @ G ) @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_eq_imp_comp
% 0.30/0.46 thf(fact_122_override__on__emptyset,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/0.46 ( ( override_on @ A @ B @ F @ G @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = F ) ).
% 0.30/0.46
% 0.30/0.46 % override_on_emptyset
% 0.30/0.46 thf(fact_123_Set_Ois__empty__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( is_empty @ A )
% 0.30/0.46 = ( ^ [A6: set @ A] :
% 0.30/0.46 ( A6
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Set.is_empty_def
% 0.30/0.46 thf(fact_124_ball__reg,axiom,
% 0.30/0.46 ! [A: $tType,R: set @ A,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ R )
% 0.30/0.46 => ( ( P @ X4 )
% 0.30/0.46 => ( Q @ X4 ) ) )
% 0.30/0.46 => ( ! [X4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ R )
% 0.30/0.46 => ( P @ X4 ) )
% 0.30/0.46 => ! [X5: A] :
% 0.30/0.46 ( ( member @ A @ X5 @ R )
% 0.30/0.46 => ( Q @ X5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ball_reg
% 0.30/0.46 thf(fact_125_rewriteR__comp__comp2,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,E2: $tType,D2: $tType,A: $tType,G: C > B,H: A > C,R1: D2 > B,R22: A > D2,F: B > E2,L: D2 > E2] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ G @ H )
% 0.30/0.46 = ( comp @ D2 @ B @ A @ R1 @ R22 ) )
% 0.30/0.46 => ( ( ( comp @ B @ E2 @ D2 @ F @ R1 )
% 0.30/0.46 = L )
% 0.30/0.46 => ( ( comp @ C @ E2 @ A @ ( comp @ B @ E2 @ C @ F @ G ) @ H )
% 0.30/0.46 = ( comp @ D2 @ E2 @ A @ L @ R22 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rewriteR_comp_comp2
% 0.30/0.46 thf(fact_126_rewriteL__comp__comp2,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,D2: $tType,E2: $tType,F: C > B,G: A > C,L1: D2 > B,L2: A > D2,H: E2 > A,R3: E2 > D2] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ F @ G )
% 0.30/0.46 = ( comp @ D2 @ B @ A @ L1 @ L2 ) )
% 0.30/0.46 => ( ( ( comp @ A @ D2 @ E2 @ L2 @ H )
% 0.30/0.46 = R3 )
% 0.30/0.46 => ( ( comp @ C @ B @ E2 @ F @ ( comp @ A @ C @ E2 @ G @ H ) )
% 0.30/0.46 = ( comp @ D2 @ B @ E2 @ L1 @ R3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rewriteL_comp_comp2
% 0.30/0.46 thf(fact_127_override__on__apply__in,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A3: A,A5: set @ A,F: A > B,G: A > B] :
% 0.30/0.46 ( ( member @ A @ A3 @ A5 )
% 0.30/0.46 => ( ( override_on @ A @ B @ F @ G @ A5 @ A3 )
% 0.30/0.46 = ( G @ A3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % override_on_apply_in
% 0.30/0.46 thf(fact_128_override__on__apply__notin,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A3: A,A5: set @ A,F: A > B,G: A > B] :
% 0.30/0.46 ( ~ ( member @ A @ A3 @ A5 )
% 0.30/0.46 => ( ( override_on @ A @ B @ F @ G @ A5 @ A3 )
% 0.30/0.46 = ( F @ A3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % override_on_apply_notin
% 0.30/0.46 thf(fact_129_override__on__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( override_on @ A @ B )
% 0.30/0.46 = ( ^ [F2: A > B,G2: A > B,A6: set @ A,A4: A] : ( if @ B @ ( member @ A @ A4 @ A6 ) @ ( G2 @ A4 ) @ ( F2 @ A4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % override_on_def
% 0.30/0.46 thf(fact_130_rewriteL__comp__comp,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,D2: $tType,F: C > B,G: A > C,L: A > B,H: D2 > A] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ F @ G )
% 0.30/0.46 = L )
% 0.30/0.46 => ( ( comp @ C @ B @ D2 @ F @ ( comp @ A @ C @ D2 @ G @ H ) )
% 0.30/0.46 = ( comp @ A @ B @ D2 @ L @ H ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rewriteL_comp_comp
% 0.30/0.46 thf(fact_131_rewriteR__comp__comp,axiom,
% 0.30/0.46 ! [C: $tType,D2: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R3: A > B,F: B > D2] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ G @ H )
% 0.30/0.46 = R3 )
% 0.30/0.46 => ( ( comp @ C @ D2 @ A @ ( comp @ B @ D2 @ C @ F @ G ) @ H )
% 0.30/0.46 = ( comp @ B @ D2 @ A @ F @ R3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rewriteR_comp_comp
% 0.30/0.46 thf(fact_132_Collect__empty__eq__bot,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o] :
% 0.30/0.46 ( ( ( collect @ A @ P )
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = ( P
% 0.30/0.46 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_empty_eq_bot
% 0.30/0.46 thf(fact_133_bot__empty__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( bot_bot @ ( A > $o ) )
% 0.30/0.46 = ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot_empty_eq
% 0.30/0.46 thf(fact_134_comp__cong,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,D2: $tType,A: $tType,E2: $tType,F: B > A,G: C > B,X: C,F3: D2 > A,G3: E2 > D2,X6: E2] :
% 0.30/0.46 ( ( ( F @ ( G @ X ) )
% 0.30/0.46 = ( F3 @ ( G3 @ X6 ) ) )
% 0.30/0.46 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 0.30/0.46 = ( comp @ D2 @ A @ E2 @ F3 @ G3 @ X6 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_cong
% 0.30/0.46 thf(fact_135_fun_Omap__comp,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,D2: $tType,G: B > C,F: A > B,V: D2 > A] :
% 0.30/0.46 ( ( comp @ B @ C @ D2 @ G @ ( comp @ A @ B @ D2 @ F @ V ) )
% 0.30/0.46 = ( comp @ A @ C @ D2 @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.map_comp
% 0.30/0.46 thf(fact_136_type__copy__map__cong0,axiom,
% 0.30/0.46 ! [B: $tType,D2: $tType,E2: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X: C,N: D2 > A,H: C > D2,F: A > E2] :
% 0.30/0.46 ( ( ( M @ ( G @ X ) )
% 0.30/0.46 = ( N @ ( H @ X ) ) )
% 0.30/0.46 => ( ( comp @ B @ E2 @ C @ ( comp @ A @ E2 @ B @ F @ M ) @ G @ X )
% 0.30/0.46 = ( comp @ D2 @ E2 @ C @ ( comp @ A @ E2 @ D2 @ F @ N ) @ H @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_copy_map_cong0
% 0.30/0.46 thf(fact_137_range__eq__singletonD,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: B > A,A3: A,X: B] :
% 0.30/0.46 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 => ( ( F @ X )
% 0.30/0.46 = A3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % range_eq_singletonD
% 0.30/0.46 thf(fact_138_singleton__insert__inj__eq_H,axiom,
% 0.30/0.46 ! [A: $tType,A3: A,A5: set @ A,B3: A] :
% 0.30/0.46 ( ( ( insert @ A @ A3 @ A5 )
% 0.30/0.46 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 = ( ( A3 = B3 )
% 0.30/0.46 & ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % singleton_insert_inj_eq'
% 0.30/0.46 thf(fact_139_singleton__insert__inj__eq,axiom,
% 0.30/0.46 ! [A: $tType,B3: A,A3: A,A5: set @ A] :
% 0.30/0.46 ( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = ( insert @ A @ A3 @ A5 ) )
% 0.30/0.46 = ( ( A3 = B3 )
% 0.30/0.46 & ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % singleton_insert_inj_eq
% 0.30/0.46 thf(fact_140_order__refl,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_refl
% 0.30/0.46 thf(fact_141_subset__antisym,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A5 )
% 0.30/0.46 => ( A5 = B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_antisym
% 0.30/0.46 thf(fact_142_subsetI,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ! [X4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ A5 )
% 0.30/0.46 => ( member @ A @ X4 @ B2 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % subsetI
% 0.30/0.46 thf(fact_143_top__apply,axiom,
% 0.30/0.46 ! [C: $tType,D2: $tType] :
% 0.30/0.46 ( ( top @ C @ ( type2 @ C ) )
% 0.30/0.46 => ( ( top_top @ ( D2 > C ) )
% 0.30/0.46 = ( ^ [X3: D2] : ( top_top @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top_apply
% 0.30/0.46 thf(fact_144_UNIV__I,axiom,
% 0.30/0.46 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UNIV_I
% 0.30/0.46 thf(fact_145_subset__empty,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = ( A5
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_empty
% 0.30/0.46 thf(fact_146_empty__subsetI,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% 0.30/0.46
% 0.30/0.46 % empty_subsetI
% 0.30/0.46 thf(fact_147_insert__subset,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ B2 )
% 0.30/0.46 = ( ( member @ A @ X @ B2 )
% 0.30/0.46 & ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_subset
% 0.30/0.46 thf(fact_148_top_Oextremum__uniqueI,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order_top @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.30/0.46 => ( A3
% 0.30/0.46 = ( top_top @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top.extremum_uniqueI
% 0.30/0.46 thf(fact_149_top_Oextremum__unique,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order_top @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.30/0.46 = ( A3
% 0.30/0.46 = ( top_top @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top.extremum_unique
% 0.30/0.46 thf(fact_150_dual__order_Oantisym,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [B3: A,A3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.46 => ( A3 = B3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.antisym
% 0.30/0.46 thf(fact_151_top__greatest,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order_top @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top_greatest
% 0.30/0.46 thf(fact_152_dual__order_Otrans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [B3: A,A3: A,C3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ C3 @ B3 )
% 0.30/0.46 => ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.trans
% 0.30/0.46 thf(fact_153_linorder__wlog,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [P: A > A > $o,A3: A,B3: A] :
% 0.30/0.46 ( ! [A7: A,B6: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A7 @ B6 )
% 0.30/0.46 => ( P @ A7 @ B6 ) )
% 0.30/0.46 => ( ! [A7: A,B6: A] :
% 0.30/0.46 ( ( P @ B6 @ A7 )
% 0.30/0.46 => ( P @ A7 @ B6 ) )
% 0.30/0.46 => ( P @ A3 @ B3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % linorder_wlog
% 0.30/0.46 thf(fact_154_dual__order_Orefl,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.refl
% 0.30/0.46 thf(fact_155_order__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A,Z3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.46 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.30/0.46 => ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_trans
% 0.30/0.46 thf(fact_156_order__class_Oorder_Oantisym,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A,B3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.30/0.46 => ( A3 = B3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_class.order.antisym
% 0.30/0.46 thf(fact_157_ord__le__eq__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A,B3: A,C3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.46 => ( ( B3 = C3 )
% 0.30/0.46 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_le_eq_trans
% 0.30/0.46 thf(fact_158_ord__eq__le__trans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A,B3: A,C3: A] :
% 0.30/0.46 ( ( A3 = B3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 0.30/0.46 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_eq_le_trans
% 0.30/0.46 thf(fact_159_antisym__conv,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [Y: A,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.46 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.46 = ( X = Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % antisym_conv
% 0.30/0.46 thf(fact_160_le__cases3,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A,Z3: A] :
% 0.30/0.46 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 0.30/0.46 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ X @ Z3 ) )
% 0.30/0.46 => ( ( ( ord_less_eq @ A @ X @ Z3 )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 0.30/0.46 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.30/0.46 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ Z3 @ X ) )
% 0.30/0.46 => ~ ( ( ord_less_eq @ A @ Z3 @ X )
% 0.30/0.46 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_cases3
% 0.30/0.46 thf(fact_161_order_Otrans,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A,B3: A,C3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 0.30/0.46 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order.trans
% 0.30/0.46 thf(fact_162_le__cases,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.30/0.46 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_cases
% 0.30/0.46 thf(fact_163_eq__refl,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( X = Y )
% 0.30/0.46 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_refl
% 0.30/0.46 thf(fact_164_linear,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.46 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % linear
% 0.30/0.46 thf(fact_165_antisym,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.46 => ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.46 => ( X = Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % antisym
% 0.30/0.46 thf(fact_166_eq__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.30/0.46 = ( ^ [X3: A,Y3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.46 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_iff
% 0.30/0.46 thf(fact_167_ord__le__eq__subst,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.46 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.46 => ! [A3: A,B3: A,F: A > B,C3: B] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.46 => ( ( ( F @ B3 )
% 0.30/0.46 = C3 )
% 0.30/0.46 => ( ! [X4: A,Y4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_le_eq_subst
% 0.30/0.46 thf(fact_168_ord__eq__le__subst,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.46 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.46 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 0.30/0.46 ( ( A3
% 0.30/0.46 = ( F @ B3 ) )
% 0.30/0.46 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 0.30/0.46 => ( ! [X4: B,Y4: B] :
% 0.30/0.46 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.30/0.46 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_eq_le_subst
% 0.30/0.46 thf(fact_169_order__subst2,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType] :
% 0.30/0.46 ( ( ( order @ C @ ( type2 @ C ) )
% 0.30/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.46 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.46 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
% 0.30/0.46 => ( ! [X4: A,Y4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.30/0.46 => ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.30/0.46 => ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_subst2
% 0.30/0.46 thf(fact_170_order__subst1,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( order @ B @ ( type2 @ B ) )
% 0.30/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.46 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.30/0.46 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 0.30/0.46 => ( ! [X4: B,Y4: B] :
% 0.30/0.46 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.30/0.46 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_subst1
% 0.30/0.46 thf(fact_171_le__fun__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.30/0.46 = ( ^ [F2: A > B,G2: A > B] :
% 0.30/0.46 ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_fun_def
% 0.30/0.46 thf(fact_172_le__funI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.46 => ! [F: A > B,G: A > B] :
% 0.30/0.46 ( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.30/0.46 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_funI
% 0.30/0.46 thf(fact_173_le__funE,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.46 => ! [F: A > B,G: A > B,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_funE
% 0.30/0.46 thf(fact_174_le__funD,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.46 => ! [F: A > B,G: A > B,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_funD
% 0.30/0.46 thf(fact_175_Collect__mono__iff,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.30/0.46 = ( ! [X3: A] :
% 0.30/0.46 ( ( P @ X3 )
% 0.30/0.46 => ( Q @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mono_iff
% 0.30/0.46 thf(fact_176_contra__subsetD,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A,C3: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ~ ( member @ A @ C3 @ B2 )
% 0.30/0.46 => ~ ( member @ A @ C3 @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % contra_subsetD
% 0.30/0.46 thf(fact_177_set__eq__subset,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ^ [Y5: set @ A,Z4: set @ A] : ( Y5 = Z4 ) )
% 0.30/0.46 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.46 & ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_eq_subset
% 0.30/0.46 thf(fact_178_subset__trans,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A,C4: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C4 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_trans
% 0.30/0.46 thf(fact_179_UNIV__witness,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ? [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UNIV_witness
% 0.30/0.46 thf(fact_180_Collect__mono,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X4: A] :
% 0.30/0.46 ( ( P @ X4 )
% 0.30/0.46 => ( Q @ X4 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mono
% 0.30/0.46 thf(fact_181_subset__refl,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% 0.30/0.46
% 0.30/0.46 % subset_refl
% 0.30/0.46 thf(fact_182_subset__UNIV,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_UNIV
% 0.30/0.46 thf(fact_183_rev__subsetD,axiom,
% 0.30/0.46 ! [A: $tType,C3: A,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( member @ A @ C3 @ A5 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( member @ A @ C3 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rev_subsetD
% 0.30/0.46 thf(fact_184_subset__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.30/0.46 ! [T: A] :
% 0.30/0.46 ( ( member @ A @ T @ A6 )
% 0.30/0.46 => ( member @ A @ T @ B7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_iff
% 0.30/0.46 thf(fact_185_set__rev__mp,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( member @ A @ X @ A5 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( member @ A @ X @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_rev_mp
% 0.30/0.46 thf(fact_186_equalityD2,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( A5 = B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ B2 @ A5 ) ) ).
% 0.30/0.46
% 0.30/0.46 % equalityD2
% 0.30/0.46 thf(fact_187_equalityD1,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( A5 = B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % equalityD1
% 0.30/0.46 thf(fact_188_subset__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.30/0.46 ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ A6 )
% 0.30/0.46 => ( member @ A @ X3 @ B7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_eq
% 0.30/0.46 thf(fact_189_equalityE,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( A5 = B2 )
% 0.30/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % equalityE
% 0.30/0.46 thf(fact_190_UNIV__eq__I,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] :
% 0.30/0.46 ( ! [X4: A] : ( member @ A @ X4 @ A5 )
% 0.30/0.46 => ( ( top_top @ ( set @ A ) )
% 0.30/0.46 = A5 ) ) ).
% 0.30/0.46
% 0.30/0.46 % UNIV_eq_I
% 0.30/0.46 thf(fact_191_subsetCE,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A,C3: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ( member @ A @ C3 @ A5 )
% 0.30/0.46 => ( member @ A @ C3 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subsetCE
% 0.30/0.46 thf(fact_192_subsetD,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A,C3: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ( member @ A @ C3 @ A5 )
% 0.30/0.46 => ( member @ A @ C3 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subsetD
% 0.30/0.46 thf(fact_193_in__mono,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ( member @ A @ X @ A5 )
% 0.30/0.46 => ( member @ A @ X @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % in_mono
% 0.30/0.46 thf(fact_194_set__mp,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ( member @ A @ X @ A5 )
% 0.30/0.46 => ( member @ A @ X @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_mp
% 0.30/0.46 thf(fact_195_insert__UNIV,axiom,
% 0.30/0.46 ! [A: $tType,X: A] :
% 0.30/0.46 ( ( insert @ A @ X @ ( top_top @ ( set @ A ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_UNIV
% 0.30/0.46 thf(fact_196_Ball__Collect,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ball @ A )
% 0.30/0.46 = ( ^ [A6: set @ A,P2: A > $o] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( collect @ A @ P2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Ball_Collect
% 0.30/0.46 thf(fact_197_subset__insertI2,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A,B3: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_insertI2
% 0.30/0.46 thf(fact_198_subset__insertI,axiom,
% 0.30/0.46 ! [A: $tType,B2: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_insertI
% 0.30/0.46 thf(fact_199_subset__insert,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ~ ( member @ A @ X @ A5 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B2 ) )
% 0.30/0.46 = ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_insert
% 0.30/0.46 thf(fact_200_insert__mono,axiom,
% 0.30/0.46 ! [A: $tType,C4: set @ A,D3: set @ A,A3: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C4 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_mono
% 0.30/0.46 thf(fact_201_empty__not__UNIV,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( bot_bot @ ( set @ A ) )
% 0.30/0.46 != ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % empty_not_UNIV
% 0.30/0.46 thf(fact_202_surjD,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 0.30/0.46 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) )
% 0.30/0.46 => ? [X4: B] :
% 0.30/0.46 ( Y
% 0.30/0.46 = ( F @ X4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surjD
% 0.30/0.46 thf(fact_203_surjE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 0.30/0.46 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) )
% 0.30/0.46 => ~ ! [X4: B] :
% 0.30/0.46 ( Y
% 0.30/0.46 != ( F @ X4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surjE
% 0.30/0.46 thf(fact_204_surjI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 0.30/0.46 ( ! [X4: A] :
% 0.30/0.46 ( ( G @ ( F @ X4 ) )
% 0.30/0.46 = X4 )
% 0.30/0.46 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surjI
% 0.30/0.46 thf(fact_205_surj__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: B > A] :
% 0.30/0.46 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) )
% 0.30/0.46 = ( ! [Y3: A] :
% 0.30/0.46 ? [X3: B] :
% 0.30/0.46 ( Y3
% 0.30/0.46 = ( F @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surj_def
% 0.30/0.46 thf(fact_206_rangeI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rangeI
% 0.30/0.46 thf(fact_207_range__eqI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B3: A,F: B > A,X: B] :
% 0.30/0.46 ( ( B3
% 0.30/0.46 = ( F @ X ) )
% 0.30/0.46 => ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % range_eqI
% 0.30/0.46 thf(fact_208_image__mono,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A5: set @ A,B2: set @ A,F: A > B] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A5 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_mono
% 0.30/0.46 thf(fact_209_image__subsetI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A5: set @ A,F: A > B,B2: set @ B] :
% 0.30/0.46 ( ! [X4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ A5 )
% 0.30/0.46 => ( member @ B @ ( F @ X4 ) @ B2 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A5 ) @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_subsetI
% 0.30/0.46 thf(fact_210_subset__imageE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A5: set @ B] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A5 ) )
% 0.30/0.46 => ~ ! [C5: set @ B] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A5 )
% 0.30/0.46 => ( B2
% 0.30/0.46 != ( image @ B @ A @ F @ C5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_imageE
% 0.30/0.46 thf(fact_211_image__subset__iff,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,A5: set @ B,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A5 ) @ B2 )
% 0.30/0.46 = ( ! [X3: B] :
% 0.30/0.46 ( ( member @ B @ X3 @ A5 )
% 0.30/0.46 => ( member @ A @ ( F @ X3 ) @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_subset_iff
% 0.30/0.46 thf(fact_212_subset__image__iff,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A5: set @ B] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A5 ) )
% 0.30/0.46 = ( ? [AA: set @ B] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ B ) @ AA @ A5 )
% 0.30/0.46 & ( B2
% 0.30/0.46 = ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_image_iff
% 0.30/0.46 thf(fact_213_bot_Oextremum__uniqueI,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.30/0.46 => ( A3
% 0.30/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot.extremum_uniqueI
% 0.30/0.46 thf(fact_214_bot_Oextremum__unique,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.30/0.46 = ( A3
% 0.30/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot.extremum_unique
% 0.30/0.46 thf(fact_215_bot_Oextremum,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot.extremum
% 0.30/0.46 thf(fact_216_pairwise__subset,axiom,
% 0.30/0.46 ! [A: $tType,P: A > A > $o,S4: set @ A,T2: set @ A] :
% 0.30/0.46 ( ( pairwise @ A @ P @ S4 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ T2 @ S4 )
% 0.30/0.46 => ( pairwise @ A @ P @ T2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pairwise_subset
% 0.30/0.46 thf(fact_217_subset__singletonD,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 => ( ( A5
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 | ( A5
% 0.30/0.46 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_singletonD
% 0.30/0.46 thf(fact_218_subset__singleton__iff,axiom,
% 0.30/0.46 ! [A: $tType,X7: set @ A,A3: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ X7 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 = ( ( X7
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 | ( X7
% 0.30/0.46 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_singleton_iff
% 0.30/0.46 thf(fact_219_comp__surj,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: A > C] :
% 0.30/0.46 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) )
% 0.30/0.46 => ( ( ( image @ A @ C @ G @ ( top_top @ ( set @ A ) ) )
% 0.30/0.46 = ( top_top @ ( set @ C ) ) )
% 0.30/0.46 => ( ( image @ B @ C @ ( comp @ A @ C @ B @ G @ F ) @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_surj
% 0.30/0.46 thf(fact_220_fun_Oinj__map__strong,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,D2: $tType,X: D2 > A,Xa2: D2 > A,F: A > B,Fa: A > B] :
% 0.30/0.46 ( ! [Z: A,Za: A] :
% 0.30/0.46 ( ( member @ A @ Z @ ( image @ D2 @ A @ X @ ( top_top @ ( set @ D2 ) ) ) )
% 0.30/0.46 => ( ( member @ A @ Za @ ( image @ D2 @ A @ Xa2 @ ( top_top @ ( set @ D2 ) ) ) )
% 0.30/0.46 => ( ( ( F @ Z )
% 0.30/0.46 = ( Fa @ Za ) )
% 0.30/0.46 => ( Z = Za ) ) ) )
% 0.30/0.46 => ( ( ( comp @ A @ B @ D2 @ F @ X )
% 0.30/0.46 = ( comp @ A @ B @ D2 @ Fa @ Xa2 ) )
% 0.30/0.46 => ( X = Xa2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.inj_map_strong
% 0.30/0.46 thf(fact_221_fun_Omap__cong0,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,D2: $tType,X: D2 > A,F: A > B,G: A > B] :
% 0.30/0.46 ( ! [Z: A] :
% 0.30/0.46 ( ( member @ A @ Z @ ( image @ D2 @ A @ X @ ( top_top @ ( set @ D2 ) ) ) )
% 0.30/0.46 => ( ( F @ Z )
% 0.30/0.46 = ( G @ Z ) ) )
% 0.30/0.46 => ( ( comp @ A @ B @ D2 @ F @ X )
% 0.30/0.46 = ( comp @ A @ B @ D2 @ G @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.map_cong0
% 0.30/0.46 thf(fact_222_fun_Omap__cong,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,D2: $tType,X: D2 > A,Ya: D2 > A,F: A > B,G: A > B] :
% 0.30/0.46 ( ( X = Ya )
% 0.30/0.46 => ( ! [Z: A] :
% 0.30/0.46 ( ( member @ A @ Z @ ( image @ D2 @ A @ Ya @ ( top_top @ ( set @ D2 ) ) ) )
% 0.30/0.46 => ( ( F @ Z )
% 0.30/0.46 = ( G @ Z ) ) )
% 0.30/0.46 => ( ( comp @ A @ B @ D2 @ F @ X )
% 0.30/0.46 = ( comp @ A @ B @ D2 @ G @ Ya ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.map_cong
% 0.30/0.46 thf(fact_223_fun_Oset__map,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,D2: $tType,F: A > B,V: D2 > A] :
% 0.30/0.46 ( ( image @ D2 @ B @ ( comp @ A @ B @ D2 @ F @ V ) @ ( top_top @ ( set @ D2 ) ) )
% 0.30/0.46 = ( image @ A @ B @ F @ ( image @ D2 @ A @ V @ ( top_top @ ( set @ D2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.set_map
% 0.30/0.46 thf(fact_224_surj__fun__eq,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,F: B > A,X7: set @ B,G12: A > C,G23: A > C] :
% 0.30/0.46 ( ( ( image @ B @ A @ F @ X7 )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) )
% 0.30/0.46 => ( ! [X4: B] :
% 0.30/0.46 ( ( member @ B @ X4 @ X7 )
% 0.30/0.46 => ( ( comp @ A @ C @ B @ G12 @ F @ X4 )
% 0.30/0.46 = ( comp @ A @ C @ B @ G23 @ F @ X4 ) ) )
% 0.30/0.46 => ( G12 = G23 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surj_fun_eq
% 0.30/0.46 thf(fact_225_insert__subsetI,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A5: set @ A,X7: set @ A] :
% 0.30/0.46 ( ( member @ A @ X @ A5 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ X7 @ A5 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X7 ) @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_subsetI
% 0.30/0.46 thf(fact_226_subset__emptyI,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] :
% 0.30/0.46 ( ! [X4: A] :
% 0.30/0.46 ~ ( member @ A @ X4 @ A5 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_emptyI
% 0.30/0.46 thf(fact_227_subset__Compl__singleton,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B3: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.46 = ( ~ ( member @ A @ B3 @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_Compl_singleton
% 0.30/0.46 thf(fact_228_ComplI,axiom,
% 0.30/0.46 ! [A: $tType,C3: A,A5: set @ A] :
% 0.30/0.46 ( ~ ( member @ A @ C3 @ A5 )
% 0.30/0.46 => ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ComplI
% 0.30/0.46 thf(fact_229_Compl__iff,axiom,
% 0.30/0.46 ! [A: $tType,C3: A,A5: set @ A] :
% 0.30/0.46 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
% 0.30/0.46 = ( ~ ( member @ A @ C3 @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Compl_iff
% 0.30/0.46 thf(fact_230_Compl__eq__Compl__iff,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ( uminus_uminus @ ( set @ A ) @ A5 )
% 0.30/0.46 = ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 0.30/0.46 = ( A5 = B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Compl_eq_Compl_iff
% 0.30/0.46 thf(fact_231_Compl__subset__Compl__iff,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 0.30/0.46 = ( ord_less_eq @ ( set @ A ) @ B2 @ A5 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Compl_subset_Compl_iff
% 0.30/0.46 thf(fact_232_Compl__anti__mono,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Compl_anti_mono
% 0.30/0.46 thf(fact_233_top__set__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( top_top @ ( set @ A ) )
% 0.30/0.46 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top_set_def
% 0.30/0.46 thf(fact_234_stream_Orel__mono,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o,Ra: A > B > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra )
% 0.30/0.46 => ( ord_less_eq @ ( ( stream170649215stream @ A ) > ( stream170649215stream @ B ) > $o ) @ ( stream686525009m_all2 @ A @ B @ R ) @ ( stream686525009m_all2 @ A @ B @ Ra ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % stream.rel_mono
% 0.30/0.46 thf(fact_235_ComplD,axiom,
% 0.30/0.46 ! [A: $tType,C3: A,A5: set @ A] :
% 0.30/0.46 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
% 0.30/0.46 => ~ ( member @ A @ C3 @ A5 ) ) ).
% 0.30/0.46
% 0.30/0.46 % ComplD
% 0.30/0.46 thf(fact_236_double__complement,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] :
% 0.30/0.46 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
% 0.30/0.46 = A5 ) ).
% 0.30/0.46
% 0.30/0.46 % double_complement
% 0.30/0.46 thf(fact_237_subset__Compl__self__eq,axiom,
% 0.30/0.46 ! [A: $tType,A5: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
% 0.30/0.46 = ( A5
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_Compl_self_eq
% 0.30/0.46 thf(fact_238_Compl__empty__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Compl_empty_eq
% 0.30/0.46 thf(fact_239_Compl__UNIV__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Compl_UNIV_eq
% 0.30/0.46 thf(fact_240_surj__Compl__image__subset,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
% 0.30/0.46 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F @ A5 ) ) @ ( image @ B @ A @ F @ ( uminus_uminus @ ( set @ B ) @ A5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surj_Compl_image_subset
% 0.30/0.46 thf(fact_241_compl__top__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ( uminus_uminus @ A @ ( top_top @ A ) )
% 0.30/0.46 = ( bot_bot @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % compl_top_eq
% 0.30/0.46 thf(fact_242_compl__bot__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
% 0.30/0.46 = ( top_top @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % compl_bot_eq
% 0.30/0.46 thf(fact_243_predicate2I,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 0.30/0.46 ( ! [X4: A,Y4: B] :
% 0.30/0.46 ( ( P @ X4 @ Y4 )
% 0.30/0.46 => ( Q @ X4 @ Y4 ) )
% 0.30/0.46 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 0.30/0.46
% 0.30/0.46 % predicate2I
% 0.30/0.46 thf(fact_244_top1I,axiom,
% 0.30/0.46 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 0.30/0.46
% 0.30/0.46 % top1I
% 0.30/0.46 thf(fact_245_rev__predicate2D,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
% 0.30/0.46 ( ( P @ X @ Y )
% 0.30/0.46 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.30/0.46 => ( Q @ X @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rev_predicate2D
% 0.30/0.46 thf(fact_246_predicate2D,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.30/0.46 => ( ( P @ X @ Y )
% 0.30/0.46 => ( Q @ X @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % predicate2D
% 0.30/0.46 thf(fact_247_refl__ge__eq,axiom,
% 0.30/0.46 ! [A: $tType,R: A > A > $o] :
% 0.30/0.46 ( ! [X4: A] : ( R @ X4 @ X4 )
% 0.30/0.46 => ( ord_less_eq @ ( A > A > $o )
% 0.30/0.46 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 0.30/0.46 @ R ) ) ).
% 0.30/0.46
% 0.30/0.46 % refl_ge_eq
% 0.30/0.46 thf(fact_248_ge__eq__refl,axiom,
% 0.30/0.46 ! [A: $tType,R: A > A > $o,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > A > $o )
% 0.30/0.46 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 0.30/0.46 @ R )
% 0.30/0.46 => ( R @ X @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % ge_eq_refl
% 0.30/0.46 thf(fact_249_predicate2D__conj,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X: A,Y: B] :
% 0.30/0.46 ( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.30/0.46 & R )
% 0.30/0.46 => ( R
% 0.30/0.46 & ( ( P @ X @ Y )
% 0.30/0.46 => ( Q @ X @ Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % predicate2D_conj
% 0.30/0.46 thf(fact_250_top__conj_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,X: A,P: $o] :
% 0.30/0.46 ( ( ( top_top @ ( A > $o ) @ X )
% 0.30/0.46 & P )
% 0.30/0.46 = P ) ).
% 0.30/0.46
% 0.30/0.46 % top_conj(1)
% 0.30/0.46 thf(fact_251_top__conj_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,P: $o,X: A] :
% 0.30/0.46 ( ( P
% 0.30/0.46 & ( top_top @ ( A > $o ) @ X ) )
% 0.30/0.46 = P ) ).
% 0.30/0.46
% 0.30/0.46 % top_conj(2)
% 0.30/0.46 thf(fact_252_inj__image__Compl__subset,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
% 0.30/0.46 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F @ A5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inj_image_Compl_subset
% 0.30/0.46 thf(fact_253_ID_Opred__set,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( bNF_id_bnf @ ( A > $o ) )
% 0.30/0.46 = ( ^ [P2: A > $o,X3: A] :
% 0.30/0.46 ! [Y3: A] :
% 0.30/0.46 ( ( member @ A @ Y3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 => ( P2 @ Y3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ID.pred_set
% 0.30/0.46 thf(fact_254_inj__on__empty,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B] : ( inj_on @ A @ B @ F @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inj_on_empty
% 0.30/0.46 thf(fact_255_inj__uminus,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A5: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A5 ) ) ).
% 0.30/0.46
% 0.30/0.46 % inj_uminus
% 0.30/0.46
% 0.30/0.46 %----Type constructors (25)
% 0.30/0.46 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 0.30/0.46 ! [A8: $tType,A9: $tType] :
% 0.30/0.46 ( ( boolean_algebra @ A9 @ ( type2 @ A9 ) )
% 0.30/0.46 => ( boolean_algebra @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.30/0.46 ! [A8: $tType,A9: $tType] :
% 0.30/0.46 ( ( order_top @ A9 @ ( type2 @ A9 ) )
% 0.30/0.46 => ( order_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.30/0.46 ! [A8: $tType,A9: $tType] :
% 0.30/0.46 ( ( order_bot @ A9 @ ( type2 @ A9 ) )
% 0.30/0.46 => ( order_bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/0.46 ! [A8: $tType,A9: $tType] :
% 0.30/0.46 ( ( preorder @ A9 @ ( type2 @ A9 ) )
% 0.30/0.46 => ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/0.46 ! [A8: $tType,A9: $tType] :
% 0.30/0.46 ( ( order @ A9 @ ( type2 @ A9 ) )
% 0.30/0.46 => ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Otop,axiom,
% 0.30/0.46 ! [A8: $tType,A9: $tType] :
% 0.30/0.46 ( ( top @ A9 @ ( type2 @ A9 ) )
% 0.30/0.46 => ( top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/0.46 ! [A8: $tType,A9: $tType] :
% 0.30/0.46 ( ( ord @ A9 @ ( type2 @ A9 ) )
% 0.30/0.46 => ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Obot,axiom,
% 0.30/0.46 ! [A8: $tType,A9: $tType] :
% 0.30/0.46 ( ( bot @ A9 @ ( type2 @ A9 ) )
% 0.30/0.46 => ( bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_1,axiom,
% 0.30/0.46 ! [A8: $tType] : ( boolean_algebra @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Oorder__top_2,axiom,
% 0.30/0.46 ! [A8: $tType] : ( order_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Oorder__bot_3,axiom,
% 0.30/0.46 ! [A8: $tType] : ( order_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.30/0.46 ! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 0.30/0.46 ! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Otop_6,axiom,
% 0.30/0.46 ! [A8: $tType] : ( top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 0.30/0.46 ! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Obot_8,axiom,
% 0.30/0.46 ! [A8: $tType] : ( bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_9,axiom,
% 0.30/0.46 boolean_algebra @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__top_10,axiom,
% 0.30/0.46 order_top @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__bot_11,axiom,
% 0.30/0.46 order_bot @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
% 0.30/0.46 preorder @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.30/0.46 linorder @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_13,axiom,
% 0.30/0.46 order @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Otop_14,axiom,
% 0.30/0.46 top @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
% 0.30/0.46 ord @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
% 0.30/0.46 bot @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 %----Helper facts (3)
% 0.30/0.46 thf(help_If_3_1_T,axiom,
% 0.30/0.46 ! [P: $o] :
% 0.30/0.46 ( ( P = $true )
% 0.30/0.46 | ( P = $false ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(help_If_2_1_T,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Y: A] :
% 0.30/0.46 ( ( if @ A @ $false @ X @ Y )
% 0.30/0.46 = Y ) ).
% 0.30/0.46
% 0.30/0.46 thf(help_If_1_1_T,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Y: A] :
% 0.30/0.46 ( ( if @ A @ $true @ X @ Y )
% 0.30/0.46 = X ) ).
% 0.30/0.46
% 0.30/0.46 %----Conjectures (1)
% 0.30/0.46 thf(conj_0,conjecture,
% 0.30/0.46 ( ( member @ a @ x @ ( stream30925839e_sset @ a @ ( stream641971652_SCons @ a @ y @ s ) ) )
% 0.30/0.46 != ( ~ ( ( x != y )
% 0.30/0.46 => ( member @ a @ x @ ( stream30925839e_sset @ a @ s ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 2.66/1.46 ---- Embedded in THF ---
% 2.66/1.46 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.66/1.46 %%% Generated on Tue Feb 24 06:11:24 EST 2026
% 2.66/1.46 %%% using '$$dhol' embedding, version 1.3.0.
% 2.66/1.46 %%% Logic specification used:
% 2.66/1.46 %%% thf(spec, logic, $$dhol).
% 2.66/1.46
% 2.66/1.46 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.e7YR8rdmNP/DTF2DTF22005.p
% 2.66/1.46 thf(ty_t_Stream__Mirabelle__hbrgyiwlrc_Ostream, type, stream170649215stream: ($tType > $tType)).
% 2.66/1.46 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.66/1.46 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.66/1.46 thf(ty_tf_a, type, a: $tType).
% 2.66/1.46 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.66/1.46 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.66/1.46 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.66/1.46 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.66/1.46 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.66/1.46 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.66/1.46 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.66/1.46 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.66/1.46 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.66/1.46 thf(sy_cl_Lattices_Oboolean__algebra, type, boolean_algebra: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.66/1.46 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.66/1.46 thf(sy_c_BNF__Composition_Oid__bnf, type, bNF_id_bnf: (!> [A:$tType]: ((A > A)))).
% 2.66/1.46 thf(sy_c_Fun_Ocomp, type, comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > B) > (A > C)))))).
% 2.66/1.46 thf(sy_c_Fun_Oinj__on, type, inj_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 2.66/1.46 thf(sy_c_Fun_Ooverride__on, type, override_on: (!> [A:$tType,B:$tType]: (((A > B) > ((A > B) > ((set @ A) > (A > B))))))).
% 2.66/1.46 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.66/1.46 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.66/1.46 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.66/1.46 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.66/1.46 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.66/1.46 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.66/1.46 thf(sy_c_Set_OBall, type, ball: (!> [A:$tType]: (((set @ A) > ((A > $o) > $o))))).
% 2.66/1.46 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.66/1.46 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.66/1.46 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.66/1.46 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.66/1.46 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.66/1.46 thf(sy_c_Set_Opairwise, type, pairwise: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > $o))))).
% 2.66/1.46 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 2.66/1.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Osmember, type, stream1586597341member: (!> [A:$tType]: ((A > ((stream170649215stream @ A) > $o))))).
% 2.66/1.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_OSCons, type, stream641971652_SCons: (!> [A:$tType]: ((A > ((stream170649215stream @ A) > (stream170649215stream @ A)))))).
% 2.66/1.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Ocase__stream, type, stream1342653232stream: (!> [A:$tType,B:$tType]: (((A > ((stream170649215stream @ A) > B)) > ((stream170649215stream @ A) > B))))).
% 2.66/1.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Ocorec__stream, type, stream660621732stream: (!> [C:$tType,A:$tType]: (((C > A) > ((C > $o) > ((C > (stream170649215stream @ A)) > ((C > C) > (C > (stream170649215stream @ A))))))))).
% 2.66/1.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Opred__stream, type, stream1153105665stream: (!> [A:$tType]: (((A > $o) > ((stream170649215stream @ A) > $o))))).
% 2.66/1.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Oshd, type, stream_Mirabelle_shd: (!> [A:$tType]: (((stream170649215stream @ A) > A)))).
% 2.66/1.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Osmap, type, stream2128578057e_smap: (!> [A:$tType,Aa:$tType]: (((A > Aa) > ((stream170649215stream @ A) > (stream170649215stream @ Aa)))))).
% 2.66/1.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Osset, type, stream30925839e_sset: (!> [A:$tType]: (((stream170649215stream @ A) > (set @ A))))).
% 2.66/1.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Ostl, type, stream_Mirabelle_stl: (!> [A:$tType]: (((stream170649215stream @ A) > (stream170649215stream @ A))))).
% 2.66/1.46 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Ostream__all2, type, stream686525009m_all2: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((stream170649215stream @ A) > ((stream170649215stream @ B) > $o)))))).
% 2.66/1.46 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.66/1.46 thf(sy_v_s, type, s: (stream170649215stream @ a)).
% 2.66/1.46 thf(sy_v_x, type, x: a).
% 2.66/1.46 thf(sy_v_y, type, y: a).
% 2.66/1.46 thf(fact_0_stream_Oinject, axiom, (! [A:$tType,X1:A,X2:(stream170649215stream @ A),Y1:A,Y2:(stream170649215stream @ A)]: ((((((stream641971652_SCons @ A) @ X1) @ X2) = (((stream641971652_SCons @ A) @ Y1) @ Y2)) = ((X1 = Y1) & (X2 = Y2)))))).
% 2.66/1.46 thf(fact_1_stream_Oset__intros_I2_J, axiom, (! [A:$tType,X:A,A2:(stream170649215stream @ A),A1:A]: (((((member @ A) @ X) @ ((stream30925839e_sset @ A) @ A2)) => (((member @ A) @ X) @ ((stream30925839e_sset @ A) @ (((stream641971652_SCons @ A) @ A1) @ A2))))))).
% 2.66/1.46 thf(fact_2_stream_Oset__intros_I1_J, axiom, (! [A:$tType,A1:A,A2:(stream170649215stream @ A)]: ((((member @ A) @ A1) @ ((stream30925839e_sset @ A) @ (((stream641971652_SCons @ A) @ A1) @ A2)))))).
% 2.66/1.46 thf(fact_3_stream_Oexhaust, axiom, (! [A:$tType,Y:(stream170649215stream @ A)]: ((~ ((! [X12:A,X22:(stream170649215stream @ A)]: ((Y != (((stream641971652_SCons @ A) @ X12) @ X22))))))))).
% 2.66/1.46 thf(fact_4_stream_Oset__cases, axiom, (! [A:$tType,E:A,A3:(stream170649215stream @ A)]: (((((member @ A) @ E) @ ((stream30925839e_sset @ A) @ A3)) => ((! [Z2:(stream170649215stream @ A)]: ((A3 != (((stream641971652_SCons @ A) @ E) @ Z2)))) => (~ ((! [Z1:A,Z2:(stream170649215stream @ A)]: (((A3 = (((stream641971652_SCons @ A) @ Z1) @ Z2)) => (~ ((((member @ A) @ E) @ ((stream30925839e_sset @ A) @ Z2)))))))))))))).
% 2.66/1.46 thf(fact_5_stream_Oset__induct, axiom, (! [A:$tType,X:A,A3:(stream170649215stream @ A),P:(A > ((stream170649215stream @ A) > $o))]: (((((member @ A) @ X) @ ((stream30925839e_sset @ A) @ A3)) => ((! [Z1:A,Z2:(stream170649215stream @ A)]: (((P @ Z1) @ (((stream641971652_SCons @ A) @ Z1) @ Z2)))) => ((! [Z1:A,Z2:(stream170649215stream @ A),Xa:A]: (((((member @ A) @ Xa) @ ((stream30925839e_sset @ A) @ Z2)) => (((P @ Xa) @ Z2) => ((P @ Xa) @ (((stream641971652_SCons @ A) @ Z1) @ Z2)))))) => ((P @ X) @ A3))))))).
% 2.66/1.46 thf(fact_6_Stream__Mirabelle__hbrgyiwlrc_Osmember__def, axiom, (! [A:$tType]: (((stream1586597341member @ A) = (^ [X3:A,S:(stream170649215stream @ A)]: ((((member @ A) @ X3) @ ((stream30925839e_sset @ A) @ S)))))))).
% 2.66/1.46 thf(fact_7_stream_Oset, axiom, (! [A:$tType,X1:A,X2:(stream170649215stream @ A)]: ((((stream30925839e_sset @ A) @ (((stream641971652_SCons @ A) @ X1) @ X2)) = (((insert @ A) @ X1) @ ((stream30925839e_sset @ A) @ X2)))))).
% 2.66/1.46 thf(fact_8_stream_Opred__inject, axiom, (! [A:$tType,P:(A > $o),A3:A,Aa2:(stream170649215stream @ A)]: (((((stream1153105665stream @ A) @ P) @ (((stream641971652_SCons @ A) @ A3) @ Aa2)) = ((P @ A3) & (((stream1153105665stream @ A) @ P) @ Aa2)))))).
% 2.66/1.46 thf(fact_9_stream_Ocase, axiom, (! [B:$tType,A:$tType,F:(A > ((stream170649215stream @ A) > B)),X1:A,X2:(stream170649215stream @ A)]: ((((((stream1342653232stream @ A) @ B) @ F) @ (((stream641971652_SCons @ A) @ X1) @ X2)) = ((F @ X1) @ X2))))).
% 2.66/1.46 thf(fact_10_stream_Ocorec__code, axiom, (! [A:$tType,C:$tType]: ((((stream660621732stream @ C) @ A) = (^ [G1:(C > A),Q2:(C > $o),G21:(C > (stream170649215stream @ A)),G22:(C > C),A4:C]: ((((stream641971652_SCons @ A) @ (G1 @ A4)) @ ((((if @ (stream170649215stream @ A)) @ (Q2 @ A4)) @ (G21 @ A4)) @ (((((((stream660621732stream @ C) @ A) @ G1) @ Q2) @ G21) @ G22) @ (G22 @ A4)))))))))).
% 2.66/1.46 thf(fact_11_stream_Opred__cong, axiom, (! [A:$tType,X:(stream170649215stream @ A),Ya:(stream170649215stream @ A),P:(A > $o),Pa:(A > $o)]: (((X = Ya) => ((! [Z:A]: (((((member @ A) @ Z) @ ((stream30925839e_sset @ A) @ Ya)) => ((P @ Z) = (Pa @ Z))))) => ((((stream1153105665stream @ A) @ P) @ X) = (((stream1153105665stream @ A) @ Pa) @ Ya))))))).
% 2.66/1.46 thf(fact_12_stream_Opred__mono__strong, axiom, (! [A:$tType,P:(A > $o),X:(stream170649215stream @ A),Pa:(A > $o)]: (((((stream1153105665stream @ A) @ P) @ X) => ((! [Z:A]: (((((member @ A) @ Z) @ ((stream30925839e_sset @ A) @ X)) => ((P @ Z) => (Pa @ Z))))) => (((stream1153105665stream @ A) @ Pa) @ X)))))).
% 2.66/1.46 thf(fact_13_stream_Osel_I1_J, axiom, (! [A:$tType,X1:A,X2:(stream170649215stream @ A)]: ((((stream_Mirabelle_shd @ A) @ (((stream641971652_SCons @ A) @ X1) @ X2)) = X1)))).
% 2.66/1.46 thf(fact_14_stream_Oset__sel_I1_J, axiom, (! [A:$tType,A3:(stream170649215stream @ A)]: ((((member @ A) @ ((stream_Mirabelle_shd @ A) @ A3)) @ ((stream30925839e_sset @ A) @ A3))))).
% 2.66/1.46 thf(fact_15_stream_Osel_I2_J, axiom, (! [A:$tType,X1:A,X2:(stream170649215stream @ A)]: ((((stream_Mirabelle_stl @ A) @ (((stream641971652_SCons @ A) @ X1) @ X2)) = X2)))).
% 2.66/1.46 thf(fact_16_stream_Oset__sel_I2_J, axiom, (! [A:$tType,X:A,A3:(stream170649215stream @ A)]: (((((member @ A) @ X) @ ((stream30925839e_sset @ A) @ ((stream_Mirabelle_stl @ A) @ A3))) => (((member @ A) @ X) @ ((stream30925839e_sset @ A) @ A3)))))).
% 2.66/1.46 thf(fact_17_stream_Omap, axiom, (! [B:$tType,A:$tType,F:(A > B),X1:A,X2:(stream170649215stream @ A)]: ((((((stream2128578057e_smap @ A) @ B) @ F) @ (((stream641971652_SCons @ A) @ X1) @ X2)) = (((stream641971652_SCons @ B) @ (F @ X1)) @ ((((stream2128578057e_smap @ A) @ B) @ F) @ X2)))))).
% 2.66/1.46 thf(fact_18_stream_Ocollapse, axiom, (! [A:$tType,Stream:(stream170649215stream @ A)]: (((((stream641971652_SCons @ A) @ ((stream_Mirabelle_shd @ A) @ Stream)) @ ((stream_Mirabelle_stl @ A) @ Stream)) = Stream)))).
% 2.66/1.46 thf(fact_19_stream_Omap__sel_I2_J, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(stream170649215stream @ A)]: ((((stream_Mirabelle_stl @ B) @ ((((stream2128578057e_smap @ A) @ B) @ F) @ A3)) = ((((stream2128578057e_smap @ A) @ B) @ F) @ ((stream_Mirabelle_stl @ A) @ A3)))))).
% 2.66/1.46 thf(fact_20_stream_Omap__sel_I1_J, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(stream170649215stream @ A)]: ((((stream_Mirabelle_shd @ B) @ ((((stream2128578057e_smap @ A) @ B) @ F) @ A3)) = (F @ ((stream_Mirabelle_shd @ A) @ A3)))))).
% 2.66/1.46 thf(fact_21_stream_Ocorec__sel_I2_J, axiom, (! [A:$tType,C:$tType,Q22:(C > $o),A3:C,G12:(C > A),G212:(C > (stream170649215stream @ A)),G222:(C > C)]: ((((Q22 @ A3) => (((stream_Mirabelle_stl @ A) @ (((((((stream660621732stream @ C) @ A) @ G12) @ Q22) @ G212) @ G222) @ A3)) = (G212 @ A3))) & ((~ ((Q22 @ A3))) => (((stream_Mirabelle_stl @ A) @ (((((((stream660621732stream @ C) @ A) @ G12) @ Q22) @ G212) @ G222) @ A3)) = (((((((stream660621732stream @ C) @ A) @ G12) @ Q22) @ G212) @ G222) @ (G222 @ A3)))))))).
% 2.66/1.46 thf(fact_22_stream_Ocorec__sel_I1_J, axiom, (! [A:$tType,C:$tType,G12:(C > A),Q22:(C > $o),G212:(C > (stream170649215stream @ A)),G222:(C > C),A3:C]: ((((stream_Mirabelle_shd @ A) @ (((((((stream660621732stream @ C) @ A) @ G12) @ Q22) @ G212) @ G222) @ A3)) = (G12 @ A3))))).
% 2.66/1.46 thf(fact_23_stream_Oexpand, axiom, (! [A:$tType,Stream:(stream170649215stream @ A),Stream2:(stream170649215stream @ A)]: ((((((stream_Mirabelle_shd @ A) @ Stream) = ((stream_Mirabelle_shd @ A) @ Stream2)) & (((stream_Mirabelle_stl @ A) @ Stream) = ((stream_Mirabelle_stl @ A) @ Stream2))) => (Stream = Stream2))))).
% 2.66/1.46 thf(fact_24_stream_Ocoinduct, axiom, (! [A:$tType,R:((stream170649215stream @ A) > ((stream170649215stream @ A) > $o)),Stream:(stream170649215stream @ A),Stream2:(stream170649215stream @ A)]: ((((R @ Stream) @ Stream2) => ((! [Stream3:(stream170649215stream @ A),Stream4:(stream170649215stream @ A)]: ((((R @ Stream3) @ Stream4) => ((((stream_Mirabelle_shd @ A) @ Stream3) = ((stream_Mirabelle_shd @ A) @ Stream4)) & ((R @ ((stream_Mirabelle_stl @ A) @ Stream3)) @ ((stream_Mirabelle_stl @ A) @ Stream4)))))) => (Stream = Stream2)))))).
% 2.66/1.46 thf(fact_25_stream_Ocase__eq__if, axiom, (! [B:$tType,A:$tType]: ((((stream1342653232stream @ A) @ B) = (^ [F2:(A > ((stream170649215stream @ A) > B)),Stream5:(stream170649215stream @ A)]: (((F2 @ ((stream_Mirabelle_shd @ A) @ Stream5)) @ ((stream_Mirabelle_stl @ A) @ Stream5)))))))).
% 2.66/1.46 thf(fact_26_stream_Ocoinduct__strong, axiom, (! [A:$tType,R:((stream170649215stream @ A) > ((stream170649215stream @ A) > $o)),Stream:(stream170649215stream @ A),Stream2:(stream170649215stream @ A)]: ((((R @ Stream) @ Stream2) => ((! [Stream3:(stream170649215stream @ A),Stream4:(stream170649215stream @ A)]: ((((R @ Stream3) @ Stream4) => ((((stream_Mirabelle_shd @ A) @ Stream3) = ((stream_Mirabelle_shd @ A) @ Stream4)) & (((R @ ((stream_Mirabelle_stl @ A) @ Stream3)) @ ((stream_Mirabelle_stl @ A) @ Stream4)) | (((stream_Mirabelle_stl @ A) @ Stream3) = ((stream_Mirabelle_stl @ A) @ Stream4))))))) => (Stream = Stream2)))))).
% 2.66/1.46 thf(fact_27_stream_Ocorec__disc, axiom, (! [A:$tType,C:$tType]: ((((stream660621732stream @ C) @ A) = ((stream660621732stream @ C) @ A))))).
% 2.66/1.46 thf(fact_28_stream_Osplit__sel__asm, axiom, (! [B:$tType,A:$tType,P:(B > $o),F:(A > ((stream170649215stream @ A) > B)),Stream:(stream170649215stream @ A)]: (((P @ ((((stream1342653232stream @ A) @ B) @ F) @ Stream)) = (~ (((Stream = (((stream641971652_SCons @ A) @ ((stream_Mirabelle_shd @ A) @ Stream)) @ ((stream_Mirabelle_stl @ A) @ Stream))) & (~ ((P @ ((F @ ((stream_Mirabelle_shd @ A) @ Stream)) @ ((stream_Mirabelle_stl @ A) @ Stream)))))))))))).
% 2.66/1.46 thf(fact_29_stream_Osplit__sel, axiom, (! [B:$tType,A:$tType,P:(B > $o),F:(A > ((stream170649215stream @ A) > B)),Stream:(stream170649215stream @ A)]: (((P @ ((((stream1342653232stream @ A) @ B) @ F) @ Stream)) = ((Stream = (((stream641971652_SCons @ A) @ ((stream_Mirabelle_shd @ A) @ Stream)) @ ((stream_Mirabelle_stl @ A) @ Stream))) => (P @ ((F @ ((stream_Mirabelle_shd @ A) @ Stream)) @ ((stream_Mirabelle_stl @ A) @ Stream)))))))).
% 2.66/1.46 thf(fact_30_stream_Oexhaust__sel, axiom, (! [A:$tType,Stream:(stream170649215stream @ A)]: ((Stream = (((stream641971652_SCons @ A) @ ((stream_Mirabelle_shd @ A) @ Stream)) @ ((stream_Mirabelle_stl @ A) @ Stream)))))).
% 2.66/1.46 thf(fact_31_stream_Oinj__map__strong, axiom, (! [B:$tType,A:$tType,X:(stream170649215stream @ A),Xa2:(stream170649215stream @ A),F:(A > B),Fa:(A > B)]: (((! [Z:A,Za:A]: (((((member @ A) @ Z) @ ((stream30925839e_sset @ A) @ X)) => ((((member @ A) @ Za) @ ((stream30925839e_sset @ A) @ Xa2)) => (((F @ Z) = (Fa @ Za)) => (Z = Za)))))) => ((((((stream2128578057e_smap @ A) @ B) @ F) @ X) = ((((stream2128578057e_smap @ A) @ B) @ Fa) @ Xa2)) => (X = Xa2)))))).
% 2.66/1.46 thf(fact_32_stream_Omap__cong0, axiom, (! [B:$tType,A:$tType,X:(stream170649215stream @ A),F:(A > B),G:(A > B)]: (((! [Z:A]: (((((member @ A) @ Z) @ ((stream30925839e_sset @ A) @ X)) => ((F @ Z) = (G @ Z))))) => (((((stream2128578057e_smap @ A) @ B) @ F) @ X) = ((((stream2128578057e_smap @ A) @ B) @ G) @ X)))))).
% 2.66/1.46 thf(fact_33_stream_Omap__cong, axiom, (! [B:$tType,A:$tType,X:(stream170649215stream @ A),Ya:(stream170649215stream @ A),F:(A > B),G:(A > B)]: (((X = Ya) => ((! [Z:A]: (((((member @ A) @ Z) @ ((stream30925839e_sset @ A) @ Ya)) => ((F @ Z) = (G @ Z))))) => (((((stream2128578057e_smap @ A) @ B) @ F) @ X) = ((((stream2128578057e_smap @ A) @ B) @ G) @ Ya))))))).
% 2.66/1.46 thf(fact_34_insertCI, axiom, (! [A:$tType,A3:A,B2:(set @ A),B3:A]: ((((~ ((((member @ A) @ A3) @ B2))) => (A3 = B3)) => (((member @ A) @ A3) @ (((insert @ A) @ B3) @ B2)))))).
% 2.66/1.46 thf(fact_35_insert__iff, axiom, (! [A:$tType,A3:A,B3:A,A5:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B3) @ A5)) = ((A3 = B3) | (((member @ A) @ A3) @ A5)))))).
% 2.66/1.46 thf(fact_36_insert__absorb2, axiom, (! [A:$tType,X:A,A5:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ X) @ A5)) = (((insert @ A) @ X) @ A5))))).
% 2.66/1.46 thf(fact_37_insertE, axiom, (! [A:$tType,A3:A,B3:A,A5:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B3) @ A5)) => ((A3 != B3) => (((member @ A) @ A3) @ A5)))))).
% 2.66/1.46 thf(fact_38_insertI1, axiom, (! [A:$tType,A3:A,B2:(set @ A)]: ((((member @ A) @ A3) @ (((insert @ A) @ A3) @ B2))))).
% 2.66/1.46 thf(fact_39_insertI2, axiom, (! [A:$tType,A3:A,B2:(set @ A),B3:A]: (((((member @ A) @ A3) @ B2) => (((member @ A) @ A3) @ (((insert @ A) @ B3) @ B2)))))).
% 2.66/1.46 thf(fact_40_Set_Oset__insert, axiom, (! [A:$tType,X:A,A5:(set @ A)]: (((((member @ A) @ X) @ A5) => (~ ((! [B4:(set @ A)]: (((A5 = (((insert @ A) @ X) @ B4)) => (((member @ A) @ X) @ B4)))))))))).
% 2.66/1.46 thf(fact_41_insert__ident, axiom, (! [A:$tType,X:A,A5:(set @ A),B2:(set @ A)]: (((~ ((((member @ A) @ X) @ A5))) => ((~ ((((member @ A) @ X) @ B2))) => (((((insert @ A) @ X) @ A5) = (((insert @ A) @ X) @ B2)) = (A5 = B2))))))).
% 2.66/1.46 thf(fact_42_insert__absorb, axiom, (! [A:$tType,A3:A,A5:(set @ A)]: (((((member @ A) @ A3) @ A5) => ((((insert @ A) @ A3) @ A5) = A5))))).
% 2.66/1.46 thf(fact_43_insert__eq__iff, axiom, (! [A:$tType,A3:A,A5:(set @ A),B3:A,B2:(set @ A)]: (((~ ((((member @ A) @ A3) @ A5))) => ((~ ((((member @ A) @ B3) @ B2))) => (((((insert @ A) @ A3) @ A5) = (((insert @ A) @ B3) @ B2)) = (((A3 = B3) => (A5 = B2)) & ((A3 != B3) => (? [C2:(set @ A)]: (((A5 = (((insert @ A) @ B3) @ C2)) & ((~ ((((member @ A) @ B3) @ C2))) & ((B2 = (((insert @ A) @ A3) @ C2)) & (~ ((((member @ A) @ A3) @ C2)))))))))))))))).
% 2.66/1.46 thf(fact_44_mk__disjoint__insert, axiom, (! [A:$tType,A3:A,A5:(set @ A)]: (((((member @ A) @ A3) @ A5) => (? [B4:(set @ A)]: (((A5 = (((insert @ A) @ A3) @ B4)) & (~ ((((member @ A) @ A3) @ B4)))))))))).
% 2.66/1.46 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 2.66/1.46 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A5:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A5)))) = A5)))).
% 2.66/1.46 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) = (Q @ X4)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.66/1.46 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X4:A]: (((F @ X4) = (G @ X4)))) => (F = G))))).
% 2.66/1.46 thf(fact_49_insert__commute, axiom, (! [A:$tType,X:A,Y:A,A5:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ Y) @ A5)) = (((insert @ A) @ Y) @ (((insert @ A) @ X) @ A5)))))).
% 2.66/1.46 thf(fact_50_stream_Orel__coinduct, axiom, (! [A:$tType,B:$tType,P:((stream170649215stream @ A) > ((stream170649215stream @ B) > $o)),X:(stream170649215stream @ A),Y:(stream170649215stream @ B),R:(A > (B > $o))]: ((((P @ X) @ Y) => ((! [Stream3:(stream170649215stream @ A),Stream4:(stream170649215stream @ B)]: ((((P @ Stream3) @ Stream4) => (((R @ ((stream_Mirabelle_shd @ A) @ Stream3)) @ ((stream_Mirabelle_shd @ B) @ Stream4)) & ((P @ ((stream_Mirabelle_stl @ A) @ Stream3)) @ ((stream_Mirabelle_stl @ B) @ Stream4)))))) => (((((stream686525009m_all2 @ A) @ B) @ R) @ X) @ Y)))))).
% 2.66/1.46 thf(fact_51_stream_Orel__sel, axiom, (! [B:$tType,A:$tType]: ((((stream686525009m_all2 @ A) @ B) = (^ [R2:(A > (B > $o)),A4:(stream170649215stream @ A),B5:(stream170649215stream @ B)]: ((((R2 @ ((stream_Mirabelle_shd @ A) @ A4)) @ ((stream_Mirabelle_shd @ B) @ B5)) & (((((stream686525009m_all2 @ A) @ B) @ R2) @ ((stream_Mirabelle_stl @ A) @ A4)) @ ((stream_Mirabelle_stl @ B) @ B5))))))))).
% 2.66/1.46 thf(fact_52_stream_Opred__set, axiom, (! [A:$tType]: (((stream1153105665stream @ A) = (^ [P2:(A > $o),X3:(stream170649215stream @ A)]: ((! [Y3:A]: (((((member @ A) @ Y3) @ ((stream30925839e_sset @ A) @ X3)) => (P2 @ Y3)))))))))).
% 2.66/1.46 thf(fact_53_stream_Opred__map, axiom, (! [B:$tType,A:$tType,Q:(B > $o),F:(A > B),X:(stream170649215stream @ A)]: (((((stream1153105665stream @ B) @ Q) @ ((((stream2128578057e_smap @ A) @ B) @ F) @ X)) = (((stream1153105665stream @ A) @ (((((comp @ B) @ $o) @ A) @ Q) @ F)) @ X))))).
% 2.66/1.46 thf(fact_54_stream_Oset__map, axiom, (! [B:$tType,A:$tType,F:(A > B),V:(stream170649215stream @ A)]: ((((stream30925839e_sset @ B) @ ((((stream2128578057e_smap @ A) @ B) @ F) @ V)) = ((((image @ A) @ B) @ F) @ ((stream30925839e_sset @ A) @ V)))))).
% 2.66/1.46 thf(fact_55_pairwise__insert, axiom, (! [A:$tType,R3:(A > (A > $o)),X:A,S2:(set @ A)]: (((((pairwise @ A) @ R3) @ (((insert @ A) @ X) @ S2)) = ((! [Y3:A]: ((((((member @ A) @ Y3) @ S2) & (Y3 != X)) => (((R3 @ X) @ Y3) & ((R3 @ Y3) @ X))))) & (((pairwise @ A) @ R3) @ S2)))))).
% 2.66/1.46 thf(fact_56_stream_Orel__inject, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),X1:A,X2:(stream170649215stream @ A),Y1:B,Y2:(stream170649215stream @ B)]: (((((((stream686525009m_all2 @ A) @ B) @ R) @ (((stream641971652_SCons @ A) @ X1) @ X2)) @ (((stream641971652_SCons @ B) @ Y1) @ Y2)) = (((R @ X1) @ Y1) & (((((stream686525009m_all2 @ A) @ B) @ R) @ X2) @ Y2)))))).
% 2.66/1.46 thf(fact_57_singletonI, axiom, (! [A:$tType,A3:A]: ((((member @ A) @ A3) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))).
% 2.66/1.46 thf(fact_58_image__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),X:B,A5:(set @ B)]: (((B3 = (F @ X)) => ((((member @ B) @ X) @ A5) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ A5))))))).
% 2.66/1.46 thf(fact_59_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 2.66/1.46 thf(fact_60_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 2.66/1.46 thf(fact_61_all__not__in__conv, axiom, (! [A:$tType,A5:(set @ A)]: (((! [X3:A]: ((~ ((((member @ A) @ X3) @ A5))))) = (A5 = (bot_bot @ (set @ A))))))).
% 2.66/1.46 thf(fact_62_empty__iff, axiom, (! [A:$tType,C3:A]: ((~ ((((member @ A) @ C3) @ (bot_bot @ (set @ A)))))))).
% 2.66/1.46 thf(fact_63_image__is__empty, axiom, (! [A:$tType,B:$tType,F:(B > A),A5:(set @ B)]: (((((((image @ B) @ A) @ F) @ A5) = (bot_bot @ (set @ A))) = (A5 = (bot_bot @ (set @ B))))))).
% 2.66/1.46 thf(fact_64_empty__is__image, axiom, (! [A:$tType,B:$tType,F:(B > A),A5:(set @ B)]: ((((bot_bot @ (set @ A)) = ((((image @ B) @ A) @ F) @ A5)) = (A5 = (bot_bot @ (set @ B))))))).
% 2.66/1.46 thf(fact_65_image__empty, axiom, (! [B:$tType,A:$tType,F:(B > A)]: ((((((image @ B) @ A) @ F) @ (bot_bot @ (set @ B))) = (bot_bot @ (set @ A)))))).
% 2.66/1.46 thf(fact_66_image__insert, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:B,B2:(set @ B)]: ((((((image @ B) @ A) @ F) @ (((insert @ B) @ A3) @ B2)) = (((insert @ A) @ (F @ A3)) @ ((((image @ B) @ A) @ F) @ B2)))))).
% 2.66/1.46 thf(fact_67_insert__image, axiom, (! [B:$tType,A:$tType,X:A,A5:(set @ A),F:(A > B)]: (((((member @ A) @ X) @ A5) => ((((insert @ B) @ (F @ X)) @ ((((image @ A) @ B) @ F) @ A5)) = ((((image @ A) @ B) @ F) @ A5)))))).
% 2.66/1.46 thf(fact_68_ball__empty, axiom, (! [A:$tType,P:(A > $o),X5:A]: (((((member @ A) @ X5) @ (bot_bot @ (set @ A))) => (P @ X5))))).
% 2.66/1.46 thf(fact_69_pairwise__empty, axiom, (! [A:$tType,P:(A > (A > $o))]: ((((pairwise @ A) @ P) @ (bot_bot @ (set @ A)))))).
% 2.66/1.46 thf(fact_70_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X:A,A5:(set @ A),B3:B,F:(A > B)]: (((((member @ A) @ X) @ A5) => ((B3 = (F @ X)) => (((member @ B) @ B3) @ ((((image @ A) @ B) @ F) @ A5))))))).
% 2.66/1.46 thf(fact_71_pairwise__def, axiom, (! [A:$tType]: (((pairwise @ A) = (^ [R2:(A > (A > $o)),S3:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ S3) => (! [Y3:A]: (((((member @ A) @ Y3) @ S3) => ((X3 != Y3) => ((R2 @ X3) @ Y3)))))))))))))).
% 2.66/1.46 thf(fact_72_ball__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A5:(set @ B),P:(A > $o)]: (((! [X4:A]: (((((member @ A) @ X4) @ ((((image @ B) @ A) @ F) @ A5)) => (P @ X4)))) => (! [X5:B]: (((((member @ B) @ X5) @ A5) => (P @ (F @ X5))))))))).
% 2.66/1.46 thf(fact_73_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N:(set @ A),F:(A > B),G:(A > B)]: (((M = N) => ((! [X4:A]: (((((member @ A) @ X4) @ N) => ((F @ X4) = (G @ X4))))) => (((((image @ A) @ B) @ F) @ M) = ((((image @ A) @ B) @ G) @ N))))))).
% 2.66/1.46 thf(fact_74_ex__in__conv, axiom, (! [A:$tType,A5:(set @ A)]: (((? [X3:A]: ((((member @ A) @ X3) @ A5))) = (A5 != (bot_bot @ (set @ A))))))).
% 2.66/1.46 thf(fact_75_bex__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A5:(set @ B),P:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F) @ A5)) & (P @ X5)))) => (? [X4:B]: (((((member @ B) @ X4) @ A5) & (P @ (F @ X4))))))))).
% 2.66/1.46 thf(fact_76_image__iff, axiom, (! [A:$tType,B:$tType,Z3:A,F:(B > A),A5:(set @ B)]: (((((member @ A) @ Z3) @ ((((image @ B) @ A) @ F) @ A5)) = (? [X3:B]: (((((member @ B) @ X3) @ A5) & (Z3 = (F @ X3))))))))).
% 2.66/1.46 thf(fact_77_equals0I, axiom, (! [A:$tType,A5:(set @ A)]: (((! [Y4:A]: ((~ ((((member @ A) @ Y4) @ A5))))) => (A5 = (bot_bot @ (set @ A))))))).
% 2.66/1.46 thf(fact_78_equals0D, axiom, (! [A:$tType,A5:(set @ A),A3:A]: (((A5 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A3) @ A5))))))).
% 2.66/1.46 thf(fact_79_Ball__def, axiom, (! [A:$tType]: (((ball @ A) = (^ [A6:(set @ A),P2:(A > $o)]: ((! [X3:A]: (((((member @ A) @ X3) @ A6) => (P2 @ X3)))))))))).
% 2.66/1.46 thf(fact_80_imageI, axiom, (! [B:$tType,A:$tType,X:A,A5:(set @ A),F:(A > B)]: (((((member @ A) @ X) @ A5) => (((member @ B) @ (F @ X)) @ ((((image @ A) @ B) @ F) @ A5)))))).
% 2.66/1.46 thf(fact_81_emptyE, axiom, (! [A:$tType,A3:A]: ((~ ((((member @ A) @ A3) @ (bot_bot @ (set @ A)))))))).
% 2.66/1.46 thf(fact_82_stream_Orel__eq, axiom, (! [A:$tType]: (((((stream686525009m_all2 @ A) @ A) @ (^ [Y5:A,Z4:A]: ((Y5 = Z4)))) = (^ [Y5:(stream170649215stream @ A),Z4:(stream170649215stream @ A)]: ((Y5 = Z4))))))).
% 2.66/1.46 thf(fact_83_stream_Orel__refl, axiom, (! [B:$tType,Ra:(B > (B > $o)),X:(stream170649215stream @ B)]: (((! [X4:B]: (((Ra @ X4) @ X4))) => (((((stream686525009m_all2 @ B) @ B) @ Ra) @ X) @ X))))).
% 2.66/1.46 thf(fact_84_pairwise__singleton, axiom, (! [A:$tType,P:(A > (A > $o)),A5:A]: ((((pairwise @ A) @ P) @ (((insert @ A) @ A5) @ (bot_bot @ (set @ A))))))).
% 2.66/1.46 thf(fact_85_stream_Omap__o__corec, axiom, (! [A:$tType,B:$tType,C:$tType,F:(A > B),G:(C > A),Ga:(C > $o),Gb:(C > (stream170649215stream @ A)),Gc:(C > C)]: (((((((comp @ (stream170649215stream @ A)) @ (stream170649215stream @ B)) @ C) @ (((stream2128578057e_smap @ A) @ B) @ F)) @ ((((((stream660621732stream @ C) @ A) @ G) @ Ga) @ Gb) @ Gc)) = ((((((stream660621732stream @ C) @ B) @ (((((comp @ A) @ B) @ C) @ F) @ G)) @ Ga) @ (((((comp @ (stream170649215stream @ A)) @ (stream170649215stream @ B)) @ C) @ (((stream2128578057e_smap @ A) @ B) @ F)) @ Gb)) @ Gc))))).
% 2.66/1.46 thf(fact_86_stream_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,G:(B > C),F:(A > B),V:(stream170649215stream @ A)]: ((((((stream2128578057e_smap @ B) @ C) @ G) @ ((((stream2128578057e_smap @ A) @ B) @ F) @ V)) = ((((stream2128578057e_smap @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ V))))).
% 2.66/1.46 thf(fact_87_singletonD, axiom, (! [A:$tType,B3:A,A3:A]: (((((member @ A) @ B3) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) => (B3 = A3))))).
% 2.66/1.46 thf(fact_88_singleton__iff, axiom, (! [A:$tType,B3:A,A3:A]: (((((member @ A) @ B3) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) = (B3 = A3))))).
% 2.66/1.46 thf(fact_89_doubleton__eq__iff, axiom, (! [A:$tType,A3:A,B3:A,C3:A,D:A]: ((((((insert @ A) @ A3) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C3) @ (((insert @ A) @ D) @ (bot_bot @ (set @ A))))) = (((A3 = C3) & (B3 = D)) | ((A3 = D) & (B3 = C3))))))).
% 2.66/1.46 thf(fact_90_insert__not__empty, axiom, (! [A:$tType,A3:A,A5:(set @ A)]: (((((insert @ A) @ A3) @ A5) != (bot_bot @ (set @ A)))))).
% 2.66/1.46 thf(fact_91_singleton__inject, axiom, (! [A:$tType,A3:A,B3:A]: ((((((insert @ A) @ A3) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) => (A3 = B3))))).
% 2.66/1.46 thf(fact_92_stream_Orel__refl__strong, axiom, (! [A:$tType,X:(stream170649215stream @ A),Ra:(A > (A > $o))]: (((! [Z:A]: (((((member @ A) @ Z) @ ((stream30925839e_sset @ A) @ X)) => ((Ra @ Z) @ Z)))) => (((((stream686525009m_all2 @ A) @ A) @ Ra) @ X) @ X))))).
% 2.66/1.46 thf(fact_93_stream_Orel__mono__strong, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),X:(stream170649215stream @ A),Y:(stream170649215stream @ B),Ra:(A > (B > $o))]: (((((((stream686525009m_all2 @ A) @ B) @ R) @ X) @ Y) => ((! [Z:A,Yb:B]: (((((member @ A) @ Z) @ ((stream30925839e_sset @ A) @ X)) => ((((member @ B) @ Yb) @ ((stream30925839e_sset @ B) @ Y)) => (((R @ Z) @ Yb) => ((Ra @ Z) @ Yb)))))) => (((((stream686525009m_all2 @ A) @ B) @ Ra) @ X) @ Y)))))).
% 2.66/1.46 thf(fact_94_stream_Orel__cong, axiom, (! [A:$tType,B:$tType,X:(stream170649215stream @ A),Ya:(stream170649215stream @ A),Y:(stream170649215stream @ B),Xa2:(stream170649215stream @ B),R:(A > (B > $o)),Ra:(A > (B > $o))]: (((X = Ya) => ((Y = Xa2) => ((! [Z:A,Yb:B]: (((((member @ A) @ Z) @ ((stream30925839e_sset @ A) @ Ya)) => ((((member @ B) @ Yb) @ ((stream30925839e_sset @ B) @ Xa2)) => (((R @ Z) @ Yb) = ((Ra @ Z) @ Yb)))))) => ((((((stream686525009m_all2 @ A) @ B) @ R) @ X) @ Y) = (((((stream686525009m_all2 @ A) @ B) @ Ra) @ Ya) @ Xa2)))))))).
% 2.66/1.46 thf(fact_95_stream_Orel__intros, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),X1:A,Y1:B,X2:(stream170649215stream @ A),Y2:(stream170649215stream @ B)]: ((((R @ X1) @ Y1) => ((((((stream686525009m_all2 @ A) @ B) @ R) @ X2) @ Y2) => (((((stream686525009m_all2 @ A) @ B) @ R) @ (((stream641971652_SCons @ A) @ X1) @ X2)) @ (((stream641971652_SCons @ B) @ Y1) @ Y2))))))).
% 2.66/1.46 thf(fact_96_stream_Orel__cases, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),A3:(stream170649215stream @ A),B3:(stream170649215stream @ B)]: (((((((stream686525009m_all2 @ A) @ B) @ R) @ A3) @ B3) => (~ ((! [X1a:A,X2a:(stream170649215stream @ A)]: (((A3 = (((stream641971652_SCons @ A) @ X1a) @ X2a)) => (! [Y1a:B,Y2a:(stream170649215stream @ B)]: (((B3 = (((stream641971652_SCons @ B) @ Y1a) @ Y2a)) => (((R @ X1a) @ Y1a) => (~ ((((((stream686525009m_all2 @ A) @ B) @ R) @ X2a) @ Y2a)))))))))))))))).
% 2.66/1.46 thf(fact_97_Ball__image__comp, axiom, (! [A:$tType,B:$tType,F:(B > A),A5:(set @ B),G:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((((image @ B) @ A) @ F) @ A5)) => (G @ X3)))) = (! [X3:B]: (((((member @ B) @ X3) @ A5) => ((((((comp @ A) @ $o) @ B) @ G) @ F) @ X3)))))))).
% 2.66/1.46 thf(fact_98_not__False__in__image__Ball, axiom, (! [A:$tType,P:(A > $o),A5:(set @ A)]: (((~ ((((member @ $o) @ $false) @ ((((image @ A) @ $o) @ P) @ A5)))) = (! [X3:A]: (((((member @ A) @ X3) @ A5) => (P @ X3)))))))).
% 2.66/1.46 thf(fact_99_comp__apply, axiom, (! [C:$tType,A:$tType,B:$tType]: (((((comp @ B) @ A) @ C) = (^ [F2:(B > A),G2:(C > B),X3:C]: ((F2 @ (G2 @ X3)))))))).
% 2.66/1.46 thf(fact_100_the__elem__eq, axiom, (! [A:$tType,X:A]: ((((the_elem @ A) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = X)))).
% 2.66/1.46 thf(fact_101_bot__apply, axiom, (! [C:$tType,D2:$tType]: ((((bot @ C) @ (type2 @ C)) => ((bot_bot @ (D2 > C)) = (^ [X3:D2]: ((bot_bot @ C)))))))).
% 2.66/1.46 thf(fact_102_is__singletonI, axiom, (! [A:$tType,X:A]: (((is_singleton @ A) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))).
% 2.66/1.46 thf(fact_103_image__comp, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > A),G:(C > B),R3:(set @ C)]: ((((((image @ B) @ A) @ F) @ ((((image @ C) @ B) @ G) @ R3)) = ((((image @ C) @ A) @ (((((comp @ B) @ A) @ C) @ F) @ G)) @ R3))))).
% 2.66/1.46 thf(fact_104_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.66/1.46 thf(fact_105_is__singleton__the__elem, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A6:(set @ A)]: ((A6 = (((insert @ A) @ ((the_elem @ A) @ A6)) @ (bot_bot @ (set @ A)))))))))).
% 2.66/1.46 thf(fact_106_is__singletonI_H, axiom, (! [A:$tType,A5:(set @ A)]: (((A5 != (bot_bot @ (set @ A))) => ((! [X4:A,Y4:A]: (((((member @ A) @ X4) @ A5) => ((((member @ A) @ Y4) @ A5) => (X4 = Y4))))) => ((is_singleton @ A) @ A5)))))).
% 2.66/1.46 thf(fact_107_Sup_OSUP__cong, axiom, (! [A:$tType,B:$tType,A5:(set @ B),B2:(set @ B),C4:(B > A),D3:(B > A),Sup:((set @ A) > A)]: (((A5 = B2) => ((! [X4:B]: (((((member @ B) @ X4) @ B2) => ((C4 @ X4) = (D3 @ X4))))) => ((Sup @ ((((image @ B) @ A) @ C4) @ A5)) = (Sup @ ((((image @ B) @ A) @ D3) @ B2)))))))).
% 2.66/1.46 thf(fact_108_Inf_OINF__cong, axiom, (! [A:$tType,B:$tType,A5:(set @ B),B2:(set @ B),C4:(B > A),D3:(B > A),Inf:((set @ A) > A)]: (((A5 = B2) => ((! [X4:B]: (((((member @ B) @ X4) @ B2) => ((C4 @ X4) = (D3 @ X4))))) => ((Inf @ ((((image @ B) @ A) @ C4) @ A5)) = (Inf @ ((((image @ B) @ A) @ D3) @ B2)))))))).
% 2.66/1.46 thf(fact_109_bot__fun__def, axiom, (! [B:$tType,A:$tType]: ((((bot @ B) @ (type2 @ B)) => ((bot_bot @ (A > B)) = (^ [X3:A]: ((bot_bot @ B)))))))).
% 2.66/1.46 thf(fact_110_comp__apply__eq, axiom, (! [B:$tType,D2:$tType,A:$tType,C:$tType,F:(B > A),G:(C > B),X:C,H:(D2 > A),K:(C > D2)]: ((((F @ (G @ X)) = (H @ (K @ X))) => (((((((comp @ B) @ A) @ C) @ F) @ G) @ X) = ((((((comp @ D2) @ A) @ C) @ H) @ K) @ X)))))).
% 2.66/1.46 thf(fact_111_comp__eq__dest__lhs, axiom, (! [C:$tType,B:$tType,A:$tType,A3:(C > B),B3:(A > C),C3:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A3) @ B3) = C3) => ((A3 @ (B3 @ V)) = (C3 @ V)))))).
% 2.66/1.46 thf(fact_112_comp__eq__elim, axiom, (! [C:$tType,B:$tType,D2:$tType,A:$tType,A3:(C > B),B3:(A > C),C3:(D2 > B),D:(A > D2)]: ((((((((comp @ C) @ B) @ A) @ A3) @ B3) = (((((comp @ D2) @ B) @ A) @ C3) @ D)) => (! [V2:A]: (((A3 @ (B3 @ V2)) = (C3 @ (D @ V2))))))))).
% 2.66/1.46 thf(fact_113_comp__eq__dest, axiom, (! [C:$tType,B:$tType,D2:$tType,A:$tType,A3:(C > B),B3:(A > C),C3:(D2 > B),D:(A > D2),V:A]: ((((((((comp @ C) @ B) @ A) @ A3) @ B3) = (((((comp @ D2) @ B) @ A) @ C3) @ D)) => ((A3 @ (B3 @ V)) = (C3 @ (D @ V))))))).
% 2.66/1.46 thf(fact_114_comp__assoc, axiom, (! [B:$tType,D2:$tType,C:$tType,A:$tType,F:(D2 > B),G:(C > D2),H:(A > C)]: (((((((comp @ C) @ B) @ A) @ (((((comp @ D2) @ B) @ C) @ F) @ G)) @ H) = (((((comp @ D2) @ B) @ A) @ F) @ (((((comp @ C) @ D2) @ A) @ G) @ H)))))).
% 2.66/1.46 thf(fact_115_comp__def, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((comp @ B) @ C) @ A) = (^ [F2:(B > C),G2:(A > B),X3:A]: ((F2 @ (G2 @ X3)))))))).
% 2.66/1.46 thf(fact_116_is__singletonE, axiom, (! [A:$tType,A5:(set @ A)]: ((((is_singleton @ A) @ A5) => (~ ((! [X4:A]: ((A5 != (((insert @ A) @ X4) @ (bot_bot @ (set @ A)))))))))))).
% 2.66/1.46 thf(fact_117_is__singleton__def, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A6:(set @ A)]: ((? [X3:A]: ((A6 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))).
% 2.66/1.46 thf(fact_118_the__elem__image__unique, axiom, (! [B:$tType,A:$tType,A5:(set @ A),F:(A > B),X:A]: (((A5 != (bot_bot @ (set @ A))) => ((! [Y4:A]: (((((member @ A) @ Y4) @ A5) => ((F @ Y4) = (F @ X))))) => (((the_elem @ B) @ ((((image @ A) @ B) @ F) @ A5)) = (F @ X))))))).
% 2.66/1.46 thf(fact_119_Sup_OSUP__image, axiom, (! [B:$tType,A:$tType,C:$tType,Sup:((set @ A) > A),G:(B > A),F:(C > B),A5:(set @ C)]: (((Sup @ ((((image @ B) @ A) @ G) @ ((((image @ C) @ B) @ F) @ A5))) = (Sup @ ((((image @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G) @ F)) @ A5)))))).
% 2.66/1.46 thf(fact_120_Inf_OINF__image, axiom, (! [B:$tType,A:$tType,C:$tType,Inf:((set @ A) > A),G:(B > A),F:(C > B),A5:(set @ C)]: (((Inf @ ((((image @ B) @ A) @ G) @ ((((image @ C) @ B) @ F) @ A5))) = (Inf @ ((((image @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G) @ F)) @ A5)))))).
% 2.66/1.46 thf(fact_121_image__eq__imp__comp, axiom, (! [B:$tType,A:$tType,D2:$tType,C:$tType,F:(B > A),A5:(set @ B),G:(C > A),B2:(set @ C),H:(A > D2)]: (((((((image @ B) @ A) @ F) @ A5) = ((((image @ C) @ A) @ G) @ B2)) => (((((image @ B) @ D2) @ (((((comp @ A) @ D2) @ B) @ H) @ F)) @ A5) = ((((image @ C) @ D2) @ (((((comp @ A) @ D2) @ C) @ H) @ G)) @ B2)))))).
% 2.66/1.46 thf(fact_122_override__on__emptyset, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((((((override_on @ A) @ B) @ F) @ G) @ (bot_bot @ (set @ A))) = F)))).
% 2.66/1.46 thf(fact_123_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A6:(set @ A)]: ((A6 = (bot_bot @ (set @ A))))))))).
% 2.66/1.46 thf(fact_124_ball__reg, axiom, (! [A:$tType,R:(set @ A),P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((((member @ A) @ X4) @ R) => ((P @ X4) => (Q @ X4))))) => ((! [X4:A]: (((((member @ A) @ X4) @ R) => (P @ X4)))) => (! [X5:A]: (((((member @ A) @ X5) @ R) => (Q @ X5))))))))).
% 2.66/1.46 thf(fact_125_rewriteR__comp__comp2, axiom, (! [C:$tType,B:$tType,E2:$tType,D2:$tType,A:$tType,G:(C > B),H:(A > C),R1:(D2 > B),R22:(A > D2),F:(B > E2),L:(D2 > E2)]: ((((((((comp @ C) @ B) @ A) @ G) @ H) = (((((comp @ D2) @ B) @ A) @ R1) @ R22)) => (((((((comp @ B) @ E2) @ D2) @ F) @ R1) = L) => ((((((comp @ C) @ E2) @ A) @ (((((comp @ B) @ E2) @ C) @ F) @ G)) @ H) = (((((comp @ D2) @ E2) @ A) @ L) @ R22))))))).
% 2.66/1.46 thf(fact_126_rewriteL__comp__comp2, axiom, (! [A:$tType,C:$tType,B:$tType,D2:$tType,E2:$tType,F:(C > B),G:(A > C),L1:(D2 > B),L2:(A > D2),H:(E2 > A),R3:(E2 > D2)]: ((((((((comp @ C) @ B) @ A) @ F) @ G) = (((((comp @ D2) @ B) @ A) @ L1) @ L2)) => (((((((comp @ A) @ D2) @ E2) @ L2) @ H) = R3) => ((((((comp @ C) @ B) @ E2) @ F) @ (((((comp @ A) @ C) @ E2) @ G) @ H)) = (((((comp @ D2) @ B) @ E2) @ L1) @ R3))))))).
% 2.66/1.46 thf(fact_127_override__on__apply__in, axiom, (! [B:$tType,A:$tType,A3:A,A5:(set @ A),F:(A > B),G:(A > B)]: (((((member @ A) @ A3) @ A5) => (((((((override_on @ A) @ B) @ F) @ G) @ A5) @ A3) = (G @ A3)))))).
% 2.66/1.46 thf(fact_128_override__on__apply__notin, axiom, (! [B:$tType,A:$tType,A3:A,A5:(set @ A),F:(A > B),G:(A > B)]: (((~ ((((member @ A) @ A3) @ A5))) => (((((((override_on @ A) @ B) @ F) @ G) @ A5) @ A3) = (F @ A3)))))).
% 2.66/1.46 thf(fact_129_override__on__def, axiom, (! [B:$tType,A:$tType]: ((((override_on @ A) @ B) = (^ [F2:(A > B),G2:(A > B),A6:(set @ A),A4:A]: (((((if @ B) @ (((member @ A) @ A4) @ A6)) @ (G2 @ A4)) @ (F2 @ A4)))))))).
% 2.66/1.46 thf(fact_130_rewriteL__comp__comp, axiom, (! [C:$tType,B:$tType,A:$tType,D2:$tType,F:(C > B),G:(A > C),L:(A > B),H:(D2 > A)]: ((((((((comp @ C) @ B) @ A) @ F) @ G) = L) => ((((((comp @ C) @ B) @ D2) @ F) @ (((((comp @ A) @ C) @ D2) @ G) @ H)) = (((((comp @ A) @ B) @ D2) @ L) @ H)))))).
% 2.66/1.46 thf(fact_131_rewriteR__comp__comp, axiom, (! [C:$tType,D2:$tType,B:$tType,A:$tType,G:(C > B),H:(A > C),R3:(A > B),F:(B > D2)]: ((((((((comp @ C) @ B) @ A) @ G) @ H) = R3) => ((((((comp @ C) @ D2) @ A) @ (((((comp @ B) @ D2) @ C) @ F) @ G)) @ H) = (((((comp @ B) @ D2) @ A) @ F) @ R3)))))).
% 2.66/1.46 thf(fact_132_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 2.66/1.46 thf(fact_133_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X3:A]: ((((member @ A) @ X3) @ (bot_bot @ (set @ A))))))))).
% 2.66/1.46 thf(fact_134_comp__cong, axiom, (! [C:$tType,B:$tType,D2:$tType,A:$tType,E2:$tType,F:(B > A),G:(C > B),X:C,F3:(D2 > A),G3:(E2 > D2),X6:E2]: ((((F @ (G @ X)) = (F3 @ (G3 @ X6))) => (((((((comp @ B) @ A) @ C) @ F) @ G) @ X) = ((((((comp @ D2) @ A) @ E2) @ F3) @ G3) @ X6)))))).
% 2.66/1.46 thf(fact_135_fun_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,D2:$tType,G:(B > C),F:(A > B),V:(D2 > A)]: (((((((comp @ B) @ C) @ D2) @ G) @ (((((comp @ A) @ B) @ D2) @ F) @ V)) = (((((comp @ A) @ C) @ D2) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ V))))).
% 2.66/1.46 thf(fact_136_type__copy__map__cong0, axiom, (! [B:$tType,D2:$tType,E2:$tType,A:$tType,C:$tType,M:(B > A),G:(C > B),X:C,N:(D2 > A),H:(C > D2),F:(A > E2)]: ((((M @ (G @ X)) = (N @ (H @ X))) => (((((((comp @ B) @ E2) @ C) @ (((((comp @ A) @ E2) @ B) @ F) @ M)) @ G) @ X) = ((((((comp @ D2) @ E2) @ C) @ (((((comp @ A) @ E2) @ D2) @ F) @ N)) @ H) @ X)))))).
% 2.66/1.46 thf(fact_137_range__eq__singletonD, axiom, (! [B:$tType,A:$tType,F:(B > A),A3:A,X:B]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) => ((F @ X) = A3))))).
% 2.66/1.46 thf(fact_138_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A3:A,A5:(set @ A),B3:A]: ((((((insert @ A) @ A3) @ A5) = (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) = ((A3 = B3) & (((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A))))))))).
% 2.66/1.46 thf(fact_139_singleton__insert__inj__eq, axiom, (! [A:$tType,B3:A,A3:A,A5:(set @ A)]: ((((((insert @ A) @ B3) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A3) @ A5)) = ((A3 = B3) & (((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A))))))))).
% 2.66/1.46 thf(fact_140_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.66/1.46 thf(fact_141_subset__antisym, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A5) => (A5 = B2)))))).
% 2.66/1.46 thf(fact_142_subsetI, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A)]: (((! [X4:A]: (((((member @ A) @ X4) @ A5) => (((member @ A) @ X4) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A5) @ B2))))).
% 2.66/1.46 thf(fact_143_top__apply, axiom, (! [C:$tType,D2:$tType]: ((((top @ C) @ (type2 @ C)) => ((top_top @ (D2 > C)) = (^ [X3:D2]: ((top_top @ C)))))))).
% 2.66/1.46 thf(fact_144_UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.66/1.46 thf(fact_145_subset__empty, axiom, (! [A:$tType,A5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ (bot_bot @ (set @ A))) = (A5 = (bot_bot @ (set @ A))))))).
% 2.66/1.46 thf(fact_146_empty__subsetI, axiom, (! [A:$tType,A5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A5)))).
% 2.66/1.46 thf(fact_147_insert__subset, axiom, (! [A:$tType,X:A,A5:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X) @ A5)) @ B2) = ((((member @ A) @ X) @ B2) & (((ord_less_eq @ (set @ A)) @ A5) @ B2)))))).
% 2.66/1.46 thf(fact_148_top_Oextremum__uniqueI, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) => (A3 = (top_top @ A))))))))).
% 2.66/1.46 thf(fact_149_top_Oextremum__unique, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) = (A3 = (top_top @ A))))))))).
% 2.66/1.46 thf(fact_150_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ A3) @ B3) => (A3 = B3))))))))).
% 2.66/1.46 thf(fact_151_top__greatest, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ (top_top @ A)))))))).
% 2.66/1.46 thf(fact_152_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A3:A,C3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C3) @ B3) => (((ord_less_eq @ A) @ C3) @ A3))))))))).
% 2.66/1.46 thf(fact_153_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A3:A,B3:A]: (((! [A7:A,B6:A]: (((((ord_less_eq @ A) @ A7) @ B6) => ((P @ A7) @ B6)))) => ((! [A7:A,B6:A]: ((((P @ B6) @ A7) => ((P @ A7) @ B6)))) => ((P @ A3) @ B3))))))))).
% 2.66/1.46 thf(fact_154_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 2.66/1.46 thf(fact_155_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z3) => (((ord_less_eq @ A) @ X) @ Z3))))))))).
% 2.66/1.46 thf(fact_156_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ A3) => (A3 = B3))))))))).
% 2.66/1.46 thf(fact_157_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((B3 = C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 2.66/1.46 thf(fact_158_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C3:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ B3) @ C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 2.66/1.46 thf(fact_159_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.66/1.46 thf(fact_160_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z3)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z3)))) => (((((ord_less_eq @ A) @ X) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ Y)))) => (((((ord_less_eq @ A) @ Z3) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ X)))) => (~ (((((ord_less_eq @ A) @ Z3) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 2.66/1.46 thf(fact_161_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 2.66/1.46 thf(fact_162_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.66/1.46 thf(fact_163_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.66/1.46 thf(fact_164_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.66/1.46 thf(fact_165_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.66/1.46 thf(fact_166_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X3))))))))).
% 2.66/1.46 thf(fact_167_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A3:A,B3:A,F:(A > B),C3:B]: (((((ord_less_eq @ A) @ A3) @ B3) => (((F @ B3) = C3) => ((! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => (((ord_less_eq @ B) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less_eq @ B) @ (F @ A3)) @ C3)))))))))).
% 2.66/1.46 thf(fact_168_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A3:A,F:(B > A),B3:B,C3:B]: (((A3 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C3) => ((! [X4:B,Y4:B]: (((((ord_less_eq @ B) @ X4) @ Y4) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A3) @ (F @ C3))))))))))).
% 2.66/1.46 thf(fact_169_order__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,B3:A,F:(A > C),C3:C]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C3) => ((! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => (((ord_less_eq @ C) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less_eq @ C) @ (F @ A3)) @ C3)))))))))).
% 2.66/1.46 thf(fact_170_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,F:(B > A),B3:B,C3:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C3) => ((! [X4:B,Y4:B]: (((((ord_less_eq @ B) @ X4) @ Y4) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A3) @ (F @ C3))))))))))).
% 2.66/1.46 thf(fact_171_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F2 @ X3)) @ (G2 @ X3))))))))))).
% 2.66/1.46 thf(fact_172_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B)]: (((! [X4:A]: ((((ord_less_eq @ B) @ (F @ X4)) @ (G @ X4)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.66/1.46 thf(fact_173_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))))))))).
% 2.66/1.46 thf(fact_174_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))))))))).
% 2.66/1.46 thf(fact_175_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X3:A]: (((P @ X3) => (Q @ X3)))))))).
% 2.66/1.46 thf(fact_176_contra__subsetD, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => ((~ ((((member @ A) @ C3) @ B2))) => (~ ((((member @ A) @ C3) @ A5)))))))).
% 2.66/1.46 thf(fact_177_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y5:(set @ A),Z4:(set @ A)]: ((Y5 = Z4))) = (^ [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) & (((ord_less_eq @ (set @ A)) @ B7) @ A6)))))))).
% 2.66/1.46 thf(fact_178_subset__trans, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C4) => (((ord_less_eq @ (set @ A)) @ A5) @ C4)))))).
% 2.66/1.46 thf(fact_179_UNIV__witness, axiom, (! [A:$tType]: ((? [X4:A]: ((((member @ A) @ X4) @ (top_top @ (set @ A)))))))).
% 2.66/1.46 thf(fact_180_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) => (Q @ X4)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.66/1.46 thf(fact_181_subset__refl, axiom, (! [A:$tType,A5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A5) @ A5)))).
% 2.66/1.46 thf(fact_182_subset__UNIV, axiom, (! [A:$tType,A5:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A5) @ (top_top @ (set @ A)))))).
% 2.66/1.46 thf(fact_183_rev__subsetD, axiom, (! [A:$tType,C3:A,A5:(set @ A),B2:(set @ A)]: (((((member @ A) @ C3) @ A5) => ((((ord_less_eq @ (set @ A)) @ A5) @ B2) => (((member @ A) @ C3) @ B2)))))).
% 2.66/1.46 thf(fact_184_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: ((! [T:A]: (((((member @ A) @ T) @ A6) => (((member @ A) @ T) @ B7)))))))))).
% 2.66/1.46 thf(fact_185_set__rev__mp, axiom, (! [A:$tType,X:A,A5:(set @ A),B2:(set @ A)]: (((((member @ A) @ X) @ A5) => ((((ord_less_eq @ (set @ A)) @ A5) @ B2) => (((member @ A) @ X) @ B2)))))).
% 2.66/1.46 thf(fact_186_equalityD2, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A)]: (((A5 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A5))))).
% 2.66/1.46 thf(fact_187_equalityD1, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A)]: (((A5 = B2) => (((ord_less_eq @ (set @ A)) @ A5) @ B2))))).
% 2.66/1.46 thf(fact_188_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((member @ A) @ X3) @ B7)))))))))).
% 2.66/1.46 thf(fact_189_equalityE, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A)]: (((A5 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A5)))))))))).
% 2.66/1.46 thf(fact_190_UNIV__eq__I, axiom, (! [A:$tType,A5:(set @ A)]: (((! [X4:A]: ((((member @ A) @ X4) @ A5))) => ((top_top @ (set @ A)) = A5))))).
% 2.66/1.46 thf(fact_191_subsetCE, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => ((((member @ A) @ C3) @ A5) => (((member @ A) @ C3) @ B2)))))).
% 2.66/1.46 thf(fact_192_subsetD, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => ((((member @ A) @ C3) @ A5) => (((member @ A) @ C3) @ B2)))))).
% 2.66/1.46 thf(fact_193_in__mono, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => ((((member @ A) @ X) @ A5) => (((member @ A) @ X) @ B2)))))).
% 2.66/1.46 thf(fact_194_set__mp, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => ((((member @ A) @ X) @ A5) => (((member @ A) @ X) @ B2)))))).
% 2.66/1.46 thf(fact_195_insert__UNIV, axiom, (! [A:$tType,X:A]: (((((insert @ A) @ X) @ (top_top @ (set @ A))) = (top_top @ (set @ A)))))).
% 2.66/1.46 thf(fact_196_Ball__Collect, axiom, (! [A:$tType]: (((ball @ A) = (^ [A6:(set @ A),P2:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ A6) @ ((collect @ A) @ P2)))))))).
% 2.66/1.46 thf(fact_197_subset__insertI2, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A),B3:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => (((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ B3) @ B2)))))).
% 2.66/1.46 thf(fact_198_subset__insertI, axiom, (! [A:$tType,B2:(set @ A),A3:A]: ((((ord_less_eq @ (set @ A)) @ B2) @ (((insert @ A) @ A3) @ B2))))).
% 2.66/1.46 thf(fact_199_subset__insert, axiom, (! [A:$tType,X:A,A5:(set @ A),B2:(set @ A)]: (((~ ((((member @ A) @ X) @ A5))) => ((((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ X) @ B2)) = (((ord_less_eq @ (set @ A)) @ A5) @ B2)))))).
% 2.66/1.46 thf(fact_200_insert__mono, axiom, (! [A:$tType,C4:(set @ A),D3:(set @ A),A3:A]: (((((ord_less_eq @ (set @ A)) @ C4) @ D3) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A3) @ C4)) @ (((insert @ A) @ A3) @ D3)))))).
% 2.66/1.46 thf(fact_201_empty__not__UNIV, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) != (top_top @ (set @ A)))))).
% 2.66/1.46 thf(fact_202_surjD, axiom, (! [A:$tType,B:$tType,F:(B > A),Y:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X4:B]: ((Y = (F @ X4)))))))).
% 2.66/1.46 thf(fact_203_surjE, axiom, (! [A:$tType,B:$tType,F:(B > A),Y:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X4:B]: ((Y != (F @ X4)))))))))).
% 2.66/1.46 thf(fact_204_surjI, axiom, (! [B:$tType,A:$tType,G:(B > A),F:(A > B)]: (((! [X4:A]: (((G @ (F @ X4)) = X4))) => (((((image @ B) @ A) @ G) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.66/1.46 thf(fact_205_surj__def, axiom, (! [B:$tType,A:$tType,F:(B > A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y3:A]: ((? [X3:B]: ((Y3 = (F @ X3)))))))))).
% 2.66/1.46 thf(fact_206_rangeI, axiom, (! [A:$tType,B:$tType,F:(B > A),X:B]: ((((member @ A) @ (F @ X)) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B))))))).
% 2.66/1.46 thf(fact_207_range__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),X:B]: (((B3 = (F @ X)) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))))))).
% 2.66/1.46 thf(fact_208_image__mono, axiom, (! [B:$tType,A:$tType,A5:(set @ A),B2:(set @ A),F:(A > B)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A5)) @ ((((image @ A) @ B) @ F) @ B2)))))).
% 2.66/1.46 thf(fact_209_image__subsetI, axiom, (! [A:$tType,B:$tType,A5:(set @ A),F:(A > B),B2:(set @ B)]: (((! [X4:A]: (((((member @ A) @ X4) @ A5) => (((member @ B) @ (F @ X4)) @ B2)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A5)) @ B2))))).
% 2.66/1.46 thf(fact_210_subset__imageE, axiom, (! [A:$tType,B:$tType,B2:(set @ A),F:(B > A),A5:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B2) @ ((((image @ B) @ A) @ F) @ A5)) => (~ ((! [C5:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C5) @ A5) => (B2 != ((((image @ B) @ A) @ F) @ C5))))))))))).
% 2.66/1.46 thf(fact_211_image__subset__iff, axiom, (! [A:$tType,B:$tType,F:(B > A),A5:(set @ B),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A5)) @ B2) = (! [X3:B]: (((((member @ B) @ X3) @ A5) => (((member @ A) @ (F @ X3)) @ B2)))))))).
% 2.66/1.46 thf(fact_212_subset__image__iff, axiom, (! [A:$tType,B:$tType,B2:(set @ A),F:(B > A),A5:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B2) @ ((((image @ B) @ A) @ F) @ A5)) = (? [AA:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ AA) @ A5) & (B2 = ((((image @ B) @ A) @ F) @ AA))))))))).
% 2.66/1.46 thf(fact_213_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) => (A3 = (bot_bot @ A))))))))).
% 2.66/1.46 thf(fact_214_bot_Oextremum__unique, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) = (A3 = (bot_bot @ A))))))))).
% 2.66/1.46 thf(fact_215_bot_Oextremum, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A3:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A3))))))).
% 2.66/1.46 thf(fact_216_pairwise__subset, axiom, (! [A:$tType,P:(A > (A > $o)),S4:(set @ A),T2:(set @ A)]: (((((pairwise @ A) @ P) @ S4) => ((((ord_less_eq @ (set @ A)) @ T2) @ S4) => (((pairwise @ A) @ P) @ T2)))))).
% 2.66/1.46 thf(fact_217_subset__singletonD, axiom, (! [A:$tType,A5:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) => ((A5 = (bot_bot @ (set @ A))) | (A5 = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))).
% 2.66/1.46 thf(fact_218_subset__singleton__iff, axiom, (! [A:$tType,X7:(set @ A),A3:A]: (((((ord_less_eq @ (set @ A)) @ X7) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) = ((X7 = (bot_bot @ (set @ A))) | (X7 = (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))))).
% 2.66/1.46 thf(fact_219_comp__surj, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > A),G:(A > C)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((((((image @ A) @ C) @ G) @ (top_top @ (set @ A))) = (top_top @ (set @ C))) => (((((image @ B) @ C) @ (((((comp @ A) @ C) @ B) @ G) @ F)) @ (top_top @ (set @ B))) = (top_top @ (set @ C)))))))).
% 2.66/1.46 thf(fact_220_fun_Oinj__map__strong, axiom, (! [B:$tType,A:$tType,D2:$tType,X:(D2 > A),Xa2:(D2 > A),F:(A > B),Fa:(A > B)]: (((! [Z:A,Za:A]: (((((member @ A) @ Z) @ ((((image @ D2) @ A) @ X) @ (top_top @ (set @ D2)))) => ((((member @ A) @ Za) @ ((((image @ D2) @ A) @ Xa2) @ (top_top @ (set @ D2)))) => (((F @ Z) = (Fa @ Za)) => (Z = Za)))))) => (((((((comp @ A) @ B) @ D2) @ F) @ X) = (((((comp @ A) @ B) @ D2) @ Fa) @ Xa2)) => (X = Xa2)))))).
% 2.66/1.46 thf(fact_221_fun_Omap__cong0, axiom, (! [B:$tType,A:$tType,D2:$tType,X:(D2 > A),F:(A > B),G:(A > B)]: (((! [Z:A]: (((((member @ A) @ Z) @ ((((image @ D2) @ A) @ X) @ (top_top @ (set @ D2)))) => ((F @ Z) = (G @ Z))))) => ((((((comp @ A) @ B) @ D2) @ F) @ X) = (((((comp @ A) @ B) @ D2) @ G) @ X)))))).
% 2.66/1.46 thf(fact_222_fun_Omap__cong, axiom, (! [B:$tType,A:$tType,D2:$tType,X:(D2 > A),Ya:(D2 > A),F:(A > B),G:(A > B)]: (((X = Ya) => ((! [Z:A]: (((((member @ A) @ Z) @ ((((image @ D2) @ A) @ Ya) @ (top_top @ (set @ D2)))) => ((F @ Z) = (G @ Z))))) => ((((((comp @ A) @ B) @ D2) @ F) @ X) = (((((comp @ A) @ B) @ D2) @ G) @ Ya))))))).
% 2.66/1.46 thf(fact_223_fun_Oset__map, axiom, (! [B:$tType,A:$tType,D2:$tType,F:(A > B),V:(D2 > A)]: ((((((image @ D2) @ B) @ (((((comp @ A) @ B) @ D2) @ F) @ V)) @ (top_top @ (set @ D2))) = ((((image @ A) @ B) @ F) @ ((((image @ D2) @ A) @ V) @ (top_top @ (set @ D2)))))))).
% 2.66/1.46 thf(fact_224_surj__fun__eq, axiom, (! [B:$tType,C:$tType,A:$tType,F:(B > A),X7:(set @ B),G12:(A > C),G23:(A > C)]: (((((((image @ B) @ A) @ F) @ X7) = (top_top @ (set @ A))) => ((! [X4:B]: (((((member @ B) @ X4) @ X7) => (((((((comp @ A) @ C) @ B) @ G12) @ F) @ X4) = ((((((comp @ A) @ C) @ B) @ G23) @ F) @ X4))))) => (G12 = G23)))))).
% 2.66/1.46 thf(fact_225_insert__subsetI, axiom, (! [A:$tType,X:A,A5:(set @ A),X7:(set @ A)]: (((((member @ A) @ X) @ A5) => ((((ord_less_eq @ (set @ A)) @ X7) @ A5) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X) @ X7)) @ A5)))))).
% 2.66/1.46 thf(fact_226_subset__emptyI, axiom, (! [A:$tType,A5:(set @ A)]: (((! [X4:A]: ((~ ((((member @ A) @ X4) @ A5))))) => (((ord_less_eq @ (set @ A)) @ A5) @ (bot_bot @ (set @ A))))))).
% 2.66/1.46 thf(fact_227_subset__Compl__singleton, axiom, (! [A:$tType,A5:(set @ A),B3:A]: (((((ord_less_eq @ (set @ A)) @ A5) @ ((uminus_uminus @ (set @ A)) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A))))) = (~ ((((member @ A) @ B3) @ A5))))))).
% 2.66/1.46 thf(fact_228_ComplI, axiom, (! [A:$tType,C3:A,A5:(set @ A)]: (((~ ((((member @ A) @ C3) @ A5))) => (((member @ A) @ C3) @ ((uminus_uminus @ (set @ A)) @ A5)))))).
% 2.66/1.46 thf(fact_229_Compl__iff, axiom, (! [A:$tType,C3:A,A5:(set @ A)]: (((((member @ A) @ C3) @ ((uminus_uminus @ (set @ A)) @ A5)) = (~ ((((member @ A) @ C3) @ A5))))))).
% 2.66/1.46 thf(fact_230_Compl__eq__Compl__iff, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A)]: (((((uminus_uminus @ (set @ A)) @ A5) = ((uminus_uminus @ (set @ A)) @ B2)) = (A5 = B2))))).
% 2.66/1.46 thf(fact_231_Compl__subset__Compl__iff, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A5)) @ ((uminus_uminus @ (set @ A)) @ B2)) = (((ord_less_eq @ (set @ A)) @ B2) @ A5))))).
% 2.66/1.46 thf(fact_232_Compl__anti__mono, axiom, (! [A:$tType,A5:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B2) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ B2)) @ ((uminus_uminus @ (set @ A)) @ A5)))))).
% 2.66/1.46 thf(fact_233_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.66/1.46 thf(fact_234_stream_Orel__mono, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o)),Ra:(A > (B > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ R) @ Ra) => (((ord_less_eq @ ((stream170649215stream @ A) > ((stream170649215stream @ B) > $o))) @ (((stream686525009m_all2 @ A) @ B) @ R)) @ (((stream686525009m_all2 @ A) @ B) @ Ra)))))).
% 2.66/1.46 thf(fact_235_ComplD, axiom, (! [A:$tType,C3:A,A5:(set @ A)]: (((((member @ A) @ C3) @ ((uminus_uminus @ (set @ A)) @ A5)) => (~ ((((member @ A) @ C3) @ A5))))))).
% 2.66/1.46 thf(fact_236_double__complement, axiom, (! [A:$tType,A5:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A5)) = A5)))).
% 2.66/1.46 thf(fact_237_subset__Compl__self__eq, axiom, (! [A:$tType,A5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ ((uminus_uminus @ (set @ A)) @ A5)) = (A5 = (bot_bot @ (set @ A))))))).
% 2.66/1.46 thf(fact_238_Compl__empty__eq, axiom, (! [A:$tType]: ((((uminus_uminus @ (set @ A)) @ (bot_bot @ (set @ A))) = (top_top @ (set @ A)))))).
% 2.66/1.46 thf(fact_239_Compl__UNIV__eq, axiom, (! [A:$tType]: ((((uminus_uminus @ (set @ A)) @ (top_top @ (set @ A))) = (bot_bot @ (set @ A)))))).
% 2.66/1.46 thf(fact_240_surj__Compl__image__subset, axiom, (! [A:$tType,B:$tType,F:(B > A),A5:(set @ B)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A5))) @ ((((image @ B) @ A) @ F) @ ((uminus_uminus @ (set @ B)) @ A5))))))).
% 2.66/1.46 thf(fact_241_compl__top__eq, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (((uminus_uminus @ A) @ (top_top @ A)) = (bot_bot @ A)))))).
% 2.66/1.46 thf(fact_242_compl__bot__eq, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (((uminus_uminus @ A) @ (bot_bot @ A)) = (top_top @ A)))))).
% 2.66/1.46 thf(fact_243_predicate2I, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Q:(A > (B > $o))]: (((! [X4:A,Y4:B]: ((((P @ X4) @ Y4) => ((Q @ X4) @ Y4)))) => (((ord_less_eq @ (A > (B > $o))) @ P) @ Q))))).
% 2.66/1.46 thf(fact_244_top1I, axiom, (! [A:$tType,X:A]: (((top_top @ (A > $o)) @ X)))).
% 2.66/1.46 thf(fact_245_rev__predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),X:A,Y:B,Q:(A > (B > $o))]: ((((P @ X) @ Y) => ((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => ((Q @ X) @ Y)))))).
% 2.66/1.46 thf(fact_246_predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Q:(A > (B > $o)),X:A,Y:B]: (((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => (((P @ X) @ Y) => ((Q @ X) @ Y)))))).
% 2.66/1.46 thf(fact_247_refl__ge__eq, axiom, (! [A:$tType,R:(A > (A > $o))]: (((! [X4:A]: (((R @ X4) @ X4))) => (((ord_less_eq @ (A > (A > $o))) @ (^ [Y5:A,Z4:A]: ((Y5 = Z4)))) @ R))))).
% 2.66/1.46 thf(fact_248_ge__eq__refl, axiom, (! [A:$tType,R:(A > (A > $o)),X:A]: (((((ord_less_eq @ (A > (A > $o))) @ (^ [Y5:A,Z4:A]: ((Y5 = Z4)))) @ R) => ((R @ X) @ X))))).
% 2.66/1.46 thf(fact_249_predicate2D__conj, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Q:(A > (B > $o)),R:$o,X:A,Y:B]: ((((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) & R) => (R & (((P @ X) @ Y) => ((Q @ X) @ Y))))))).
% 2.66/1.46 thf(fact_250_top__conj_I1_J, axiom, (! [A:$tType,X:A,P:$o]: (((((top_top @ (A > $o)) @ X) & P) = P)))).
% 2.66/1.46 thf(fact_251_top__conj_I2_J, axiom, (! [A:$tType,P:$o,X:A]: (((P & ((top_top @ (A > $o)) @ X)) = P)))).
% 2.66/1.46 thf(fact_252_inj__image__Compl__subset, axiom, (! [B:$tType,A:$tType,F:(A > B),A5:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ ((uminus_uminus @ (set @ A)) @ A5))) @ ((uminus_uminus @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A5))))))).
% 2.66/1.46 thf(fact_253_ID_Opred__set, axiom, (! [A:$tType]: (((bNF_id_bnf @ (A > $o)) = (^ [P2:(A > $o),X3:A]: ((! [Y3:A]: (((((member @ A) @ Y3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) => (P2 @ Y3)))))))))).
% 2.66/1.46 thf(fact_254_inj__on__empty, axiom, (! [B:$tType,A:$tType,F:(A > B)]: (((((inj_on @ A) @ B) @ F) @ (bot_bot @ (set @ A)))))).
% 2.66/1.46 thf(fact_255_inj__uminus, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A5:(set @ A)]: (((((inj_on @ A) @ A) @ (uminus_uminus @ A)) @ A5))))))).
% 2.66/1.46 thf(tcon_fun___Lattices_Oboolean__algebra, axiom, (! [A8:$tType,A9:$tType]: ((((boolean_algebra @ A9) @ (type2 @ A9)) => ((boolean_algebra @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.66/1.46 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A8:$tType,A9:$tType]: ((((order_top @ A9) @ (type2 @ A9)) => ((order_top @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.66/1.46 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A8:$tType,A9:$tType]: ((((order_bot @ A9) @ (type2 @ A9)) => ((order_bot @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.66/1.46 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: ((((preorder @ A9) @ (type2 @ A9)) => ((preorder @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.66/1.46 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: ((((order @ A9) @ (type2 @ A9)) => ((order @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.66/1.46 thf(tcon_fun___Orderings_Otop, axiom, (! [A8:$tType,A9:$tType]: ((((top @ A9) @ (type2 @ A9)) => ((top @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.66/1.46 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: ((((ord @ A9) @ (type2 @ A9)) => ((ord @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.66/1.46 thf(tcon_fun___Orderings_Obot, axiom, (! [A8:$tType,A9:$tType]: ((((bot @ A9) @ (type2 @ A9)) => ((bot @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.66/1.46 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_1, axiom, (! [A8:$tType]: (((boolean_algebra @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.66/1.46 thf(tcon_Set_Oset___Orderings_Oorder__top_2, axiom, (! [A8:$tType]: (((order_top @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.66/1.46 thf(tcon_Set_Oset___Orderings_Oorder__bot_3, axiom, (! [A8:$tType]: (((order_bot @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.66/1.46 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A8:$tType]: (((preorder @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.66/1.46 thf(tcon_Set_Oset___Orderings_Oorder_5, axiom, (! [A8:$tType]: (((order @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.66/1.46 thf(tcon_Set_Oset___Orderings_Otop_6, axiom, (! [A8:$tType]: (((top @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.66/1.46 thf(tcon_Set_Oset___Orderings_Oord_7, axiom, (! [A8:$tType]: (((ord @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.66/1.46 thf(tcon_Set_Oset___Orderings_Obot_8, axiom, (! [A8:$tType]: (((bot @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.66/1.46 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_9, axiom, ((boolean_algebra @ $o) @ (type2 @ $o))).
% 2.66/1.46 thf(tcon_HOL_Obool___Orderings_Oorder__top_10, axiom, ((order_top @ $o) @ (type2 @ $o))).
% 2.66/1.46 thf(tcon_HOL_Obool___Orderings_Oorder__bot_11, axiom, ((order_bot @ $o) @ (type2 @ $o))).
% 2.66/1.46 thf(tcon_HOL_Obool___Orderings_Opreorder_12, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.66/1.46 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.66/1.46 thf(tcon_HOL_Obool___Orderings_Oorder_13, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.66/1.46 thf(tcon_HOL_Obool___Orderings_Otop_14, axiom, ((top @ $o) @ (type2 @ $o))).
% 2.66/1.46 thf(tcon_HOL_Obool___Orderings_Oord_15, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.66/1.46 thf(tcon_HOL_Obool___Orderings_Obot_16, axiom, ((bot @ $o) @ (type2 @ $o))).
% 2.66/1.46 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.66/1.46 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.66/1.46 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.66/1.46 thf(conj_0, conjecture, ((((member @ a) @ x) @ ((stream30925839e_sset @ a) @ (((stream641971652_SCons @ a) @ y) @ s))) != (~ (((x != y) => (((member @ a) @ x) @ ((stream30925839e_sset @ a) @ s))))))).
% 2.66/1.46 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.e7YR8rdmNP/DTF2DTF22005.p
% 2.66/1.46 ------------------------
% 2.90/1.51 ---- Cleaned THF ---
% 2.90/1.51 thf(ty_t_Stream__Mirabelle__hbrgyiwlrc_Ostream,type,
% 2.90/1.51 stream170649215stream: $tType > $tType ).
% 2.90/1.51
% 2.90/1.51 thf(ty_t_Set_Oset,type,
% 2.90/1.51 set: $tType > $tType ).
% 2.90/1.51
% 2.90/1.51 thf(ty_t_itself,type,
% 2.90/1.51 itself: $tType > $tType ).
% 2.90/1.51
% 2.90/1.51 thf(ty_tf_a,type,
% 2.90/1.51 a: $tType ).
% 2.90/1.51
% 2.90/1.51 thf(sy_cl_HOL_Otype,type,
% 2.90/1.51 type:
% 2.90/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_cl_Orderings_Obot,type,
% 2.90/1.51 bot:
% 2.90/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_cl_Orderings_Oord,type,
% 2.90/1.51 ord:
% 2.90/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_cl_Orderings_Otop,type,
% 2.90/1.51 top:
% 2.90/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_cl_Orderings_Oorder,type,
% 2.90/1.51 order:
% 2.90/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_cl_Orderings_Olinorder,type,
% 2.90/1.51 linorder:
% 2.90/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_cl_Orderings_Opreorder,type,
% 2.90/1.51 preorder:
% 2.90/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.90/1.51 order_bot:
% 2.90/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_cl_Orderings_Oorder__top,type,
% 2.90/1.51 order_top:
% 2.90/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 2.90/1.51 boolean_algebra:
% 2.90/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.90/1.51 ordered_ab_group_add:
% 2.90/1.51 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_BNF__Composition_Oid__bnf,type,
% 2.90/1.51 bNF_id_bnf:
% 2.90/1.51 !>[A: $tType] : ( A > A ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Fun_Ocomp,type,
% 2.90/1.51 comp:
% 2.90/1.51 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Fun_Oinj__on,type,
% 2.90/1.51 inj_on:
% 2.90/1.51 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Fun_Ooverride__on,type,
% 2.90/1.51 override_on:
% 2.90/1.51 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 2.90/1.51 uminus_uminus:
% 2.90/1.51 !>[A: $tType] : ( A > A ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_If,type,
% 2.90/1.51 if:
% 2.90/1.51 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.90/1.51 bot_bot:
% 2.90/1.51 !>[A: $tType] : A ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.90/1.51 ord_less_eq:
% 2.90/1.51 !>[A: $tType] : ( A > A > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.90/1.51 top_top:
% 2.90/1.51 !>[A: $tType] : A ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Pure_Otype,type,
% 2.90/1.51 type2:
% 2.90/1.51 !>[A: $tType] : ( itself @ A ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Set_OBall,type,
% 2.90/1.51 ball:
% 2.90/1.51 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Set_OCollect,type,
% 2.90/1.51 collect:
% 2.90/1.51 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Set_Oimage,type,
% 2.90/1.51 image:
% 2.90/1.51 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Set_Oinsert,type,
% 2.90/1.51 insert:
% 2.90/1.51 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Set_Ois__empty,type,
% 2.90/1.51 is_empty:
% 2.90/1.51 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Set_Ois__singleton,type,
% 2.90/1.51 is_singleton:
% 2.90/1.51 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Set_Opairwise,type,
% 2.90/1.51 pairwise:
% 2.90/1.51 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Set_Othe__elem,type,
% 2.90/1.51 the_elem:
% 2.90/1.51 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Osmember,type,
% 2.90/1.51 stream1586597341member:
% 2.90/1.51 !>[A: $tType] : ( A > ( stream170649215stream @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_OSCons,type,
% 2.90/1.51 stream641971652_SCons:
% 2.90/1.51 !>[A: $tType] : ( A > ( stream170649215stream @ A ) > ( stream170649215stream @ A ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Ocase__stream,type,
% 2.90/1.51 stream1342653232stream:
% 2.90/1.51 !>[A: $tType,B: $tType] : ( ( A > ( stream170649215stream @ A ) > B ) > ( stream170649215stream @ A ) > B ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Ocorec__stream,type,
% 2.90/1.51 stream660621732stream:
% 2.90/1.51 !>[C: $tType,A: $tType] : ( ( C > A ) > ( C > $o ) > ( C > ( stream170649215stream @ A ) ) > ( C > C ) > C > ( stream170649215stream @ A ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Opred__stream,type,
% 2.90/1.51 stream1153105665stream:
% 2.90/1.51 !>[A: $tType] : ( ( A > $o ) > ( stream170649215stream @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Oshd,type,
% 2.90/1.51 stream_Mirabelle_shd:
% 2.90/1.51 !>[A: $tType] : ( ( stream170649215stream @ A ) > A ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Osmap,type,
% 2.90/1.51 stream2128578057e_smap:
% 2.90/1.51 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( stream170649215stream @ A ) > ( stream170649215stream @ Aa ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Osset,type,
% 2.90/1.51 stream30925839e_sset:
% 2.90/1.51 !>[A: $tType] : ( ( stream170649215stream @ A ) > ( set @ A ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Ostl,type,
% 2.90/1.51 stream_Mirabelle_stl:
% 2.90/1.51 !>[A: $tType] : ( ( stream170649215stream @ A ) > ( stream170649215stream @ A ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Ostream_Ostream__all2,type,
% 2.90/1.51 stream686525009m_all2:
% 2.90/1.51 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( stream170649215stream @ A ) > ( stream170649215stream @ B ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_c_member,type,
% 2.90/1.51 member:
% 2.90/1.51 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(sy_v_s,type,
% 2.90/1.51 s: stream170649215stream @ a ).
% 2.90/1.51
% 2.90/1.51 thf(sy_v_x,type,
% 2.90/1.51 x: a ).
% 2.90/1.51
% 2.90/1.51 thf(sy_v_y,type,
% 2.90/1.51 y: a ).
% 2.90/1.51
% 2.90/1.51 thf(fact_0_stream_Oinject,axiom,
% 2.90/1.51 ! [A: $tType,X1: A,X2: stream170649215stream @ A,Y1: A,Y2: stream170649215stream @ A] :
% 2.90/1.51 ( ( ( stream641971652_SCons @ A @ X1 @ X2 )
% 2.90/1.51 = ( stream641971652_SCons @ A @ Y1 @ Y2 ) )
% 2.90/1.51 = ( ( X1 = Y1 )
% 2.90/1.51 & ( X2 = Y2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_1_stream_Oset__intros_I2_J,axiom,
% 2.90/1.51 ! [A: $tType,X: A,A2: stream170649215stream @ A,A1: A] :
% 2.90/1.51 ( ( member @ A @ X @ ( stream30925839e_sset @ A @ A2 ) )
% 2.90/1.51 => ( member @ A @ X @ ( stream30925839e_sset @ A @ ( stream641971652_SCons @ A @ A1 @ A2 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_2_stream_Oset__intros_I1_J,axiom,
% 2.90/1.51 ! [A: $tType,A1: A,A2: stream170649215stream @ A] : ( member @ A @ A1 @ ( stream30925839e_sset @ A @ ( stream641971652_SCons @ A @ A1 @ A2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_3_stream_Oexhaust,axiom,
% 2.90/1.51 ! [A: $tType,Y: stream170649215stream @ A] :
% 2.90/1.51 ~ ! [X12: A,X22: stream170649215stream @ A] :
% 2.90/1.51 ( Y
% 2.90/1.51 != ( stream641971652_SCons @ A @ X12 @ X22 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_4_stream_Oset__cases,axiom,
% 2.90/1.51 ! [A: $tType,E: A,A3: stream170649215stream @ A] :
% 2.90/1.51 ( ( member @ A @ E @ ( stream30925839e_sset @ A @ A3 ) )
% 2.90/1.51 => ( ! [Z2: stream170649215stream @ A] :
% 2.90/1.51 ( A3
% 2.90/1.51 != ( stream641971652_SCons @ A @ E @ Z2 ) )
% 2.90/1.51 => ~ ! [Z1: A,Z2: stream170649215stream @ A] :
% 2.90/1.51 ( ( A3
% 2.90/1.51 = ( stream641971652_SCons @ A @ Z1 @ Z2 ) )
% 2.90/1.51 => ~ ( member @ A @ E @ ( stream30925839e_sset @ A @ Z2 ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_5_stream_Oset__induct,axiom,
% 2.90/1.51 ! [A: $tType,X: A,A3: stream170649215stream @ A,P: A > ( stream170649215stream @ A ) > $o] :
% 2.90/1.51 ( ( member @ A @ X @ ( stream30925839e_sset @ A @ A3 ) )
% 2.90/1.51 => ( ! [Z1: A,Z2: stream170649215stream @ A] : ( P @ Z1 @ ( stream641971652_SCons @ A @ Z1 @ Z2 ) )
% 2.90/1.51 => ( ! [Z1: A,Z2: stream170649215stream @ A,Xa: A] :
% 2.90/1.51 ( ( member @ A @ Xa @ ( stream30925839e_sset @ A @ Z2 ) )
% 2.90/1.51 => ( ( P @ Xa @ Z2 )
% 2.90/1.51 => ( P @ Xa @ ( stream641971652_SCons @ A @ Z1 @ Z2 ) ) ) )
% 2.90/1.51 => ( P @ X @ A3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_6_Stream__Mirabelle__hbrgyiwlrc_Osmember__def,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( stream1586597341member @ A )
% 2.90/1.51 = ( ^ [X3: A,S: stream170649215stream @ A] : ( member @ A @ X3 @ ( stream30925839e_sset @ A @ S ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_7_stream_Oset,axiom,
% 2.90/1.51 ! [A: $tType,X1: A,X2: stream170649215stream @ A] :
% 2.90/1.51 ( ( stream30925839e_sset @ A @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
% 2.90/1.51 = ( insert @ A @ X1 @ ( stream30925839e_sset @ A @ X2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_8_stream_Opred__inject,axiom,
% 2.90/1.51 ! [A: $tType,P: A > $o,A3: A,Aa2: stream170649215stream @ A] :
% 2.90/1.51 ( ( stream1153105665stream @ A @ P @ ( stream641971652_SCons @ A @ A3 @ Aa2 ) )
% 2.90/1.51 = ( ( P @ A3 )
% 2.90/1.51 & ( stream1153105665stream @ A @ P @ Aa2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_9_stream_Ocase,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,F: A > ( stream170649215stream @ A ) > B,X1: A,X2: stream170649215stream @ A] :
% 2.90/1.51 ( ( stream1342653232stream @ A @ B @ F @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
% 2.90/1.51 = ( F @ X1 @ X2 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_10_stream_Ocorec__code,axiom,
% 2.90/1.51 ! [A: $tType,C: $tType] :
% 2.90/1.51 ( ( stream660621732stream @ C @ A )
% 2.90/1.51 = ( ^ [G1: C > A,Q2: C > $o,G21: C > ( stream170649215stream @ A ),G22: C > C,A4: C] : ( stream641971652_SCons @ A @ ( G1 @ A4 ) @ ( if @ ( stream170649215stream @ A ) @ ( Q2 @ A4 ) @ ( G21 @ A4 ) @ ( stream660621732stream @ C @ A @ G1 @ Q2 @ G21 @ G22 @ ( G22 @ A4 ) ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_11_stream_Opred__cong,axiom,
% 2.90/1.51 ! [A: $tType,X: stream170649215stream @ A,Ya: stream170649215stream @ A,P: A > $o,Pa: A > $o] :
% 2.90/1.51 ( ( X = Ya )
% 2.90/1.51 => ( ! [Z: A] :
% 2.90/1.51 ( ( member @ A @ Z @ ( stream30925839e_sset @ A @ Ya ) )
% 2.90/1.51 => ( ( P @ Z )
% 2.90/1.51 = ( Pa @ Z ) ) )
% 2.90/1.51 => ( ( stream1153105665stream @ A @ P @ X )
% 2.90/1.51 = ( stream1153105665stream @ A @ Pa @ Ya ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_12_stream_Opred__mono__strong,axiom,
% 2.90/1.51 ! [A: $tType,P: A > $o,X: stream170649215stream @ A,Pa: A > $o] :
% 2.90/1.51 ( ( stream1153105665stream @ A @ P @ X )
% 2.90/1.51 => ( ! [Z: A] :
% 2.90/1.51 ( ( member @ A @ Z @ ( stream30925839e_sset @ A @ X ) )
% 2.90/1.51 => ( ( P @ Z )
% 2.90/1.51 => ( Pa @ Z ) ) )
% 2.90/1.51 => ( stream1153105665stream @ A @ Pa @ X ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_13_stream_Osel_I1_J,axiom,
% 2.90/1.51 ! [A: $tType,X1: A,X2: stream170649215stream @ A] :
% 2.90/1.51 ( ( stream_Mirabelle_shd @ A @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
% 2.90/1.51 = X1 ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_14_stream_Oset__sel_I1_J,axiom,
% 2.90/1.51 ! [A: $tType,A3: stream170649215stream @ A] : ( member @ A @ ( stream_Mirabelle_shd @ A @ A3 ) @ ( stream30925839e_sset @ A @ A3 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_15_stream_Osel_I2_J,axiom,
% 2.90/1.51 ! [A: $tType,X1: A,X2: stream170649215stream @ A] :
% 2.90/1.51 ( ( stream_Mirabelle_stl @ A @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
% 2.90/1.51 = X2 ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_16_stream_Oset__sel_I2_J,axiom,
% 2.90/1.51 ! [A: $tType,X: A,A3: stream170649215stream @ A] :
% 2.90/1.51 ( ( member @ A @ X @ ( stream30925839e_sset @ A @ ( stream_Mirabelle_stl @ A @ A3 ) ) )
% 2.90/1.51 => ( member @ A @ X @ ( stream30925839e_sset @ A @ A3 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_17_stream_Omap,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,F: A > B,X1: A,X2: stream170649215stream @ A] :
% 2.90/1.51 ( ( stream2128578057e_smap @ A @ B @ F @ ( stream641971652_SCons @ A @ X1 @ X2 ) )
% 2.90/1.51 = ( stream641971652_SCons @ B @ ( F @ X1 ) @ ( stream2128578057e_smap @ A @ B @ F @ X2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_18_stream_Ocollapse,axiom,
% 2.90/1.51 ! [A: $tType,Stream: stream170649215stream @ A] :
% 2.90/1.51 ( ( stream641971652_SCons @ A @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) )
% 2.90/1.51 = Stream ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_19_stream_Omap__sel_I2_J,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,F: A > B,A3: stream170649215stream @ A] :
% 2.90/1.51 ( ( stream_Mirabelle_stl @ B @ ( stream2128578057e_smap @ A @ B @ F @ A3 ) )
% 2.90/1.51 = ( stream2128578057e_smap @ A @ B @ F @ ( stream_Mirabelle_stl @ A @ A3 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_20_stream_Omap__sel_I1_J,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,F: A > B,A3: stream170649215stream @ A] :
% 2.90/1.51 ( ( stream_Mirabelle_shd @ B @ ( stream2128578057e_smap @ A @ B @ F @ A3 ) )
% 2.90/1.51 = ( F @ ( stream_Mirabelle_shd @ A @ A3 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_21_stream_Ocorec__sel_I2_J,axiom,
% 2.90/1.51 ! [A: $tType,C: $tType,Q22: C > $o,A3: C,G12: C > A,G212: C > ( stream170649215stream @ A ),G222: C > C] :
% 2.90/1.51 ( ( ( Q22 @ A3 )
% 2.90/1.51 => ( ( stream_Mirabelle_stl @ A @ ( stream660621732stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ A3 ) )
% 2.90/1.51 = ( G212 @ A3 ) ) )
% 2.90/1.51 & ( ~ ( Q22 @ A3 )
% 2.90/1.51 => ( ( stream_Mirabelle_stl @ A @ ( stream660621732stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ A3 ) )
% 2.90/1.51 = ( stream660621732stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ ( G222 @ A3 ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_22_stream_Ocorec__sel_I1_J,axiom,
% 2.90/1.51 ! [A: $tType,C: $tType,G12: C > A,Q22: C > $o,G212: C > ( stream170649215stream @ A ),G222: C > C,A3: C] :
% 2.90/1.51 ( ( stream_Mirabelle_shd @ A @ ( stream660621732stream @ C @ A @ G12 @ Q22 @ G212 @ G222 @ A3 ) )
% 2.90/1.51 = ( G12 @ A3 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_23_stream_Oexpand,axiom,
% 2.90/1.51 ! [A: $tType,Stream: stream170649215stream @ A,Stream2: stream170649215stream @ A] :
% 2.90/1.51 ( ( ( ( stream_Mirabelle_shd @ A @ Stream )
% 2.90/1.51 = ( stream_Mirabelle_shd @ A @ Stream2 ) )
% 2.90/1.51 & ( ( stream_Mirabelle_stl @ A @ Stream )
% 2.90/1.51 = ( stream_Mirabelle_stl @ A @ Stream2 ) ) )
% 2.90/1.51 => ( Stream = Stream2 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_24_stream_Ocoinduct,axiom,
% 2.90/1.51 ! [A: $tType,R: ( stream170649215stream @ A ) > ( stream170649215stream @ A ) > $o,Stream: stream170649215stream @ A,Stream2: stream170649215stream @ A] :
% 2.90/1.51 ( ( R @ Stream @ Stream2 )
% 2.90/1.51 => ( ! [Stream3: stream170649215stream @ A,Stream4: stream170649215stream @ A] :
% 2.90/1.51 ( ( R @ Stream3 @ Stream4 )
% 2.90/1.51 => ( ( ( stream_Mirabelle_shd @ A @ Stream3 )
% 2.90/1.51 = ( stream_Mirabelle_shd @ A @ Stream4 ) )
% 2.90/1.51 & ( R @ ( stream_Mirabelle_stl @ A @ Stream3 ) @ ( stream_Mirabelle_stl @ A @ Stream4 ) ) ) )
% 2.90/1.51 => ( Stream = Stream2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_25_stream_Ocase__eq__if,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType] :
% 2.90/1.51 ( ( stream1342653232stream @ A @ B )
% 2.90/1.51 = ( ^ [F2: A > ( stream170649215stream @ A ) > B,Stream5: stream170649215stream @ A] : ( F2 @ ( stream_Mirabelle_shd @ A @ Stream5 ) @ ( stream_Mirabelle_stl @ A @ Stream5 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_26_stream_Ocoinduct__strong,axiom,
% 2.90/1.51 ! [A: $tType,R: ( stream170649215stream @ A ) > ( stream170649215stream @ A ) > $o,Stream: stream170649215stream @ A,Stream2: stream170649215stream @ A] :
% 2.90/1.51 ( ( R @ Stream @ Stream2 )
% 2.90/1.51 => ( ! [Stream3: stream170649215stream @ A,Stream4: stream170649215stream @ A] :
% 2.90/1.51 ( ( R @ Stream3 @ Stream4 )
% 2.90/1.51 => ( ( ( stream_Mirabelle_shd @ A @ Stream3 )
% 2.90/1.51 = ( stream_Mirabelle_shd @ A @ Stream4 ) )
% 2.90/1.51 & ( ( R @ ( stream_Mirabelle_stl @ A @ Stream3 ) @ ( stream_Mirabelle_stl @ A @ Stream4 ) )
% 2.90/1.51 | ( ( stream_Mirabelle_stl @ A @ Stream3 )
% 2.90/1.51 = ( stream_Mirabelle_stl @ A @ Stream4 ) ) ) ) )
% 2.90/1.51 => ( Stream = Stream2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_27_stream_Ocorec__disc,axiom,
% 2.90/1.51 ! [A: $tType,C: $tType] :
% 2.90/1.51 ( ( stream660621732stream @ C @ A )
% 2.90/1.51 = ( stream660621732stream @ C @ A ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_28_stream_Osplit__sel__asm,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,P: B > $o,F: A > ( stream170649215stream @ A ) > B,Stream: stream170649215stream @ A] :
% 2.90/1.51 ( ( P @ ( stream1342653232stream @ A @ B @ F @ Stream ) )
% 2.90/1.51 = ( ~ ( ( Stream
% 2.90/1.51 = ( stream641971652_SCons @ A @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) )
% 2.90/1.51 & ~ ( P @ ( F @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_29_stream_Osplit__sel,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,P: B > $o,F: A > ( stream170649215stream @ A ) > B,Stream: stream170649215stream @ A] :
% 2.90/1.51 ( ( P @ ( stream1342653232stream @ A @ B @ F @ Stream ) )
% 2.90/1.51 = ( ( Stream
% 2.90/1.51 = ( stream641971652_SCons @ A @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) )
% 2.90/1.51 => ( P @ ( F @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_30_stream_Oexhaust__sel,axiom,
% 2.90/1.51 ! [A: $tType,Stream: stream170649215stream @ A] :
% 2.90/1.51 ( Stream
% 2.90/1.51 = ( stream641971652_SCons @ A @ ( stream_Mirabelle_shd @ A @ Stream ) @ ( stream_Mirabelle_stl @ A @ Stream ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_31_stream_Oinj__map__strong,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,X: stream170649215stream @ A,Xa2: stream170649215stream @ A,F: A > B,Fa: A > B] :
% 2.90/1.51 ( ! [Z: A,Za: A] :
% 2.90/1.51 ( ( member @ A @ Z @ ( stream30925839e_sset @ A @ X ) )
% 2.90/1.51 => ( ( member @ A @ Za @ ( stream30925839e_sset @ A @ Xa2 ) )
% 2.90/1.51 => ( ( ( F @ Z )
% 2.90/1.51 = ( Fa @ Za ) )
% 2.90/1.51 => ( Z = Za ) ) ) )
% 2.90/1.51 => ( ( ( stream2128578057e_smap @ A @ B @ F @ X )
% 2.90/1.51 = ( stream2128578057e_smap @ A @ B @ Fa @ Xa2 ) )
% 2.90/1.51 => ( X = Xa2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_32_stream_Omap__cong0,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,X: stream170649215stream @ A,F: A > B,G: A > B] :
% 2.90/1.51 ( ! [Z: A] :
% 2.90/1.51 ( ( member @ A @ Z @ ( stream30925839e_sset @ A @ X ) )
% 2.90/1.51 => ( ( F @ Z )
% 2.90/1.51 = ( G @ Z ) ) )
% 2.90/1.51 => ( ( stream2128578057e_smap @ A @ B @ F @ X )
% 2.90/1.51 = ( stream2128578057e_smap @ A @ B @ G @ X ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_33_stream_Omap__cong,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,X: stream170649215stream @ A,Ya: stream170649215stream @ A,F: A > B,G: A > B] :
% 2.90/1.51 ( ( X = Ya )
% 2.90/1.51 => ( ! [Z: A] :
% 2.90/1.51 ( ( member @ A @ Z @ ( stream30925839e_sset @ A @ Ya ) )
% 2.90/1.51 => ( ( F @ Z )
% 2.90/1.51 = ( G @ Z ) ) )
% 2.90/1.51 => ( ( stream2128578057e_smap @ A @ B @ F @ X )
% 2.90/1.51 = ( stream2128578057e_smap @ A @ B @ G @ Ya ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_34_insertCI,axiom,
% 2.90/1.51 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 2.90/1.51 ( ( ~ ( member @ A @ A3 @ B2 )
% 2.90/1.51 => ( A3 = B3 ) )
% 2.90/1.51 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_35_insert__iff,axiom,
% 2.90/1.51 ! [A: $tType,A3: A,B3: A,A5: set @ A] :
% 2.90/1.51 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A5 ) )
% 2.90/1.51 = ( ( A3 = B3 )
% 2.90/1.51 | ( member @ A @ A3 @ A5 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_36_insert__absorb2,axiom,
% 2.90/1.51 ! [A: $tType,X: A,A5: set @ A] :
% 2.90/1.51 ( ( insert @ A @ X @ ( insert @ A @ X @ A5 ) )
% 2.90/1.51 = ( insert @ A @ X @ A5 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_37_insertE,axiom,
% 2.90/1.51 ! [A: $tType,A3: A,B3: A,A5: set @ A] :
% 2.90/1.51 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A5 ) )
% 2.90/1.51 => ( ( A3 != B3 )
% 2.90/1.51 => ( member @ A @ A3 @ A5 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_38_insertI1,axiom,
% 2.90/1.51 ! [A: $tType,A3: A,B2: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B2 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_39_insertI2,axiom,
% 2.90/1.51 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 2.90/1.51 ( ( member @ A @ A3 @ B2 )
% 2.90/1.51 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_40_Set_Oset__insert,axiom,
% 2.90/1.51 ! [A: $tType,X: A,A5: set @ A] :
% 2.90/1.51 ( ( member @ A @ X @ A5 )
% 2.90/1.51 => ~ ! [B4: set @ A] :
% 2.90/1.51 ( ( A5
% 2.90/1.51 = ( insert @ A @ X @ B4 ) )
% 2.90/1.51 => ( member @ A @ X @ B4 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_41_insert__ident,axiom,
% 2.90/1.51 ! [A: $tType,X: A,A5: set @ A,B2: set @ A] :
% 2.90/1.51 ( ~ ( member @ A @ X @ A5 )
% 2.90/1.51 => ( ~ ( member @ A @ X @ B2 )
% 2.90/1.51 => ( ( ( insert @ A @ X @ A5 )
% 2.90/1.51 = ( insert @ A @ X @ B2 ) )
% 2.90/1.51 = ( A5 = B2 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_42_insert__absorb,axiom,
% 2.90/1.51 ! [A: $tType,A3: A,A5: set @ A] :
% 2.90/1.51 ( ( member @ A @ A3 @ A5 )
% 2.90/1.51 => ( ( insert @ A @ A3 @ A5 )
% 2.90/1.51 = A5 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_43_insert__eq__iff,axiom,
% 2.90/1.51 ! [A: $tType,A3: A,A5: set @ A,B3: A,B2: set @ A] :
% 2.90/1.51 ( ~ ( member @ A @ A3 @ A5 )
% 2.90/1.51 => ( ~ ( member @ A @ B3 @ B2 )
% 2.90/1.51 => ( ( ( insert @ A @ A3 @ A5 )
% 2.90/1.51 = ( insert @ A @ B3 @ B2 ) )
% 2.90/1.51 = ( ( ( A3 = B3 )
% 2.90/1.51 => ( A5 = B2 ) )
% 2.90/1.51 & ( ( A3 != B3 )
% 2.90/1.51 => ? [C2: set @ A] :
% 2.90/1.51 ( ( A5
% 2.90/1.51 = ( insert @ A @ B3 @ C2 ) )
% 2.90/1.51 & ~ ( member @ A @ B3 @ C2 )
% 2.90/1.51 & ( B2
% 2.90/1.51 = ( insert @ A @ A3 @ C2 ) )
% 2.90/1.51 & ~ ( member @ A @ A3 @ C2 ) ) ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_44_mk__disjoint__insert,axiom,
% 2.90/1.51 ! [A: $tType,A3: A,A5: set @ A] :
% 2.90/1.51 ( ( member @ A @ A3 @ A5 )
% 2.90/1.51 => ? [B4: set @ A] :
% 2.90/1.51 ( ( A5
% 2.90/1.51 = ( insert @ A @ A3 @ B4 ) )
% 2.90/1.51 & ~ ( member @ A @ A3 @ B4 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_45_mem__Collect__eq,axiom,
% 2.90/1.51 ! [A: $tType,A3: A,P: A > $o] :
% 2.90/1.51 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 2.90/1.51 = ( P @ A3 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_46_Collect__mem__eq,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A] :
% 2.90/1.51 ( ( collect @ A
% 2.90/1.51 @ ^ [X3: A] : ( member @ A @ X3 @ A5 ) )
% 2.90/1.51 = A5 ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_47_Collect__cong,axiom,
% 2.90/1.51 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.90/1.51 ( ! [X4: A] :
% 2.90/1.51 ( ( P @ X4 )
% 2.90/1.51 = ( Q @ X4 ) )
% 2.90/1.51 => ( ( collect @ A @ P )
% 2.90/1.51 = ( collect @ A @ Q ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_48_ext,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.90/1.51 ( ! [X4: A] :
% 2.90/1.51 ( ( F @ X4 )
% 2.90/1.51 = ( G @ X4 ) )
% 2.90/1.51 => ( F = G ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_49_insert__commute,axiom,
% 2.90/1.51 ! [A: $tType,X: A,Y: A,A5: set @ A] :
% 2.90/1.51 ( ( insert @ A @ X @ ( insert @ A @ Y @ A5 ) )
% 2.90/1.51 = ( insert @ A @ Y @ ( insert @ A @ X @ A5 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_50_stream_Orel__coinduct,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,P: ( stream170649215stream @ A ) > ( stream170649215stream @ B ) > $o,X: stream170649215stream @ A,Y: stream170649215stream @ B,R: A > B > $o] :
% 2.90/1.51 ( ( P @ X @ Y )
% 2.90/1.51 => ( ! [Stream3: stream170649215stream @ A,Stream4: stream170649215stream @ B] :
% 2.90/1.51 ( ( P @ Stream3 @ Stream4 )
% 2.90/1.51 => ( ( R @ ( stream_Mirabelle_shd @ A @ Stream3 ) @ ( stream_Mirabelle_shd @ B @ Stream4 ) )
% 2.90/1.51 & ( P @ ( stream_Mirabelle_stl @ A @ Stream3 ) @ ( stream_Mirabelle_stl @ B @ Stream4 ) ) ) )
% 2.90/1.51 => ( stream686525009m_all2 @ A @ B @ R @ X @ Y ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_51_stream_Orel__sel,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType] :
% 2.90/1.51 ( ( stream686525009m_all2 @ A @ B )
% 2.90/1.51 = ( ^ [R2: A > B > $o,A4: stream170649215stream @ A,B5: stream170649215stream @ B] :
% 2.90/1.51 ( ( R2 @ ( stream_Mirabelle_shd @ A @ A4 ) @ ( stream_Mirabelle_shd @ B @ B5 ) )
% 2.90/1.51 & ( stream686525009m_all2 @ A @ B @ R2 @ ( stream_Mirabelle_stl @ A @ A4 ) @ ( stream_Mirabelle_stl @ B @ B5 ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_52_stream_Opred__set,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( stream1153105665stream @ A )
% 2.90/1.51 = ( ^ [P2: A > $o,X3: stream170649215stream @ A] :
% 2.90/1.51 ! [Y3: A] :
% 2.90/1.51 ( ( member @ A @ Y3 @ ( stream30925839e_sset @ A @ X3 ) )
% 2.90/1.51 => ( P2 @ Y3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_53_stream_Opred__map,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: stream170649215stream @ A] :
% 2.90/1.51 ( ( stream1153105665stream @ B @ Q @ ( stream2128578057e_smap @ A @ B @ F @ X ) )
% 2.90/1.51 = ( stream1153105665stream @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_54_stream_Oset__map,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,F: A > B,V: stream170649215stream @ A] :
% 2.90/1.51 ( ( stream30925839e_sset @ B @ ( stream2128578057e_smap @ A @ B @ F @ V ) )
% 2.90/1.51 = ( image @ A @ B @ F @ ( stream30925839e_sset @ A @ V ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_55_pairwise__insert,axiom,
% 2.90/1.51 ! [A: $tType,R3: A > A > $o,X: A,S2: set @ A] :
% 2.90/1.51 ( ( pairwise @ A @ R3 @ ( insert @ A @ X @ S2 ) )
% 2.90/1.51 = ( ! [Y3: A] :
% 2.90/1.51 ( ( ( member @ A @ Y3 @ S2 )
% 2.90/1.51 & ( Y3 != X ) )
% 2.90/1.51 => ( ( R3 @ X @ Y3 )
% 2.90/1.51 & ( R3 @ Y3 @ X ) ) )
% 2.90/1.51 & ( pairwise @ A @ R3 @ S2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_56_stream_Orel__inject,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,R: A > B > $o,X1: A,X2: stream170649215stream @ A,Y1: B,Y2: stream170649215stream @ B] :
% 2.90/1.51 ( ( stream686525009m_all2 @ A @ B @ R @ ( stream641971652_SCons @ A @ X1 @ X2 ) @ ( stream641971652_SCons @ B @ Y1 @ Y2 ) )
% 2.90/1.51 = ( ( R @ X1 @ Y1 )
% 2.90/1.51 & ( stream686525009m_all2 @ A @ B @ R @ X2 @ Y2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_57_singletonI,axiom,
% 2.90/1.51 ! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_58_image__eqI,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,B3: A,F: B > A,X: B,A5: set @ B] :
% 2.90/1.51 ( ( B3
% 2.90/1.51 = ( F @ X ) )
% 2.90/1.51 => ( ( member @ B @ X @ A5 )
% 2.90/1.51 => ( member @ A @ B3 @ ( image @ B @ A @ F @ A5 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_59_empty__Collect__eq,axiom,
% 2.90/1.51 ! [A: $tType,P: A > $o] :
% 2.90/1.51 ( ( ( bot_bot @ ( set @ A ) )
% 2.90/1.51 = ( collect @ A @ P ) )
% 2.90/1.51 = ( ! [X3: A] :
% 2.90/1.51 ~ ( P @ X3 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_60_Collect__empty__eq,axiom,
% 2.90/1.51 ! [A: $tType,P: A > $o] :
% 2.90/1.51 ( ( ( collect @ A @ P )
% 2.90/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.90/1.51 = ( ! [X3: A] :
% 2.90/1.51 ~ ( P @ X3 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_61_all__not__in__conv,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A] :
% 2.90/1.51 ( ( ! [X3: A] :
% 2.90/1.51 ~ ( member @ A @ X3 @ A5 ) )
% 2.90/1.51 = ( A5
% 2.90/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_62_empty__iff,axiom,
% 2.90/1.51 ! [A: $tType,C3: A] :
% 2.90/1.51 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_63_image__is__empty,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
% 2.90/1.51 ( ( ( image @ B @ A @ F @ A5 )
% 2.90/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.90/1.51 = ( A5
% 2.90/1.51 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_64_empty__is__image,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
% 2.90/1.51 ( ( ( bot_bot @ ( set @ A ) )
% 2.90/1.51 = ( image @ B @ A @ F @ A5 ) )
% 2.90/1.51 = ( A5
% 2.90/1.51 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_65_image__empty,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,F: B > A] :
% 2.90/1.51 ( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
% 2.90/1.51 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_66_image__insert,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,F: B > A,A3: B,B2: set @ B] :
% 2.90/1.51 ( ( image @ B @ A @ F @ ( insert @ B @ A3 @ B2 ) )
% 2.90/1.51 = ( insert @ A @ ( F @ A3 ) @ ( image @ B @ A @ F @ B2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_67_insert__image,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,X: A,A5: set @ A,F: A > B] :
% 2.90/1.51 ( ( member @ A @ X @ A5 )
% 2.90/1.51 => ( ( insert @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A5 ) )
% 2.90/1.51 = ( image @ A @ B @ F @ A5 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_68_ball__empty,axiom,
% 2.90/1.51 ! [A: $tType,P: A > $o,X5: A] :
% 2.90/1.51 ( ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
% 2.90/1.51 => ( P @ X5 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_69_pairwise__empty,axiom,
% 2.90/1.51 ! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_70_rev__image__eqI,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,X: A,A5: set @ A,B3: B,F: A > B] :
% 2.90/1.51 ( ( member @ A @ X @ A5 )
% 2.90/1.51 => ( ( B3
% 2.90/1.51 = ( F @ X ) )
% 2.90/1.51 => ( member @ B @ B3 @ ( image @ A @ B @ F @ A5 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_71_pairwise__def,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( pairwise @ A )
% 2.90/1.51 = ( ^ [R2: A > A > $o,S3: set @ A] :
% 2.90/1.51 ! [X3: A] :
% 2.90/1.51 ( ( member @ A @ X3 @ S3 )
% 2.90/1.51 => ! [Y3: A] :
% 2.90/1.51 ( ( member @ A @ Y3 @ S3 )
% 2.90/1.51 => ( ( X3 != Y3 )
% 2.90/1.51 => ( R2 @ X3 @ Y3 ) ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_72_ball__imageD,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: A > $o] :
% 2.90/1.51 ( ! [X4: A] :
% 2.90/1.51 ( ( member @ A @ X4 @ ( image @ B @ A @ F @ A5 ) )
% 2.90/1.51 => ( P @ X4 ) )
% 2.90/1.51 => ! [X5: B] :
% 2.90/1.51 ( ( member @ B @ X5 @ A5 )
% 2.90/1.51 => ( P @ ( F @ X5 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_73_image__cong,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G: A > B] :
% 2.90/1.51 ( ( M = N )
% 2.90/1.51 => ( ! [X4: A] :
% 2.90/1.51 ( ( member @ A @ X4 @ N )
% 2.90/1.51 => ( ( F @ X4 )
% 2.90/1.51 = ( G @ X4 ) ) )
% 2.90/1.51 => ( ( image @ A @ B @ F @ M )
% 2.90/1.51 = ( image @ A @ B @ G @ N ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_74_ex__in__conv,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A] :
% 2.90/1.51 ( ( ? [X3: A] : ( member @ A @ X3 @ A5 ) )
% 2.90/1.51 = ( A5
% 2.90/1.51 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_75_bex__imageD,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: A > $o] :
% 2.90/1.51 ( ? [X5: A] :
% 2.90/1.51 ( ( member @ A @ X5 @ ( image @ B @ A @ F @ A5 ) )
% 2.90/1.51 & ( P @ X5 ) )
% 2.90/1.51 => ? [X4: B] :
% 2.90/1.51 ( ( member @ B @ X4 @ A5 )
% 2.90/1.51 & ( P @ ( F @ X4 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_76_image__iff,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,Z3: A,F: B > A,A5: set @ B] :
% 2.90/1.51 ( ( member @ A @ Z3 @ ( image @ B @ A @ F @ A5 ) )
% 2.90/1.51 = ( ? [X3: B] :
% 2.90/1.51 ( ( member @ B @ X3 @ A5 )
% 2.90/1.51 & ( Z3
% 2.90/1.51 = ( F @ X3 ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_77_equals0I,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A] :
% 2.90/1.51 ( ! [Y4: A] :
% 2.90/1.51 ~ ( member @ A @ Y4 @ A5 )
% 2.90/1.51 => ( A5
% 2.90/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_78_equals0D,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,A3: A] :
% 2.90/1.51 ( ( A5
% 2.90/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.90/1.51 => ~ ( member @ A @ A3 @ A5 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_79_Ball__def,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( ball @ A )
% 2.90/1.51 = ( ^ [A6: set @ A,P2: A > $o] :
% 2.90/1.51 ! [X3: A] :
% 2.90/1.51 ( ( member @ A @ X3 @ A6 )
% 2.90/1.51 => ( P2 @ X3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_80_imageI,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,X: A,A5: set @ A,F: A > B] :
% 2.90/1.51 ( ( member @ A @ X @ A5 )
% 2.90/1.51 => ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A5 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_81_emptyE,axiom,
% 2.90/1.51 ! [A: $tType,A3: A] :
% 2.90/1.51 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_82_stream_Orel__eq,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( stream686525009m_all2 @ A @ A
% 2.90/1.51 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 2.90/1.51 = ( ^ [Y5: stream170649215stream @ A,Z4: stream170649215stream @ A] : ( Y5 = Z4 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_83_stream_Orel__refl,axiom,
% 2.90/1.51 ! [B: $tType,Ra: B > B > $o,X: stream170649215stream @ B] :
% 2.90/1.51 ( ! [X4: B] : ( Ra @ X4 @ X4 )
% 2.90/1.51 => ( stream686525009m_all2 @ B @ B @ Ra @ X @ X ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_84_pairwise__singleton,axiom,
% 2.90/1.51 ! [A: $tType,P: A > A > $o,A5: A] : ( pairwise @ A @ P @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_85_stream_Omap__o__corec,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > A,Ga: C > $o,Gb: C > ( stream170649215stream @ A ),Gc: C > C] :
% 2.90/1.51 ( ( comp @ ( stream170649215stream @ A ) @ ( stream170649215stream @ B ) @ C @ ( stream2128578057e_smap @ A @ B @ F ) @ ( stream660621732stream @ C @ A @ G @ Ga @ Gb @ Gc ) )
% 2.90/1.51 = ( stream660621732stream @ C @ B @ ( comp @ A @ B @ C @ F @ G ) @ Ga @ ( comp @ ( stream170649215stream @ A ) @ ( stream170649215stream @ B ) @ C @ ( stream2128578057e_smap @ A @ B @ F ) @ Gb ) @ Gc ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_86_stream_Omap__comp,axiom,
% 2.90/1.51 ! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: stream170649215stream @ A] :
% 2.90/1.51 ( ( stream2128578057e_smap @ B @ C @ G @ ( stream2128578057e_smap @ A @ B @ F @ V ) )
% 2.90/1.51 = ( stream2128578057e_smap @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_87_singletonD,axiom,
% 2.90/1.51 ! [A: $tType,B3: A,A3: A] :
% 2.90/1.51 ( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.90/1.51 => ( B3 = A3 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_88_singleton__iff,axiom,
% 2.90/1.51 ! [A: $tType,B3: A,A3: A] :
% 2.90/1.51 ( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.90/1.51 = ( B3 = A3 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_89_doubleton__eq__iff,axiom,
% 2.90/1.51 ! [A: $tType,A3: A,B3: A,C3: A,D: A] :
% 2.90/1.51 ( ( ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.90/1.51 = ( insert @ A @ C3 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.90/1.51 = ( ( ( A3 = C3 )
% 2.90/1.51 & ( B3 = D ) )
% 2.90/1.51 | ( ( A3 = D )
% 2.90/1.51 & ( B3 = C3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_90_insert__not__empty,axiom,
% 2.90/1.51 ! [A: $tType,A3: A,A5: set @ A] :
% 2.90/1.51 ( ( insert @ A @ A3 @ A5 )
% 2.90/1.51 != ( bot_bot @ ( set @ A ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_91_singleton__inject,axiom,
% 2.90/1.51 ! [A: $tType,A3: A,B3: A] :
% 2.90/1.51 ( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.90/1.51 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.90/1.51 => ( A3 = B3 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_92_stream_Orel__refl__strong,axiom,
% 2.90/1.51 ! [A: $tType,X: stream170649215stream @ A,Ra: A > A > $o] :
% 2.90/1.51 ( ! [Z: A] :
% 2.90/1.51 ( ( member @ A @ Z @ ( stream30925839e_sset @ A @ X ) )
% 2.90/1.51 => ( Ra @ Z @ Z ) )
% 2.90/1.51 => ( stream686525009m_all2 @ A @ A @ Ra @ X @ X ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_93_stream_Orel__mono__strong,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,R: A > B > $o,X: stream170649215stream @ A,Y: stream170649215stream @ B,Ra: A > B > $o] :
% 2.90/1.51 ( ( stream686525009m_all2 @ A @ B @ R @ X @ Y )
% 2.90/1.51 => ( ! [Z: A,Yb: B] :
% 2.90/1.51 ( ( member @ A @ Z @ ( stream30925839e_sset @ A @ X ) )
% 2.90/1.51 => ( ( member @ B @ Yb @ ( stream30925839e_sset @ B @ Y ) )
% 2.90/1.51 => ( ( R @ Z @ Yb )
% 2.90/1.51 => ( Ra @ Z @ Yb ) ) ) )
% 2.90/1.51 => ( stream686525009m_all2 @ A @ B @ Ra @ X @ Y ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_94_stream_Orel__cong,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,X: stream170649215stream @ A,Ya: stream170649215stream @ A,Y: stream170649215stream @ B,Xa2: stream170649215stream @ B,R: A > B > $o,Ra: A > B > $o] :
% 2.90/1.51 ( ( X = Ya )
% 2.90/1.51 => ( ( Y = Xa2 )
% 2.90/1.51 => ( ! [Z: A,Yb: B] :
% 2.90/1.51 ( ( member @ A @ Z @ ( stream30925839e_sset @ A @ Ya ) )
% 2.90/1.51 => ( ( member @ B @ Yb @ ( stream30925839e_sset @ B @ Xa2 ) )
% 2.90/1.51 => ( ( R @ Z @ Yb )
% 2.90/1.51 = ( Ra @ Z @ Yb ) ) ) )
% 2.90/1.51 => ( ( stream686525009m_all2 @ A @ B @ R @ X @ Y )
% 2.90/1.51 = ( stream686525009m_all2 @ A @ B @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_95_stream_Orel__intros,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,R: A > B > $o,X1: A,Y1: B,X2: stream170649215stream @ A,Y2: stream170649215stream @ B] :
% 2.90/1.51 ( ( R @ X1 @ Y1 )
% 2.90/1.51 => ( ( stream686525009m_all2 @ A @ B @ R @ X2 @ Y2 )
% 2.90/1.51 => ( stream686525009m_all2 @ A @ B @ R @ ( stream641971652_SCons @ A @ X1 @ X2 ) @ ( stream641971652_SCons @ B @ Y1 @ Y2 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_96_stream_Orel__cases,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,R: A > B > $o,A3: stream170649215stream @ A,B3: stream170649215stream @ B] :
% 2.90/1.51 ( ( stream686525009m_all2 @ A @ B @ R @ A3 @ B3 )
% 2.90/1.51 => ~ ! [X1a: A,X2a: stream170649215stream @ A] :
% 2.90/1.51 ( ( A3
% 2.90/1.51 = ( stream641971652_SCons @ A @ X1a @ X2a ) )
% 2.90/1.51 => ! [Y1a: B,Y2a: stream170649215stream @ B] :
% 2.90/1.51 ( ( B3
% 2.90/1.51 = ( stream641971652_SCons @ B @ Y1a @ Y2a ) )
% 2.90/1.51 => ( ( R @ X1a @ Y1a )
% 2.90/1.51 => ~ ( stream686525009m_all2 @ A @ B @ R @ X2a @ Y2a ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_97_Ball__image__comp,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,F: B > A,A5: set @ B,G: A > $o] :
% 2.90/1.51 ( ( ! [X3: A] :
% 2.90/1.51 ( ( member @ A @ X3 @ ( image @ B @ A @ F @ A5 ) )
% 2.90/1.51 => ( G @ X3 ) ) )
% 2.90/1.51 = ( ! [X3: B] :
% 2.90/1.51 ( ( member @ B @ X3 @ A5 )
% 2.90/1.51 => ( comp @ A @ $o @ B @ G @ F @ X3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_98_not__False__in__image__Ball,axiom,
% 2.90/1.51 ! [A: $tType,P: A > $o,A5: set @ A] :
% 2.90/1.51 ( ( ~ ( member @ $o @ $false @ ( image @ A @ $o @ P @ A5 ) ) )
% 2.90/1.51 = ( ! [X3: A] :
% 2.90/1.51 ( ( member @ A @ X3 @ A5 )
% 2.90/1.51 => ( P @ X3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_99_comp__apply,axiom,
% 2.90/1.51 ! [C: $tType,A: $tType,B: $tType] :
% 2.90/1.51 ( ( comp @ B @ A @ C )
% 2.90/1.51 = ( ^ [F2: B > A,G2: C > B,X3: C] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_100_the__elem__eq,axiom,
% 2.90/1.51 ! [A: $tType,X: A] :
% 2.90/1.51 ( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 2.90/1.51 = X ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_101_bot__apply,axiom,
% 2.90/1.51 ! [C: $tType,D2: $tType] :
% 2.90/1.51 ( ( bot @ C @ ( type2 @ C ) )
% 2.90/1.51 => ( ( bot_bot @ ( D2 > C ) )
% 2.90/1.51 = ( ^ [X3: D2] : ( bot_bot @ C ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_102_is__singletonI,axiom,
% 2.90/1.51 ! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_103_image__comp,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,R3: set @ C] :
% 2.90/1.51 ( ( image @ B @ A @ F @ ( image @ C @ B @ G @ R3 ) )
% 2.90/1.51 = ( image @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ R3 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_104_bot__set__def,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( bot_bot @ ( set @ A ) )
% 2.90/1.51 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_105_is__singleton__the__elem,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( is_singleton @ A )
% 2.90/1.51 = ( ^ [A6: set @ A] :
% 2.90/1.51 ( A6
% 2.90/1.51 = ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_106_is__singletonI_H,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A] :
% 2.90/1.51 ( ( A5
% 2.90/1.51 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.51 => ( ! [X4: A,Y4: A] :
% 2.90/1.51 ( ( member @ A @ X4 @ A5 )
% 2.90/1.51 => ( ( member @ A @ Y4 @ A5 )
% 2.90/1.51 => ( X4 = Y4 ) ) )
% 2.90/1.51 => ( is_singleton @ A @ A5 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_107_Sup_OSUP__cong,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,A5: set @ B,B2: set @ B,C4: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 2.90/1.51 ( ( A5 = B2 )
% 2.90/1.51 => ( ! [X4: B] :
% 2.90/1.51 ( ( member @ B @ X4 @ B2 )
% 2.90/1.51 => ( ( C4 @ X4 )
% 2.90/1.51 = ( D3 @ X4 ) ) )
% 2.90/1.51 => ( ( Sup @ ( image @ B @ A @ C4 @ A5 ) )
% 2.90/1.51 = ( Sup @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_108_Inf_OINF__cong,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,A5: set @ B,B2: set @ B,C4: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 2.90/1.51 ( ( A5 = B2 )
% 2.90/1.51 => ( ! [X4: B] :
% 2.90/1.51 ( ( member @ B @ X4 @ B2 )
% 2.90/1.51 => ( ( C4 @ X4 )
% 2.90/1.51 = ( D3 @ X4 ) ) )
% 2.90/1.51 => ( ( Inf @ ( image @ B @ A @ C4 @ A5 ) )
% 2.90/1.51 = ( Inf @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_109_bot__fun__def,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType] :
% 2.90/1.51 ( ( bot @ B @ ( type2 @ B ) )
% 2.90/1.51 => ( ( bot_bot @ ( A > B ) )
% 2.90/1.51 = ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_110_comp__apply__eq,axiom,
% 2.90/1.51 ! [B: $tType,D2: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H: D2 > A,K: C > D2] :
% 2.90/1.51 ( ( ( F @ ( G @ X ) )
% 2.90/1.51 = ( H @ ( K @ X ) ) )
% 2.90/1.51 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 2.90/1.51 = ( comp @ D2 @ A @ C @ H @ K @ X ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_111_comp__eq__dest__lhs,axiom,
% 2.90/1.51 ! [C: $tType,B: $tType,A: $tType,A3: C > B,B3: A > C,C3: A > B,V: A] :
% 2.90/1.51 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 2.90/1.51 = C3 )
% 2.90/1.51 => ( ( A3 @ ( B3 @ V ) )
% 2.90/1.51 = ( C3 @ V ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_112_comp__eq__elim,axiom,
% 2.90/1.51 ! [C: $tType,B: $tType,D2: $tType,A: $tType,A3: C > B,B3: A > C,C3: D2 > B,D: A > D2] :
% 2.90/1.51 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 2.90/1.51 = ( comp @ D2 @ B @ A @ C3 @ D ) )
% 2.90/1.51 => ! [V2: A] :
% 2.90/1.51 ( ( A3 @ ( B3 @ V2 ) )
% 2.90/1.51 = ( C3 @ ( D @ V2 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_113_comp__eq__dest,axiom,
% 2.90/1.51 ! [C: $tType,B: $tType,D2: $tType,A: $tType,A3: C > B,B3: A > C,C3: D2 > B,D: A > D2,V: A] :
% 2.90/1.51 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 2.90/1.51 = ( comp @ D2 @ B @ A @ C3 @ D ) )
% 2.90/1.51 => ( ( A3 @ ( B3 @ V ) )
% 2.90/1.51 = ( C3 @ ( D @ V ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_114_comp__assoc,axiom,
% 2.90/1.51 ! [B: $tType,D2: $tType,C: $tType,A: $tType,F: D2 > B,G: C > D2,H: A > C] :
% 2.90/1.51 ( ( comp @ C @ B @ A @ ( comp @ D2 @ B @ C @ F @ G ) @ H )
% 2.90/1.51 = ( comp @ D2 @ B @ A @ F @ ( comp @ C @ D2 @ A @ G @ H ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_115_comp__def,axiom,
% 2.90/1.51 ! [A: $tType,C: $tType,B: $tType] :
% 2.90/1.51 ( ( comp @ B @ C @ A )
% 2.90/1.51 = ( ^ [F2: B > C,G2: A > B,X3: A] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_116_is__singletonE,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A] :
% 2.90/1.51 ( ( is_singleton @ A @ A5 )
% 2.90/1.51 => ~ ! [X4: A] :
% 2.90/1.51 ( A5
% 2.90/1.51 != ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_117_is__singleton__def,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( is_singleton @ A )
% 2.90/1.51 = ( ^ [A6: set @ A] :
% 2.90/1.51 ? [X3: A] :
% 2.90/1.51 ( A6
% 2.90/1.51 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_118_the__elem__image__unique,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,A5: set @ A,F: A > B,X: A] :
% 2.90/1.51 ( ( A5
% 2.90/1.51 != ( bot_bot @ ( set @ A ) ) )
% 2.90/1.51 => ( ! [Y4: A] :
% 2.90/1.51 ( ( member @ A @ Y4 @ A5 )
% 2.90/1.51 => ( ( F @ Y4 )
% 2.90/1.51 = ( F @ X ) ) )
% 2.90/1.51 => ( ( the_elem @ B @ ( image @ A @ B @ F @ A5 ) )
% 2.90/1.51 = ( F @ X ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_119_Sup_OSUP__image,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A5: set @ C] :
% 2.90/1.51 ( ( Sup @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A5 ) ) )
% 2.90/1.51 = ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A5 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_120_Inf_OINF__image,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A5: set @ C] :
% 2.90/1.51 ( ( Inf @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A5 ) ) )
% 2.90/1.51 = ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A5 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_121_image__eq__imp__comp,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,D2: $tType,C: $tType,F: B > A,A5: set @ B,G: C > A,B2: set @ C,H: A > D2] :
% 2.90/1.51 ( ( ( image @ B @ A @ F @ A5 )
% 2.90/1.51 = ( image @ C @ A @ G @ B2 ) )
% 2.90/1.51 => ( ( image @ B @ D2 @ ( comp @ A @ D2 @ B @ H @ F ) @ A5 )
% 2.90/1.51 = ( image @ C @ D2 @ ( comp @ A @ D2 @ C @ H @ G ) @ B2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_122_override__on__emptyset,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.90/1.51 ( ( override_on @ A @ B @ F @ G @ ( bot_bot @ ( set @ A ) ) )
% 2.90/1.51 = F ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_123_Set_Ois__empty__def,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( is_empty @ A )
% 2.90/1.51 = ( ^ [A6: set @ A] :
% 2.90/1.51 ( A6
% 2.90/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_124_ball__reg,axiom,
% 2.90/1.51 ! [A: $tType,R: set @ A,P: A > $o,Q: A > $o] :
% 2.90/1.51 ( ! [X4: A] :
% 2.90/1.51 ( ( member @ A @ X4 @ R )
% 2.90/1.51 => ( ( P @ X4 )
% 2.90/1.51 => ( Q @ X4 ) ) )
% 2.90/1.51 => ( ! [X4: A] :
% 2.90/1.51 ( ( member @ A @ X4 @ R )
% 2.90/1.51 => ( P @ X4 ) )
% 2.90/1.51 => ! [X5: A] :
% 2.90/1.51 ( ( member @ A @ X5 @ R )
% 2.90/1.51 => ( Q @ X5 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_125_rewriteR__comp__comp2,axiom,
% 2.90/1.51 ! [C: $tType,B: $tType,E2: $tType,D2: $tType,A: $tType,G: C > B,H: A > C,R1: D2 > B,R22: A > D2,F: B > E2,L: D2 > E2] :
% 2.90/1.51 ( ( ( comp @ C @ B @ A @ G @ H )
% 2.90/1.51 = ( comp @ D2 @ B @ A @ R1 @ R22 ) )
% 2.90/1.51 => ( ( ( comp @ B @ E2 @ D2 @ F @ R1 )
% 2.90/1.51 = L )
% 2.90/1.51 => ( ( comp @ C @ E2 @ A @ ( comp @ B @ E2 @ C @ F @ G ) @ H )
% 2.90/1.51 = ( comp @ D2 @ E2 @ A @ L @ R22 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_126_rewriteL__comp__comp2,axiom,
% 2.90/1.51 ! [A: $tType,C: $tType,B: $tType,D2: $tType,E2: $tType,F: C > B,G: A > C,L1: D2 > B,L2: A > D2,H: E2 > A,R3: E2 > D2] :
% 2.90/1.51 ( ( ( comp @ C @ B @ A @ F @ G )
% 2.90/1.51 = ( comp @ D2 @ B @ A @ L1 @ L2 ) )
% 2.90/1.51 => ( ( ( comp @ A @ D2 @ E2 @ L2 @ H )
% 2.90/1.51 = R3 )
% 2.90/1.51 => ( ( comp @ C @ B @ E2 @ F @ ( comp @ A @ C @ E2 @ G @ H ) )
% 2.90/1.51 = ( comp @ D2 @ B @ E2 @ L1 @ R3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_127_override__on__apply__in,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,A3: A,A5: set @ A,F: A > B,G: A > B] :
% 2.90/1.51 ( ( member @ A @ A3 @ A5 )
% 2.90/1.51 => ( ( override_on @ A @ B @ F @ G @ A5 @ A3 )
% 2.90/1.51 = ( G @ A3 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_128_override__on__apply__notin,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,A3: A,A5: set @ A,F: A > B,G: A > B] :
% 2.90/1.51 ( ~ ( member @ A @ A3 @ A5 )
% 2.90/1.51 => ( ( override_on @ A @ B @ F @ G @ A5 @ A3 )
% 2.90/1.51 = ( F @ A3 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_129_override__on__def,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType] :
% 2.90/1.51 ( ( override_on @ A @ B )
% 2.90/1.51 = ( ^ [F2: A > B,G2: A > B,A6: set @ A,A4: A] : ( if @ B @ ( member @ A @ A4 @ A6 ) @ ( G2 @ A4 ) @ ( F2 @ A4 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_130_rewriteL__comp__comp,axiom,
% 2.90/1.51 ! [C: $tType,B: $tType,A: $tType,D2: $tType,F: C > B,G: A > C,L: A > B,H: D2 > A] :
% 2.90/1.51 ( ( ( comp @ C @ B @ A @ F @ G )
% 2.90/1.51 = L )
% 2.90/1.51 => ( ( comp @ C @ B @ D2 @ F @ ( comp @ A @ C @ D2 @ G @ H ) )
% 2.90/1.51 = ( comp @ A @ B @ D2 @ L @ H ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_131_rewriteR__comp__comp,axiom,
% 2.90/1.51 ! [C: $tType,D2: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R3: A > B,F: B > D2] :
% 2.90/1.51 ( ( ( comp @ C @ B @ A @ G @ H )
% 2.90/1.51 = R3 )
% 2.90/1.51 => ( ( comp @ C @ D2 @ A @ ( comp @ B @ D2 @ C @ F @ G ) @ H )
% 2.90/1.51 = ( comp @ B @ D2 @ A @ F @ R3 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_132_Collect__empty__eq__bot,axiom,
% 2.90/1.51 ! [A: $tType,P: A > $o] :
% 2.90/1.51 ( ( ( collect @ A @ P )
% 2.90/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.90/1.51 = ( P
% 2.90/1.51 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_133_bot__empty__eq,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( bot_bot @ ( A > $o ) )
% 2.90/1.51 = ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_134_comp__cong,axiom,
% 2.90/1.51 ! [C: $tType,B: $tType,D2: $tType,A: $tType,E2: $tType,F: B > A,G: C > B,X: C,F3: D2 > A,G3: E2 > D2,X6: E2] :
% 2.90/1.51 ( ( ( F @ ( G @ X ) )
% 2.90/1.51 = ( F3 @ ( G3 @ X6 ) ) )
% 2.90/1.51 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 2.90/1.51 = ( comp @ D2 @ A @ E2 @ F3 @ G3 @ X6 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_135_fun_Omap__comp,axiom,
% 2.90/1.51 ! [B: $tType,C: $tType,A: $tType,D2: $tType,G: B > C,F: A > B,V: D2 > A] :
% 2.90/1.51 ( ( comp @ B @ C @ D2 @ G @ ( comp @ A @ B @ D2 @ F @ V ) )
% 2.90/1.51 = ( comp @ A @ C @ D2 @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_136_type__copy__map__cong0,axiom,
% 2.90/1.51 ! [B: $tType,D2: $tType,E2: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X: C,N: D2 > A,H: C > D2,F: A > E2] :
% 2.90/1.51 ( ( ( M @ ( G @ X ) )
% 2.90/1.51 = ( N @ ( H @ X ) ) )
% 2.90/1.51 => ( ( comp @ B @ E2 @ C @ ( comp @ A @ E2 @ B @ F @ M ) @ G @ X )
% 2.90/1.51 = ( comp @ D2 @ E2 @ C @ ( comp @ A @ E2 @ D2 @ F @ N ) @ H @ X ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_137_range__eq__singletonD,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,F: B > A,A3: A,X: B] :
% 2.90/1.51 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.90/1.51 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.90/1.51 => ( ( F @ X )
% 2.90/1.51 = A3 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_138_singleton__insert__inj__eq_H,axiom,
% 2.90/1.51 ! [A: $tType,A3: A,A5: set @ A,B3: A] :
% 2.90/1.51 ( ( ( insert @ A @ A3 @ A5 )
% 2.90/1.51 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.90/1.51 = ( ( A3 = B3 )
% 2.90/1.51 & ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_139_singleton__insert__inj__eq,axiom,
% 2.90/1.51 ! [A: $tType,B3: A,A3: A,A5: set @ A] :
% 2.90/1.51 ( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
% 2.90/1.51 = ( insert @ A @ A3 @ A5 ) )
% 2.90/1.51 = ( ( A3 = B3 )
% 2.90/1.51 & ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_140_order__refl,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_141_subset__antisym,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.90/1.51 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A5 )
% 2.90/1.51 => ( A5 = B2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_142_subsetI,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 2.90/1.51 ( ! [X4: A] :
% 2.90/1.51 ( ( member @ A @ X4 @ A5 )
% 2.90/1.51 => ( member @ A @ X4 @ B2 ) )
% 2.90/1.51 => ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_143_top__apply,axiom,
% 2.90/1.51 ! [C: $tType,D2: $tType] :
% 2.90/1.51 ( ( top @ C @ ( type2 @ C ) )
% 2.90/1.51 => ( ( top_top @ ( D2 > C ) )
% 2.90/1.51 = ( ^ [X3: D2] : ( top_top @ C ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_144_UNIV__I,axiom,
% 2.90/1.51 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_145_subset__empty,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
% 2.90/1.51 = ( A5
% 2.90/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_146_empty__subsetI,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_147_insert__subset,axiom,
% 2.90/1.51 ! [A: $tType,X: A,A5: set @ A,B2: set @ A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ B2 )
% 2.90/1.51 = ( ( member @ A @ X @ B2 )
% 2.90/1.51 & ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_148_top_Oextremum__uniqueI,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( order_top @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [A3: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 2.90/1.51 => ( A3
% 2.90/1.51 = ( top_top @ A ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_149_top_Oextremum__unique,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( order_top @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [A3: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 2.90/1.51 = ( A3
% 2.90/1.51 = ( top_top @ A ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_150_dual__order_Oantisym,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [B3: A,A3: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.90/1.51 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.51 => ( A3 = B3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_151_top__greatest,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( order_top @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_152_dual__order_Otrans,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [B3: A,A3: A,C3: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.90/1.51 => ( ( ord_less_eq @ A @ C3 @ B3 )
% 2.90/1.51 => ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_153_linorder__wlog,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [P: A > A > $o,A3: A,B3: A] :
% 2.90/1.51 ( ! [A7: A,B6: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ A7 @ B6 )
% 2.90/1.51 => ( P @ A7 @ B6 ) )
% 2.90/1.51 => ( ! [A7: A,B6: A] :
% 2.90/1.51 ( ( P @ B6 @ A7 )
% 2.90/1.51 => ( P @ A7 @ B6 ) )
% 2.90/1.51 => ( P @ A3 @ B3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_154_dual__order_Orefl,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_155_order__trans,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [X: A,Y: A,Z3: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.51 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 2.90/1.51 => ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_156_order__class_Oorder_Oantisym,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [A3: A,B3: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.51 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.90/1.51 => ( A3 = B3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_157_ord__le__eq__trans,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( ord @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [A3: A,B3: A,C3: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.51 => ( ( B3 = C3 )
% 2.90/1.51 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_158_ord__eq__le__trans,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( ord @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [A3: A,B3: A,C3: A] :
% 2.90/1.51 ( ( A3 = B3 )
% 2.90/1.51 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 2.90/1.51 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_159_antisym__conv,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [Y: A,X: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.51 => ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.51 = ( X = Y ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_160_le__cases3,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [X: A,Y: A,Z3: A] :
% 2.90/1.51 ( ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.51 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 2.90/1.51 => ( ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.51 => ~ ( ord_less_eq @ A @ X @ Z3 ) )
% 2.90/1.51 => ( ( ( ord_less_eq @ A @ X @ Z3 )
% 2.90/1.51 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 2.90/1.51 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 2.90/1.51 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 2.90/1.51 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 2.90/1.51 => ~ ( ord_less_eq @ A @ Z3 @ X ) )
% 2.90/1.51 => ~ ( ( ord_less_eq @ A @ Z3 @ X )
% 2.90/1.51 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_161_order_Otrans,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [A3: A,B3: A,C3: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.51 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 2.90/1.51 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_162_le__cases,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [X: A,Y: A] :
% 2.90/1.51 ( ~ ( ord_less_eq @ A @ X @ Y )
% 2.90/1.51 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_163_eq__refl,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [X: A,Y: A] :
% 2.90/1.51 ( ( X = Y )
% 2.90/1.51 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_164_linear,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [X: A,Y: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.51 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_165_antisym,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [X: A,Y: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.90/1.51 => ( ( ord_less_eq @ A @ Y @ X )
% 2.90/1.51 => ( X = Y ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_166_eq__iff,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.90/1.51 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 2.90/1.51 = ( ^ [X3: A,Y3: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.90/1.51 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_167_ord__le__eq__subst,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType] :
% 2.90/1.51 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.51 & ( ord @ A @ ( type2 @ A ) ) )
% 2.90/1.51 => ! [A3: A,B3: A,F: A > B,C3: B] :
% 2.90/1.51 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.51 => ( ( ( F @ B3 )
% 2.90/1.51 = C3 )
% 2.90/1.51 => ( ! [X4: A,Y4: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 2.90/1.51 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.90/1.51 => ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_168_ord__eq__le__subst,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType] :
% 2.90/1.51 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.51 & ( ord @ A @ ( type2 @ A ) ) )
% 2.90/1.51 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 2.90/1.51 ( ( A3
% 2.90/1.51 = ( F @ B3 ) )
% 2.90/1.51 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 2.90/1.51 => ( ! [X4: B,Y4: B] :
% 2.90/1.51 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 2.90/1.51 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.90/1.51 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_169_order__subst2,axiom,
% 2.90/1.51 ! [A: $tType,C: $tType] :
% 2.90/1.51 ( ( ( order @ C @ ( type2 @ C ) )
% 2.90/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.90/1.51 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 2.90/1.51 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.90/1.51 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
% 2.90/1.51 => ( ! [X4: A,Y4: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 2.90/1.51 => ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.90/1.51 => ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_170_order__subst1,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType] :
% 2.90/1.51 ( ( ( order @ B @ ( type2 @ B ) )
% 2.90/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.90/1.51 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 2.90/1.51 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 2.90/1.51 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 2.90/1.51 => ( ! [X4: B,Y4: B] :
% 2.90/1.51 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 2.90/1.51 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 2.90/1.51 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_171_le__fun__def,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType] :
% 2.90/1.51 ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.51 => ( ( ord_less_eq @ ( A > B ) )
% 2.90/1.51 = ( ^ [F2: A > B,G2: A > B] :
% 2.90/1.51 ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_172_le__funI,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType] :
% 2.90/1.51 ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.51 => ! [F: A > B,G: A > B] :
% 2.90/1.51 ( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
% 2.90/1.51 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_173_le__funE,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType] :
% 2.90/1.51 ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.51 => ! [F: A > B,G: A > B,X: A] :
% 2.90/1.51 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.90/1.51 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_174_le__funD,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType] :
% 2.90/1.51 ( ( ord @ B @ ( type2 @ B ) )
% 2.90/1.51 => ! [F: A > B,G: A > B,X: A] :
% 2.90/1.51 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.90/1.51 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_175_Collect__mono__iff,axiom,
% 2.90/1.51 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.90/1.51 = ( ! [X3: A] :
% 2.90/1.51 ( ( P @ X3 )
% 2.90/1.51 => ( Q @ X3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_176_contra__subsetD,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,B2: set @ A,C3: A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.90/1.51 => ( ~ ( member @ A @ C3 @ B2 )
% 2.90/1.51 => ~ ( member @ A @ C3 @ A5 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_177_set__eq__subset,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( ^ [Y5: set @ A,Z4: set @ A] : ( Y5 = Z4 ) )
% 2.90/1.51 = ( ^ [A6: set @ A,B7: set @ A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 2.90/1.51 & ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_178_subset__trans,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,B2: set @ A,C4: set @ A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.90/1.51 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C4 )
% 2.90/1.51 => ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_179_UNIV__witness,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ? [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_180_Collect__mono,axiom,
% 2.90/1.51 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.90/1.51 ( ! [X4: A] :
% 2.90/1.51 ( ( P @ X4 )
% 2.90/1.51 => ( Q @ X4 ) )
% 2.90/1.51 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_181_subset__refl,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_182_subset__UNIV,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_183_rev__subsetD,axiom,
% 2.90/1.51 ! [A: $tType,C3: A,A5: set @ A,B2: set @ A] :
% 2.90/1.51 ( ( member @ A @ C3 @ A5 )
% 2.90/1.51 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.90/1.51 => ( member @ A @ C3 @ B2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_184_subset__iff,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) )
% 2.90/1.51 = ( ^ [A6: set @ A,B7: set @ A] :
% 2.90/1.51 ! [T: A] :
% 2.90/1.51 ( ( member @ A @ T @ A6 )
% 2.90/1.51 => ( member @ A @ T @ B7 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_185_set__rev__mp,axiom,
% 2.90/1.51 ! [A: $tType,X: A,A5: set @ A,B2: set @ A] :
% 2.90/1.51 ( ( member @ A @ X @ A5 )
% 2.90/1.51 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.90/1.51 => ( member @ A @ X @ B2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_186_equalityD2,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 2.90/1.51 ( ( A5 = B2 )
% 2.90/1.51 => ( ord_less_eq @ ( set @ A ) @ B2 @ A5 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_187_equalityD1,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 2.90/1.51 ( ( A5 = B2 )
% 2.90/1.51 => ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_188_subset__eq,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) )
% 2.90/1.51 = ( ^ [A6: set @ A,B7: set @ A] :
% 2.90/1.51 ! [X3: A] :
% 2.90/1.51 ( ( member @ A @ X3 @ A6 )
% 2.90/1.51 => ( member @ A @ X3 @ B7 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_189_equalityE,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 2.90/1.51 ( ( A5 = B2 )
% 2.90/1.51 => ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.90/1.51 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A5 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_190_UNIV__eq__I,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A] :
% 2.90/1.51 ( ! [X4: A] : ( member @ A @ X4 @ A5 )
% 2.90/1.51 => ( ( top_top @ ( set @ A ) )
% 2.90/1.51 = A5 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_191_subsetCE,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,B2: set @ A,C3: A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.90/1.51 => ( ( member @ A @ C3 @ A5 )
% 2.90/1.51 => ( member @ A @ C3 @ B2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_192_subsetD,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,B2: set @ A,C3: A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.90/1.51 => ( ( member @ A @ C3 @ A5 )
% 2.90/1.51 => ( member @ A @ C3 @ B2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_193_in__mono,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,B2: set @ A,X: A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.90/1.51 => ( ( member @ A @ X @ A5 )
% 2.90/1.51 => ( member @ A @ X @ B2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_194_set__mp,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,B2: set @ A,X: A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.90/1.51 => ( ( member @ A @ X @ A5 )
% 2.90/1.51 => ( member @ A @ X @ B2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_195_insert__UNIV,axiom,
% 2.90/1.51 ! [A: $tType,X: A] :
% 2.90/1.51 ( ( insert @ A @ X @ ( top_top @ ( set @ A ) ) )
% 2.90/1.51 = ( top_top @ ( set @ A ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_196_Ball__Collect,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( ball @ A )
% 2.90/1.51 = ( ^ [A6: set @ A,P2: A > $o] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( collect @ A @ P2 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_197_subset__insertI2,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,B2: set @ A,B3: A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.90/1.51 => ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_198_subset__insertI,axiom,
% 2.90/1.51 ! [A: $tType,B2: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ B2 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_199_subset__insert,axiom,
% 2.90/1.51 ! [A: $tType,X: A,A5: set @ A,B2: set @ A] :
% 2.90/1.51 ( ~ ( member @ A @ X @ A5 )
% 2.90/1.51 => ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B2 ) )
% 2.90/1.51 = ( ord_less_eq @ ( set @ A ) @ A5 @ B2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_200_insert__mono,axiom,
% 2.90/1.51 ! [A: $tType,C4: set @ A,D3: set @ A,A3: A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
% 2.90/1.51 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C4 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_201_empty__not__UNIV,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( bot_bot @ ( set @ A ) )
% 2.90/1.51 != ( top_top @ ( set @ A ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_202_surjD,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 2.90/1.51 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.90/1.51 = ( top_top @ ( set @ A ) ) )
% 2.90/1.51 => ? [X4: B] :
% 2.90/1.51 ( Y
% 2.90/1.51 = ( F @ X4 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_203_surjE,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 2.90/1.51 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.90/1.51 = ( top_top @ ( set @ A ) ) )
% 2.90/1.51 => ~ ! [X4: B] :
% 2.90/1.51 ( Y
% 2.90/1.51 != ( F @ X4 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_204_surjI,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 2.90/1.51 ( ! [X4: A] :
% 2.90/1.51 ( ( G @ ( F @ X4 ) )
% 2.90/1.51 = X4 )
% 2.90/1.51 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 2.90/1.51 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_205_surj__def,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,F: B > A] :
% 2.90/1.51 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.90/1.51 = ( top_top @ ( set @ A ) ) )
% 2.90/1.51 = ( ! [Y3: A] :
% 2.90/1.51 ? [X3: B] :
% 2.90/1.51 ( Y3
% 2.90/1.51 = ( F @ X3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_206_rangeI,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_207_range__eqI,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,B3: A,F: B > A,X: B] :
% 2.90/1.51 ( ( B3
% 2.90/1.51 = ( F @ X ) )
% 2.90/1.51 => ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_208_image__mono,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,A5: set @ A,B2: set @ A,F: A > B] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.90/1.51 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A5 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_209_image__subsetI,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,A5: set @ A,F: A > B,B2: set @ B] :
% 2.90/1.51 ( ! [X4: A] :
% 2.90/1.51 ( ( member @ A @ X4 @ A5 )
% 2.90/1.51 => ( member @ B @ ( F @ X4 ) @ B2 ) )
% 2.90/1.51 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A5 ) @ B2 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_210_subset__imageE,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A5: set @ B] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A5 ) )
% 2.90/1.51 => ~ ! [C5: set @ B] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A5 )
% 2.90/1.51 => ( B2
% 2.90/1.51 != ( image @ B @ A @ F @ C5 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_211_image__subset__iff,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,F: B > A,A5: set @ B,B2: set @ A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A5 ) @ B2 )
% 2.90/1.51 = ( ! [X3: B] :
% 2.90/1.51 ( ( member @ B @ X3 @ A5 )
% 2.90/1.51 => ( member @ A @ ( F @ X3 ) @ B2 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_212_subset__image__iff,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A5: set @ B] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A5 ) )
% 2.90/1.51 = ( ? [AA: set @ B] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ B ) @ AA @ A5 )
% 2.90/1.51 & ( B2
% 2.90/1.51 = ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_213_bot_Oextremum__uniqueI,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [A3: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 2.90/1.51 => ( A3
% 2.90/1.51 = ( bot_bot @ A ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_214_bot_Oextremum__unique,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [A3: A] :
% 2.90/1.51 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 2.90/1.51 = ( A3
% 2.90/1.51 = ( bot_bot @ A ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_215_bot_Oextremum,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_216_pairwise__subset,axiom,
% 2.90/1.51 ! [A: $tType,P: A > A > $o,S4: set @ A,T2: set @ A] :
% 2.90/1.51 ( ( pairwise @ A @ P @ S4 )
% 2.90/1.51 => ( ( ord_less_eq @ ( set @ A ) @ T2 @ S4 )
% 2.90/1.51 => ( pairwise @ A @ P @ T2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_217_subset__singletonD,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,X: A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 2.90/1.51 => ( ( A5
% 2.90/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.90/1.51 | ( A5
% 2.90/1.51 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_218_subset__singleton__iff,axiom,
% 2.90/1.51 ! [A: $tType,X7: set @ A,A3: A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ X7 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.90/1.51 = ( ( X7
% 2.90/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.90/1.51 | ( X7
% 2.90/1.51 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_219_comp__surj,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: A > C] :
% 2.90/1.51 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.90/1.51 = ( top_top @ ( set @ A ) ) )
% 2.90/1.51 => ( ( ( image @ A @ C @ G @ ( top_top @ ( set @ A ) ) )
% 2.90/1.51 = ( top_top @ ( set @ C ) ) )
% 2.90/1.51 => ( ( image @ B @ C @ ( comp @ A @ C @ B @ G @ F ) @ ( top_top @ ( set @ B ) ) )
% 2.90/1.51 = ( top_top @ ( set @ C ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_220_fun_Oinj__map__strong,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,D2: $tType,X: D2 > A,Xa2: D2 > A,F: A > B,Fa: A > B] :
% 2.90/1.51 ( ! [Z: A,Za: A] :
% 2.90/1.51 ( ( member @ A @ Z @ ( image @ D2 @ A @ X @ ( top_top @ ( set @ D2 ) ) ) )
% 2.90/1.51 => ( ( member @ A @ Za @ ( image @ D2 @ A @ Xa2 @ ( top_top @ ( set @ D2 ) ) ) )
% 2.90/1.51 => ( ( ( F @ Z )
% 2.90/1.51 = ( Fa @ Za ) )
% 2.90/1.51 => ( Z = Za ) ) ) )
% 2.90/1.51 => ( ( ( comp @ A @ B @ D2 @ F @ X )
% 2.90/1.51 = ( comp @ A @ B @ D2 @ Fa @ Xa2 ) )
% 2.90/1.51 => ( X = Xa2 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_221_fun_Omap__cong0,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,D2: $tType,X: D2 > A,F: A > B,G: A > B] :
% 2.90/1.51 ( ! [Z: A] :
% 2.90/1.51 ( ( member @ A @ Z @ ( image @ D2 @ A @ X @ ( top_top @ ( set @ D2 ) ) ) )
% 2.90/1.51 => ( ( F @ Z )
% 2.90/1.51 = ( G @ Z ) ) )
% 2.90/1.51 => ( ( comp @ A @ B @ D2 @ F @ X )
% 2.90/1.51 = ( comp @ A @ B @ D2 @ G @ X ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_222_fun_Omap__cong,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,D2: $tType,X: D2 > A,Ya: D2 > A,F: A > B,G: A > B] :
% 2.90/1.51 ( ( X = Ya )
% 2.90/1.51 => ( ! [Z: A] :
% 2.90/1.51 ( ( member @ A @ Z @ ( image @ D2 @ A @ Ya @ ( top_top @ ( set @ D2 ) ) ) )
% 2.90/1.51 => ( ( F @ Z )
% 2.90/1.51 = ( G @ Z ) ) )
% 2.90/1.51 => ( ( comp @ A @ B @ D2 @ F @ X )
% 2.90/1.51 = ( comp @ A @ B @ D2 @ G @ Ya ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_223_fun_Oset__map,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,D2: $tType,F: A > B,V: D2 > A] :
% 2.90/1.51 ( ( image @ D2 @ B @ ( comp @ A @ B @ D2 @ F @ V ) @ ( top_top @ ( set @ D2 ) ) )
% 2.90/1.51 = ( image @ A @ B @ F @ ( image @ D2 @ A @ V @ ( top_top @ ( set @ D2 ) ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_224_surj__fun__eq,axiom,
% 2.90/1.51 ! [B: $tType,C: $tType,A: $tType,F: B > A,X7: set @ B,G12: A > C,G23: A > C] :
% 2.90/1.51 ( ( ( image @ B @ A @ F @ X7 )
% 2.90/1.51 = ( top_top @ ( set @ A ) ) )
% 2.90/1.51 => ( ! [X4: B] :
% 2.90/1.51 ( ( member @ B @ X4 @ X7 )
% 2.90/1.51 => ( ( comp @ A @ C @ B @ G12 @ F @ X4 )
% 2.90/1.51 = ( comp @ A @ C @ B @ G23 @ F @ X4 ) ) )
% 2.90/1.51 => ( G12 = G23 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_225_insert__subsetI,axiom,
% 2.90/1.51 ! [A: $tType,X: A,A5: set @ A,X7: set @ A] :
% 2.90/1.51 ( ( member @ A @ X @ A5 )
% 2.90/1.51 => ( ( ord_less_eq @ ( set @ A ) @ X7 @ A5 )
% 2.90/1.51 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X7 ) @ A5 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_226_subset__emptyI,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A] :
% 2.90/1.51 ( ! [X4: A] :
% 2.90/1.51 ~ ( member @ A @ X4 @ A5 )
% 2.90/1.51 => ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_227_subset__Compl__singleton,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,B3: A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.90/1.51 = ( ~ ( member @ A @ B3 @ A5 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_228_ComplI,axiom,
% 2.90/1.51 ! [A: $tType,C3: A,A5: set @ A] :
% 2.90/1.51 ( ~ ( member @ A @ C3 @ A5 )
% 2.90/1.51 => ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_229_Compl__iff,axiom,
% 2.90/1.51 ! [A: $tType,C3: A,A5: set @ A] :
% 2.90/1.51 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
% 2.90/1.51 = ( ~ ( member @ A @ C3 @ A5 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_230_Compl__eq__Compl__iff,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 2.90/1.51 ( ( ( uminus_uminus @ ( set @ A ) @ A5 )
% 2.90/1.51 = ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 2.90/1.51 = ( A5 = B2 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_231_Compl__subset__Compl__iff,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 2.90/1.51 = ( ord_less_eq @ ( set @ A ) @ B2 @ A5 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_232_Compl__anti__mono,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A,B2: set @ A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B2 )
% 2.90/1.51 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_233_top__set__def,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( top_top @ ( set @ A ) )
% 2.90/1.51 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_234_stream_Orel__mono,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,R: A > B > $o,Ra: A > B > $o] :
% 2.90/1.51 ( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra )
% 2.90/1.51 => ( ord_less_eq @ ( ( stream170649215stream @ A ) > ( stream170649215stream @ B ) > $o ) @ ( stream686525009m_all2 @ A @ B @ R ) @ ( stream686525009m_all2 @ A @ B @ Ra ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_235_ComplD,axiom,
% 2.90/1.51 ! [A: $tType,C3: A,A5: set @ A] :
% 2.90/1.51 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
% 2.90/1.51 => ~ ( member @ A @ C3 @ A5 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_236_double__complement,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A] :
% 2.90/1.51 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
% 2.90/1.51 = A5 ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_237_subset__Compl__self__eq,axiom,
% 2.90/1.51 ! [A: $tType,A5: set @ A] :
% 2.90/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
% 2.90/1.51 = ( A5
% 2.90/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_238_Compl__empty__eq,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
% 2.90/1.51 = ( top_top @ ( set @ A ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_239_Compl__UNIV__eq,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
% 2.90/1.51 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_240_surj__Compl__image__subset,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
% 2.90/1.51 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.90/1.51 = ( top_top @ ( set @ A ) ) )
% 2.90/1.51 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F @ A5 ) ) @ ( image @ B @ A @ F @ ( uminus_uminus @ ( set @ B ) @ A5 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_241_compl__top__eq,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.90/1.51 => ( ( uminus_uminus @ A @ ( top_top @ A ) )
% 2.90/1.51 = ( bot_bot @ A ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_242_compl__bot__eq,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.90/1.51 => ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
% 2.90/1.51 = ( top_top @ A ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_243_predicate2I,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 2.90/1.51 ( ! [X4: A,Y4: B] :
% 2.90/1.51 ( ( P @ X4 @ Y4 )
% 2.90/1.51 => ( Q @ X4 @ Y4 ) )
% 2.90/1.51 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_244_top1I,axiom,
% 2.90/1.51 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_245_rev__predicate2D,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
% 2.90/1.51 ( ( P @ X @ Y )
% 2.90/1.51 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 2.90/1.51 => ( Q @ X @ Y ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_246_predicate2D,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
% 2.90/1.51 ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 2.90/1.51 => ( ( P @ X @ Y )
% 2.90/1.51 => ( Q @ X @ Y ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_247_refl__ge__eq,axiom,
% 2.90/1.51 ! [A: $tType,R: A > A > $o] :
% 2.90/1.51 ( ! [X4: A] : ( R @ X4 @ X4 )
% 2.90/1.51 => ( ord_less_eq @ ( A > A > $o )
% 2.90/1.51 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 2.90/1.51 @ R ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_248_ge__eq__refl,axiom,
% 2.90/1.51 ! [A: $tType,R: A > A > $o,X: A] :
% 2.90/1.51 ( ( ord_less_eq @ ( A > A > $o )
% 2.90/1.51 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 2.90/1.51 @ R )
% 2.90/1.51 => ( R @ X @ X ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_249_predicate2D__conj,axiom,
% 2.90/1.51 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X: A,Y: B] :
% 2.90/1.51 ( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 2.90/1.51 & R )
% 2.90/1.51 => ( R
% 2.90/1.51 & ( ( P @ X @ Y )
% 2.90/1.51 => ( Q @ X @ Y ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_250_top__conj_I1_J,axiom,
% 2.90/1.51 ! [A: $tType,X: A,P: $o] :
% 2.90/1.51 ( ( ( top_top @ ( A > $o ) @ X )
% 2.90/1.51 & P )
% 2.90/1.51 = P ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_251_top__conj_I2_J,axiom,
% 2.90/1.51 ! [A: $tType,P: $o,X: A] :
% 2.90/1.51 ( ( P
% 2.90/1.51 & ( top_top @ ( A > $o ) @ X ) )
% 2.90/1.51 = P ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_252_inj__image__Compl__subset,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
% 2.90/1.51 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.90/1.51 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F @ A5 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_253_ID_Opred__set,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( bNF_id_bnf @ ( A > $o ) )
% 2.90/1.51 = ( ^ [P2: A > $o,X3: A] :
% 2.90/1.51 ! [Y3: A] :
% 2.90/1.51 ( ( member @ A @ Y3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.90/1.51 => ( P2 @ Y3 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_254_inj__on__empty,axiom,
% 2.90/1.51 ! [B: $tType,A: $tType,F: A > B] : ( inj_on @ A @ B @ F @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(fact_255_inj__uminus,axiom,
% 2.90/1.51 ! [A: $tType] :
% 2.90/1.51 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.90/1.51 => ! [A5: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A5 ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 2.90/1.51 ! [A8: $tType,A9: $tType] :
% 2.90/1.51 ( ( boolean_algebra @ A9 @ ( type2 @ A9 ) )
% 2.90/1.51 => ( boolean_algebra @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 2.90/1.51 ! [A8: $tType,A9: $tType] :
% 2.90/1.51 ( ( order_top @ A9 @ ( type2 @ A9 ) )
% 2.90/1.51 => ( order_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.90/1.51 ! [A8: $tType,A9: $tType] :
% 2.90/1.51 ( ( order_bot @ A9 @ ( type2 @ A9 ) )
% 2.90/1.51 => ( order_bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.90/1.51 ! [A8: $tType,A9: $tType] :
% 2.90/1.51 ( ( preorder @ A9 @ ( type2 @ A9 ) )
% 2.90/1.51 => ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.90/1.51 ! [A8: $tType,A9: $tType] :
% 2.90/1.51 ( ( order @ A9 @ ( type2 @ A9 ) )
% 2.90/1.51 => ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_fun___Orderings_Otop,axiom,
% 2.90/1.51 ! [A8: $tType,A9: $tType] :
% 2.90/1.51 ( ( top @ A9 @ ( type2 @ A9 ) )
% 2.90/1.51 => ( top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_fun___Orderings_Oord,axiom,
% 2.90/1.51 ! [A8: $tType,A9: $tType] :
% 2.90/1.51 ( ( ord @ A9 @ ( type2 @ A9 ) )
% 2.90/1.51 => ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_fun___Orderings_Obot,axiom,
% 2.90/1.51 ! [A8: $tType,A9: $tType] :
% 2.90/1.51 ( ( bot @ A9 @ ( type2 @ A9 ) )
% 2.90/1.51 => ( bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_1,axiom,
% 2.90/1.51 ! [A8: $tType] : ( boolean_algebra @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_Set_Oset___Orderings_Oorder__top_2,axiom,
% 2.90/1.51 ! [A8: $tType] : ( order_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_Set_Oset___Orderings_Oorder__bot_3,axiom,
% 2.90/1.51 ! [A8: $tType] : ( order_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 2.90/1.51 ! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 2.90/1.51 ! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_Set_Oset___Orderings_Otop_6,axiom,
% 2.90/1.51 ! [A8: $tType] : ( top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 2.90/1.51 ! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_Set_Oset___Orderings_Obot_8,axiom,
% 2.90/1.51 ! [A8: $tType] : ( bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_9,axiom,
% 2.90/1.51 boolean_algebra @ $o @ ( type2 @ $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_HOL_Obool___Orderings_Oorder__top_10,axiom,
% 2.90/1.51 order_top @ $o @ ( type2 @ $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_HOL_Obool___Orderings_Oorder__bot_11,axiom,
% 2.90/1.51 order_bot @ $o @ ( type2 @ $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
% 2.90/1.51 preorder @ $o @ ( type2 @ $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.90/1.51 linorder @ $o @ ( type2 @ $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_HOL_Obool___Orderings_Oorder_13,axiom,
% 2.90/1.51 order @ $o @ ( type2 @ $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_HOL_Obool___Orderings_Otop_14,axiom,
% 2.90/1.51 top @ $o @ ( type2 @ $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
% 2.90/1.51 ord @ $o @ ( type2 @ $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
% 2.90/1.51 bot @ $o @ ( type2 @ $o ) ).
% 2.90/1.51
% 2.90/1.51 thf(help_If_3_1_T,axiom,
% 2.90/1.51 ! [P: $o] :
% 2.90/1.51 ( ( P = $true )
% 2.90/1.51 | ( P = $false ) ) ).
% 2.90/1.51
% 2.90/1.51 thf(help_If_2_1_T,axiom,
% 2.90/1.51 ! [A: $tType,X: A,Y: A] :
% 2.90/1.51 ( ( if @ A @ $false @ X @ Y )
% 2.90/1.51 = Y ) ).
% 2.90/1.51
% 2.90/1.51 thf(help_If_1_1_T,axiom,
% 2.90/1.51 ! [A: $tType,X: A,Y: A] :
% 2.90/1.51 ( ( if @ A @ $true @ X @ Y )
% 2.90/1.51 = X ) ).
% 2.90/1.51
% 2.90/1.51 thf(conj_0,conjecture,
% 2.90/1.51 ( ( member @ a @ x @ ( stream30925839e_sset @ a @ ( stream641971652_SCons @ a @ y @ s ) ) )
% 2.90/1.51 != ( ~ ( ( x != y )
% 2.90/1.51 => ( member @ a @ x @ ( stream30925839e_sset @ a @ s ) ) ) ) ) ).
% 2.90/1.51 ------------------------
% 2.90/1.53 % (22140)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_22005 for (2999ds/4Mi)
% 2.90/1.53 % (22145)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22005 for (2999ds/18Mi)
% 2.90/1.53 % (22143)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_22005 for (2999ds/2Mi)
% 2.90/1.53 % (22144)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_22005 for (2999ds/275Mi)
% 2.90/1.53 % (22140)Instruction limit reached!
% 2.90/1.53 % (22140)------------------------------
% 2.90/1.53 % (22140)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.53 % (22140)Termination reason: Unknown
% 2.90/1.53 % (22140)Termination phase: shuffling
% 2.90/1.53
% 2.90/1.53 % (22140)Memory used [KB]: 1407
% 2.90/1.53 % (22140)Time elapsed: 0.003 s
% 2.90/1.53 % (22140)Instructions burned: 5 (million)
% 2.90/1.53 % (22140)------------------------------
% 2.90/1.53 % (22140)------------------------------
% 2.90/1.53 % (22143)Instruction limit reached!
% 2.90/1.53 % (22143)------------------------------
% 2.90/1.53 % (22143)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.53 % (22143)Termination reason: Unknown
% 2.90/1.53 % (22143)Termination phase: shuffling
% 2.90/1.53
% 2.90/1.53 % (22143)Memory used [KB]: 1407
% 2.90/1.53 % (22143)Time elapsed: 0.002 s
% 2.90/1.53 % (22143)Instructions burned: 3 (million)
% 2.90/1.53 % (22143)------------------------------
% 2.90/1.53 % (22143)------------------------------
% 2.90/1.54 % (22145)Instruction limit reached!
% 2.90/1.54 % (22145)------------------------------
% 2.90/1.54 % (22145)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.54 % (22145)Termination reason: Unknown
% 2.90/1.54 % (22145)Termination phase: shuffling
% 2.90/1.54
% 2.90/1.54 % (22145)Memory used [KB]: 1791
% 2.90/1.54 % (22145)Time elapsed: 0.007 s
% 2.90/1.54 % (22145)Instructions burned: 18 (million)
% 2.90/1.54 % (22145)------------------------------
% 2.90/1.54 % (22145)------------------------------
% 2.90/1.54 % (22139)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22005 for (2999ds/183Mi)
% 2.90/1.54 % (22141)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_22005 for (2999ds/27Mi)
% 2.90/1.54 % (22142)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22005 for (2999ds/2Mi)
% 2.90/1.54 % (22142)Instruction limit reached!
% 2.90/1.54 % (22142)------------------------------
% 2.90/1.54 % (22142)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.54 % (22142)Termination reason: Unknown
% 2.90/1.54 % (22142)Termination phase: shuffling
% 2.90/1.54
% 2.90/1.54 % (22142)Memory used [KB]: 1407
% 2.90/1.54 % (22142)Time elapsed: 0.004 s
% 2.90/1.54 % (22142)Instructions burned: 3 (million)
% 2.90/1.54 % (22142)------------------------------
% 2.90/1.54 % (22142)------------------------------
% 2.90/1.54 % (22146)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22005 for (2999ds/3Mi)
% 2.90/1.54 % (22147)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_22005 for (2999ds/37Mi)
% 2.90/1.54 % (22146)Instruction limit reached!
% 2.90/1.54 % (22146)------------------------------
% 2.90/1.54 % (22146)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.54 % (22146)Termination reason: Unknown
% 2.90/1.54 % (22146)Termination phase: shuffling
% 2.90/1.54
% 2.90/1.54 % (22146)Memory used [KB]: 1407
% 2.90/1.54 % (22146)Time elapsed: 0.002 s
% 2.90/1.54 % (22146)Instructions burned: 3 (million)
% 2.90/1.54 % (22146)------------------------------
% 2.90/1.54 % (22146)------------------------------
% 2.90/1.55 % (22148)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_22005 for (2999ds/15Mi)
% 2.90/1.55 % (22148)Instruction limit reached!
% 2.90/1.55 % (22148)------------------------------
% 2.90/1.55 % (22148)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.55 % (22148)Termination reason: Unknown
% 2.90/1.55 % (22148)Termination phase: shuffling
% 2.90/1.55
% 2.90/1.55 % (22148)Memory used [KB]: 1663
% 2.90/1.55 % (22148)Time elapsed: 0.006 s
% 2.90/1.55 % (22148)Instructions burned: 15 (million)
% 2.90/1.55 % (22148)------------------------------
% 2.90/1.55 % (22148)------------------------------
% 2.90/1.55 % (22150)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_22005 for (2999ds/1041Mi)
% 2.90/1.55 % (22144)First to succeed.
% 2.90/1.56 % (22147)Instruction limit reached!
% 2.90/1.56 % (22147)------------------------------
% 2.90/1.56 % (22147)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.56 % (22147)Termination reason: Unknown
% 2.90/1.56 % (22147)Termination phase: shuffling
% 2.90/1.56
% 2.90/1.56 % (22147)Memory used [KB]: 2174
% 2.90/1.56 % (22147)Time elapsed: 0.013 s
% 2.90/1.56 % (22147)Instructions burned: 37 (million)
% 2.90/1.56 % (22147)------------------------------
% 2.90/1.56 % (22147)------------------------------
% 2.90/1.56 % (22144)Refutation found. Thanks to Tanya!
% 2.90/1.56 % SZS status Theorem for DTF2THF_22005
% 2.90/1.56 % SZS output start Proof for DTF2THF_22005
% 2.90/1.56 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.90/1.56 thf(type_def_7, type, set: ($tType) > $tType).
% 2.90/1.56 thf(type_def_8, type, stream170649215stream: ($tType) > $tType).
% 2.90/1.56 thf(type_def_9, type, a: $tType).
% 2.90/1.56 thf(func_def_0, type, stream170649215stream: ($tType) > $tType).
% 2.90/1.56 thf(func_def_1, type, set: ($tType) > $tType).
% 2.90/1.56 thf(func_def_2, type, itself: ($tType) > $tType).
% 2.90/1.56 thf(func_def_3, type, a: $tType).
% 2.90/1.56 thf(func_def_4, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.56 thf(func_def_5, type, bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.56 thf(func_def_6, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.56 thf(func_def_7, type, top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.56 thf(func_def_8, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.56 thf(func_def_9, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.56 thf(func_def_10, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.56 thf(func_def_11, type, order_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.56 thf(func_def_12, type, order_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.56 thf(func_def_13, type, boolean_algebra: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.56 thf(func_def_14, type, ordered_ab_group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.90/1.56 thf(func_def_15, type, bNF_id_bnf: !>[X0: $tType]:(X0 > X0)).
% 2.90/1.56 thf(func_def_16, type, comp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X0) > X2 > X1)).
% 2.90/1.56 thf(func_def_17, type, inj_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 2.90/1.56 thf(func_def_18, type, override_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > X1) > set @ X0 > X0 > X1)).
% 2.90/1.56 thf(func_def_19, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 2.90/1.56 thf(func_def_20, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.90/1.56 thf(func_def_21, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.90/1.56 thf(func_def_22, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.90/1.56 thf(func_def_23, type, top_top: !>[X0: $tType]:(X0)).
% 2.90/1.56 thf(func_def_24, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.90/1.56 thf(func_def_25, type, ball: !>[X0: $tType]:(set @ X0 > (X0 > $o) > $o)).
% 2.90/1.56 thf(func_def_26, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.90/1.56 thf(func_def_27, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 2.90/1.56 thf(func_def_28, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 2.90/1.56 thf(func_def_29, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 2.90/1.56 thf(func_def_30, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 2.90/1.56 thf(func_def_31, type, pairwise: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > $o)).
% 2.90/1.56 thf(func_def_32, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 2.90/1.56 thf(func_def_33, type, stream1586597341member: !>[X0: $tType]:(X0 > stream170649215stream @ X0 > $o)).
% 2.90/1.56 thf(func_def_34, type, stream641971652_SCons: !>[X0: $tType]:(X0 > stream170649215stream @ X0 > stream170649215stream @ X0)).
% 2.90/1.56 thf(func_def_35, type, stream1342653232stream: !>[X0: $tType, X1: $tType]:((X0 > stream170649215stream @ X0 > X1) > stream170649215stream @ X0 > X1)).
% 2.90/1.56 thf(func_def_36, type, stream660621732stream: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > (X0 > stream170649215stream @ X1) > (X0 > X0) > X0 > stream170649215stream @ X1)).
% 2.90/1.56 thf(func_def_37, type, stream1153105665stream: !>[X0: $tType]:((X0 > $o) > stream170649215stream @ X0 > $o)).
% 2.90/1.56 thf(func_def_38, type, stream_Mirabelle_shd: !>[X0: $tType]:(stream170649215stream @ X0 > X0)).
% 2.90/1.56 thf(func_def_39, type, stream2128578057e_smap: !>[X0: $tType, X1: $tType]:((X0 > X1) > stream170649215stream @ X0 > stream170649215stream @ X1)).
% 2.90/1.56 thf(func_def_40, type, stream30925839e_sset: !>[X0: $tType]:(stream170649215stream @ X0 > set @ X0)).
% 2.90/1.56 thf(func_def_41, type, stream_Mirabelle_stl: !>[X0: $tType]:(stream170649215stream @ X0 > stream170649215stream @ X0)).
% 2.90/1.56 thf(func_def_42, type, stream686525009m_all2: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > stream170649215stream @ X0 > stream170649215stream @ X1 > $o)).
% 2.90/1.56 thf(func_def_43, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.90/1.56 thf(func_def_44, type, s: stream170649215stream @ a).
% 2.90/1.56 thf(func_def_45, type, x: a).
% 2.90/1.56 thf(func_def_46, type, y: a).
% 2.90/1.56 thf(func_def_62, type, sP0: !>[X0: $tType]:(X0 > X0 > set @ X0 > set @ X0 > $o)).
% 2.90/1.56 thf(func_def_63, type, sP1: !>[X0: $tType]:(X0 > set @ X0 > X0 > set @ X0 > $o)).
% 2.90/1.56 thf(func_def_64, type, sK2: !>[X0: $tType]:(set @ X0 > X0 > set @ X0)).
% 2.90/1.56 thf(func_def_65, type, sK3: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > set @ X0 > X0)).
% 2.90/1.56 thf(func_def_66, type, sK4: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.90/1.56 thf(func_def_67, type, sK5: !>[X0: $tType]:(stream170649215stream @ X0 > X0 > X0)).
% 2.90/1.56 thf(func_def_68, type, sK6: !>[X0: $tType]:(stream170649215stream @ X0 > X0 > stream170649215stream @ X0)).
% 2.90/1.56 thf(func_def_69, type, sK7: !>[X0: $tType]:(stream170649215stream @ X0 > X0 > stream170649215stream @ X0)).
% 2.90/1.56 thf(func_def_70, type, sK8: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.90/1.56 thf(func_def_71, type, sK9: !>[X0: $tType]:(X0 > X0 > set @ X0 > set @ X0 > set @ X0)).
% 2.90/1.56 thf(func_def_72, type, sK10: !>[X0: $tType]:(stream170649215stream @ X0 > X0)).
% 2.90/1.56 thf(func_def_73, type, sK11: !>[X0: $tType]:(stream170649215stream @ X0 > stream170649215stream @ X0)).
% 2.90/1.56 thf(func_def_74, type, sK12: !>[X0: $tType]:((X0 > stream170649215stream @ X0 > $o) > X0)).
% 2.90/1.56 thf(func_def_75, type, sK13: !>[X0: $tType]:((X0 > stream170649215stream @ X0 > $o) > stream170649215stream @ X0)).
% 2.90/1.56 thf(func_def_76, type, sK14: !>[X0: $tType]:((X0 > stream170649215stream @ X0 > $o) > X0)).
% 2.90/1.56 thf(func_def_77, type, sK15: !>[X0: $tType]:((X0 > stream170649215stream @ X0 > $o) > stream170649215stream @ X0)).
% 2.90/1.56 thf(func_def_78, type, sK16: !>[X0: $tType]:((X0 > stream170649215stream @ X0 > $o) > X0)).
% 2.90/1.56 thf(func_def_79, type, sK17: !>[X0: $tType]:(set @ X0 > X0 > set @ X0)).
% 2.90/1.56 thf(func_def_81, type, ph19: !>[X0: $tType]:(X0)).
% 2.90/1.56 thf(f986,plain,(
% 2.90/1.56 $false),
% 2.90/1.56 inference(avatar_sat_refutation,[status(thm)],[f952,f955,f971,f985])).
% 2.90/1.56 thf(f985,plain,(
% 2.90/1.56 ~spl18_1 | spl18_2),
% 2.90/1.56 inference(avatar_contradiction_clause,[status(thm)],[f984])).
% 2.90/1.56 thf(f984,plain,(
% 2.90/1.56 $false | (~spl18_1 | spl18_2)),
% 2.90/1.56 inference(subsumption_resolution,[status(thm)],[f983,f940])).
% 2.90/1.56 thf(f940,plain,(
% 2.90/1.56 ( ! [X0 : $tType,X2 : X0,X3 : set @ X0] : (((member @ X0 @ X2 @ (insert @ X0 @ X2 @ X3)) = $true)) )),
% 2.90/1.56 inference(equality_resolution,[status(thm)],[f905])).
% 2.90/1.56 thf(f905,plain,(
% 2.90/1.56 ( ! [X0 : $tType,X2 : X0,X3 : set @ X0,X1 : X0] : (((member @ X0 @ X1 @ (insert @ X0 @ X2 @ X3)) = $true) | (X1 != X2)) )),
% 2.90/1.56 inference(cnf_transformation,[status(thm)],[f828])).
% 2.90/1.56 thf(f828,plain,(
% 2.90/1.56 ! [X0 : $tType,X1 : X0,X2 : X0,X3 : set @ X0] : ((((member @ X0 @ X1 @ (insert @ X0 @ X2 @ X3)) = $true) | (((member @ X0 @ X1 @ X3) != $true) & (X1 != X2))) & (((member @ X0 @ X1 @ X3) = $true) | (X1 = X2) | ((member @ X0 @ X1 @ (insert @ X0 @ X2 @ X3)) != $true)))),
% 2.90/1.56 inference(flattening,[status(thm)],[f827])).
% 2.90/1.56 thf(f827,plain,(
% 2.90/1.56 ! [X0 : $tType,X1 : X0,X2 : X0,X3 : set @ X0] : ((((member @ X0 @ X1 @ (insert @ X0 @ X2 @ X3)) = $true) | (((member @ X0 @ X1 @ X3) != $true) & (X1 != X2))) & ((((member @ X0 @ X1 @ X3) = $true) | (X1 = X2)) | ((member @ X0 @ X1 @ (insert @ X0 @ X2 @ X3)) != $true)))),
% 2.90/1.56 inference(nnf_transformation,[status(thm)],[f511])).
% 2.90/1.56 thf(f511,plain,(
% 2.90/1.56 ! [X0 : $tType,X1 : X0,X2 : X0,X3 : set @ X0] : (((member @ X0 @ X1 @ (insert @ X0 @ X2 @ X3)) = $true) <=> (((member @ X0 @ X1 @ X3) = $true) | (X1 = X2)))),
% 2.90/1.56 inference(fool_elimination,[status(thm)],[f510])).
% 2.90/1.56 thf(f510,plain,(
% 2.90/1.56 ! [X0 : $tType,X1 : X0,X2 : X0,X3 : set @ X0] : ((member @ X0 @ X1 @ X3) | (X1 = X2) = (member @ X0 @ X1 @ (insert @ X0 @ X2 @ X3)))),
% 2.90/1.56 inference(rectify,[status(thm)],[f36])).
% 2.90/1.56 thf(f36,axiom,(
% 2.90/1.56 ! [X0 : $tType,X15 : X0,X48 : X0,X49 : set @ X0] : ((member @ X0 @ X15 @ X49) | (X15 = X48) = (member @ X0 @ X15 @ (insert @ X0 @ X48 @ X49)))),
% 2.90/1.56 file('/export/starexec/sandbox2/tmp/tmp.e7YR8rdmNP/DTF2THF_22005.p',fact_35_insert__iff)).
% 2.90/1.56 thf(f983,plain,(
% 2.90/1.56 ((member @ a @ x @ (insert @ a @ x @ (stream30925839e_sset @ a @ s))) != $true) | (~spl18_1 | spl18_2)),
% 2.90/1.56 inference(forward_demodulation,[status(thm)],[f951,f946])).
% 2.90/1.56 thf(f946,plain,(
% 2.90/1.56 (x = y) | ~spl18_1),
% 2.90/1.56 inference(avatar_component_clause,[status(thm)],[f945])).
% 2.90/1.56 thf(f945,definition,(
% 2.90/1.56 spl18_1 <=> (x = y)),
% 2.90/1.56 introduced(definition,[new_symbols(naming,[spl18_1])],[avatar_definition])).
% 2.90/1.56 thf(f951,plain,(
% 2.90/1.56 ((member @ a @ x @ (insert @ a @ y @ (stream30925839e_sset @ a @ s))) != $true) | spl18_2),
% 2.90/1.56 inference(avatar_component_clause,[status(thm)],[f949])).
% 2.90/1.56 thf(f949,definition,(
% 2.90/1.56 spl18_2 <=> ((member @ a @ x @ (insert @ a @ y @ (stream30925839e_sset @ a @ s))) = $true)),
% 2.90/1.56 introduced(definition,[new_symbols(naming,[spl18_2])],[avatar_definition])).
% 2.90/1.56 thf(f971,plain,(
% 2.90/1.56 spl18_1 | ~spl18_2),
% 2.90/1.56 inference(avatar_contradiction_clause,[status(thm)],[f970])).
% 2.90/1.56 thf(f970,plain,(
% 2.90/1.56 $false | (spl18_1 | ~spl18_2)),
% 2.90/1.56 inference(subsumption_resolution,[status(thm)],[f969,f947])).
% 2.90/1.56 thf(f947,plain,(
% 2.90/1.56 (x != y) | spl18_1),
% 2.90/1.56 inference(avatar_component_clause,[status(thm)],[f945])).
% 2.90/1.56 thf(f969,plain,(
% 2.90/1.56 (x = y) | ~spl18_2),
% 2.90/1.56 inference(subsumption_resolution,[status(thm)],[f964,f956])).
% 2.90/1.56 thf(f956,plain,(
% 2.90/1.56 ((member @ a @ x @ (stream30925839e_sset @ a @ s)) != $true)),
% 2.90/1.56 inference(subsumption_resolution,[status(thm)],[f876,f842])).
% 2.90/1.56 thf(f842,plain,(
% 2.90/1.56 ( ! [X0 : $tType,X2 : stream170649215stream @ X0,X3 : X0,X1 : X0] : (((member @ X0 @ X1 @ (stream30925839e_sset @ X0 @ (stream641971652_SCons @ X0 @ X3 @ X2))) = $true) | ((member @ X0 @ X1 @ (stream30925839e_sset @ X0 @ X2)) != $true)) )),
% 2.90/1.56 inference(cnf_transformation,[status(thm)],[f762])).
% 2.90/1.56 thf(f762,plain,(
% 2.90/1.56 ! [X0 : $tType,X1 : X0,X2 : stream170649215stream @ X0,X3 : X0] : (((member @ X0 @ X1 @ (stream30925839e_sset @ X0 @ X2)) != $true) | ((member @ X0 @ X1 @ (stream30925839e_sset @ X0 @ (stream641971652_SCons @ X0 @ X3 @ X2))) = $true))),
% 2.90/1.56 inference(ennf_transformation,[status(thm)],[f703])).
% 2.90/1.56 thf(f703,plain,(
% 2.90/1.56 ! [X0 : $tType,X1 : X0,X2 : stream170649215stream @ X0,X3 : X0] : (((member @ X0 @ X1 @ (stream30925839e_sset @ X0 @ X2)) = $true) => ((member @ X0 @ X1 @ (stream30925839e_sset @ X0 @ (stream641971652_SCons @ X0 @ X3 @ X2))) = $true))),
% 2.90/1.56 inference(fool_elimination,[status(thm)],[f702])).
% 2.90/1.56 thf(f702,plain,(
% 2.90/1.56 ! [X0 : $tType,X1 : X0,X2 : stream170649215stream @ X0,X3 : X0] : ((member @ X0 @ X1 @ (stream30925839e_sset @ X0 @ X2)) => (member @ X0 @ X1 @ (stream30925839e_sset @ X0 @ (stream641971652_SCons @ X0 @ X3 @ X2))))),
% 2.90/1.56 inference(rectify,[status(thm)],[f2])).
% 2.90/1.56 thf(f2,axiom,(
% 2.90/1.56 ! [X0 : $tType,X8 : X0,X9 : stream170649215stream @ X0,X10 : X0] : ((member @ X0 @ X8 @ (stream30925839e_sset @ X0 @ X9)) => (member @ X0 @ X8 @ (stream30925839e_sset @ X0 @ (stream641971652_SCons @ X0 @ X10 @ X9))))),
% 2.90/1.56 file('/export/starexec/sandbox2/tmp/tmp.e7YR8rdmNP/DTF2THF_22005.p',fact_1_stream_Oset__intros_I2_J)).
% 2.90/1.56 thf(f876,plain,(
% 2.90/1.56 ((member @ a @ x @ (stream30925839e_sset @ a @ s)) != $true) | ((member @ a @ x @ (stream30925839e_sset @ a @ (stream641971652_SCons @ a @ y @ s))) != $true)),
% 2.90/1.56 inference(cnf_transformation,[status(thm)],[f812])).
% 2.90/1.56 thf(f812,plain,(
% 2.90/1.56 (((x != y) & ((member @ a @ x @ (stream30925839e_sset @ a @ s)) != $true)) | ((member @ a @ x @ (stream30925839e_sset @ a @ (stream641971652_SCons @ a @ y @ s))) != $true)) & (((member @ a @ x @ (stream30925839e_sset @ a @ (stream641971652_SCons @ a @ y @ s))) = $true) | (x = y) | ((member @ a @ x @ (stream30925839e_sset @ a @ s)) = $true))),
% 2.90/1.56 inference(flattening,[status(thm)],[f811])).
% 2.90/1.56 thf(f811,plain,(
% 2.90/1.56 (((x != y) & ((member @ a @ x @ (stream30925839e_sset @ a @ s)) != $true)) | ((member @ a @ x @ (stream30925839e_sset @ a @ (stream641971652_SCons @ a @ y @ s))) != $true)) & (((member @ a @ x @ (stream30925839e_sset @ a @ (stream641971652_SCons @ a @ y @ s))) = $true) | ((x = y) | ((member @ a @ x @ (stream30925839e_sset @ a @ s)) = $true)))),
% 2.90/1.56 inference(nnf_transformation,[status(thm)],[f773])).
% 2.90/1.56 thf(f773,plain,(
% 2.90/1.56 ((x != y) & ((member @ a @ x @ (stream30925839e_sset @ a @ s)) != $true)) <=> ((member @ a @ x @ (stream30925839e_sset @ a @ (stream641971652_SCons @ a @ y @ s))) = $true)),
% 2.90/1.56 inference(ennf_transformation,[status(thm)],[f649])).
% 2.90/1.56 thf(f649,plain,(
% 2.90/1.56 ~(((member @ a @ x @ (stream30925839e_sset @ a @ (stream641971652_SCons @ a @ y @ s))) = $true) <~> ~((x != y) => ((member @ a @ x @ (stream30925839e_sset @ a @ s)) = $true)))),
% 2.90/1.56 inference(fool_elimination,[status(thm)],[f648])).
% 2.90/1.56 thf(f648,plain,(
% 2.90/1.56 ~(~((x != y) => (member @ a @ x @ (stream30925839e_sset @ a @ s))) != (member @ a @ x @ (stream30925839e_sset @ a @ (stream641971652_SCons @ a @ y @ s))))),
% 2.90/1.56 inference(rectify,[status(thm)],[f286])).
% 2.90/1.56 thf(f286,negated_conjecture,(
% 2.90/1.56 ~(~((x != y) => (member @ a @ x @ (stream30925839e_sset @ a @ s))) != (member @ a @ x @ (stream30925839e_sset @ a @ (stream641971652_SCons @ a @ y @ s))))),
% 2.90/1.56 inference(negated_conjecture,[status(cth)],[f285])).
% 2.90/1.56 thf(f285,conjecture,(
% 2.90/1.56 (~((x != y) => (member @ a @ x @ (stream30925839e_sset @ a @ s))) != (member @ a @ x @ (stream30925839e_sset @ a @ (stream641971652_SCons @ a @ y @ s))))),
% 2.90/1.56 file('/export/starexec/sandbox2/tmp/tmp.e7YR8rdmNP/DTF2THF_22005.p',conj_0)).
% 2.90/1.56 thf(f964,plain,(
% 2.90/1.56 ((member @ a @ x @ (stream30925839e_sset @ a @ s)) = $true) | (x = y) | ~spl18_2),
% 2.90/1.56 inference(trivial_inequality_removal,[status(thm)],[f960])).
% 2.90/1.56 thf(f960,plain,(
% 2.90/1.56 ((member @ a @ x @ (stream30925839e_sset @ a @ s)) = $true) | (x = y) | ($true != $true) | ~spl18_2),
% 2.90/1.56 inference(superposition,[status(thm)],[f860,f950])).
% 2.90/1.56 thf(f950,plain,(
% 2.90/1.56 ((member @ a @ x @ (insert @ a @ y @ (stream30925839e_sset @ a @ s))) = $true) | ~spl18_2),
% 2.90/1.56 inference(avatar_component_clause,[status(thm)],[f949])).
% 2.90/1.56 thf(f860,plain,(
% 2.90/1.56 ( ! [X0 : $tType,X2 : X0,X3 : set @ X0,X1 : X0] : (((member @ X0 @ X1 @ (insert @ X0 @ X2 @ X3)) != $true) | ((member @ X0 @ X1 @ X3) = $true) | (X1 = X2)) )),
% 2.90/1.56 inference(cnf_transformation,[status(thm)],[f760])).
% 2.90/1.56 thf(f760,plain,(
% 2.90/1.56 ! [X0 : $tType,X1 : X0,X2 : X0,X3 : set @ X0] : (((member @ X0 @ X1 @ (insert @ X0 @ X2 @ X3)) != $true) | ((member @ X0 @ X1 @ X3) = $true) | (X1 = X2))),
% 2.90/1.56 inference(flattening,[status(thm)],[f759])).
% 2.90/1.56 thf(f759,plain,(
% 2.90/1.56 ! [X0 : $tType,X1 : X0,X2 : X0,X3 : set @ X0] : ((((member @ X0 @ X1 @ X3) = $true) | (X1 = X2)) | ((member @ X0 @ X1 @ (insert @ X0 @ X2 @ X3)) != $true))),
% 2.90/1.56 inference(ennf_transformation,[status(thm)],[f675])).
% 2.90/1.56 thf(f675,plain,(
% 2.90/1.56 ! [X0 : $tType,X1 : X0,X2 : X0,X3 : set @ X0] : (((member @ X0 @ X1 @ (insert @ X0 @ X2 @ X3)) = $true) => ((X1 != X2) => ((member @ X0 @ X1 @ X3) = $true)))),
% 2.90/1.56 inference(fool_elimination,[status(thm)],[f674])).
% 2.90/1.56 thf(f674,plain,(
% 2.90/1.56 ! [X0 : $tType,X1 : X0,X2 : X0,X3 : set @ X0] : ((member @ X0 @ X1 @ (insert @ X0 @ X2 @ X3)) => ((X1 != X2) => (member @ X0 @ X1 @ X3)))),
% 2.90/1.56 inference(rectify,[status(thm)],[f38])).
% 2.90/1.56 thf(f38,axiom,(
% 2.90/1.56 ! [X0 : $tType,X15 : X0,X48 : X0,X49 : set @ X0] : ((member @ X0 @ X15 @ (insert @ X0 @ X48 @ X49)) => ((X15 != X48) => (member @ X0 @ X15 @ X49)))),
% 2.90/1.56 file('/export/starexec/sandbox2/tmp/tmp.e7YR8rdmNP/DTF2THF_22005.p',fact_37_insertE)).
% 2.90/1.56 thf(f955,plain,(
% 2.90/1.56 spl18_2 | spl18_1),
% 2.90/1.56 inference(avatar_split_clause,[status(thm)],[f954,f945,f949])).
% 2.90/1.56 thf(f954,plain,(
% 2.90/1.56 (x = y) | ((member @ a @ x @ (insert @ a @ y @ (stream30925839e_sset @ a @ s))) = $true)),
% 2.90/1.56 inference(forward_demodulation,[status(thm)],[f953,f899])).
% 2.90/1.56 thf(f899,plain,(
% 2.90/1.56 ( ! [X0 : $tType,X2 : stream170649215stream @ X0,X1 : X0] : (((stream30925839e_sset @ X0 @ (stream641971652_SCons @ X0 @ X1 @ X2)) = (insert @ X0 @ X1 @ (stream30925839e_sset @ X0 @ X2)))) )),
% 2.90/1.56 inference(cnf_transformation,[status(thm)],[f743])).
% 2.90/1.56 thf(f743,plain,(
% 2.90/1.56 ! [X0 : $tType,X1 : X0,X2 : stream170649215stream @ X0] : ((stream30925839e_sset @ X0 @ (stream641971652_SCons @ X0 @ X1 @ X2)) = (insert @ X0 @ X1 @ (stream30925839e_sset @ X0 @ X2)))),
% 2.90/1.56 inference(rectify,[status(thm)],[f8])).
% 2.90/1.56 thf(f8,axiom,(
% 2.90/1.56 ! [X0 : $tType,X4 : X0,X5 : stream170649215stream @ X0] : ((stream30925839e_sset @ X0 @ (stream641971652_SCons @ X0 @ X4 @ X5)) = (insert @ X0 @ X4 @ (stream30925839e_sset @ X0 @ X5)))),
% 2.90/1.56 file('/export/starexec/sandbox2/tmp/tmp.e7YR8rdmNP/DTF2THF_22005.p',fact_7_stream_Oset)).
% 2.90/1.56 thf(f953,plain,(
% 2.90/1.56 (x = y) | ((member @ a @ x @ (stream30925839e_sset @ a @ (stream641971652_SCons @ a @ y @ s))) = $true)),
% 2.90/1.56 inference(subsumption_resolution,[status(thm)],[f875,f842])).
% 2.90/1.56 thf(f875,plain,(
% 2.90/1.56 ((member @ a @ x @ (stream30925839e_sset @ a @ (stream641971652_SCons @ a @ y @ s))) = $true) | ((member @ a @ x @ (stream30925839e_sset @ a @ s)) = $true) | (x = y)),
% 2.90/1.56 inference(cnf_transformation,[status(thm)],[f812])).
% 2.90/1.56 thf(f952,plain,(
% 2.90/1.56 ~spl18_1 | ~spl18_2),
% 2.90/1.56 inference(avatar_split_clause,[status(thm)],[f943,f949,f945])).
% 2.90/1.56 thf(f943,plain,(
% 2.90/1.56 (x != y) | ((member @ a @ x @ (insert @ a @ y @ (stream30925839e_sset @ a @ s))) != $true)),
% 2.90/1.56 inference(forward_demodulation,[status(thm)],[f877,f899])).
% 2.90/1.56 thf(f877,plain,(
% 2.90/1.56 ((member @ a @ x @ (stream30925839e_sset @ a @ (stream641971652_SCons @ a @ y @ s))) != $true) | (x != y)),
% 2.90/1.56 inference(cnf_transformation,[status(thm)],[f812])).
% 2.90/1.56 % SZS output end Proof for DTF2THF_22005
% 2.90/1.56 % (22144)------------------------------
% 2.90/1.56 % (22144)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.56 % (22144)Termination reason: Refutation
% 2.90/1.56
% 2.90/1.56 % (22144)Memory used [KB]: 6652
% 2.90/1.56 % (22144)Time elapsed: 0.023 s
% 2.90/1.56 % (22144)Instructions burned: 62 (million)
% 2.90/1.56 % (22144)------------------------------
% 2.90/1.56 % (22144)------------------------------
% 2.90/1.56 % (22138)Success in time 0.03 s
%------------------------------------------------------------------------------