%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP144^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n025.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:54 AM UTC 2026
% Result : Theorem 3.79s 1.91s
% Output : Refutation 3.79s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : ITP144^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.10 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.31 % Computer : n025.cluster.edu
% 0.15/0.31 % Model : x86_64 x86_64
% 0.15/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.31 % Memory : 8042.1875MB
% 0.15/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.31 % CPULimit : 300
% 0.15/0.31 % WCLimit : 300
% 0.15/0.31 % DateTime : Tue Feb 24 08:27:57 EST 2026
% 0.15/0.31 % CPUTime :
% 0.15/0.31 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.42 ---- Original DTF file ---
% 0.20/0.42 thf(spec,logic,$$dhol).
% 0.20/0.42 %------------------------------------------------------------------------------
% 0.20/0.42 % File : ITP144^2 : TPTP v9.2.1. Released v7.5.0.
% 0.20/0.42 % Domain : Interactive Theorem Proving
% 0.20/0.42 % Problem : Sledgehammer PHoareTotal problem prob_493__3263728_1
% 0.20/0.42 % Version : Especial.
% 0.20/0.42 % English :
% 0.20/0.42
% 0.20/0.42 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.42 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.20/0.42 % Source : [Des21]
% 0.20/0.42 % Names : PHoareTotal/prob_493__3263728_1 [Des21]
% 0.20/0.42
% 0.20/0.42 % Status : Theorem
% 0.20/0.42 % Rating : 0.00 v7.5.0
% 0.20/0.42 % Syntax : Number of formulae : 315 ( 117 unt; 45 typ; 0 def)
% 0.20/0.42 % Number of atoms : 617 ( 354 equ; 0 cnn)
% 0.20/0.42 % Maximal formula atoms : 7 ( 2 avg)
% 0.20/0.42 % Number of connectives : 3330 ( 68 ~; 6 |; 57 &;2948 @)
% 0.20/0.42 % ( 0 <=>; 251 =>; 0 <=; 0 <~>)
% 0.20/0.42 % Maximal formula depth : 20 ( 8 avg)
% 0.20/0.42 % Number of types : 3 ( 2 usr)
% 0.20/0.42 % Number of type conns : 409 ( 409 >; 0 *; 0 +; 0 <<)
% 0.20/0.42 % Number of symbols : 46 ( 43 usr; 2 con; 0-9 aty)
% 0.20/0.42 % Number of variables : 1203 ( 121 ^; 990 !; 30 ?;1203 :)
% 0.20/0.42 % ( 62 !>; 0 ?*; 0 @-; 0 @+)
% 0.20/0.42 % SPC : TH1_THM_EQU_NAR
% 0.20/0.42
% 0.20/0.42 % Comments : This file was generated by Sledgehammer 2021-02-23 16:19:11.981
% 0.20/0.42 %------------------------------------------------------------------------------
% 0.20/0.42 % Could-be-implicit typings (4)
% 0.20/0.42 thf(ty_t_Product__Type_Oprod,type,
% 0.20/0.42 product_prod: $tType > $tType > $tType ).
% 0.20/0.42
% 0.20/0.42 thf(ty_t_Set_Oset,type,
% 0.20/0.42 set: $tType > $tType ).
% 0.20/0.42
% 0.20/0.42 thf(ty_tf_b,type,
% 0.20/0.42 b: $tType ).
% 0.20/0.42
% 0.20/0.42 thf(ty_tf_a,type,
% 0.20/0.42 a: $tType ).
% 0.20/0.42
% 0.20/0.42 % Explicit typings (41)
% 0.20/0.42 thf(sy_cl_HOL_Otype,type,
% 0.20/0.42 type:
% 0.20/0.42 !>[A: $tType] : $o ).
% 0.20/0.42
% 0.20/0.42 thf(sy_cl_Orderings_Obot,type,
% 0.20/0.42 bot:
% 0.20/0.42 !>[A: $tType] : $o ).
% 0.20/0.42
% 0.20/0.42 thf(sy_cl_Orderings_Oord,type,
% 0.20/0.42 ord:
% 0.20/0.42 !>[A: $tType] : $o ).
% 0.20/0.42
% 0.20/0.42 thf(sy_cl_Orderings_Otop,type,
% 0.20/0.42 top:
% 0.20/0.42 !>[A: $tType] : $o ).
% 0.20/0.42
% 0.20/0.42 thf(sy_cl_Groups_Oab__group__add,type,
% 0.20/0.42 ab_group_add:
% 0.20/0.42 !>[A: $tType] : $o ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_BNF__Def_Opick__middlep,type,
% 0.20/0.42 bNF_pick_middlep:
% 0.20/0.42 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_BNF__Greatest__Fixpoint_OfromCard,type,
% 0.20/0.42 bNF_Gr932048252omCard:
% 0.20/0.42 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > B > A ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2p,type,
% 0.20/0.42 bNF_Greatest_image2p:
% 0.20/0.42 !>[C: $tType,A: $tType,D: $tType,B: $tType] : ( ( C > A ) > ( D > B ) > ( C > D > $o ) > A > B > $o ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_BNF__Greatest__Fixpoint_OtoCard,type,
% 0.20/0.42 bNF_Greatest_toCard:
% 0.20/0.42 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > A > B ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
% 0.20/0.42 bNF_Gr1978856366d_pred:
% 0.20/0.42 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > $o ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_BNF__Greatest__Fixpoint_Ouniv,type,
% 0.20/0.42 bNF_Greatest_univ:
% 0.20/0.42 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Fun_Obij__betw,type,
% 0.20/0.42 bij_betw:
% 0.20/0.42 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Fun_Ofun__upd,type,
% 0.20/0.42 fun_upd:
% 0.20/0.42 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Fun_Oinj__on,type,
% 0.20/0.42 inj_on:
% 0.20/0.42 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.20/0.42 minus_minus:
% 0.20/0.42 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Hilbert__Choice_OEps,type,
% 0.20/0.42 hilbert_Eps:
% 0.20/0.42 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Hilbert__Choice_Obijection,type,
% 0.20/0.42 hilbert_bijection:
% 0.20/0.42 !>[A: $tType] : ( ( A > A ) > $o ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Hilbert__Choice_Oinv__into,type,
% 0.20/0.42 hilbert_inv_into:
% 0.20/0.42 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_If,type,
% 0.20/0.42 if:
% 0.20/0.42 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Lattices__Big_Oord__class_Oarg__max,type,
% 0.20/0.42 lattices_ord_arg_max:
% 0.20/0.42 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Lattices__Big_Oord__class_Oarg__max__on,type,
% 0.20/0.42 lattic1534546155max_on:
% 0.20/0.42 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Lattices__Big_Oord__class_Ois__arg__max,type,
% 0.20/0.42 lattic2130478198rg_max:
% 0.20/0.42 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Lifting_Orel__pred__comp,type,
% 0.20/0.42 rel_pred_comp:
% 0.20/0.42 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( B > $o ) > A > $o ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.20/0.42 bot_bot:
% 0.20/0.42 !>[A: $tType] : A ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.20/0.42 top_top:
% 0.20/0.42 !>[A: $tType] : A ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Product__Type_Obool_Ocase__bool,type,
% 0.20/0.42 product_case_bool:
% 0.20/0.42 !>[A: $tType] : ( A > A > $o > A ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Quotient_OQuotient3,type,
% 0.20/0.42 quotient3:
% 0.20/0.42 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Quotient_Oquot__type,type,
% 0.20/0.42 quot_type:
% 0.20/0.42 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > B ) > ( B > ( set @ A ) ) > $o ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Quotient_Oquot__type_Oabs,type,
% 0.20/0.42 quot_abs:
% 0.20/0.42 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > B ) > A > B ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Quotient_Oquot__type_Orep,type,
% 0.20/0.42 quot_rep:
% 0.20/0.42 !>[B: $tType,A: $tType] : ( ( B > ( set @ A ) ) > B > A ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Relation_Oinv__imagep,type,
% 0.20/0.42 inv_imagep:
% 0.20/0.42 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Set_OCollect,type,
% 0.20/0.42 collect:
% 0.20/0.42 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Set_Oimage,type,
% 0.20/0.42 image:
% 0.20/0.42 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Set_Oinsert,type,
% 0.20/0.42 insert:
% 0.20/0.42 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Set_Ois__empty,type,
% 0.20/0.42 is_empty:
% 0.20/0.42 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Set_Ois__singleton,type,
% 0.20/0.42 is_singleton:
% 0.20/0.42 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.20/0.42
% 0.20/0.42 thf(sy_c_Set_Oremove,type,
% 0.20/0.42 remove:
% 0.20/0.42 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.20/0.43
% 0.20/0.43 thf(sy_c_Set_Othe__elem,type,
% 0.20/0.43 the_elem:
% 0.20/0.43 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.20/0.43
% 0.20/0.43 thf(sy_c_member,type,
% 0.20/0.43 member:
% 0.20/0.43 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.20/0.43
% 0.20/0.43 thf(sy_v_P,type,
% 0.20/0.43 p: a > $o ).
% 0.20/0.43
% 0.20/0.43 thf(sy_v_Q,type,
% 0.20/0.43 q: a > b > $o ).
% 0.20/0.43
% 0.20/0.43 % Relevant facts (256)
% 0.20/0.43 thf(fact_0_some__equality,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o,A2: A] :
% 0.20/0.43 ( ( P @ A2 )
% 0.20/0.43 => ( ! [X: A] :
% 0.20/0.43 ( ( P @ X )
% 0.20/0.43 => ( X = A2 ) )
% 0.20/0.43 => ( ( hilbert_Eps @ A @ P )
% 0.20/0.43 = A2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % some_equality
% 0.20/0.43 thf(fact_1_some__eq__trivial,axiom,
% 0.20/0.43 ! [A: $tType,X2: A] :
% 0.20/0.43 ( ( hilbert_Eps @ A
% 0.20/0.43 @ ^ [Y: A] : ( Y = X2 ) )
% 0.20/0.43 = X2 ) ).
% 0.20/0.43
% 0.20/0.43 % some_eq_trivial
% 0.20/0.43 thf(fact_2_some__sym__eq__trivial,axiom,
% 0.20/0.43 ! [A: $tType,X2: A] :
% 0.20/0.43 ( ( hilbert_Eps @ A
% 0.20/0.43 @ ( ^ [Y2: A,Z: A] : ( Y2 = Z )
% 0.20/0.43 @ X2 ) )
% 0.20/0.43 = X2 ) ).
% 0.20/0.43
% 0.20/0.43 % some_sym_eq_trivial
% 0.20/0.43 thf(fact_3_verit__sko__ex_H,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o,A3: $o] :
% 0.20/0.43 ( ( ( P @ ( hilbert_Eps @ A @ P ) )
% 0.20/0.43 = A3 )
% 0.20/0.43 => ( ( ? [X3: A] : ( P @ X3 ) )
% 0.20/0.43 = A3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % verit_sko_ex'
% 0.20/0.43 thf(fact_4_verit__sko__forall,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( ^ [P2: A > $o] :
% 0.20/0.43 ! [X3: A] : ( P2 @ X3 ) )
% 0.20/0.43 = ( ^ [P3: A > $o] :
% 0.20/0.43 ( P3
% 0.20/0.43 @ ( hilbert_Eps @ A
% 0.20/0.43 @ ^ [X4: A] :
% 0.20/0.43 ~ ( P3 @ X4 ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % verit_sko_forall
% 0.20/0.43 thf(fact_5_someI2,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
% 0.20/0.43 ( ( P @ A2 )
% 0.20/0.43 => ( ! [X: A] :
% 0.20/0.43 ( ( P @ X )
% 0.20/0.43 => ( Q @ X ) )
% 0.20/0.43 => ( Q @ ( hilbert_Eps @ A @ P ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % someI2
% 0.20/0.43 thf(fact_6_verit__sko__forall_H,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o,A3: $o] :
% 0.20/0.43 ( ( ( P
% 0.20/0.43 @ ( hilbert_Eps @ A
% 0.20/0.43 @ ^ [X4: A] :
% 0.20/0.43 ~ ( P @ X4 ) ) )
% 0.20/0.43 = A3 )
% 0.20/0.43 => ( ( ! [X3: A] : ( P @ X3 ) )
% 0.20/0.43 = A3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % verit_sko_forall'
% 0.20/0.43 thf(fact_7_verit__sko__forall_H_H,axiom,
% 0.20/0.43 ! [A: $tType,B2: A,A3: A,P: A > $o] :
% 0.20/0.43 ( ( B2 = A3 )
% 0.20/0.43 => ( ( ( hilbert_Eps @ A @ P )
% 0.20/0.43 = A3 )
% 0.20/0.43 = ( ( hilbert_Eps @ A @ P )
% 0.20/0.43 = B2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % verit_sko_forall''
% 0.20/0.43 thf(fact_8_someI__ex,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o] :
% 0.20/0.43 ( ? [X_1: A] : ( P @ X_1 )
% 0.20/0.43 => ( P @ ( hilbert_Eps @ A @ P ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % someI_ex
% 0.20/0.43 thf(fact_9_someI2__ex,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.20/0.43 ( ? [X_1: A] : ( P @ X_1 )
% 0.20/0.43 => ( ! [X: A] :
% 0.20/0.43 ( ( P @ X )
% 0.20/0.43 => ( Q @ X ) )
% 0.20/0.43 => ( Q @ ( hilbert_Eps @ A @ P ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % someI2_ex
% 0.20/0.43 thf(fact_10_someI2__bex,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] :
% 0.20/0.43 ( ? [X5: A] :
% 0.20/0.43 ( ( member @ A @ X5 @ A3 )
% 0.20/0.43 & ( P @ X5 ) )
% 0.20/0.43 => ( ! [X: A] :
% 0.20/0.43 ( ( ( member @ A @ X @ A3 )
% 0.20/0.43 & ( P @ X ) )
% 0.20/0.43 => ( Q @ X ) )
% 0.20/0.43 => ( Q
% 0.20/0.43 @ ( hilbert_Eps @ A
% 0.20/0.43 @ ^ [X4: A] :
% 0.20/0.43 ( ( member @ A @ X4 @ A3 )
% 0.20/0.43 & ( P @ X4 ) ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % someI2_bex
% 0.20/0.43 thf(fact_11_some__eq__ex,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o] :
% 0.20/0.43 ( ( P @ ( hilbert_Eps @ A @ P ) )
% 0.20/0.43 = ( ? [X3: A] : ( P @ X3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % some_eq_ex
% 0.20/0.43 thf(fact_12_some__eq__imp,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o,A2: A,B3: A] :
% 0.20/0.43 ( ( ( hilbert_Eps @ A @ P )
% 0.20/0.43 = A2 )
% 0.20/0.43 => ( ( P @ B3 )
% 0.20/0.43 => ( P @ A2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % some_eq_imp
% 0.20/0.43 thf(fact_13_tfl__some,axiom,
% 0.20/0.43 ! [A: $tType,P4: A > $o,X5: A] :
% 0.20/0.43 ( ( P4 @ X5 )
% 0.20/0.43 => ( P4 @ ( hilbert_Eps @ A @ P4 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % tfl_some
% 0.20/0.43 thf(fact_14_Eps__cong,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.20/0.43 ( ! [X: A] :
% 0.20/0.43 ( ( P @ X )
% 0.20/0.43 = ( Q @ X ) )
% 0.20/0.43 => ( ( hilbert_Eps @ A @ P )
% 0.20/0.43 = ( hilbert_Eps @ A @ Q ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Eps_cong
% 0.20/0.43 thf(fact_15_someI,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o,X2: A] :
% 0.20/0.43 ( ( P @ X2 )
% 0.20/0.43 => ( P @ ( hilbert_Eps @ A @ P ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % someI
% 0.20/0.43 thf(fact_16_verit__sko__forall__indirect2,axiom,
% 0.20/0.43 ! [A: $tType,X2: A,P: A > $o,P5: A > $o] :
% 0.20/0.43 ( ( X2
% 0.20/0.43 = ( hilbert_Eps @ A
% 0.20/0.43 @ ^ [X4: A] :
% 0.20/0.43 ~ ( P @ X4 ) ) )
% 0.20/0.43 => ( ! [X: A] :
% 0.20/0.43 ( ( P @ X )
% 0.20/0.43 = ( P5 @ X ) )
% 0.20/0.43 => ( ( ! [X3: A] : ( P5 @ X3 ) )
% 0.20/0.43 = ( P @ X2 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % verit_sko_forall_indirect2
% 0.20/0.43 thf(fact_17_verit__sko__forall__indirect,axiom,
% 0.20/0.43 ! [A: $tType,X2: A,P: A > $o] :
% 0.20/0.43 ( ( X2
% 0.20/0.43 = ( hilbert_Eps @ A
% 0.20/0.43 @ ^ [X4: A] :
% 0.20/0.43 ~ ( P @ X4 ) ) )
% 0.20/0.43 => ( ( ! [X3: A] : ( P @ X3 ) )
% 0.20/0.43 = ( P @ X2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % verit_sko_forall_indirect
% 0.20/0.43 thf(fact_18_some1__equality,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o,A2: A] :
% 0.20/0.43 ( ? [X5: A] :
% 0.20/0.43 ( ( P @ X5 )
% 0.20/0.43 & ! [Y3: A] :
% 0.20/0.43 ( ( P @ Y3 )
% 0.20/0.43 => ( Y3 = X5 ) ) )
% 0.20/0.43 => ( ( P @ A2 )
% 0.20/0.43 => ( ( hilbert_Eps @ A @ P )
% 0.20/0.43 = A2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % some1_equality
% 0.20/0.43 thf(fact_19_verit__sko__ex__indirect2,axiom,
% 0.20/0.43 ! [A: $tType,X2: A,P: A > $o,P5: A > $o] :
% 0.20/0.43 ( ( X2
% 0.20/0.43 = ( hilbert_Eps @ A @ P ) )
% 0.20/0.43 => ( ! [X: A] :
% 0.20/0.43 ( ( P @ X )
% 0.20/0.43 = ( P5 @ X ) )
% 0.20/0.43 => ( ( ? [X3: A] : ( P5 @ X3 ) )
% 0.20/0.43 = ( P @ X2 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % verit_sko_ex_indirect2
% 0.20/0.43 thf(fact_20_verit__sko__ex__indirect,axiom,
% 0.20/0.43 ! [A: $tType,X2: A,P: A > $o] :
% 0.20/0.43 ( ( X2
% 0.20/0.43 = ( hilbert_Eps @ A @ P ) )
% 0.20/0.43 => ( ( ? [X3: A] : ( P @ X3 ) )
% 0.20/0.43 = ( P @ X2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % verit_sko_ex_indirect
% 0.20/0.43 thf(fact_21_Nitpick_OEps__psimp,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o,X2: A,Y4: A] :
% 0.20/0.43 ( ( P @ X2 )
% 0.20/0.43 => ( ~ ( P @ Y4 )
% 0.20/0.43 => ( ( ( hilbert_Eps @ A @ P )
% 0.20/0.43 = Y4 )
% 0.20/0.43 => ( ( hilbert_Eps @ A @ P )
% 0.20/0.43 = X2 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Nitpick.Eps_psimp
% 0.20/0.43 thf(fact_22_univ__def,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType] :
% 0.20/0.43 ( ( bNF_Greatest_univ @ B @ A )
% 0.20/0.43 = ( ^ [F: B > A,X6: set @ B] :
% 0.20/0.43 ( F
% 0.20/0.43 @ ( hilbert_Eps @ B
% 0.20/0.43 @ ^ [X4: B] : ( member @ B @ X4 @ X6 ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % univ_def
% 0.20/0.43 thf(fact_23_pick__middlep__def,axiom,
% 0.20/0.43 ! [C: $tType,A: $tType,B: $tType] :
% 0.20/0.43 ( ( bNF_pick_middlep @ B @ A @ C )
% 0.20/0.43 = ( ^ [P3: B > A > $o,Q2: A > C > $o,A4: B,C2: C] :
% 0.20/0.43 ( hilbert_Eps @ A
% 0.20/0.43 @ ^ [B4: A] :
% 0.20/0.43 ( ( P3 @ A4 @ B4 )
% 0.20/0.43 & ( Q2 @ B4 @ C2 ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % pick_middlep_def
% 0.20/0.43 thf(fact_24_some__in__eq,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A] :
% 0.20/0.43 ( ( member @ A
% 0.20/0.43 @ ( hilbert_Eps @ A
% 0.20/0.43 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 0.20/0.43 @ A3 )
% 0.20/0.43 = ( A3
% 0.20/0.43 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % some_in_eq
% 0.20/0.43 thf(fact_25_inv__into__def2,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType] :
% 0.20/0.43 ( ( hilbert_inv_into @ A @ B )
% 0.20/0.43 = ( ^ [A5: set @ A,F: A > B,X4: B] :
% 0.20/0.43 ( hilbert_Eps @ A
% 0.20/0.43 @ ^ [Y: A] :
% 0.20/0.43 ( ( member @ A @ Y @ A5 )
% 0.20/0.43 & ( ( F @ Y )
% 0.20/0.43 = X4 ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inv_into_def2
% 0.20/0.43 thf(fact_26_inv__into__def,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType] :
% 0.20/0.43 ( ( hilbert_inv_into @ A @ B )
% 0.20/0.43 = ( ^ [A5: set @ A,F: A > B,X4: B] :
% 0.20/0.43 ( hilbert_Eps @ A
% 0.20/0.43 @ ^ [Y: A] :
% 0.20/0.43 ( ( member @ A @ Y @ A5 )
% 0.20/0.43 & ( ( F @ Y )
% 0.20/0.43 = X4 ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inv_into_def
% 0.20/0.43 thf(fact_27_quot__type_Osome__collect,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,R: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A ),R2: A] :
% 0.20/0.43 ( ( quot_type @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( ( R @ R2 @ R2 )
% 0.20/0.43 => ( ( R
% 0.20/0.43 @ ( hilbert_Eps @ A
% 0.20/0.43 @ ^ [X4: A] : ( member @ A @ X4 @ ( collect @ A @ ( R @ R2 ) ) ) ) )
% 0.20/0.43 = ( R @ R2 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % quot_type.some_collect
% 0.20/0.43 thf(fact_28_fromCard__def,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType] :
% 0.20/0.43 ( ( bNF_Gr932048252omCard @ A @ B )
% 0.20/0.43 = ( ^ [A5: set @ A,R3: set @ ( product_prod @ B @ B ),K: B] :
% 0.20/0.43 ( hilbert_Eps @ A
% 0.20/0.43 @ ^ [B4: A] :
% 0.20/0.43 ( ( member @ A @ B4 @ A5 )
% 0.20/0.43 & ( ( bNF_Greatest_toCard @ A @ B @ A5 @ R3 @ B4 )
% 0.20/0.43 = K ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % fromCard_def
% 0.20/0.43 thf(fact_29_arg__max__def,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType] :
% 0.20/0.43 ( ( ord @ A )
% 0.20/0.43 => ( ( lattices_ord_arg_max @ B @ A )
% 0.20/0.43 = ( ^ [F: B > A,P3: B > $o] : ( hilbert_Eps @ B @ ( lattic2130478198rg_max @ B @ A @ F @ P3 ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % arg_max_def
% 0.20/0.43 thf(fact_30_case__bool__if,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( product_case_bool @ A )
% 0.20/0.43 = ( ^ [X4: A,Y: A,P3: $o] : ( if @ A @ P3 @ X4 @ Y ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % case_bool_if
% 0.20/0.43 thf(fact_31_exE__some,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o,C3: A] :
% 0.20/0.43 ( ? [X_1: A] : ( P @ X_1 )
% 0.20/0.43 => ( ( C3
% 0.20/0.43 = ( hilbert_Eps @ A @ P ) )
% 0.20/0.43 => ( P @ C3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % exE_some
% 0.20/0.43 thf(fact_32_quot__type_Orep__inverse,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,R: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A ),X2: B] :
% 0.20/0.43 ( ( quot_type @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( ( Abs @ ( Rep @ X2 ) )
% 0.20/0.43 = X2 ) ) ).
% 0.20/0.43
% 0.20/0.43 % quot_type.rep_inverse
% 0.20/0.43 thf(fact_33_quot__type_Oabs__inverse,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,R: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A ),C3: set @ A] :
% 0.20/0.43 ( ( quot_type @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( ? [X5: A] :
% 0.20/0.43 ( ( R @ X5 @ X5 )
% 0.20/0.43 & ( C3
% 0.20/0.43 = ( collect @ A @ ( R @ X5 ) ) ) )
% 0.20/0.43 => ( ( Rep @ ( Abs @ C3 ) )
% 0.20/0.43 = C3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % quot_type.abs_inverse
% 0.20/0.43 thf(fact_34_quot__type_Orep__inject,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,R: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A ),X2: B,Y4: B] :
% 0.20/0.43 ( ( quot_type @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( ( ( Rep @ X2 )
% 0.20/0.43 = ( Rep @ Y4 ) )
% 0.20/0.43 = ( X2 = Y4 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % quot_type.rep_inject
% 0.20/0.43 thf(fact_35_quot__type_Orep__prop,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,R: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A ),Y4: B] :
% 0.20/0.43 ( ( quot_type @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ? [X: A] :
% 0.20/0.43 ( ( R @ X @ X )
% 0.20/0.43 & ( ( Rep @ Y4 )
% 0.20/0.43 = ( collect @ A @ ( R @ X ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % quot_type.rep_prop
% 0.20/0.43 thf(fact_36_arg__max__on__def,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType] :
% 0.20/0.43 ( ( ord @ A )
% 0.20/0.43 => ( ( lattic1534546155max_on @ B @ A )
% 0.20/0.43 = ( ^ [F: B > A,S: set @ B] :
% 0.20/0.43 ( lattices_ord_arg_max @ B @ A @ F
% 0.20/0.43 @ ^ [X4: B] : ( member @ B @ X4 @ S ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % arg_max_on_def
% 0.20/0.43 thf(fact_37_empty__Collect__eq,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o] :
% 0.20/0.43 ( ( ( bot_bot @ ( set @ A ) )
% 0.20/0.43 = ( collect @ A @ P ) )
% 0.20/0.43 = ( ! [X4: A] :
% 0.20/0.43 ~ ( P @ X4 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % empty_Collect_eq
% 0.20/0.43 thf(fact_38_Collect__empty__eq,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o] :
% 0.20/0.43 ( ( ( collect @ A @ P )
% 0.20/0.43 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.43 = ( ! [X4: A] :
% 0.20/0.43 ~ ( P @ X4 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Collect_empty_eq
% 0.20/0.43 thf(fact_39_all__not__in__conv,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A] :
% 0.20/0.43 ( ( ! [X4: A] :
% 0.20/0.43 ~ ( member @ A @ X4 @ A3 ) )
% 0.20/0.43 = ( A3
% 0.20/0.43 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % all_not_in_conv
% 0.20/0.43 thf(fact_40_empty__iff,axiom,
% 0.20/0.43 ! [A: $tType,C3: A] :
% 0.20/0.43 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % empty_iff
% 0.20/0.43 thf(fact_41_bot__apply,axiom,
% 0.20/0.43 ! [C: $tType,D: $tType] :
% 0.20/0.43 ( ( bot @ C )
% 0.20/0.43 => ( ( bot_bot @ ( D > C ) )
% 0.20/0.43 = ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bot_apply
% 0.20/0.43 thf(fact_42_quot__type_Orep__def,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,R: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A ),A2: B] :
% 0.20/0.43 ( ( quot_type @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( ( quot_rep @ B @ A @ Rep @ A2 )
% 0.20/0.43 = ( hilbert_Eps @ A
% 0.20/0.43 @ ^ [X4: A] : ( member @ A @ X4 @ ( Rep @ A2 ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % quot_type.rep_def
% 0.20/0.43 thf(fact_43_toCard__def,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType] :
% 0.20/0.43 ( ( bNF_Greatest_toCard @ A @ B )
% 0.20/0.43 = ( ^ [A5: set @ A,R3: set @ ( product_prod @ B @ B )] : ( hilbert_Eps @ ( A > B ) @ ( bNF_Gr1978856366d_pred @ A @ B @ A5 @ R3 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % toCard_def
% 0.20/0.43 thf(fact_44_bool_Osplit__sel__asm,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o,F1: A,F2: A,Bool: $o] :
% 0.20/0.43 ( ( P @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 0.20/0.43 = ( ~ ( ( Bool
% 0.20/0.43 & ~ ( P @ F1 ) )
% 0.20/0.43 | ( ~ Bool
% 0.20/0.43 & ~ ( P @ F2 ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bool.split_sel_asm
% 0.20/0.43 thf(fact_45_mem__Collect__eq,axiom,
% 0.20/0.43 ! [A: $tType,A2: A,P: A > $o] :
% 0.20/0.43 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.20/0.43 = ( P @ A2 ) ) ).
% 0.20/0.43
% 0.20/0.43 % mem_Collect_eq
% 0.20/0.43 thf(fact_46_Collect__mem__eq,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A] :
% 0.20/0.43 ( ( collect @ A
% 0.20/0.43 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 0.20/0.43 = A3 ) ).
% 0.20/0.43
% 0.20/0.43 % Collect_mem_eq
% 0.20/0.43 thf(fact_47_Collect__cong,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.20/0.43 ( ! [X: A] :
% 0.20/0.43 ( ( P @ X )
% 0.20/0.43 = ( Q @ X ) )
% 0.20/0.43 => ( ( collect @ A @ P )
% 0.20/0.43 = ( collect @ A @ Q ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Collect_cong
% 0.20/0.43 thf(fact_48_ext,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,G: A > B] :
% 0.20/0.43 ( ! [X: A] :
% 0.20/0.43 ( ( F3 @ X )
% 0.20/0.43 = ( G @ X ) )
% 0.20/0.43 => ( F3 = G ) ) ).
% 0.20/0.43
% 0.20/0.43 % ext
% 0.20/0.43 thf(fact_49_bot__set__def,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( bot_bot @ ( set @ A ) )
% 0.20/0.43 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bot_set_def
% 0.20/0.43 thf(fact_50_quot__type_Orep_Ocong,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType] :
% 0.20/0.43 ( ( quot_rep @ B @ A )
% 0.20/0.43 = ( quot_rep @ B @ A ) ) ).
% 0.20/0.43
% 0.20/0.43 % quot_type.rep.cong
% 0.20/0.43 thf(fact_51_bot__fun__def,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType] :
% 0.20/0.43 ( ( bot @ B )
% 0.20/0.43 => ( ( bot_bot @ ( A > B ) )
% 0.20/0.43 = ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bot_fun_def
% 0.20/0.43 thf(fact_52_emptyE,axiom,
% 0.20/0.43 ! [A: $tType,A2: A] :
% 0.20/0.43 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % emptyE
% 0.20/0.43 thf(fact_53_equals0D,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A,A2: A] :
% 0.20/0.43 ( ( A3
% 0.20/0.43 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.43 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % equals0D
% 0.20/0.43 thf(fact_54_equals0I,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A] :
% 0.20/0.43 ( ! [Y3: A] :
% 0.20/0.43 ~ ( member @ A @ Y3 @ A3 )
% 0.20/0.43 => ( A3
% 0.20/0.43 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % equals0I
% 0.20/0.43 thf(fact_55_ex__in__conv,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A] :
% 0.20/0.43 ( ( ? [X4: A] : ( member @ A @ X4 @ A3 ) )
% 0.20/0.43 = ( A3
% 0.20/0.43 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % ex_in_conv
% 0.20/0.43 thf(fact_56_empty__def,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( bot_bot @ ( set @ A ) )
% 0.20/0.43 = ( collect @ A
% 0.20/0.43 @ ^ [X4: A] : $false ) ) ).
% 0.20/0.43
% 0.20/0.43 % empty_def
% 0.20/0.43 thf(fact_57_old_Obool_Osimps_I4_J,axiom,
% 0.20/0.43 ! [A: $tType,F1: A,F2: A] :
% 0.20/0.43 ( ( product_case_bool @ A @ F1 @ F2 @ $false )
% 0.20/0.43 = F2 ) ).
% 0.20/0.43
% 0.20/0.43 % old.bool.simps(4)
% 0.20/0.43 thf(fact_58_old_Obool_Osimps_I3_J,axiom,
% 0.20/0.43 ! [A: $tType,F1: A,F2: A] :
% 0.20/0.43 ( ( product_case_bool @ A @ F1 @ F2 @ $true )
% 0.20/0.43 = F1 ) ).
% 0.20/0.43
% 0.20/0.43 % old.bool.simps(3)
% 0.20/0.43 thf(fact_59_bool_Osplit__sel,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o,F1: A,F2: A,Bool: $o] :
% 0.20/0.43 ( ( P @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 0.20/0.43 = ( ( Bool
% 0.20/0.43 => ( P @ F1 ) )
% 0.20/0.43 & ( ~ Bool
% 0.20/0.43 => ( P @ F2 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bool.split_sel
% 0.20/0.43 thf(fact_60_bool_Ocase__eq__if,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( product_case_bool @ A )
% 0.20/0.43 = ( ^ [F12: A,F22: A,Bool2: $o] : ( if @ A @ Bool2 @ F12 @ F22 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bool.case_eq_if
% 0.20/0.43 thf(fact_61_bool_Ocase__distrib,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,H: A > B,F1: A,F2: A,Bool: $o] :
% 0.20/0.43 ( ( H @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 0.20/0.43 = ( product_case_bool @ B @ ( H @ F1 ) @ ( H @ F2 ) @ Bool ) ) ).
% 0.20/0.43
% 0.20/0.43 % bool.case_distrib
% 0.20/0.43 thf(fact_62_Set_Ois__empty__def,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( is_empty @ A )
% 0.20/0.43 = ( ^ [A5: set @ A] :
% 0.20/0.43 ( A5
% 0.20/0.43 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Set.is_empty_def
% 0.20/0.43 thf(fact_63_quot__type_Oabs__def,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,R: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A ),X2: A] :
% 0.20/0.43 ( ( quot_type @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( ( quot_abs @ A @ B @ R @ Abs @ X2 )
% 0.20/0.43 = ( Abs @ ( collect @ A @ ( R @ X2 ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % quot_type.abs_def
% 0.20/0.43 thf(fact_64_image2p__def,axiom,
% 0.20/0.43 ! [D: $tType,B: $tType,A: $tType,C: $tType] :
% 0.20/0.43 ( ( bNF_Greatest_image2p @ C @ A @ D @ B )
% 0.20/0.43 = ( ^ [F: C > A,G2: D > B,R4: C > D > $o,X4: A,Y: B] :
% 0.20/0.43 ? [X7: C,Y5: D] :
% 0.20/0.43 ( ( R4 @ X7 @ Y5 )
% 0.20/0.43 & ( ( F @ X7 )
% 0.20/0.43 = X4 )
% 0.20/0.43 & ( ( G2 @ Y5 )
% 0.20/0.43 = Y ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % image2p_def
% 0.20/0.43 thf(fact_65_Collect__empty__eq__bot,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o] :
% 0.20/0.43 ( ( ( collect @ A @ P )
% 0.20/0.43 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.43 = ( P
% 0.20/0.43 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Collect_empty_eq_bot
% 0.20/0.43 thf(fact_66_bot__empty__eq,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( bot_bot @ ( A > $o ) )
% 0.20/0.43 = ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bot_empty_eq
% 0.20/0.43 thf(fact_67_inv__def,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: B > A] :
% 0.20/0.43 ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 )
% 0.20/0.43 = ( ^ [Y: A] :
% 0.20/0.43 ( hilbert_Eps @ B
% 0.20/0.43 @ ^ [X4: B] :
% 0.20/0.43 ( ( F3 @ X4 )
% 0.20/0.43 = Y ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inv_def
% 0.20/0.43 thf(fact_68_rel__pred__comp__def,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType] :
% 0.20/0.43 ( ( rel_pred_comp @ A @ B )
% 0.20/0.43 = ( ^ [R4: A > B > $o,P3: B > $o,X4: A] :
% 0.20/0.43 ? [Y: B] :
% 0.20/0.43 ( ( R4 @ X4 @ Y )
% 0.20/0.43 & ( P3 @ Y ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % rel_pred_comp_def
% 0.20/0.43 thf(fact_69_is__singletonI_H,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A] :
% 0.20/0.43 ( ( A3
% 0.20/0.43 != ( bot_bot @ ( set @ A ) ) )
% 0.20/0.43 => ( ! [X: A,Y3: A] :
% 0.20/0.43 ( ( member @ A @ X @ A3 )
% 0.20/0.43 => ( ( member @ A @ Y3 @ A3 )
% 0.20/0.43 => ( X = Y3 ) ) )
% 0.20/0.43 => ( is_singleton @ A @ A3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % is_singletonI'
% 0.20/0.43 thf(fact_70_top__apply,axiom,
% 0.20/0.43 ! [C: $tType,D: $tType] :
% 0.20/0.43 ( ( top @ C )
% 0.20/0.43 => ( ( top_top @ ( D > C ) )
% 0.20/0.43 = ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % top_apply
% 0.20/0.43 thf(fact_71_UNIV__I,axiom,
% 0.20/0.43 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % UNIV_I
% 0.20/0.43 thf(fact_72_Collect__const,axiom,
% 0.20/0.43 ! [A: $tType,P: $o] :
% 0.20/0.43 ( ( P
% 0.20/0.43 => ( ( collect @ A
% 0.20/0.43 @ ^ [S2: A] : P )
% 0.20/0.43 = ( top_top @ ( set @ A ) ) ) )
% 0.20/0.43 & ( ~ P
% 0.20/0.43 => ( ( collect @ A
% 0.20/0.43 @ ^ [S2: A] : P )
% 0.20/0.43 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Collect_const
% 0.20/0.43 thf(fact_73_inv__identity,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) )
% 0.20/0.43 @ ^ [A4: A] : A4 )
% 0.20/0.43 = ( ^ [A4: A] : A4 ) ) ).
% 0.20/0.43
% 0.20/0.43 % inv_identity
% 0.20/0.43 thf(fact_74_UNIV__def,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( top_top @ ( set @ A ) )
% 0.20/0.43 = ( collect @ A
% 0.20/0.43 @ ^ [X4: A] : $true ) ) ).
% 0.20/0.43
% 0.20/0.43 % UNIV_def
% 0.20/0.43 thf(fact_75_UNIV__eq__I,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A] :
% 0.20/0.43 ( ! [X: A] : ( member @ A @ X @ A3 )
% 0.20/0.43 => ( ( top_top @ ( set @ A ) )
% 0.20/0.43 = A3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % UNIV_eq_I
% 0.20/0.43 thf(fact_76_UNIV__witness,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ? [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % UNIV_witness
% 0.20/0.43 thf(fact_77_image2pE,axiom,
% 0.20/0.43 ! [D: $tType,B: $tType,A: $tType,C: $tType,F3: A > B,G: C > D,R: A > C > $o,Fx: B,Gy: D] :
% 0.20/0.43 ( ( bNF_Greatest_image2p @ A @ B @ C @ D @ F3 @ G @ R @ Fx @ Gy )
% 0.20/0.43 => ~ ! [X: A] :
% 0.20/0.43 ( ( Fx
% 0.20/0.43 = ( F3 @ X ) )
% 0.20/0.43 => ! [Y3: C] :
% 0.20/0.43 ( ( Gy
% 0.20/0.43 = ( G @ Y3 ) )
% 0.20/0.43 => ~ ( R @ X @ Y3 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % image2pE
% 0.20/0.43 thf(fact_78_image2pI,axiom,
% 0.20/0.43 ! [A: $tType,C: $tType,D: $tType,B: $tType,R: A > B > $o,X2: A,Y4: B,F3: A > C,G: B > D] :
% 0.20/0.43 ( ( R @ X2 @ Y4 )
% 0.20/0.43 => ( bNF_Greatest_image2p @ A @ C @ B @ D @ F3 @ G @ R @ ( F3 @ X2 ) @ ( G @ Y4 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % image2pI
% 0.20/0.43 thf(fact_79_quot__type_Oabs_Ocong,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType] :
% 0.20/0.43 ( ( quot_abs @ A @ B )
% 0.20/0.43 = ( quot_abs @ A @ B ) ) ).
% 0.20/0.43
% 0.20/0.43 % quot_type.abs.cong
% 0.20/0.43 thf(fact_80_empty__not__UNIV,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( bot_bot @ ( set @ A ) )
% 0.20/0.43 != ( top_top @ ( set @ A ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % empty_not_UNIV
% 0.20/0.43 thf(fact_81_inv__equality,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,G: B > A,F3: A > B] :
% 0.20/0.43 ( ! [X: A] :
% 0.20/0.43 ( ( G @ ( F3 @ X ) )
% 0.20/0.43 = X )
% 0.20/0.43 => ( ! [Y3: B] :
% 0.20/0.43 ( ( F3 @ ( G @ Y3 ) )
% 0.20/0.43 = Y3 )
% 0.20/0.43 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
% 0.20/0.43 = G ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inv_equality
% 0.20/0.43 thf(fact_82_iso__tuple__UNIV__I,axiom,
% 0.20/0.43 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % iso_tuple_UNIV_I
% 0.20/0.43 thf(fact_83_quot__type_OQuotient,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,R: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A )] :
% 0.20/0.43 ( ( quot_type @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( quotient3 @ A @ B @ R @ ( quot_abs @ A @ B @ R @ Abs ) @ ( quot_rep @ B @ A @ Rep ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % quot_type.Quotient
% 0.20/0.43 thf(fact_84_in__inv__imagep,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType] :
% 0.20/0.43 ( ( inv_imagep @ A @ B )
% 0.20/0.43 = ( ^ [R3: A > A > $o,F: B > A,X4: B,Y: B] : ( R3 @ ( F @ X4 ) @ ( F @ Y ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % in_inv_imagep
% 0.20/0.43 thf(fact_85_is__singletonI,axiom,
% 0.20/0.43 ! [A: $tType,X2: A] : ( is_singleton @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % is_singletonI
% 0.20/0.43 thf(fact_86_bijection_Oinv__right__eq__iff,axiom,
% 0.20/0.43 ! [A: $tType,F3: A > A,B3: A,A2: A] :
% 0.20/0.43 ( ( hilbert_bijection @ A @ F3 )
% 0.20/0.43 => ( ( B3
% 0.20/0.43 = ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ A2 ) )
% 0.20/0.43 = ( ( F3 @ B3 )
% 0.20/0.43 = A2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijection.inv_right_eq_iff
% 0.20/0.43 thf(fact_87_bijection_Oinv__left__eq__iff,axiom,
% 0.20/0.43 ! [A: $tType,F3: A > A,A2: A,B3: A] :
% 0.20/0.43 ( ( hilbert_bijection @ A @ F3 )
% 0.20/0.43 => ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ A2 )
% 0.20/0.43 = B3 )
% 0.20/0.43 = ( ( F3 @ B3 )
% 0.20/0.43 = A2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijection.inv_left_eq_iff
% 0.20/0.43 thf(fact_88_bijection_Oeq__inv__iff,axiom,
% 0.20/0.43 ! [A: $tType,F3: A > A,A2: A,B3: A] :
% 0.20/0.43 ( ( hilbert_bijection @ A @ F3 )
% 0.20/0.43 => ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ A2 )
% 0.20/0.43 = ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ B3 ) )
% 0.20/0.43 = ( A2 = B3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijection.eq_inv_iff
% 0.20/0.43 thf(fact_89_insert__absorb2,axiom,
% 0.20/0.43 ! [A: $tType,X2: A,A3: set @ A] :
% 0.20/0.43 ( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A3 ) )
% 0.20/0.43 = ( insert @ A @ X2 @ A3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % insert_absorb2
% 0.20/0.43 thf(fact_90_insert__iff,axiom,
% 0.20/0.43 ! [A: $tType,A2: A,B3: A,A3: set @ A] :
% 0.20/0.43 ( ( member @ A @ A2 @ ( insert @ A @ B3 @ A3 ) )
% 0.20/0.43 = ( ( A2 = B3 )
% 0.20/0.43 | ( member @ A @ A2 @ A3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % insert_iff
% 0.20/0.43 thf(fact_91_insertCI,axiom,
% 0.20/0.43 ! [A: $tType,A2: A,B2: set @ A,B3: A] :
% 0.20/0.43 ( ( ~ ( member @ A @ A2 @ B2 )
% 0.20/0.43 => ( A2 = B3 ) )
% 0.20/0.43 => ( member @ A @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % insertCI
% 0.20/0.43 thf(fact_92_singletonI,axiom,
% 0.20/0.43 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % singletonI
% 0.20/0.43 thf(fact_93_singleton__conv,axiom,
% 0.20/0.43 ! [A: $tType,A2: A] :
% 0.20/0.43 ( ( collect @ A
% 0.20/0.43 @ ^ [X4: A] : ( X4 = A2 ) )
% 0.20/0.43 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % singleton_conv
% 0.20/0.43 thf(fact_94_singleton__conv2,axiom,
% 0.20/0.43 ! [A: $tType,A2: A] :
% 0.20/0.43 ( ( collect @ A
% 0.20/0.43 @ ( ^ [Y2: A,Z: A] : ( Y2 = Z )
% 0.20/0.43 @ A2 ) )
% 0.20/0.43 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % singleton_conv2
% 0.20/0.43 thf(fact_95_Quotient3__rep__reflp,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A2: B] :
% 0.20/0.43 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( R @ ( Rep @ A2 ) @ ( Rep @ A2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Quotient3_rep_reflp
% 0.20/0.43 thf(fact_96_Quotient3__rep__abs,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R2: A] :
% 0.20/0.43 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( ( R @ R2 @ R2 )
% 0.20/0.43 => ( R @ ( Rep @ ( Abs @ R2 ) ) @ R2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Quotient3_rep_abs
% 0.20/0.43 thf(fact_97_Quotient3__rel__rep,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A2: B,B3: B] :
% 0.20/0.43 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( ( R @ ( Rep @ A2 ) @ ( Rep @ B3 ) )
% 0.20/0.43 = ( A2 = B3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Quotient3_rel_rep
% 0.20/0.43 thf(fact_98_Quotient3__rel__abs,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R2: A,S3: A] :
% 0.20/0.43 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( ( R @ R2 @ S3 )
% 0.20/0.43 => ( ( Abs @ R2 )
% 0.20/0.43 = ( Abs @ S3 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Quotient3_rel_abs
% 0.20/0.43 thf(fact_99_Quotient3__abs__rep,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A2: B] :
% 0.20/0.43 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( ( Abs @ ( Rep @ A2 ) )
% 0.20/0.43 = A2 ) ) ).
% 0.20/0.43
% 0.20/0.43 % Quotient3_abs_rep
% 0.20/0.43 thf(fact_100_rep__abs__rsp__left,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
% 0.20/0.43 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( ( R @ X1 @ X22 )
% 0.20/0.43 => ( R @ ( Rep @ ( Abs @ X1 ) ) @ X22 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % rep_abs_rsp_left
% 0.20/0.43 thf(fact_101_Quotient3__refl2,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R2: A,S3: A] :
% 0.20/0.43 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( ( R @ R2 @ S3 )
% 0.20/0.43 => ( R @ S3 @ S3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Quotient3_refl2
% 0.20/0.43 thf(fact_102_Quotient3__refl1,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R2: A,S3: A] :
% 0.20/0.43 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( ( R @ R2 @ S3 )
% 0.20/0.43 => ( R @ R2 @ R2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Quotient3_refl1
% 0.20/0.43 thf(fact_103_Quotient3__rel,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R2: A,S3: A] :
% 0.20/0.43 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( ( ( R @ R2 @ R2 )
% 0.20/0.43 & ( R @ S3 @ S3 )
% 0.20/0.43 & ( ( Abs @ R2 )
% 0.20/0.43 = ( Abs @ S3 ) ) )
% 0.20/0.43 = ( R @ R2 @ S3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Quotient3_rel
% 0.20/0.43 thf(fact_104_Quotient3__def,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType] :
% 0.20/0.43 ( ( quotient3 @ A @ B )
% 0.20/0.43 = ( ^ [R4: A > A > $o,Abs2: A > B,Rep2: B > A] :
% 0.20/0.43 ( ! [A4: B] :
% 0.20/0.43 ( ( Abs2 @ ( Rep2 @ A4 ) )
% 0.20/0.43 = A4 )
% 0.20/0.43 & ! [A4: B] : ( R4 @ ( Rep2 @ A4 ) @ ( Rep2 @ A4 ) )
% 0.20/0.43 & ! [R3: A,S2: A] :
% 0.20/0.43 ( ( R4 @ R3 @ S2 )
% 0.20/0.43 = ( ( R4 @ R3 @ R3 )
% 0.20/0.43 & ( R4 @ S2 @ S2 )
% 0.20/0.43 & ( ( Abs2 @ R3 )
% 0.20/0.43 = ( Abs2 @ S2 ) ) ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Quotient3_def
% 0.20/0.43 thf(fact_105_rep__abs__rsp,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
% 0.20/0.43 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( ( R @ X1 @ X22 )
% 0.20/0.43 => ( R @ X1 @ ( Rep @ ( Abs @ X22 ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % rep_abs_rsp
% 0.20/0.43 thf(fact_106_equals__rsp,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,Xa: A,Xb: A,Ya: A,Yb: A] :
% 0.20/0.43 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.20/0.43 => ( ( R @ Xa @ Xb )
% 0.20/0.43 => ( ( R @ Ya @ Yb )
% 0.20/0.43 => ( ( R @ Xa @ Ya )
% 0.20/0.43 = ( R @ Xb @ Yb ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % equals_rsp
% 0.20/0.43 thf(fact_107_Quotient3I,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,Abs: B > A,Rep: A > B,R: B > B > $o] :
% 0.20/0.43 ( ! [A6: A] :
% 0.20/0.43 ( ( Abs @ ( Rep @ A6 ) )
% 0.20/0.43 = A6 )
% 0.20/0.43 => ( ! [A6: A] : ( R @ ( Rep @ A6 ) @ ( Rep @ A6 ) )
% 0.20/0.43 => ( ! [R5: B,S4: B] :
% 0.20/0.43 ( ( R @ R5 @ S4 )
% 0.20/0.43 = ( ( R @ R5 @ R5 )
% 0.20/0.43 & ( R @ S4 @ S4 )
% 0.20/0.43 & ( ( Abs @ R5 )
% 0.20/0.43 = ( Abs @ S4 ) ) ) )
% 0.20/0.43 => ( quotient3 @ B @ A @ R @ Abs @ Rep ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Quotient3I
% 0.20/0.43 thf(fact_108_cond__prs,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,R: A > A > $o,Absf: A > B,Repf: B > A,A2: $o,B3: B,C3: B] :
% 0.20/0.43 ( ( quotient3 @ A @ B @ R @ Absf @ Repf )
% 0.20/0.43 => ( ( A2
% 0.20/0.43 => ( ( Absf @ ( if @ A @ A2 @ ( Repf @ B3 ) @ ( Repf @ C3 ) ) )
% 0.20/0.43 = B3 ) )
% 0.20/0.43 & ( ~ A2
% 0.20/0.43 => ( ( Absf @ ( if @ A @ A2 @ ( Repf @ B3 ) @ ( Repf @ C3 ) ) )
% 0.20/0.43 = C3 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % cond_prs
% 0.20/0.43 thf(fact_109_bijection_Oeq__iff,axiom,
% 0.20/0.43 ! [A: $tType,F3: A > A,A2: A,B3: A] :
% 0.20/0.43 ( ( hilbert_bijection @ A @ F3 )
% 0.20/0.43 => ( ( ( F3 @ A2 )
% 0.20/0.43 = ( F3 @ B3 ) )
% 0.20/0.43 = ( A2 = B3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijection.eq_iff
% 0.20/0.43 thf(fact_110_bijection_OeqI,axiom,
% 0.20/0.43 ! [A: $tType,F3: A > A,A2: A,B3: A] :
% 0.20/0.43 ( ( hilbert_bijection @ A @ F3 )
% 0.20/0.43 => ( ( ( F3 @ A2 )
% 0.20/0.43 = ( F3 @ B3 ) )
% 0.20/0.43 => ( A2 = B3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijection.eqI
% 0.20/0.43 thf(fact_111_mk__disjoint__insert,axiom,
% 0.20/0.43 ! [A: $tType,A2: A,A3: set @ A] :
% 0.20/0.43 ( ( member @ A @ A2 @ A3 )
% 0.20/0.43 => ? [B5: set @ A] :
% 0.20/0.43 ( ( A3
% 0.20/0.43 = ( insert @ A @ A2 @ B5 ) )
% 0.20/0.43 & ~ ( member @ A @ A2 @ B5 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % mk_disjoint_insert
% 0.20/0.43 thf(fact_112_insert__commute,axiom,
% 0.20/0.43 ! [A: $tType,X2: A,Y4: A,A3: set @ A] :
% 0.20/0.43 ( ( insert @ A @ X2 @ ( insert @ A @ Y4 @ A3 ) )
% 0.20/0.43 = ( insert @ A @ Y4 @ ( insert @ A @ X2 @ A3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % insert_commute
% 0.20/0.43 thf(fact_113_insert__Collect,axiom,
% 0.20/0.43 ! [A: $tType,A2: A,P: A > $o] :
% 0.20/0.43 ( ( insert @ A @ A2 @ ( collect @ A @ P ) )
% 0.20/0.43 = ( collect @ A
% 0.20/0.43 @ ^ [U: A] :
% 0.20/0.43 ( ( U != A2 )
% 0.20/0.43 => ( P @ U ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % insert_Collect
% 0.20/0.43 thf(fact_114_insert__eq__iff,axiom,
% 0.20/0.43 ! [A: $tType,A2: A,A3: set @ A,B3: A,B2: set @ A] :
% 0.20/0.43 ( ~ ( member @ A @ A2 @ A3 )
% 0.20/0.43 => ( ~ ( member @ A @ B3 @ B2 )
% 0.20/0.43 => ( ( ( insert @ A @ A2 @ A3 )
% 0.20/0.43 = ( insert @ A @ B3 @ B2 ) )
% 0.20/0.43 = ( ( ( A2 = B3 )
% 0.20/0.43 => ( A3 = B2 ) )
% 0.20/0.43 & ( ( A2 != B3 )
% 0.20/0.43 => ? [C4: set @ A] :
% 0.20/0.43 ( ( A3
% 0.20/0.43 = ( insert @ A @ B3 @ C4 ) )
% 0.20/0.43 & ~ ( member @ A @ B3 @ C4 )
% 0.20/0.43 & ( B2
% 0.20/0.43 = ( insert @ A @ A2 @ C4 ) )
% 0.20/0.43 & ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % insert_eq_iff
% 0.20/0.43 thf(fact_115_insert__absorb,axiom,
% 0.20/0.43 ! [A: $tType,A2: A,A3: set @ A] :
% 0.20/0.43 ( ( member @ A @ A2 @ A3 )
% 0.20/0.43 => ( ( insert @ A @ A2 @ A3 )
% 0.20/0.43 = A3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % insert_absorb
% 0.20/0.43 thf(fact_116_insert__ident,axiom,
% 0.20/0.43 ! [A: $tType,X2: A,A3: set @ A,B2: set @ A] :
% 0.20/0.43 ( ~ ( member @ A @ X2 @ A3 )
% 0.20/0.43 => ( ~ ( member @ A @ X2 @ B2 )
% 0.20/0.43 => ( ( ( insert @ A @ X2 @ A3 )
% 0.20/0.43 = ( insert @ A @ X2 @ B2 ) )
% 0.20/0.43 = ( A3 = B2 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % insert_ident
% 0.20/0.43 thf(fact_117_insert__compr,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( insert @ A )
% 0.20/0.43 = ( ^ [A4: A,B6: set @ A] :
% 0.20/0.43 ( collect @ A
% 0.20/0.43 @ ^ [X4: A] :
% 0.20/0.43 ( ( X4 = A4 )
% 0.20/0.43 | ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % insert_compr
% 0.20/0.43 thf(fact_118_Set_Oset__insert,axiom,
% 0.20/0.43 ! [A: $tType,X2: A,A3: set @ A] :
% 0.20/0.43 ( ( member @ A @ X2 @ A3 )
% 0.20/0.43 => ~ ! [B5: set @ A] :
% 0.20/0.43 ( ( A3
% 0.20/0.43 = ( insert @ A @ X2 @ B5 ) )
% 0.20/0.43 => ( member @ A @ X2 @ B5 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Set.set_insert
% 0.20/0.43 thf(fact_119_insertI2,axiom,
% 0.20/0.43 ! [A: $tType,A2: A,B2: set @ A,B3: A] :
% 0.20/0.43 ( ( member @ A @ A2 @ B2 )
% 0.20/0.43 => ( member @ A @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % insertI2
% 0.20/0.43 thf(fact_120_insertI1,axiom,
% 0.20/0.43 ! [A: $tType,A2: A,B2: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B2 ) ) ).
% 0.20/0.43
% 0.20/0.43 % insertI1
% 0.20/0.43 thf(fact_121_insertE,axiom,
% 0.20/0.43 ! [A: $tType,A2: A,B3: A,A3: set @ A] :
% 0.20/0.43 ( ( member @ A @ A2 @ ( insert @ A @ B3 @ A3 ) )
% 0.20/0.43 => ( ( A2 != B3 )
% 0.20/0.43 => ( member @ A @ A2 @ A3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % insertE
% 0.20/0.43 thf(fact_122_top__empty__eq,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( top_top @ ( A > $o ) )
% 0.20/0.43 = ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % top_empty_eq
% 0.20/0.43 thf(fact_123_top__set__def,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( top_top @ ( set @ A ) )
% 0.20/0.43 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % top_set_def
% 0.20/0.43 thf(fact_124_singleton__inject,axiom,
% 0.20/0.43 ! [A: $tType,A2: A,B3: A] :
% 0.20/0.43 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.20/0.43 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.43 => ( A2 = B3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % singleton_inject
% 0.20/0.43 thf(fact_125_insert__not__empty,axiom,
% 0.20/0.43 ! [A: $tType,A2: A,A3: set @ A] :
% 0.20/0.43 ( ( insert @ A @ A2 @ A3 )
% 0.20/0.43 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % insert_not_empty
% 0.20/0.43 thf(fact_126_doubleton__eq__iff,axiom,
% 0.20/0.43 ! [A: $tType,A2: A,B3: A,C3: A,D2: A] :
% 0.20/0.43 ( ( ( insert @ A @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.43 = ( insert @ A @ C3 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.20/0.43 = ( ( ( A2 = C3 )
% 0.20/0.43 & ( B3 = D2 ) )
% 0.20/0.43 | ( ( A2 = D2 )
% 0.20/0.43 & ( B3 = C3 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % doubleton_eq_iff
% 0.20/0.43 thf(fact_127_singleton__iff,axiom,
% 0.20/0.43 ! [A: $tType,B3: A,A2: A] :
% 0.20/0.43 ( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.43 = ( B3 = A2 ) ) ).
% 0.20/0.43
% 0.20/0.43 % singleton_iff
% 0.20/0.43 thf(fact_128_singletonD,axiom,
% 0.20/0.43 ! [A: $tType,B3: A,A2: A] :
% 0.20/0.43 ( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.43 => ( B3 = A2 ) ) ).
% 0.20/0.43
% 0.20/0.43 % singletonD
% 0.20/0.43 thf(fact_129_insert__UNIV,axiom,
% 0.20/0.43 ! [A: $tType,X2: A] :
% 0.20/0.43 ( ( insert @ A @ X2 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 = ( top_top @ ( set @ A ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % insert_UNIV
% 0.20/0.43 thf(fact_130_Collect__conv__if,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o,A2: A] :
% 0.20/0.43 ( ( ( P @ A2 )
% 0.20/0.43 => ( ( collect @ A
% 0.20/0.43 @ ^ [X4: A] :
% 0.20/0.43 ( ( X4 = A2 )
% 0.20/0.43 & ( P @ X4 ) ) )
% 0.20/0.43 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.20/0.43 & ( ~ ( P @ A2 )
% 0.20/0.43 => ( ( collect @ A
% 0.20/0.43 @ ^ [X4: A] :
% 0.20/0.43 ( ( X4 = A2 )
% 0.20/0.43 & ( P @ X4 ) ) )
% 0.20/0.43 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Collect_conv_if
% 0.20/0.43 thf(fact_131_Collect__conv__if2,axiom,
% 0.20/0.43 ! [A: $tType,P: A > $o,A2: A] :
% 0.20/0.43 ( ( ( P @ A2 )
% 0.20/0.43 => ( ( collect @ A
% 0.20/0.43 @ ^ [X4: A] :
% 0.20/0.43 ( ( A2 = X4 )
% 0.20/0.43 & ( P @ X4 ) ) )
% 0.20/0.43 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.20/0.43 & ( ~ ( P @ A2 )
% 0.20/0.43 => ( ( collect @ A
% 0.20/0.43 @ ^ [X4: A] :
% 0.20/0.43 ( ( A2 = X4 )
% 0.20/0.43 & ( P @ X4 ) ) )
% 0.20/0.43 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Collect_conv_if2
% 0.20/0.43 thf(fact_132_inv__imagep__def,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType] :
% 0.20/0.43 ( ( inv_imagep @ B @ A )
% 0.20/0.43 = ( ^ [R3: B > B > $o,F: A > B,X4: A,Y: A] : ( R3 @ ( F @ X4 ) @ ( F @ Y ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inv_imagep_def
% 0.20/0.43 thf(fact_133_is__singleton__def,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( is_singleton @ A )
% 0.20/0.43 = ( ^ [A5: set @ A] :
% 0.20/0.43 ? [X4: A] :
% 0.20/0.43 ( A5
% 0.20/0.43 = ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % is_singleton_def
% 0.20/0.43 thf(fact_134_is__singletonE,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A] :
% 0.20/0.43 ( ( is_singleton @ A @ A3 )
% 0.20/0.43 => ~ ! [X: A] :
% 0.20/0.43 ( A3
% 0.20/0.43 != ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % is_singletonE
% 0.20/0.43 thf(fact_135_bijection_Oeq__invI,axiom,
% 0.20/0.43 ! [A: $tType,F3: A > A,A2: A,B3: A] :
% 0.20/0.43 ( ( hilbert_bijection @ A @ F3 )
% 0.20/0.43 => ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ A2 )
% 0.20/0.43 = ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ B3 ) )
% 0.20/0.43 => ( A2 = B3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijection.eq_invI
% 0.20/0.43 thf(fact_136_bijection_Oinv__left,axiom,
% 0.20/0.43 ! [A: $tType,F3: A > A,A2: A] :
% 0.20/0.43 ( ( hilbert_bijection @ A @ F3 )
% 0.20/0.43 => ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ ( F3 @ A2 ) )
% 0.20/0.43 = A2 ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijection.inv_left
% 0.20/0.43 thf(fact_137_bijection_Oinv__right,axiom,
% 0.20/0.43 ! [A: $tType,F3: A > A,A2: A] :
% 0.20/0.43 ( ( hilbert_bijection @ A @ F3 )
% 0.20/0.43 => ( ( F3 @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ A2 ) )
% 0.20/0.43 = A2 ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijection.inv_right
% 0.20/0.43 thf(fact_138_is__singleton__the__elem,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( is_singleton @ A )
% 0.20/0.43 = ( ^ [A5: set @ A] :
% 0.20/0.43 ( A5
% 0.20/0.43 = ( insert @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % is_singleton_the_elem
% 0.20/0.43 thf(fact_139_the__elem__eq,axiom,
% 0.20/0.43 ! [A: $tType,X2: A] :
% 0.20/0.43 ( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.43 = X2 ) ).
% 0.20/0.43
% 0.20/0.43 % the_elem_eq
% 0.20/0.43 thf(fact_140_range__constant,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,X2: A] :
% 0.20/0.43 ( ( image @ B @ A
% 0.20/0.43 @ ^ [Uu: B] : X2
% 0.20/0.43 @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % range_constant
% 0.20/0.43 thf(fact_141_bijection_Osurj__inv,axiom,
% 0.20/0.43 ! [A: $tType,F3: A > A] :
% 0.20/0.43 ( ( hilbert_bijection @ A @ F3 )
% 0.20/0.43 => ( ( image @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijection.surj_inv
% 0.20/0.43 thf(fact_142_bijection_Oinj__inv,axiom,
% 0.20/0.43 ! [A: $tType,F3: A > A] :
% 0.20/0.43 ( ( hilbert_bijection @ A @ F3 )
% 0.20/0.43 => ( inj_on @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijection.inj_inv
% 0.20/0.43 thf(fact_143_bijection_Obij__inv,axiom,
% 0.20/0.43 ! [A: $tType,F3: A > A] :
% 0.20/0.43 ( ( hilbert_bijection @ A @ F3 )
% 0.20/0.43 => ( bij_betw @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijection.bij_inv
% 0.20/0.43 thf(fact_144_image__eqI,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,B3: A,F3: B > A,X2: B,A3: set @ B] :
% 0.20/0.43 ( ( B3
% 0.20/0.43 = ( F3 @ X2 ) )
% 0.20/0.43 => ( ( member @ B @ X2 @ A3 )
% 0.20/0.43 => ( member @ A @ B3 @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % image_eqI
% 0.20/0.43 thf(fact_145_image__ident,axiom,
% 0.20/0.43 ! [A: $tType,Y6: set @ A] :
% 0.20/0.43 ( ( image @ A @ A
% 0.20/0.43 @ ^ [X4: A] : X4
% 0.20/0.43 @ Y6 )
% 0.20/0.43 = Y6 ) ).
% 0.20/0.43
% 0.20/0.43 % image_ident
% 0.20/0.43 thf(fact_146_top1I,axiom,
% 0.20/0.43 ! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% 0.20/0.43
% 0.20/0.43 % top1I
% 0.20/0.43 thf(fact_147_image__is__empty,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
% 0.20/0.43 ( ( ( image @ B @ A @ F3 @ A3 )
% 0.20/0.43 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.43 = ( A3
% 0.20/0.43 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % image_is_empty
% 0.20/0.43 thf(fact_148_empty__is__image,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
% 0.20/0.43 ( ( ( bot_bot @ ( set @ A ) )
% 0.20/0.43 = ( image @ B @ A @ F3 @ A3 ) )
% 0.20/0.43 = ( A3
% 0.20/0.43 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % empty_is_image
% 0.20/0.43 thf(fact_149_image__empty,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: B > A] :
% 0.20/0.43 ( ( image @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) )
% 0.20/0.43 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % image_empty
% 0.20/0.43 thf(fact_150_image__insert,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: B > A,A2: B,B2: set @ B] :
% 0.20/0.43 ( ( image @ B @ A @ F3 @ ( insert @ B @ A2 @ B2 ) )
% 0.20/0.43 = ( insert @ A @ ( F3 @ A2 ) @ ( image @ B @ A @ F3 @ B2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % image_insert
% 0.20/0.43 thf(fact_151_insert__image,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,X2: A,A3: set @ A,F3: A > B] :
% 0.20/0.43 ( ( member @ A @ X2 @ A3 )
% 0.20/0.43 => ( ( insert @ B @ ( F3 @ X2 ) @ ( image @ A @ B @ F3 @ A3 ) )
% 0.20/0.43 = ( image @ A @ B @ F3 @ A3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % insert_image
% 0.20/0.43 thf(fact_152_inv__into__f__f,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X2: A] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ A3 )
% 0.20/0.43 => ( ( member @ A @ X2 @ A3 )
% 0.20/0.43 => ( ( hilbert_inv_into @ A @ B @ A3 @ F3 @ ( F3 @ X2 ) )
% 0.20/0.43 = X2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inv_into_f_f
% 0.20/0.43 thf(fact_153_inj__imp__bij__betw__inv,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,M: set @ A] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( image @ A @ B @ F3 @ M ) @ M ) ) ).
% 0.20/0.43
% 0.20/0.43 % inj_imp_bij_betw_inv
% 0.20/0.43 thf(fact_154_inj__transfer,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,P: A > $o,X2: A] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( ! [Y3: B] :
% 0.20/0.43 ( ( member @ B @ Y3 @ ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) )
% 0.20/0.43 => ( P @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 @ Y3 ) ) )
% 0.20/0.43 => ( P @ X2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inj_transfer
% 0.20/0.43 thf(fact_155_image__inv__f__f,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( image @ A @ B @ F3 @ A3 ) )
% 0.20/0.43 = A3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % image_inv_f_f
% 0.20/0.43 thf(fact_156_inj__imp__surj__inv,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inj_imp_surj_inv
% 0.20/0.43 thf(fact_157_surj__imp__inj__inv,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: B > A] :
% 0.20/0.43 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 = ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % surj_imp_inj_inv
% 0.20/0.43 thf(fact_158_bijI_H,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: A > B] :
% 0.20/0.43 ( ! [X: A,Y3: A] :
% 0.20/0.43 ( ( ( F3 @ X )
% 0.20/0.43 = ( F3 @ Y3 ) )
% 0.20/0.43 = ( X = Y3 ) )
% 0.20/0.43 => ( ! [Y3: B] :
% 0.20/0.43 ? [X5: A] :
% 0.20/0.43 ( Y3
% 0.20/0.43 = ( F3 @ X5 ) )
% 0.20/0.43 => ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijI'
% 0.20/0.43 thf(fact_159_imageE,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,B3: A,F3: B > A,A3: set @ B] :
% 0.20/0.43 ( ( member @ A @ B3 @ ( image @ B @ A @ F3 @ A3 ) )
% 0.20/0.43 => ~ ! [X: B] :
% 0.20/0.43 ( ( B3
% 0.20/0.43 = ( F3 @ X ) )
% 0.20/0.43 => ~ ( member @ B @ X @ A3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % imageE
% 0.20/0.43 thf(fact_160_imageI,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,X2: A,A3: set @ A,F3: A > B] :
% 0.20/0.43 ( ( member @ A @ X2 @ A3 )
% 0.20/0.43 => ( member @ B @ ( F3 @ X2 ) @ ( image @ A @ B @ F3 @ A3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % imageI
% 0.20/0.43 thf(fact_161_image__iff,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,Z2: A,F3: B > A,A3: set @ B] :
% 0.20/0.43 ( ( member @ A @ Z2 @ ( image @ B @ A @ F3 @ A3 ) )
% 0.20/0.43 = ( ? [X4: B] :
% 0.20/0.43 ( ( member @ B @ X4 @ A3 )
% 0.20/0.43 & ( Z2
% 0.20/0.43 = ( F3 @ X4 ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % image_iff
% 0.20/0.43 thf(fact_162_bex__imageD,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P: A > $o] :
% 0.20/0.43 ( ? [X5: A] :
% 0.20/0.43 ( ( member @ A @ X5 @ ( image @ B @ A @ F3 @ A3 ) )
% 0.20/0.43 & ( P @ X5 ) )
% 0.20/0.43 => ? [X: B] :
% 0.20/0.43 ( ( member @ B @ X @ A3 )
% 0.20/0.43 & ( P @ ( F3 @ X ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bex_imageD
% 0.20/0.43 thf(fact_163_image__cong,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F3: A > B,G: A > B] :
% 0.20/0.43 ( ( M = N )
% 0.20/0.43 => ( ! [X: A] :
% 0.20/0.43 ( ( member @ A @ X @ N )
% 0.20/0.43 => ( ( F3 @ X )
% 0.20/0.43 = ( G @ X ) ) )
% 0.20/0.43 => ( ( image @ A @ B @ F3 @ M )
% 0.20/0.43 = ( image @ A @ B @ G @ N ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % image_cong
% 0.20/0.43 thf(fact_164_ball__imageD,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P: A > $o] :
% 0.20/0.43 ( ! [X: A] :
% 0.20/0.43 ( ( member @ A @ X @ ( image @ B @ A @ F3 @ A3 ) )
% 0.20/0.43 => ( P @ X ) )
% 0.20/0.43 => ! [X5: B] :
% 0.20/0.43 ( ( member @ B @ X5 @ A3 )
% 0.20/0.43 => ( P @ ( F3 @ X5 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % ball_imageD
% 0.20/0.43 thf(fact_165_image__image,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,C: $tType,F3: B > A,G: C > B,A3: set @ C] :
% 0.20/0.43 ( ( image @ B @ A @ F3 @ ( image @ C @ B @ G @ A3 ) )
% 0.20/0.43 = ( image @ C @ A
% 0.20/0.43 @ ^ [X4: C] : ( F3 @ ( G @ X4 ) )
% 0.20/0.43 @ A3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % image_image
% 0.20/0.43 thf(fact_166_rev__image__eqI,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,X2: A,A3: set @ A,B3: B,F3: A > B] :
% 0.20/0.43 ( ( member @ A @ X2 @ A3 )
% 0.20/0.43 => ( ( B3
% 0.20/0.43 = ( F3 @ X2 ) )
% 0.20/0.43 => ( member @ B @ B3 @ ( image @ A @ B @ F3 @ A3 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % rev_image_eqI
% 0.20/0.43 thf(fact_167_Compr__image__eq,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P: A > $o] :
% 0.20/0.43 ( ( collect @ A
% 0.20/0.43 @ ^ [X4: A] :
% 0.20/0.43 ( ( member @ A @ X4 @ ( image @ B @ A @ F3 @ A3 ) )
% 0.20/0.43 & ( P @ X4 ) ) )
% 0.20/0.43 = ( image @ B @ A @ F3
% 0.20/0.43 @ ( collect @ B
% 0.20/0.43 @ ^ [X4: B] :
% 0.20/0.43 ( ( member @ B @ X4 @ A3 )
% 0.20/0.43 & ( P @ ( F3 @ X4 ) ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Compr_image_eq
% 0.20/0.43 thf(fact_168_the__elem__image__unique,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,A3: set @ A,F3: A > B,X2: A] :
% 0.20/0.43 ( ( A3
% 0.20/0.43 != ( bot_bot @ ( set @ A ) ) )
% 0.20/0.43 => ( ! [Y3: A] :
% 0.20/0.43 ( ( member @ A @ Y3 @ A3 )
% 0.20/0.43 => ( ( F3 @ Y3 )
% 0.20/0.43 = ( F3 @ X2 ) ) )
% 0.20/0.43 => ( ( the_elem @ B @ ( image @ A @ B @ F3 @ A3 ) )
% 0.20/0.43 = ( F3 @ X2 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % the_elem_image_unique
% 0.20/0.43 thf(fact_169_rangeI,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: B > A,X2: B] : ( member @ A @ ( F3 @ X2 ) @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % rangeI
% 0.20/0.43 thf(fact_170_range__eqI,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,B3: A,F3: B > A,X2: B] :
% 0.20/0.43 ( ( B3
% 0.20/0.43 = ( F3 @ X2 ) )
% 0.20/0.43 => ( member @ A @ B3 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % range_eqI
% 0.20/0.43 thf(fact_171_bij__image__Collect__eq,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: A > B,P: A > $o] :
% 0.20/0.43 ( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 => ( ( image @ A @ B @ F3 @ ( collect @ A @ P ) )
% 0.20/0.43 = ( collect @ B
% 0.20/0.43 @ ^ [Y: B] : ( P @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 @ Y ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bij_image_Collect_eq
% 0.20/0.43 thf(fact_172_bij__betw__inv__into,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,B2: set @ B] :
% 0.20/0.43 ( ( bij_betw @ A @ B @ F3 @ A3 @ B2 )
% 0.20/0.43 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A3 @ F3 ) @ B2 @ A3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % bij_betw_inv_into
% 0.20/0.43 thf(fact_173_inv__into__inv__into__eq,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,A7: set @ B,A2: A] :
% 0.20/0.43 ( ( bij_betw @ A @ B @ F3 @ A3 @ A7 )
% 0.20/0.43 => ( ( member @ A @ A2 @ A3 )
% 0.20/0.43 => ( ( hilbert_inv_into @ B @ A @ A7 @ ( hilbert_inv_into @ A @ B @ A3 @ F3 ) @ A2 )
% 0.20/0.43 = ( F3 @ A2 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inv_into_inv_into_eq
% 0.20/0.43 thf(fact_174_bij__betw__inv__into__left,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,A7: set @ B,A2: A] :
% 0.20/0.43 ( ( bij_betw @ A @ B @ F3 @ A3 @ A7 )
% 0.20/0.43 => ( ( member @ A @ A2 @ A3 )
% 0.20/0.43 => ( ( hilbert_inv_into @ A @ B @ A3 @ F3 @ ( F3 @ A2 ) )
% 0.20/0.43 = A2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bij_betw_inv_into_left
% 0.20/0.43 thf(fact_175_bij__betw__inv__into__right,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: A > B,A3: set @ A,A7: set @ B,A8: B] :
% 0.20/0.43 ( ( bij_betw @ A @ B @ F3 @ A3 @ A7 )
% 0.20/0.43 => ( ( member @ B @ A8 @ A7 )
% 0.20/0.43 => ( ( F3 @ ( hilbert_inv_into @ A @ B @ A3 @ F3 @ A8 ) )
% 0.20/0.43 = A8 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bij_betw_inv_into_right
% 0.20/0.43 thf(fact_176_inv__into__f__eq,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X2: A,Y4: B] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ A3 )
% 0.20/0.43 => ( ( member @ A @ X2 @ A3 )
% 0.20/0.43 => ( ( ( F3 @ X2 )
% 0.20/0.43 = Y4 )
% 0.20/0.43 => ( ( hilbert_inv_into @ A @ B @ A3 @ F3 @ Y4 )
% 0.20/0.43 = X2 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inv_into_f_eq
% 0.20/0.43 thf(fact_177_f__inv__into__f,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,Y4: A,F3: B > A,A3: set @ B] :
% 0.20/0.43 ( ( member @ A @ Y4 @ ( image @ B @ A @ F3 @ A3 ) )
% 0.20/0.43 => ( ( F3 @ ( hilbert_inv_into @ B @ A @ A3 @ F3 @ Y4 ) )
% 0.20/0.43 = Y4 ) ) ).
% 0.20/0.43
% 0.20/0.43 % f_inv_into_f
% 0.20/0.43 thf(fact_178_inv__into__into,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,X2: A,F3: B > A,A3: set @ B] :
% 0.20/0.43 ( ( member @ A @ X2 @ ( image @ B @ A @ F3 @ A3 ) )
% 0.20/0.43 => ( member @ B @ ( hilbert_inv_into @ B @ A @ A3 @ F3 @ X2 ) @ A3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % inv_into_into
% 0.20/0.43 thf(fact_179_inv__into__injective,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,A3: set @ A,F3: A > B,X2: B,Y4: B] :
% 0.20/0.43 ( ( ( hilbert_inv_into @ A @ B @ A3 @ F3 @ X2 )
% 0.20/0.43 = ( hilbert_inv_into @ A @ B @ A3 @ F3 @ Y4 ) )
% 0.20/0.43 => ( ( member @ B @ X2 @ ( image @ A @ B @ F3 @ A3 ) )
% 0.20/0.43 => ( ( member @ B @ Y4 @ ( image @ A @ B @ F3 @ A3 ) )
% 0.20/0.43 => ( X2 = Y4 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inv_into_injective
% 0.20/0.43 thf(fact_180_setcompr__eq__image,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: B > A,P: B > $o] :
% 0.20/0.43 ( ( collect @ A
% 0.20/0.43 @ ^ [Uu: A] :
% 0.20/0.43 ? [X4: B] :
% 0.20/0.43 ( ( Uu
% 0.20/0.43 = ( F3 @ X4 ) )
% 0.20/0.43 & ( P @ X4 ) ) )
% 0.20/0.43 = ( image @ B @ A @ F3 @ ( collect @ B @ P ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % setcompr_eq_image
% 0.20/0.43 thf(fact_181_Setcompr__eq__image,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
% 0.20/0.43 ( ( collect @ A
% 0.20/0.43 @ ^ [Uu: A] :
% 0.20/0.43 ? [X4: B] :
% 0.20/0.43 ( ( Uu
% 0.20/0.43 = ( F3 @ X4 ) )
% 0.20/0.43 & ( member @ B @ X4 @ A3 ) ) )
% 0.20/0.43 = ( image @ B @ A @ F3 @ A3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % Setcompr_eq_image
% 0.20/0.43 thf(fact_182_rangeE,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,B3: A,F3: B > A] :
% 0.20/0.43 ( ( member @ A @ B3 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
% 0.20/0.43 => ~ ! [X: B] :
% 0.20/0.43 ( B3
% 0.20/0.43 != ( F3 @ X ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % rangeE
% 0.20/0.43 thf(fact_183_range__composition,axiom,
% 0.20/0.43 ! [A: $tType,C: $tType,B: $tType,F3: C > A,G: B > C] :
% 0.20/0.43 ( ( image @ B @ A
% 0.20/0.43 @ ^ [X4: B] : ( F3 @ ( G @ X4 ) )
% 0.20/0.43 @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 = ( image @ C @ A @ F3 @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % range_composition
% 0.20/0.43 thf(fact_184_bij__imp__bij__inv,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B] :
% 0.20/0.43 ( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bij_imp_bij_inv
% 0.20/0.43 thf(fact_185_bij__inv__eq__iff,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,P6: A > B,X2: A,Y4: B] :
% 0.20/0.43 ( ( bij_betw @ A @ B @ P6 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 => ( ( X2
% 0.20/0.43 = ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P6 @ Y4 ) )
% 0.20/0.43 = ( ( P6 @ X2 )
% 0.20/0.43 = Y4 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bij_inv_eq_iff
% 0.20/0.43 thf(fact_186_inv__inv__eq,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B] :
% 0.20/0.43 ( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 => ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) )
% 0.20/0.43 = F3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % inv_inv_eq
% 0.20/0.43 thf(fact_187_inj__imp__inv__eq,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: A > B,G: B > A] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( ! [X: B] :
% 0.20/0.43 ( ( F3 @ ( G @ X ) )
% 0.20/0.43 = X )
% 0.20/0.43 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
% 0.20/0.43 = G ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inj_imp_inv_eq
% 0.20/0.43 thf(fact_188_inv__f__eq,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,X2: A,Y4: B] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( ( ( F3 @ X2 )
% 0.20/0.43 = Y4 )
% 0.20/0.43 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 @ Y4 )
% 0.20/0.43 = X2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inv_f_eq
% 0.20/0.43 thf(fact_189_inv__f__f,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,X2: A] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 @ ( F3 @ X2 ) )
% 0.20/0.43 = X2 ) ) ).
% 0.20/0.43
% 0.20/0.43 % inv_f_f
% 0.20/0.43 thf(fact_190_bijection_Ointro,axiom,
% 0.20/0.43 ! [A: $tType,F3: A > A] :
% 0.20/0.43 ( ( bij_betw @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( hilbert_bijection @ A @ F3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijection.intro
% 0.20/0.43 thf(fact_191_bijection__def,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( hilbert_bijection @ A )
% 0.20/0.43 = ( ^ [F: A > A] : ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijection_def
% 0.20/0.43 thf(fact_192_bijection_Obij,axiom,
% 0.20/0.43 ! [A: $tType,F3: A > A] :
% 0.20/0.43 ( ( hilbert_bijection @ A @ F3 )
% 0.20/0.43 => ( bij_betw @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijection.bij
% 0.20/0.43 thf(fact_193_surj__imp__inv__eq,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: B > A,G: A > B] :
% 0.20/0.43 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 = ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( ! [X: B] :
% 0.20/0.43 ( ( G @ ( F3 @ X ) )
% 0.20/0.43 = X )
% 0.20/0.43 => ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 )
% 0.20/0.43 = G ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % surj_imp_inv_eq
% 0.20/0.43 thf(fact_194_image__f__inv__f,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: B > A,A3: set @ A] :
% 0.20/0.43 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 = ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( ( image @ B @ A @ F3 @ ( image @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 ) @ A3 ) )
% 0.20/0.43 = A3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % image_f_inv_f
% 0.20/0.43 thf(fact_195_surj__iff__all,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: B > A] :
% 0.20/0.43 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 = ( top_top @ ( set @ A ) ) )
% 0.20/0.43 = ( ! [X4: A] :
% 0.20/0.43 ( ( F3 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 @ X4 ) )
% 0.20/0.43 = X4 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % surj_iff_all
% 0.20/0.43 thf(fact_196_surj__f__inv__f,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: B > A,Y4: A] :
% 0.20/0.43 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 = ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( ( F3 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 @ Y4 ) )
% 0.20/0.43 = Y4 ) ) ).
% 0.20/0.43
% 0.20/0.43 % surj_f_inv_f
% 0.20/0.43 thf(fact_197_bijection_Oinj,axiom,
% 0.20/0.43 ! [A: $tType,F3: A > A] :
% 0.20/0.43 ( ( hilbert_bijection @ A @ F3 )
% 0.20/0.43 => ( inj_on @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijection.inj
% 0.20/0.43 thf(fact_198_bijection_Osurj,axiom,
% 0.20/0.43 ! [A: $tType,F3: A > A] :
% 0.20/0.43 ( ( hilbert_bijection @ A @ F3 )
% 0.20/0.43 => ( ( image @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijection.surj
% 0.20/0.43 thf(fact_199_image__constant,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,X2: A,A3: set @ A,C3: B] :
% 0.20/0.43 ( ( member @ A @ X2 @ A3 )
% 0.20/0.43 => ( ( image @ A @ B
% 0.20/0.43 @ ^ [X4: A] : C3
% 0.20/0.43 @ A3 )
% 0.20/0.43 = ( insert @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % image_constant
% 0.20/0.43 thf(fact_200_image__constant__conv,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,A3: set @ B,C3: A] :
% 0.20/0.43 ( ( ( A3
% 0.20/0.43 = ( bot_bot @ ( set @ B ) ) )
% 0.20/0.43 => ( ( image @ B @ A
% 0.20/0.43 @ ^ [X4: B] : C3
% 0.20/0.43 @ A3 )
% 0.20/0.43 = ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.43 & ( ( A3
% 0.20/0.43 != ( bot_bot @ ( set @ B ) ) )
% 0.20/0.43 => ( ( image @ B @ A
% 0.20/0.43 @ ^ [X4: B] : C3
% 0.20/0.43 @ A3 )
% 0.20/0.43 = ( insert @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % image_constant_conv
% 0.20/0.43 thf(fact_201_full__SetCompr__eq,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: B > A] :
% 0.20/0.43 ( ( collect @ A
% 0.20/0.43 @ ^ [U: A] :
% 0.20/0.43 ? [X4: B] :
% 0.20/0.43 ( U
% 0.20/0.43 = ( F3 @ X4 ) ) )
% 0.20/0.43 = ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % full_SetCompr_eq
% 0.20/0.43 thf(fact_202_range__eq__singletonD,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: B > A,A2: A,X2: B] :
% 0.20/0.43 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.43 => ( ( F3 @ X2 )
% 0.20/0.43 = A2 ) ) ).
% 0.20/0.43
% 0.20/0.43 % range_eq_singletonD
% 0.20/0.43 thf(fact_203_inj__on__empty,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B] : ( inj_on @ A @ B @ F3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inj_on_empty
% 0.20/0.43 thf(fact_204_bij__def,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: A > B] :
% 0.20/0.43 ( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 = ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 & ( ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 = ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bij_def
% 0.20/0.43 thf(fact_205_bijI,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: A > B] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( ( ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 = ( top_top @ ( set @ B ) ) )
% 0.20/0.43 => ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bijI
% 0.20/0.43 thf(fact_206_bij__is__inj,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B] :
% 0.20/0.43 ( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 => ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bij_is_inj
% 0.20/0.43 thf(fact_207_inj__on__id2,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A] :
% 0.20/0.43 ( inj_on @ A @ A
% 0.20/0.43 @ ^ [X4: A] : X4
% 0.20/0.43 @ A3 ) ).
% 0.20/0.43
% 0.20/0.43 % inj_on_id2
% 0.20/0.43 thf(fact_208_surjD,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: B > A,Y4: A] :
% 0.20/0.43 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 = ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ? [X: B] :
% 0.20/0.43 ( Y4
% 0.20/0.43 = ( F3 @ X ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % surjD
% 0.20/0.43 thf(fact_209_surjE,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: B > A,Y4: A] :
% 0.20/0.43 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 = ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ~ ! [X: B] :
% 0.20/0.43 ( Y4
% 0.20/0.43 != ( F3 @ X ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % surjE
% 0.20/0.43 thf(fact_210_surjI,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,G: B > A,F3: A > B] :
% 0.20/0.43 ( ! [X: A] :
% 0.20/0.43 ( ( G @ ( F3 @ X ) )
% 0.20/0.43 = X )
% 0.20/0.43 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % surjI
% 0.20/0.43 thf(fact_211_surj__def,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: B > A] :
% 0.20/0.43 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 = ( top_top @ ( set @ A ) ) )
% 0.20/0.43 = ( ! [Y: A] :
% 0.20/0.43 ? [X4: B] :
% 0.20/0.43 ( Y
% 0.20/0.43 = ( F3 @ X4 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % surj_def
% 0.20/0.43 thf(fact_212_injD,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,X2: A,Y4: A] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( ( ( F3 @ X2 )
% 0.20/0.43 = ( F3 @ Y4 ) )
% 0.20/0.43 => ( X2 = Y4 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % injD
% 0.20/0.43 thf(fact_213_injI,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B] :
% 0.20/0.43 ( ! [X: A,Y3: A] :
% 0.20/0.43 ( ( ( F3 @ X )
% 0.20/0.43 = ( F3 @ Y3 ) )
% 0.20/0.43 => ( X = Y3 ) )
% 0.20/0.43 => ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % injI
% 0.20/0.43 thf(fact_214_inj__eq,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,X2: A,Y4: A] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( ( ( F3 @ X2 )
% 0.20/0.43 = ( F3 @ Y4 ) )
% 0.20/0.43 = ( X2 = Y4 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inj_eq
% 0.20/0.43 thf(fact_215_inj__def,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 = ( ! [X4: A,Y: A] :
% 0.20/0.43 ( ( ( F3 @ X4 )
% 0.20/0.43 = ( F3 @ Y ) )
% 0.20/0.43 => ( X4 = Y ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inj_def
% 0.20/0.43 thf(fact_216_bij__betw__empty2,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A] :
% 0.20/0.43 ( ( bij_betw @ A @ B @ F3 @ A3 @ ( bot_bot @ ( set @ B ) ) )
% 0.20/0.43 => ( A3
% 0.20/0.43 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bij_betw_empty2
% 0.20/0.43 thf(fact_217_bij__betw__empty1,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: A > B,A3: set @ B] :
% 0.20/0.43 ( ( bij_betw @ A @ B @ F3 @ ( bot_bot @ ( set @ A ) ) @ A3 )
% 0.20/0.43 => ( A3
% 0.20/0.43 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bij_betw_empty1
% 0.20/0.43 thf(fact_218_bij__pointE,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,Y4: B] :
% 0.20/0.43 ( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 => ~ ! [X: A] :
% 0.20/0.43 ( ( Y4
% 0.20/0.43 = ( F3 @ X ) )
% 0.20/0.43 => ~ ! [X8: A] :
% 0.20/0.43 ( ( Y4
% 0.20/0.43 = ( F3 @ X8 ) )
% 0.20/0.43 => ( X8 = X ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bij_pointE
% 0.20/0.43 thf(fact_219_inj__singleton,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A] :
% 0.20/0.43 ( inj_on @ A @ ( set @ A )
% 0.20/0.43 @ ^ [X4: A] : ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
% 0.20/0.43 @ A3 ) ).
% 0.20/0.43
% 0.20/0.43 % inj_singleton
% 0.20/0.43 thf(fact_220_inj__fun,axiom,
% 0.20/0.43 ! [B: $tType,C: $tType,A: $tType,F3: A > B] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( inj_on @ A @ ( C > B )
% 0.20/0.43 @ ^ [X4: A,Y: C] : ( F3 @ X4 )
% 0.20/0.43 @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inj_fun
% 0.20/0.43 thf(fact_221_inj__img__insertE,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X2: B,B2: set @ B] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ A3 )
% 0.20/0.43 => ( ~ ( member @ B @ X2 @ B2 )
% 0.20/0.43 => ( ( ( insert @ B @ X2 @ B2 )
% 0.20/0.43 = ( image @ A @ B @ F3 @ A3 ) )
% 0.20/0.43 => ~ ! [X9: A,A9: set @ A] :
% 0.20/0.43 ( ~ ( member @ A @ X9 @ A9 )
% 0.20/0.43 => ( ( A3
% 0.20/0.43 = ( insert @ A @ X9 @ A9 ) )
% 0.20/0.43 => ( ( X2
% 0.20/0.43 = ( F3 @ X9 ) )
% 0.20/0.43 => ( B2
% 0.20/0.43 != ( image @ A @ B @ F3 @ A9 ) ) ) ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inj_img_insertE
% 0.20/0.43 thf(fact_222_range__ex1__eq,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,B3: B] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( ( member @ B @ B3 @ ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) )
% 0.20/0.43 = ( ? [X4: A] :
% 0.20/0.43 ( ( B3
% 0.20/0.43 = ( F3 @ X4 ) )
% 0.20/0.43 & ! [Y: A] :
% 0.20/0.43 ( ( B3
% 0.20/0.43 = ( F3 @ Y ) )
% 0.20/0.43 => ( Y = X4 ) ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % range_ex1_eq
% 0.20/0.43 thf(fact_223_inj__image__eq__iff,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,B2: set @ A] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( ( ( image @ A @ B @ F3 @ A3 )
% 0.20/0.43 = ( image @ A @ B @ F3 @ B2 ) )
% 0.20/0.43 = ( A3 = B2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inj_image_eq_iff
% 0.20/0.43 thf(fact_224_inj__image__mem__iff,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,A2: A,A3: set @ A] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( ( member @ B @ ( F3 @ A2 ) @ ( image @ A @ B @ F3 @ A3 ) )
% 0.20/0.43 = ( member @ A @ A2 @ A3 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inj_image_mem_iff
% 0.20/0.43 thf(fact_225_bij__is__surj,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: A > B] :
% 0.20/0.43 ( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 => ( ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 = ( top_top @ ( set @ B ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bij_is_surj
% 0.20/0.43 thf(fact_226_bij__betw__imp__surj,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,F3: A > B,A3: set @ A] :
% 0.20/0.43 ( ( bij_betw @ A @ B @ F3 @ A3 @ ( top_top @ ( set @ B ) ) )
% 0.20/0.43 => ( ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 = ( top_top @ ( set @ B ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % bij_betw_imp_surj
% 0.20/0.43 thf(fact_227_inj__on__insert,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,A2: A,A3: set @ A] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( insert @ A @ A2 @ A3 ) )
% 0.20/0.43 = ( ( inj_on @ A @ B @ F3 @ A3 )
% 0.20/0.43 & ~ ( member @ B @ ( F3 @ A2 ) @ ( image @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inj_on_insert
% 0.20/0.43 thf(fact_228_top__conj_I2_J,axiom,
% 0.20/0.43 ! [A: $tType,P: $o,X2: A] :
% 0.20/0.43 ( ( P
% 0.20/0.43 & ( top_top @ ( A > $o ) @ X2 ) )
% 0.20/0.43 = P ) ).
% 0.20/0.43
% 0.20/0.43 % top_conj(2)
% 0.20/0.43 thf(fact_229_DiffI,axiom,
% 0.20/0.43 ! [A: $tType,C3: A,A3: set @ A,B2: set @ A] :
% 0.20/0.43 ( ( member @ A @ C3 @ A3 )
% 0.20/0.43 => ( ~ ( member @ A @ C3 @ B2 )
% 0.20/0.43 => ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % DiffI
% 0.20/0.43 thf(fact_230_Diff__iff,axiom,
% 0.20/0.43 ! [A: $tType,C3: A,A3: set @ A,B2: set @ A] :
% 0.20/0.43 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
% 0.20/0.43 = ( ( member @ A @ C3 @ A3 )
% 0.20/0.43 & ~ ( member @ A @ C3 @ B2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Diff_iff
% 0.20/0.43 thf(fact_231_Diff__idemp,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.20/0.43 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ B2 )
% 0.20/0.43 = ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ).
% 0.20/0.43
% 0.20/0.43 % Diff_idemp
% 0.20/0.43 thf(fact_232_Diff__empty,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A] :
% 0.20/0.43 ( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.20/0.43 = A3 ) ).
% 0.20/0.43
% 0.20/0.43 % Diff_empty
% 0.20/0.43 thf(fact_233_empty__Diff,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A] :
% 0.20/0.43 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
% 0.20/0.43 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % empty_Diff
% 0.20/0.43 thf(fact_234_Diff__cancel,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A] :
% 0.20/0.43 ( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
% 0.20/0.43 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Diff_cancel
% 0.20/0.43 thf(fact_235_insert__Diff1,axiom,
% 0.20/0.43 ! [A: $tType,X2: A,B2: set @ A,A3: set @ A] :
% 0.20/0.43 ( ( member @ A @ X2 @ B2 )
% 0.20/0.43 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B2 )
% 0.20/0.43 = ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % insert_Diff1
% 0.20/0.43 thf(fact_236_Diff__insert0,axiom,
% 0.20/0.43 ! [A: $tType,X2: A,A3: set @ A,B2: set @ A] :
% 0.20/0.43 ( ~ ( member @ A @ X2 @ A3 )
% 0.20/0.43 => ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ B2 ) )
% 0.20/0.43 = ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Diff_insert0
% 0.20/0.43 thf(fact_237_insert__Diff__single,axiom,
% 0.20/0.43 ! [A: $tType,A2: A,A3: set @ A] :
% 0.20/0.43 ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.20/0.43 = ( insert @ A @ A2 @ A3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % insert_Diff_single
% 0.20/0.43 thf(fact_238_Diff__UNIV,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A] :
% 0.20/0.43 ( ( minus_minus @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Diff_UNIV
% 0.20/0.43 thf(fact_239_surj__diff__right,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( ab_group_add @ A )
% 0.20/0.43 => ! [A2: A] :
% 0.20/0.43 ( ( image @ A @ A
% 0.20/0.43 @ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
% 0.20/0.43 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % surj_diff_right
% 0.20/0.43 thf(fact_240_inj__diff__right,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( ab_group_add @ A )
% 0.20/0.43 => ! [A2: A] :
% 0.20/0.43 ( inj_on @ A @ A
% 0.20/0.43 @ ^ [B4: A] : ( minus_minus @ A @ B4 @ A2 )
% 0.20/0.43 @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % inj_diff_right
% 0.20/0.43 thf(fact_241_translation__subtract__diff,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( ab_group_add @ A )
% 0.20/0.43 => ! [A2: A,S3: set @ A,T: set @ A] :
% 0.20/0.43 ( ( image @ A @ A
% 0.20/0.43 @ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
% 0.20/0.43 @ ( minus_minus @ ( set @ A ) @ S3 @ T ) )
% 0.20/0.43 = ( minus_minus @ ( set @ A )
% 0.20/0.43 @ ( image @ A @ A
% 0.20/0.43 @ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
% 0.20/0.43 @ S3 )
% 0.20/0.43 @ ( image @ A @ A
% 0.20/0.43 @ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
% 0.20/0.43 @ T ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % translation_subtract_diff
% 0.20/0.43 thf(fact_242_Diff__insert__absorb,axiom,
% 0.20/0.43 ! [A: $tType,X2: A,A3: set @ A] :
% 0.20/0.43 ( ~ ( member @ A @ X2 @ A3 )
% 0.20/0.43 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.20/0.43 = A3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % Diff_insert_absorb
% 0.20/0.43 thf(fact_243_Diff__insert2,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A,A2: A,B2: set @ A] :
% 0.20/0.43 ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B2 ) )
% 0.20/0.43 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ).
% 0.20/0.43
% 0.20/0.43 % Diff_insert2
% 0.20/0.43 thf(fact_244_insert__Diff,axiom,
% 0.20/0.43 ! [A: $tType,A2: A,A3: set @ A] :
% 0.20/0.43 ( ( member @ A @ A2 @ A3 )
% 0.20/0.43 => ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.20/0.43 = A3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % insert_Diff
% 0.20/0.43 thf(fact_245_Diff__insert,axiom,
% 0.20/0.43 ! [A: $tType,A3: set @ A,A2: A,B2: set @ A] :
% 0.20/0.43 ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B2 ) )
% 0.20/0.43 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % Diff_insert
% 0.20/0.43 thf(fact_246_in__image__insert__iff,axiom,
% 0.20/0.43 ! [A: $tType,B2: set @ ( set @ A ),X2: A,A3: set @ A] :
% 0.20/0.43 ( ! [C5: set @ A] :
% 0.20/0.43 ( ( member @ ( set @ A ) @ C5 @ B2 )
% 0.20/0.43 => ~ ( member @ A @ X2 @ C5 ) )
% 0.20/0.43 => ( ( member @ ( set @ A ) @ A3 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X2 ) @ B2 ) )
% 0.20/0.43 = ( ( member @ A @ X2 @ A3 )
% 0.20/0.43 & ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % in_image_insert_iff
% 0.20/0.43 thf(fact_247_DiffE,axiom,
% 0.20/0.43 ! [A: $tType,C3: A,A3: set @ A,B2: set @ A] :
% 0.20/0.43 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
% 0.20/0.43 => ~ ( ( member @ A @ C3 @ A3 )
% 0.20/0.43 => ( member @ A @ C3 @ B2 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % DiffE
% 0.20/0.43 thf(fact_248_DiffD1,axiom,
% 0.20/0.43 ! [A: $tType,C3: A,A3: set @ A,B2: set @ A] :
% 0.20/0.43 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
% 0.20/0.43 => ( member @ A @ C3 @ A3 ) ) ).
% 0.20/0.43
% 0.20/0.43 % DiffD1
% 0.20/0.43 thf(fact_249_DiffD2,axiom,
% 0.20/0.43 ! [A: $tType,C3: A,A3: set @ A,B2: set @ A] :
% 0.20/0.43 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
% 0.20/0.43 => ~ ( member @ A @ C3 @ B2 ) ) ).
% 0.20/0.43
% 0.20/0.43 % DiffD2
% 0.20/0.43 thf(fact_250_set__diff__eq,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( minus_minus @ ( set @ A ) )
% 0.20/0.43 = ( ^ [A5: set @ A,B6: set @ A] :
% 0.20/0.43 ( collect @ A
% 0.20/0.43 @ ^ [X4: A] :
% 0.20/0.43 ( ( member @ A @ X4 @ A5 )
% 0.20/0.43 & ~ ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % set_diff_eq
% 0.20/0.43 thf(fact_251_insert__Diff__if,axiom,
% 0.20/0.43 ! [A: $tType,X2: A,B2: set @ A,A3: set @ A] :
% 0.20/0.43 ( ( ( member @ A @ X2 @ B2 )
% 0.20/0.43 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B2 )
% 0.20/0.43 = ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) )
% 0.20/0.43 & ( ~ ( member @ A @ X2 @ B2 )
% 0.20/0.43 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B2 )
% 0.20/0.43 = ( insert @ A @ X2 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % insert_Diff_if
% 0.20/0.43 thf(fact_252_image__set__diff,axiom,
% 0.20/0.43 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,B2: set @ A] :
% 0.20/0.43 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.20/0.43 => ( ( image @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
% 0.20/0.43 = ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F3 @ A3 ) @ ( image @ A @ B @ F3 @ B2 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % image_set_diff
% 0.20/0.43 thf(fact_253_top__conj_I1_J,axiom,
% 0.20/0.43 ! [A: $tType,X2: A,P: $o] :
% 0.20/0.43 ( ( ( top_top @ ( A > $o ) @ X2 )
% 0.20/0.43 & P )
% 0.20/0.43 = P ) ).
% 0.20/0.43
% 0.20/0.43 % top_conj(1)
% 0.20/0.43 thf(fact_254_remove__def,axiom,
% 0.20/0.43 ! [A: $tType] :
% 0.20/0.43 ( ( remove @ A )
% 0.20/0.43 = ( ^ [X4: A,A5: set @ A] : ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % remove_def
% 0.20/0.43 thf(fact_255_fun__upd__image,axiom,
% 0.20/0.43 ! [A: $tType,B: $tType,X2: B,A3: set @ B,F3: B > A,Y4: A] :
% 0.20/0.43 ( ( ( member @ B @ X2 @ A3 )
% 0.20/0.43 => ( ( image @ B @ A @ ( fun_upd @ B @ A @ F3 @ X2 @ Y4 ) @ A3 )
% 0.20/0.43 = ( insert @ A @ Y4 @ ( image @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
% 0.20/0.43 & ( ~ ( member @ B @ X2 @ A3 )
% 0.20/0.43 => ( ( image @ B @ A @ ( fun_upd @ B @ A @ F3 @ X2 @ Y4 ) @ A3 )
% 0.20/0.43 = ( image @ B @ A @ F3 @ A3 ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 % fun_upd_image
% 0.20/0.43
% 0.20/0.43 % Type constructors (9)
% 0.20/0.43 thf(tcon_fun___Orderings_Otop,axiom,
% 0.20/0.43 ! [A10: $tType,A11: $tType] :
% 0.20/0.43 ( ( top @ A11 )
% 0.20/0.43 => ( top @ ( A10 > A11 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 thf(tcon_fun___Orderings_Oord,axiom,
% 0.20/0.43 ! [A10: $tType,A11: $tType] :
% 0.20/0.43 ( ( ord @ A11 )
% 0.20/0.43 => ( ord @ ( A10 > A11 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 thf(tcon_fun___Orderings_Obot,axiom,
% 0.20/0.43 ! [A10: $tType,A11: $tType] :
% 0.20/0.43 ( ( bot @ A11 )
% 0.20/0.43 => ( bot @ ( A10 > A11 ) ) ) ).
% 0.20/0.43
% 0.20/0.43 thf(tcon_Set_Oset___Orderings_Otop_1,axiom,
% 0.20/0.43 ! [A10: $tType] : ( top @ ( set @ A10 ) ) ).
% 0.20/0.43
% 0.20/0.43 thf(tcon_Set_Oset___Orderings_Oord_2,axiom,
% 0.20/0.43 ! [A10: $tType] : ( ord @ ( set @ A10 ) ) ).
% 0.20/0.43
% 0.20/0.43 thf(tcon_Set_Oset___Orderings_Obot_3,axiom,
% 0.20/0.43 ! [A10: $tType] : ( bot @ ( set @ A10 ) ) ).
% 0.20/0.43
% 0.20/0.43 thf(tcon_HOL_Obool___Orderings_Otop_4,axiom,
% 0.20/0.43 top @ $o ).
% 0.20/0.43
% 0.20/0.43 thf(tcon_HOL_Obool___Orderings_Oord_5,axiom,
% 0.20/0.43 ord @ $o ).
% 0.20/0.43
% 0.20/0.43 thf(tcon_HOL_Obool___Orderings_Obot_6,axiom,
% 0.20/0.43 bot @ $o ).
% 0.20/0.43
% 0.20/0.43 % Helper facts (3)
% 0.20/0.43 thf(help_If_3_1_T,axiom,
% 0.20/0.43 ! [P: $o] :
% 0.20/0.43 ( ( P = $true )
% 0.20/0.43 | ( P = $false ) ) ).
% 0.20/0.43
% 0.20/0.43 thf(help_If_2_1_T,axiom,
% 0.20/0.43 ! [A: $tType,X2: A,Y4: A] :
% 0.20/0.43 ( ( if @ A @ $false @ X2 @ Y4 )
% 0.20/0.43 = Y4 ) ).
% 0.20/0.43
% 0.20/0.43 thf(help_If_1_1_T,axiom,
% 0.20/0.43 ! [A: $tType,X2: A,Y4: A] :
% 0.20/0.43 ( ( if @ A @ $true @ X2 @ Y4 )
% 0.20/0.43 = X2 ) ).
% 0.20/0.43
% 0.20/0.43 % Conjectures (2)
% 0.20/0.43 thf(conj_0,hypothesis,
% 0.20/0.43 ! [X5: a] :
% 0.20/0.43 ( ( p @ X5 )
% 0.20/0.43 => ? [X_12: b] : ( q @ X5 @ X_12 ) ) ).
% 0.20/0.43
% 0.20/0.43 thf(conj_1,conjecture,
% 0.20/0.43 ! [X: a] :
% 0.20/0.43 ( ~ ( p @ X )
% 0.20/0.43 | ( q @ X @ ( hilbert_Eps @ b @ ( q @ X ) ) ) ) ).
% 0.20/0.43
% 0.20/0.43 %------------------------------------------------------------------------------
% 0.20/0.43 ------------------------
% 3.45/1.78 ---- Embedded in THF ---
% 3.45/1.79 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.45/1.79 %%% Generated on Tue Feb 24 08:27:59 EST 2026
% 3.45/1.79 %%% using '$$dhol' embedding, version 1.3.0.
% 3.45/1.79 %%% Logic specification used:
% 3.45/1.79 %%% thf(spec, logic, $$dhol).
% 3.45/1.79
% 3.45/1.79 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.ttMsP055XI/DTF2DTF17675.p
% 3.45/1.79 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 3.45/1.79 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.45/1.79 thf(ty_tf_b, type, b: $tType).
% 3.45/1.79 thf(ty_tf_a, type, a: $tType).
% 3.45/1.79 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 3.45/1.79 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 3.45/1.79 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 3.45/1.79 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 3.45/1.79 thf(sy_cl_Groups_Oab__group__add, type, ab_group_add: (!> [A:$tType]: ($o))).
% 3.45/1.79 thf(sy_c_BNF__Def_Opick__middlep, type, bNF_pick_middlep: (!> [B:$tType,A:$tType,C:$tType]: (((B > (A > $o)) > ((A > (C > $o)) > (B > (C > A))))))).
% 3.45/1.79 thf(sy_c_BNF__Greatest__Fixpoint_OfromCard, type, bNF_Gr932048252omCard: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ ((product_prod @ B) @ B)) > (B > A)))))).
% 3.45/1.79 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2p, type, bNF_Greatest_image2p: (!> [C:$tType,A:$tType,D:$tType,B:$tType]: (((C > A) > ((D > B) > ((C > (D > $o)) > (A > (B > $o)))))))).
% 3.45/1.79 thf(sy_c_BNF__Greatest__Fixpoint_OtoCard, type, bNF_Greatest_toCard: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ ((product_prod @ B) @ B)) > (A > B)))))).
% 3.45/1.79 thf(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred, type, bNF_Gr1978856366d_pred: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ ((product_prod @ B) @ B)) > ((A > B) > $o)))))).
% 3.45/1.79 thf(sy_c_BNF__Greatest__Fixpoint_Ouniv, type, bNF_Greatest_univ: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > A))))).
% 3.45/1.79 thf(sy_c_Fun_Obij__betw, type, bij_betw: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > ((set @ B) > $o)))))).
% 3.45/1.79 thf(sy_c_Fun_Ofun__upd, type, fun_upd: (!> [A:$tType,B:$tType]: (((A > B) > (A > (B > (A > B))))))).
% 3.45/1.79 thf(sy_c_Fun_Oinj__on, type, inj_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 3.45/1.79 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 3.45/1.79 thf(sy_c_Hilbert__Choice_OEps, type, hilbert_Eps: (!> [A:$tType]: (((A > $o) > A)))).
% 3.45/1.79 thf(sy_c_Hilbert__Choice_Obijection, type, hilbert_bijection: (!> [A:$tType]: (((A > A) > $o)))).
% 3.45/1.79 thf(sy_c_Hilbert__Choice_Oinv__into, type, hilbert_inv_into: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (B > A)))))).
% 3.45/1.79 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 3.45/1.79 thf(sy_c_Lattices__Big_Oord__class_Oarg__max, type, lattices_ord_arg_max: (!> [B:$tType,A:$tType]: (((B > A) > ((B > $o) > B))))).
% 3.45/1.79 thf(sy_c_Lattices__Big_Oord__class_Oarg__max__on, type, lattic1534546155max_on: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > B))))).
% 3.45/1.79 thf(sy_c_Lattices__Big_Oord__class_Ois__arg__max, type, lattic2130478198rg_max: (!> [B:$tType,A:$tType]: (((B > A) > ((B > $o) > (B > $o)))))).
% 3.45/1.79 thf(sy_c_Lifting_Orel__pred__comp, type, rel_pred_comp: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((B > $o) > (A > $o)))))).
% 3.45/1.79 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 3.45/1.79 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 3.45/1.79 thf(sy_c_Product__Type_Obool_Ocase__bool, type, product_case_bool: (!> [A:$tType]: ((A > (A > ($o > A)))))).
% 3.45/1.79 thf(sy_c_Quotient_OQuotient3, type, quotient3: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > ((B > A) > $o)))))).
% 3.45/1.79 thf(sy_c_Quotient_Oquot__type, type, quot_type: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > (((set @ A) > B) > ((B > (set @ A)) > $o)))))).
% 3.45/1.79 thf(sy_c_Quotient_Oquot__type_Oabs, type, quot_abs: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > (((set @ A) > B) > (A > B)))))).
% 3.45/1.79 thf(sy_c_Quotient_Oquot__type_Orep, type, quot_rep: (!> [B:$tType,A:$tType]: (((B > (set @ A)) > (B > A))))).
% 3.45/1.79 thf(sy_c_Relation_Oinv__imagep, type, inv_imagep: (!> [B:$tType,A:$tType]: (((B > (B > $o)) > ((A > B) > (A > (A > $o))))))).
% 3.45/1.79 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.45/1.79 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 3.45/1.79 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 3.45/1.79 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 3.45/1.79 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 3.45/1.79 thf(sy_c_Set_Oremove, type, remove: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 3.45/1.79 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 3.45/1.79 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.45/1.79 thf(sy_v_P, type, p: (a > $o)).
% 3.45/1.79 thf(sy_v_Q, type, q: (a > (b > $o))).
% 3.45/1.79 thf(fact_0_some__equality, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((P @ A2) => ((! [X:A]: (((P @ X) => (X = A2)))) => (((hilbert_Eps @ A) @ P) = A2)))))).
% 3.45/1.79 thf(fact_1_some__eq__trivial, axiom, (! [A:$tType,X2:A]: ((((hilbert_Eps @ A) @ (^ [Y:A]: ((Y = X2)))) = X2)))).
% 3.45/1.79 thf(fact_2_some__sym__eq__trivial, axiom, (! [A:$tType,X2:A]: ((((hilbert_Eps @ A) @ ((^ [Y2:A,Z:A]: ((Y2 = Z))) @ X2)) = X2)))).
% 3.45/1.79 thf(fact_3_verit__sko__ex_H, axiom, (! [A:$tType,P:(A > $o),A3:$o]: ((((P @ ((hilbert_Eps @ A) @ P)) = A3) => ((? [X3:A]: ((P @ X3))) = A3))))).
% 3.45/1.79 thf(fact_4_verit__sko__forall, axiom, (! [A:$tType]: (((^ [P2:(A > $o)]: ((! [X3:A]: ((P2 @ X3))))) = (^ [P3:(A > $o)]: ((P3 @ ((hilbert_Eps @ A) @ (^ [X4:A]: ((~ ((P3 @ X4))))))))))))).
% 3.45/1.79 thf(fact_5_someI2, axiom, (! [A:$tType,P:(A > $o),A2:A,Q:(A > $o)]: (((P @ A2) => ((! [X:A]: (((P @ X) => (Q @ X)))) => (Q @ ((hilbert_Eps @ A) @ P))))))).
% 3.45/1.79 thf(fact_6_verit__sko__forall_H, axiom, (! [A:$tType,P:(A > $o),A3:$o]: ((((P @ ((hilbert_Eps @ A) @ (^ [X4:A]: ((~ ((P @ X4))))))) = A3) => ((! [X3:A]: ((P @ X3))) = A3))))).
% 3.45/1.79 thf(fact_7_verit__sko__forall_H_H, axiom, (! [A:$tType,B2:A,A3:A,P:(A > $o)]: (((B2 = A3) => ((((hilbert_Eps @ A) @ P) = A3) = (((hilbert_Eps @ A) @ P) = B2)))))).
% 3.45/1.79 thf(fact_8_someI__ex, axiom, (! [A:$tType,P:(A > $o)]: (((? [X_1:A]: ((P @ X_1))) => (P @ ((hilbert_Eps @ A) @ P)))))).
% 3.45/1.79 thf(fact_9_someI2__ex, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((? [X_1:A]: ((P @ X_1))) => ((! [X:A]: (((P @ X) => (Q @ X)))) => (Q @ ((hilbert_Eps @ A) @ P))))))).
% 3.45/1.79 thf(fact_10_someI2__bex, axiom, (! [A:$tType,A3:(set @ A),P:(A > $o),Q:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ A3) & (P @ X5)))) => ((! [X:A]: ((((((member @ A) @ X) @ A3) & (P @ X)) => (Q @ X)))) => (Q @ ((hilbert_Eps @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ A3) & (P @ X4))))))))))).
% 3.45/1.79 thf(fact_11_some__eq__ex, axiom, (! [A:$tType,P:(A > $o)]: (((P @ ((hilbert_Eps @ A) @ P)) = (? [X3:A]: ((P @ X3))))))).
% 3.45/1.79 thf(fact_12_some__eq__imp, axiom, (! [A:$tType,P:(A > $o),A2:A,B3:A]: (((((hilbert_Eps @ A) @ P) = A2) => ((P @ B3) => (P @ A2)))))).
% 3.45/1.79 thf(fact_13_tfl__some, axiom, (! [A:$tType,P4:(A > $o),X5:A]: (((P4 @ X5) => (P4 @ ((hilbert_Eps @ A) @ P4)))))).
% 3.45/1.79 thf(fact_14_Eps__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P @ X) = (Q @ X)))) => (((hilbert_Eps @ A) @ P) = ((hilbert_Eps @ A) @ Q)))))).
% 3.45/1.79 thf(fact_15_someI, axiom, (! [A:$tType,P:(A > $o),X2:A]: (((P @ X2) => (P @ ((hilbert_Eps @ A) @ P)))))).
% 3.45/1.79 thf(fact_16_verit__sko__forall__indirect2, axiom, (! [A:$tType,X2:A,P:(A > $o),P5:(A > $o)]: (((X2 = ((hilbert_Eps @ A) @ (^ [X4:A]: ((~ ((P @ X4))))))) => ((! [X:A]: (((P @ X) = (P5 @ X)))) => ((! [X3:A]: ((P5 @ X3))) = (P @ X2))))))).
% 3.45/1.79 thf(fact_17_verit__sko__forall__indirect, axiom, (! [A:$tType,X2:A,P:(A > $o)]: (((X2 = ((hilbert_Eps @ A) @ (^ [X4:A]: ((~ ((P @ X4))))))) => ((! [X3:A]: ((P @ X3))) = (P @ X2)))))).
% 3.45/1.79 thf(fact_18_some1__equality, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((? [X5:A]: (((P @ X5) & (! [Y3:A]: (((P @ Y3) => (Y3 = X5))))))) => ((P @ A2) => (((hilbert_Eps @ A) @ P) = A2)))))).
% 3.45/1.79 thf(fact_19_verit__sko__ex__indirect2, axiom, (! [A:$tType,X2:A,P:(A > $o),P5:(A > $o)]: (((X2 = ((hilbert_Eps @ A) @ P)) => ((! [X:A]: (((P @ X) = (P5 @ X)))) => ((? [X3:A]: ((P5 @ X3))) = (P @ X2))))))).
% 3.45/1.79 thf(fact_20_verit__sko__ex__indirect, axiom, (! [A:$tType,X2:A,P:(A > $o)]: (((X2 = ((hilbert_Eps @ A) @ P)) => ((? [X3:A]: ((P @ X3))) = (P @ X2)))))).
% 3.45/1.79 thf(fact_21_Nitpick_OEps__psimp, axiom, (! [A:$tType,P:(A > $o),X2:A,Y4:A]: (((P @ X2) => ((~ ((P @ Y4))) => ((((hilbert_Eps @ A) @ P) = Y4) => (((hilbert_Eps @ A) @ P) = X2))))))).
% 3.45/1.79 thf(fact_22_univ__def, axiom, (! [A:$tType,B:$tType]: ((((bNF_Greatest_univ @ B) @ A) = (^ [F:(B > A),X6:(set @ B)]: ((F @ ((hilbert_Eps @ B) @ (^ [X4:B]: ((((member @ B) @ X4) @ X6))))))))))).
% 3.45/1.79 thf(fact_23_pick__middlep__def, axiom, (! [C:$tType,A:$tType,B:$tType]: (((((bNF_pick_middlep @ B) @ A) @ C) = (^ [P3:(B > (A > $o)),Q2:(A > (C > $o)),A4:B,C2:C]: (((hilbert_Eps @ A) @ (^ [B4:A]: ((((P3 @ A4) @ B4) & ((Q2 @ B4) @ C2))))))))))).
% 3.45/1.79 thf(fact_24_some__in__eq, axiom, (! [A:$tType,A3:(set @ A)]: (((((member @ A) @ ((hilbert_Eps @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A3))))) @ A3) = (A3 != (bot_bot @ (set @ A))))))).
% 3.45/1.79 thf(fact_25_inv__into__def2, axiom, (! [B:$tType,A:$tType]: ((((hilbert_inv_into @ A) @ B) = (^ [A5:(set @ A),F:(A > B),X4:B]: (((hilbert_Eps @ A) @ (^ [Y:A]: (((((member @ A) @ Y) @ A5) & ((F @ Y) = X4))))))))))).
% 3.45/1.79 thf(fact_26_inv__into__def, axiom, (! [B:$tType,A:$tType]: ((((hilbert_inv_into @ A) @ B) = (^ [A5:(set @ A),F:(A > B),X4:B]: (((hilbert_Eps @ A) @ (^ [Y:A]: (((((member @ A) @ Y) @ A5) & ((F @ Y) = X4))))))))))).
% 3.45/1.79 thf(fact_27_quot__type_Osome__collect, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:((set @ A) > B),Rep:(B > (set @ A)),R2:A]: (((((((quot_type @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R2) @ R2) => ((R @ ((hilbert_Eps @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ ((collect @ A) @ (R @ R2))))))) = (R @ R2))))))).
% 3.45/1.79 thf(fact_28_fromCard__def, axiom, (! [B:$tType,A:$tType]: ((((bNF_Gr932048252omCard @ A) @ B) = (^ [A5:(set @ A),R3:(set @ ((product_prod @ B) @ B)),K:B]: (((hilbert_Eps @ A) @ (^ [B4:A]: (((((member @ A) @ B4) @ A5) & ((((((bNF_Greatest_toCard @ A) @ B) @ A5) @ R3) @ B4) = K))))))))))).
% 3.45/1.79 thf(fact_29_arg__max__def, axiom, (! [A:$tType,B:$tType]: (((ord @ A) => (((lattices_ord_arg_max @ B) @ A) = (^ [F:(B > A),P3:(B > $o)]: (((hilbert_Eps @ B) @ ((((lattic2130478198rg_max @ B) @ A) @ F) @ P3))))))))).
% 3.45/1.79 thf(fact_30_case__bool__if, axiom, (! [A:$tType]: (((product_case_bool @ A) = (^ [X4:A,Y:A,P3:$o]: (((((if @ A) @ P3) @ X4) @ Y))))))).
% 3.45/1.79 thf(fact_31_exE__some, axiom, (! [A:$tType,P:(A > $o),C3:A]: (((? [X_1:A]: ((P @ X_1))) => ((C3 = ((hilbert_Eps @ A) @ P)) => (P @ C3)))))).
% 3.45/1.79 thf(fact_32_quot__type_Orep__inverse, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:((set @ A) > B),Rep:(B > (set @ A)),X2:B]: (((((((quot_type @ A) @ B) @ R) @ Abs) @ Rep) => ((Abs @ (Rep @ X2)) = X2))))).
% 3.45/1.79 thf(fact_33_quot__type_Oabs__inverse, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:((set @ A) > B),Rep:(B > (set @ A)),C3:(set @ A)]: (((((((quot_type @ A) @ B) @ R) @ Abs) @ Rep) => ((? [X5:A]: ((((R @ X5) @ X5) & (C3 = ((collect @ A) @ (R @ X5)))))) => ((Rep @ (Abs @ C3)) = C3)))))).
% 3.45/1.79 thf(fact_34_quot__type_Orep__inject, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:((set @ A) > B),Rep:(B > (set @ A)),X2:B,Y4:B]: (((((((quot_type @ A) @ B) @ R) @ Abs) @ Rep) => (((Rep @ X2) = (Rep @ Y4)) = (X2 = Y4)))))).
% 3.45/1.79 thf(fact_35_quot__type_Orep__prop, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:((set @ A) > B),Rep:(B > (set @ A)),Y4:B]: (((((((quot_type @ A) @ B) @ R) @ Abs) @ Rep) => (? [X:A]: ((((R @ X) @ X) & ((Rep @ Y4) = ((collect @ A) @ (R @ X)))))))))).
% 3.45/1.79 thf(fact_36_arg__max__on__def, axiom, (! [A:$tType,B:$tType]: (((ord @ A) => (((lattic1534546155max_on @ B) @ A) = (^ [F:(B > A),S:(set @ B)]: (((((lattices_ord_arg_max @ B) @ A) @ F) @ (^ [X4:B]: ((((member @ B) @ X4) @ S))))))))))).
% 3.45/1.79 thf(fact_37_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X4:A]: ((~ ((P @ X4))))))))).
% 3.45/1.79 thf(fact_38_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X4:A]: ((~ ((P @ X4))))))))).
% 3.45/1.79 thf(fact_39_all__not__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X4:A]: ((~ ((((member @ A) @ X4) @ A3))))) = (A3 = (bot_bot @ (set @ A))))))).
% 3.45/1.79 thf(fact_40_empty__iff, axiom, (! [A:$tType,C3:A]: ((~ ((((member @ A) @ C3) @ (bot_bot @ (set @ A)))))))).
% 3.45/1.79 thf(fact_41_bot__apply, axiom, (! [C:$tType,D:$tType]: (((bot @ C) => ((bot_bot @ (D > C)) = (^ [X4:D]: ((bot_bot @ C)))))))).
% 3.45/1.79 thf(fact_42_quot__type_Orep__def, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:((set @ A) > B),Rep:(B > (set @ A)),A2:B]: (((((((quot_type @ A) @ B) @ R) @ Abs) @ Rep) => (((((quot_rep @ B) @ A) @ Rep) @ A2) = ((hilbert_Eps @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ (Rep @ A2)))))))))).
% 3.45/1.79 thf(fact_43_toCard__def, axiom, (! [B:$tType,A:$tType]: ((((bNF_Greatest_toCard @ A) @ B) = (^ [A5:(set @ A),R3:(set @ ((product_prod @ B) @ B))]: (((hilbert_Eps @ (A > B)) @ ((((bNF_Gr1978856366d_pred @ A) @ B) @ A5) @ R3)))))))).
% 3.45/1.79 thf(fact_44_bool_Osplit__sel__asm, axiom, (! [A:$tType,P:(A > $o),F1:A,F2:A,Bool:$o]: (((P @ ((((product_case_bool @ A) @ F1) @ F2) @ Bool)) = (~ (((Bool & (~ ((P @ F1)))) | ((~ (Bool)) & (~ ((P @ F2))))))))))).
% 3.45/1.79 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 3.45/1.79 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A3)))) = A3)))).
% 3.45/1.79 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P @ X) = (Q @ X)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 3.45/1.79 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F3:(A > B),G:(A > B)]: (((! [X:A]: (((F3 @ X) = (G @ X)))) => (F3 = G))))).
% 3.45/1.79 thf(fact_49_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 3.45/1.79 thf(fact_50_quot__type_Orep_Ocong, axiom, (! [A:$tType,B:$tType]: ((((quot_rep @ B) @ A) = ((quot_rep @ B) @ A))))).
% 3.45/1.79 thf(fact_51_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X4:A]: ((bot_bot @ B)))))))).
% 3.45/1.79 thf(fact_52_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 3.45/1.79 thf(fact_53_equals0D, axiom, (! [A:$tType,A3:(set @ A),A2:A]: (((A3 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A3))))))).
% 3.45/1.79 thf(fact_54_equals0I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [Y3:A]: ((~ ((((member @ A) @ Y3) @ A3))))) => (A3 = (bot_bot @ (set @ A))))))).
% 3.45/1.79 thf(fact_55_ex__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((? [X4:A]: ((((member @ A) @ X4) @ A3))) = (A3 != (bot_bot @ (set @ A))))))).
% 3.45/1.79 thf(fact_56_empty__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (^ [X4:A]: ($false))))))).
% 3.45/1.79 thf(fact_57_old_Obool_Osimps_I4_J, axiom, (! [A:$tType,F1:A,F2:A]: ((((((product_case_bool @ A) @ F1) @ F2) @ $false) = F2)))).
% 3.45/1.79 thf(fact_58_old_Obool_Osimps_I3_J, axiom, (! [A:$tType,F1:A,F2:A]: ((((((product_case_bool @ A) @ F1) @ F2) @ $true) = F1)))).
% 3.45/1.79 thf(fact_59_bool_Osplit__sel, axiom, (! [A:$tType,P:(A > $o),F1:A,F2:A,Bool:$o]: (((P @ ((((product_case_bool @ A) @ F1) @ F2) @ Bool)) = ((Bool => (P @ F1)) & ((~ (Bool)) => (P @ F2))))))).
% 3.45/1.79 thf(fact_60_bool_Ocase__eq__if, axiom, (! [A:$tType]: (((product_case_bool @ A) = (^ [F12:A,F22:A,Bool2:$o]: (((((if @ A) @ Bool2) @ F12) @ F22))))))).
% 3.45/1.79 thf(fact_61_bool_Ocase__distrib, axiom, (! [A:$tType,B:$tType,H:(A > B),F1:A,F2:A,Bool:$o]: (((H @ ((((product_case_bool @ A) @ F1) @ F2) @ Bool)) = ((((product_case_bool @ B) @ (H @ F1)) @ (H @ F2)) @ Bool))))).
% 3.45/1.79 thf(fact_62_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A5:(set @ A)]: ((A5 = (bot_bot @ (set @ A))))))))).
% 3.45/1.79 thf(fact_63_quot__type_Oabs__def, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:((set @ A) > B),Rep:(B > (set @ A)),X2:A]: (((((((quot_type @ A) @ B) @ R) @ Abs) @ Rep) => ((((((quot_abs @ A) @ B) @ R) @ Abs) @ X2) = (Abs @ ((collect @ A) @ (R @ X2)))))))).
% 3.45/1.79 thf(fact_64_image2p__def, axiom, (! [D:$tType,B:$tType,A:$tType,C:$tType]: ((((((bNF_Greatest_image2p @ C) @ A) @ D) @ B) = (^ [F:(C > A),G2:(D > B),R4:(C > (D > $o)),X4:A,Y:B]: ((? [X7:C,Y5:D]: ((((R4 @ X7) @ Y5) & (((F @ X7) = X4) & ((G2 @ Y5) = Y))))))))))).
% 3.45/1.79 thf(fact_65_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 3.45/1.79 thf(fact_66_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X4:A]: ((((member @ A) @ X4) @ (bot_bot @ (set @ A))))))))).
% 3.45/1.79 thf(fact_67_inv__def, axiom, (! [B:$tType,A:$tType,F3:(B > A)]: ((((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F3) = (^ [Y:A]: (((hilbert_Eps @ B) @ (^ [X4:B]: (((F3 @ X4) = Y)))))))))).
% 3.45/1.79 thf(fact_68_rel__pred__comp__def, axiom, (! [B:$tType,A:$tType]: ((((rel_pred_comp @ A) @ B) = (^ [R4:(A > (B > $o)),P3:(B > $o),X4:A]: ((? [Y:B]: ((((R4 @ X4) @ Y) & (P3 @ Y)))))))))).
% 3.45/1.79 thf(fact_69_is__singletonI_H, axiom, (! [A:$tType,A3:(set @ A)]: (((A3 != (bot_bot @ (set @ A))) => ((! [X:A,Y3:A]: (((((member @ A) @ X) @ A3) => ((((member @ A) @ Y3) @ A3) => (X = Y3))))) => ((is_singleton @ A) @ A3)))))).
% 3.45/1.79 thf(fact_70_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X4:D]: ((top_top @ C)))))))).
% 3.45/1.79 thf(fact_71_UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 3.45/1.79 thf(fact_72_Collect__const, axiom, (! [A:$tType,P:$o]: (((P => (((collect @ A) @ (^ [S2:A]: (P))) = (top_top @ (set @ A)))) & ((~ (P)) => (((collect @ A) @ (^ [S2:A]: (P))) = (bot_bot @ (set @ A)))))))).
% 3.45/1.79 thf(fact_73_inv__identity, axiom, (! [A:$tType]: ((((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ (^ [A4:A]: (A4))) = (^ [A4:A]: (A4)))))).
% 3.45/1.79 thf(fact_74_UNIV__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (^ [X4:A]: ($true))))))).
% 3.45/1.79 thf(fact_75_UNIV__eq__I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X:A]: ((((member @ A) @ X) @ A3))) => ((top_top @ (set @ A)) = A3))))).
% 3.45/1.79 thf(fact_76_UNIV__witness, axiom, (! [A:$tType]: ((? [X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))))).
% 3.45/1.79 thf(fact_77_image2pE, axiom, (! [D:$tType,B:$tType,A:$tType,C:$tType,F3:(A > B),G:(C > D),R:(A > (C > $o)),Fx:B,Gy:D]: (((((((((((bNF_Greatest_image2p @ A) @ B) @ C) @ D) @ F3) @ G) @ R) @ Fx) @ Gy) => (~ ((! [X:A]: (((Fx = (F3 @ X)) => (! [Y3:C]: (((Gy = (G @ Y3)) => (~ (((R @ X) @ Y3))))))))))))))).
% 3.45/1.79 thf(fact_78_image2pI, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,R:(A > (B > $o)),X2:A,Y4:B,F3:(A > C),G:(B > D)]: ((((R @ X2) @ Y4) => (((((((((bNF_Greatest_image2p @ A) @ C) @ B) @ D) @ F3) @ G) @ R) @ (F3 @ X2)) @ (G @ Y4)))))).
% 3.45/1.79 thf(fact_79_quot__type_Oabs_Ocong, axiom, (! [B:$tType,A:$tType]: ((((quot_abs @ A) @ B) = ((quot_abs @ A) @ B))))).
% 3.45/1.79 thf(fact_80_empty__not__UNIV, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) != (top_top @ (set @ A)))))).
% 3.45/1.79 thf(fact_81_inv__equality, axiom, (! [A:$tType,B:$tType,G:(B > A),F3:(A > B)]: (((! [X:A]: (((G @ (F3 @ X)) = X))) => ((! [Y3:B]: (((F3 @ (G @ Y3)) = Y3))) => (((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F3) = G)))))).
% 3.45/1.79 thf(fact_82_iso__tuple__UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 3.45/1.79 thf(fact_83_quot__type_OQuotient, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:((set @ A) > B),Rep:(B > (set @ A))]: (((((((quot_type @ A) @ B) @ R) @ Abs) @ Rep) => (((((quotient3 @ A) @ B) @ R) @ ((((quot_abs @ A) @ B) @ R) @ Abs)) @ (((quot_rep @ B) @ A) @ Rep)))))).
% 3.45/1.79 thf(fact_84_in__inv__imagep, axiom, (! [B:$tType,A:$tType]: ((((inv_imagep @ A) @ B) = (^ [R3:(A > (A > $o)),F:(B > A),X4:B,Y:B]: (((R3 @ (F @ X4)) @ (F @ Y)))))))).
% 3.45/1.79 thf(fact_85_is__singletonI, axiom, (! [A:$tType,X2:A]: (((is_singleton @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))))).
% 3.45/1.79 thf(fact_86_bijection_Oinv__right__eq__iff, axiom, (! [A:$tType,F3:(A > A),B3:A,A2:A]: ((((hilbert_bijection @ A) @ F3) => ((B3 = (((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F3) @ A2)) = ((F3 @ B3) = A2)))))).
% 3.45/1.79 thf(fact_87_bijection_Oinv__left__eq__iff, axiom, (! [A:$tType,F3:(A > A),A2:A,B3:A]: ((((hilbert_bijection @ A) @ F3) => (((((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F3) @ A2) = B3) = ((F3 @ B3) = A2)))))).
% 3.45/1.79 thf(fact_88_bijection_Oeq__inv__iff, axiom, (! [A:$tType,F3:(A > A),A2:A,B3:A]: ((((hilbert_bijection @ A) @ F3) => (((((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F3) @ A2) = (((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F3) @ B3)) = (A2 = B3)))))).
% 3.45/1.79 thf(fact_89_insert__absorb2, axiom, (! [A:$tType,X2:A,A3:(set @ A)]: (((((insert @ A) @ X2) @ (((insert @ A) @ X2) @ A3)) = (((insert @ A) @ X2) @ A3))))).
% 3.45/1.79 thf(fact_90_insert__iff, axiom, (! [A:$tType,A2:A,B3:A,A3:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B3) @ A3)) = ((A2 = B3) | (((member @ A) @ A2) @ A3)))))).
% 3.45/1.79 thf(fact_91_insertCI, axiom, (! [A:$tType,A2:A,B2:(set @ A),B3:A]: ((((~ ((((member @ A) @ A2) @ B2))) => (A2 = B3)) => (((member @ A) @ A2) @ (((insert @ A) @ B3) @ B2)))))).
% 3.45/1.79 thf(fact_92_singletonI, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 3.45/1.79 thf(fact_93_singleton__conv, axiom, (! [A:$tType,A2:A]: ((((collect @ A) @ (^ [X4:A]: ((X4 = A2)))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 3.45/1.79 thf(fact_94_singleton__conv2, axiom, (! [A:$tType,A2:A]: ((((collect @ A) @ ((^ [Y2:A,Z:A]: ((Y2 = Z))) @ A2)) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 3.45/1.79 thf(fact_95_Quotient3__rep__reflp, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),A2:B]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((R @ (Rep @ A2)) @ (Rep @ A2)))))).
% 3.45/1.79 thf(fact_96_Quotient3__rep__abs, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R2:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R2) @ R2) => ((R @ (Rep @ (Abs @ R2))) @ R2)))))).
% 3.45/1.79 thf(fact_97_Quotient3__rel__rep, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),A2:B,B3:B]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ (Rep @ A2)) @ (Rep @ B3)) = (A2 = B3)))))).
% 3.45/1.79 thf(fact_98_Quotient3__rel__abs, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R2:A,S3:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R2) @ S3) => ((Abs @ R2) = (Abs @ S3))))))).
% 3.45/1.79 thf(fact_99_Quotient3__abs__rep, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),A2:B]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((Abs @ (Rep @ A2)) = A2))))).
% 3.45/1.79 thf(fact_100_rep__abs__rsp__left, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),X1:A,X22:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ X1) @ X22) => ((R @ (Rep @ (Abs @ X1))) @ X22)))))).
% 3.45/1.79 thf(fact_101_Quotient3__refl2, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R2:A,S3:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R2) @ S3) => ((R @ S3) @ S3)))))).
% 3.45/1.79 thf(fact_102_Quotient3__refl1, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R2:A,S3:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R2) @ S3) => ((R @ R2) @ R2)))))).
% 3.45/1.79 thf(fact_103_Quotient3__rel, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R2:A,S3:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((((R @ R2) @ R2) & (((R @ S3) @ S3) & ((Abs @ R2) = (Abs @ S3)))) = ((R @ R2) @ S3)))))).
% 3.45/1.79 thf(fact_104_Quotient3__def, axiom, (! [B:$tType,A:$tType]: ((((quotient3 @ A) @ B) = (^ [R4:(A > (A > $o)),Abs2:(A > B),Rep2:(B > A)]: (((! [A4:B]: (((Abs2 @ (Rep2 @ A4)) = A4))) & ((! [A4:B]: (((R4 @ (Rep2 @ A4)) @ (Rep2 @ A4)))) & (! [R3:A,S2:A]: ((((R4 @ R3) @ S2) = (((R4 @ R3) @ R3) & (((R4 @ S2) @ S2) & ((Abs2 @ R3) = (Abs2 @ S2))))))))))))))).
% 3.45/1.79 thf(fact_105_rep__abs__rsp, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),X1:A,X22:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ X1) @ X22) => ((R @ X1) @ (Rep @ (Abs @ X22)))))))).
% 3.45/1.79 thf(fact_106_equals__rsp, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),Xa:A,Xb:A,Ya:A,Yb:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ Xa) @ Xb) => (((R @ Ya) @ Yb) => (((R @ Xa) @ Ya) = ((R @ Xb) @ Yb)))))))).
% 3.45/1.79 thf(fact_107_Quotient3I, axiom, (! [B:$tType,A:$tType,Abs:(B > A),Rep:(A > B),R:(B > (B > $o))]: (((! [A6:A]: (((Abs @ (Rep @ A6)) = A6))) => ((! [A6:A]: (((R @ (Rep @ A6)) @ (Rep @ A6)))) => ((! [R5:B,S4:B]: ((((R @ R5) @ S4) = (((R @ R5) @ R5) & (((R @ S4) @ S4) & ((Abs @ R5) = (Abs @ S4))))))) => (((((quotient3 @ B) @ A) @ R) @ Abs) @ Rep))))))).
% 3.45/1.79 thf(fact_108_cond__prs, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Absf:(A > B),Repf:(B > A),A2:$o,B3:B,C3:B]: (((((((quotient3 @ A) @ B) @ R) @ Absf) @ Repf) => ((A2 => ((Absf @ ((((if @ A) @ A2) @ (Repf @ B3)) @ (Repf @ C3))) = B3)) & ((~ (A2)) => ((Absf @ ((((if @ A) @ A2) @ (Repf @ B3)) @ (Repf @ C3))) = C3))))))).
% 3.45/1.79 thf(fact_109_bijection_Oeq__iff, axiom, (! [A:$tType,F3:(A > A),A2:A,B3:A]: ((((hilbert_bijection @ A) @ F3) => (((F3 @ A2) = (F3 @ B3)) = (A2 = B3)))))).
% 3.45/1.79 thf(fact_110_bijection_OeqI, axiom, (! [A:$tType,F3:(A > A),A2:A,B3:A]: ((((hilbert_bijection @ A) @ F3) => (((F3 @ A2) = (F3 @ B3)) => (A2 = B3)))))).
% 3.45/1.79 thf(fact_111_mk__disjoint__insert, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => (? [B5:(set @ A)]: (((A3 = (((insert @ A) @ A2) @ B5)) & (~ ((((member @ A) @ A2) @ B5)))))))))).
% 3.45/1.79 thf(fact_112_insert__commute, axiom, (! [A:$tType,X2:A,Y4:A,A3:(set @ A)]: (((((insert @ A) @ X2) @ (((insert @ A) @ Y4) @ A3)) = (((insert @ A) @ Y4) @ (((insert @ A) @ X2) @ A3)))))).
% 3.45/1.79 thf(fact_113_insert__Collect, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((insert @ A) @ A2) @ ((collect @ A) @ P)) = ((collect @ A) @ (^ [U:A]: (((U != A2) => (P @ U))))))))).
% 3.45/1.79 thf(fact_114_insert__eq__iff, axiom, (! [A:$tType,A2:A,A3:(set @ A),B3:A,B2:(set @ A)]: (((~ ((((member @ A) @ A2) @ A3))) => ((~ ((((member @ A) @ B3) @ B2))) => (((((insert @ A) @ A2) @ A3) = (((insert @ A) @ B3) @ B2)) = (((A2 = B3) => (A3 = B2)) & ((A2 != B3) => (? [C4:(set @ A)]: (((A3 = (((insert @ A) @ B3) @ C4)) & ((~ ((((member @ A) @ B3) @ C4))) & ((B2 = (((insert @ A) @ A2) @ C4)) & (~ ((((member @ A) @ A2) @ C4)))))))))))))))).
% 3.45/1.79 thf(fact_115_insert__absorb, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => ((((insert @ A) @ A2) @ A3) = A3))))).
% 3.45/1.79 thf(fact_116_insert__ident, axiom, (! [A:$tType,X2:A,A3:(set @ A),B2:(set @ A)]: (((~ ((((member @ A) @ X2) @ A3))) => ((~ ((((member @ A) @ X2) @ B2))) => (((((insert @ A) @ X2) @ A3) = (((insert @ A) @ X2) @ B2)) = (A3 = B2))))))).
% 3.45/1.79 thf(fact_117_insert__compr, axiom, (! [A:$tType]: (((insert @ A) = (^ [A4:A,B6:(set @ A)]: (((collect @ A) @ (^ [X4:A]: (((X4 = A4) | (((member @ A) @ X4) @ B6))))))))))).
% 3.45/1.79 thf(fact_118_Set_Oset__insert, axiom, (! [A:$tType,X2:A,A3:(set @ A)]: (((((member @ A) @ X2) @ A3) => (~ ((! [B5:(set @ A)]: (((A3 = (((insert @ A) @ X2) @ B5)) => (((member @ A) @ X2) @ B5)))))))))).
% 3.45/1.79 thf(fact_119_insertI2, axiom, (! [A:$tType,A2:A,B2:(set @ A),B3:A]: (((((member @ A) @ A2) @ B2) => (((member @ A) @ A2) @ (((insert @ A) @ B3) @ B2)))))).
% 3.45/1.79 thf(fact_120_insertI1, axiom, (! [A:$tType,A2:A,B2:(set @ A)]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ B2))))).
% 3.45/1.79 thf(fact_121_insertE, axiom, (! [A:$tType,A2:A,B3:A,A3:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B3) @ A3)) => ((A2 != B3) => (((member @ A) @ A2) @ A3)))))).
% 3.45/1.79 thf(fact_122_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X4:A]: ((((member @ A) @ X4) @ (top_top @ (set @ A))))))))).
% 3.45/1.79 thf(fact_123_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 3.45/1.79 thf(fact_124_singleton__inject, axiom, (! [A:$tType,A2:A,B3:A]: ((((((insert @ A) @ A2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) => (A2 = B3))))).
% 3.45/1.79 thf(fact_125_insert__not__empty, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((insert @ A) @ A2) @ A3) != (bot_bot @ (set @ A)))))).
% 3.45/1.79 thf(fact_126_doubleton__eq__iff, axiom, (! [A:$tType,A2:A,B3:A,C3:A,D2:A]: ((((((insert @ A) @ A2) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C3) @ (((insert @ A) @ D2) @ (bot_bot @ (set @ A))))) = (((A2 = C3) & (B3 = D2)) | ((A2 = D2) & (B3 = C3))))))).
% 3.45/1.79 thf(fact_127_singleton__iff, axiom, (! [A:$tType,B3:A,A2:A]: (((((member @ A) @ B3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = (B3 = A2))))).
% 3.45/1.79 thf(fact_128_singletonD, axiom, (! [A:$tType,B3:A,A2:A]: (((((member @ A) @ B3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => (B3 = A2))))).
% 3.45/1.79 thf(fact_129_insert__UNIV, axiom, (! [A:$tType,X2:A]: (((((insert @ A) @ X2) @ (top_top @ (set @ A))) = (top_top @ (set @ A)))))).
% 3.45/1.79 thf(fact_130_Collect__conv__if, axiom, (! [A:$tType,P:(A > $o),A2:A]: ((((P @ A2) => (((collect @ A) @ (^ [X4:A]: (((X4 = A2) & (P @ X4))))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A2))) => (((collect @ A) @ (^ [X4:A]: (((X4 = A2) & (P @ X4))))) = (bot_bot @ (set @ A)))))))).
% 3.45/1.79 thf(fact_131_Collect__conv__if2, axiom, (! [A:$tType,P:(A > $o),A2:A]: ((((P @ A2) => (((collect @ A) @ (^ [X4:A]: (((A2 = X4) & (P @ X4))))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A2))) => (((collect @ A) @ (^ [X4:A]: (((A2 = X4) & (P @ X4))))) = (bot_bot @ (set @ A)))))))).
% 3.45/1.79 thf(fact_132_inv__imagep__def, axiom, (! [A:$tType,B:$tType]: ((((inv_imagep @ B) @ A) = (^ [R3:(B > (B > $o)),F:(A > B),X4:A,Y:A]: (((R3 @ (F @ X4)) @ (F @ Y)))))))).
% 3.45/1.79 thf(fact_133_is__singleton__def, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A5:(set @ A)]: ((? [X4:A]: ((A5 = (((insert @ A) @ X4) @ (bot_bot @ (set @ A)))))))))))).
% 3.45/1.79 thf(fact_134_is__singletonE, axiom, (! [A:$tType,A3:(set @ A)]: ((((is_singleton @ A) @ A3) => (~ ((! [X:A]: ((A3 != (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))))).
% 3.45/1.79 thf(fact_135_bijection_Oeq__invI, axiom, (! [A:$tType,F3:(A > A),A2:A,B3:A]: ((((hilbert_bijection @ A) @ F3) => (((((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F3) @ A2) = (((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F3) @ B3)) => (A2 = B3)))))).
% 3.45/1.79 thf(fact_136_bijection_Oinv__left, axiom, (! [A:$tType,F3:(A > A),A2:A]: ((((hilbert_bijection @ A) @ F3) => ((((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F3) @ (F3 @ A2)) = A2))))).
% 3.45/1.79 thf(fact_137_bijection_Oinv__right, axiom, (! [A:$tType,F3:(A > A),A2:A]: ((((hilbert_bijection @ A) @ F3) => ((F3 @ (((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F3) @ A2)) = A2))))).
% 3.45/1.79 thf(fact_138_is__singleton__the__elem, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A5:(set @ A)]: ((A5 = (((insert @ A) @ ((the_elem @ A) @ A5)) @ (bot_bot @ (set @ A)))))))))).
% 3.45/1.79 thf(fact_139_the__elem__eq, axiom, (! [A:$tType,X2:A]: ((((the_elem @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) = X2)))).
% 3.45/1.79 thf(fact_140_range__constant, axiom, (! [B:$tType,A:$tType,X2:A]: ((((((image @ B) @ A) @ (^ [Uu:B]: (X2))) @ (top_top @ (set @ B))) = (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))))).
% 3.45/1.79 thf(fact_141_bijection_Osurj__inv, axiom, (! [A:$tType,F3:(A > A)]: ((((hilbert_bijection @ A) @ F3) => (((((image @ A) @ A) @ ((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F3)) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))).
% 3.45/1.79 thf(fact_142_bijection_Oinj__inv, axiom, (! [A:$tType,F3:(A > A)]: ((((hilbert_bijection @ A) @ F3) => ((((inj_on @ A) @ A) @ ((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F3)) @ (top_top @ (set @ A))))))).
% 3.45/1.79 thf(fact_143_bijection_Obij__inv, axiom, (! [A:$tType,F3:(A > A)]: ((((hilbert_bijection @ A) @ F3) => (((((bij_betw @ A) @ A) @ ((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F3)) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))))))).
% 3.45/1.79 thf(fact_144_image__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F3:(B > A),X2:B,A3:(set @ B)]: (((B3 = (F3 @ X2)) => ((((member @ B) @ X2) @ A3) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F3) @ A3))))))).
% 3.45/1.79 thf(fact_145_image__ident, axiom, (! [A:$tType,Y6:(set @ A)]: ((((((image @ A) @ A) @ (^ [X4:A]: (X4))) @ Y6) = Y6)))).
% 3.45/1.79 thf(fact_146_top1I, axiom, (! [A:$tType,X2:A]: (((top_top @ (A > $o)) @ X2)))).
% 3.45/1.79 thf(fact_147_image__is__empty, axiom, (! [A:$tType,B:$tType,F3:(B > A),A3:(set @ B)]: (((((((image @ B) @ A) @ F3) @ A3) = (bot_bot @ (set @ A))) = (A3 = (bot_bot @ (set @ B))))))).
% 3.45/1.79 thf(fact_148_empty__is__image, axiom, (! [A:$tType,B:$tType,F3:(B > A),A3:(set @ B)]: ((((bot_bot @ (set @ A)) = ((((image @ B) @ A) @ F3) @ A3)) = (A3 = (bot_bot @ (set @ B))))))).
% 3.45/1.79 thf(fact_149_image__empty, axiom, (! [B:$tType,A:$tType,F3:(B > A)]: ((((((image @ B) @ A) @ F3) @ (bot_bot @ (set @ B))) = (bot_bot @ (set @ A)))))).
% 3.45/1.79 thf(fact_150_image__insert, axiom, (! [A:$tType,B:$tType,F3:(B > A),A2:B,B2:(set @ B)]: ((((((image @ B) @ A) @ F3) @ (((insert @ B) @ A2) @ B2)) = (((insert @ A) @ (F3 @ A2)) @ ((((image @ B) @ A) @ F3) @ B2)))))).
% 3.45/1.79 thf(fact_151_insert__image, axiom, (! [B:$tType,A:$tType,X2:A,A3:(set @ A),F3:(A > B)]: (((((member @ A) @ X2) @ A3) => ((((insert @ B) @ (F3 @ X2)) @ ((((image @ A) @ B) @ F3) @ A3)) = ((((image @ A) @ B) @ F3) @ A3)))))).
% 3.45/1.79 thf(fact_152_inv__into__f__f, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A),X2:A]: ((((((inj_on @ A) @ B) @ F3) @ A3) => ((((member @ A) @ X2) @ A3) => ((((((hilbert_inv_into @ A) @ B) @ A3) @ F3) @ (F3 @ X2)) = X2)))))).
% 3.45/1.79 thf(fact_153_inj__imp__bij__betw__inv, axiom, (! [B:$tType,A:$tType,F3:(A > B),M:(set @ A)]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => (((((bij_betw @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F3)) @ ((((image @ A) @ B) @ F3) @ M)) @ M))))).
% 3.45/1.79 thf(fact_154_inj__transfer, axiom, (! [B:$tType,A:$tType,F3:(A > B),P:(A > $o),X2:A]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => ((! [Y3:B]: (((((member @ B) @ Y3) @ ((((image @ A) @ B) @ F3) @ (top_top @ (set @ A)))) => (P @ (((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F3) @ Y3))))) => (P @ X2)))))).
% 3.45/1.79 thf(fact_155_image__inv__f__f, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => (((((image @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F3)) @ ((((image @ A) @ B) @ F3) @ A3)) = A3))))).
% 3.45/1.79 thf(fact_156_inj__imp__surj__inv, axiom, (! [B:$tType,A:$tType,F3:(A > B)]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => (((((image @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F3)) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 3.45/1.79 thf(fact_157_surj__imp__inj__inv, axiom, (! [B:$tType,A:$tType,F3:(B > A)]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((((inj_on @ A) @ B) @ ((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F3)) @ (top_top @ (set @ A))))))).
% 3.45/1.79 thf(fact_158_bijI_H, axiom, (! [A:$tType,B:$tType,F3:(A > B)]: (((! [X:A,Y3:A]: ((((F3 @ X) = (F3 @ Y3)) = (X = Y3)))) => ((! [Y3:B]: ((? [X5:A]: ((Y3 = (F3 @ X5)))))) => (((((bij_betw @ A) @ B) @ F3) @ (top_top @ (set @ A))) @ (top_top @ (set @ B)))))))).
% 3.45/1.79 thf(fact_159_imageE, axiom, (! [A:$tType,B:$tType,B3:A,F3:(B > A),A3:(set @ B)]: (((((member @ A) @ B3) @ ((((image @ B) @ A) @ F3) @ A3)) => (~ ((! [X:B]: (((B3 = (F3 @ X)) => (~ ((((member @ B) @ X) @ A3)))))))))))).
% 3.45/1.79 thf(fact_160_imageI, axiom, (! [B:$tType,A:$tType,X2:A,A3:(set @ A),F3:(A > B)]: (((((member @ A) @ X2) @ A3) => (((member @ B) @ (F3 @ X2)) @ ((((image @ A) @ B) @ F3) @ A3)))))).
% 3.45/1.79 thf(fact_161_image__iff, axiom, (! [A:$tType,B:$tType,Z2:A,F3:(B > A),A3:(set @ B)]: (((((member @ A) @ Z2) @ ((((image @ B) @ A) @ F3) @ A3)) = (? [X4:B]: (((((member @ B) @ X4) @ A3) & (Z2 = (F3 @ X4))))))))).
% 3.45/1.79 thf(fact_162_bex__imageD, axiom, (! [A:$tType,B:$tType,F3:(B > A),A3:(set @ B),P:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F3) @ A3)) & (P @ X5)))) => (? [X:B]: (((((member @ B) @ X) @ A3) & (P @ (F3 @ X))))))))).
% 3.45/1.79 thf(fact_163_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N:(set @ A),F3:(A > B),G:(A > B)]: (((M = N) => ((! [X:A]: (((((member @ A) @ X) @ N) => ((F3 @ X) = (G @ X))))) => (((((image @ A) @ B) @ F3) @ M) = ((((image @ A) @ B) @ G) @ N))))))).
% 3.45/1.79 thf(fact_164_ball__imageD, axiom, (! [A:$tType,B:$tType,F3:(B > A),A3:(set @ B),P:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ ((((image @ B) @ A) @ F3) @ A3)) => (P @ X)))) => (! [X5:B]: (((((member @ B) @ X5) @ A3) => (P @ (F3 @ X5))))))))).
% 3.45/1.79 thf(fact_165_image__image, axiom, (! [A:$tType,B:$tType,C:$tType,F3:(B > A),G:(C > B),A3:(set @ C)]: ((((((image @ B) @ A) @ F3) @ ((((image @ C) @ B) @ G) @ A3)) = ((((image @ C) @ A) @ (^ [X4:C]: ((F3 @ (G @ X4))))) @ A3))))).
% 3.45/1.79 thf(fact_166_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X2:A,A3:(set @ A),B3:B,F3:(A > B)]: (((((member @ A) @ X2) @ A3) => ((B3 = (F3 @ X2)) => (((member @ B) @ B3) @ ((((image @ A) @ B) @ F3) @ A3))))))).
% 3.45/1.79 thf(fact_167_Compr__image__eq, axiom, (! [A:$tType,B:$tType,F3:(B > A),A3:(set @ B),P:(A > $o)]: ((((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ ((((image @ B) @ A) @ F3) @ A3)) & (P @ X4))))) = ((((image @ B) @ A) @ F3) @ ((collect @ B) @ (^ [X4:B]: (((((member @ B) @ X4) @ A3) & (P @ (F3 @ X4))))))))))).
% 3.45/1.79 thf(fact_168_the__elem__image__unique, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F3:(A > B),X2:A]: (((A3 != (bot_bot @ (set @ A))) => ((! [Y3:A]: (((((member @ A) @ Y3) @ A3) => ((F3 @ Y3) = (F3 @ X2))))) => (((the_elem @ B) @ ((((image @ A) @ B) @ F3) @ A3)) = (F3 @ X2))))))).
% 3.45/1.79 thf(fact_169_rangeI, axiom, (! [A:$tType,B:$tType,F3:(B > A),X2:B]: ((((member @ A) @ (F3 @ X2)) @ ((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))))))).
% 3.45/1.79 thf(fact_170_range__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F3:(B > A),X2:B]: (((B3 = (F3 @ X2)) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F3) @ (top_top @ (set @ B)))))))).
% 3.45/1.79 thf(fact_171_bij__image__Collect__eq, axiom, (! [A:$tType,B:$tType,F3:(A > B),P:(A > $o)]: (((((((bij_betw @ A) @ B) @ F3) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((((image @ A) @ B) @ F3) @ ((collect @ A) @ P)) = ((collect @ B) @ (^ [Y:B]: ((P @ (((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F3) @ Y)))))))))).
% 3.45/1.79 thf(fact_172_bij__betw__inv__into, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A),B2:(set @ B)]: (((((((bij_betw @ A) @ B) @ F3) @ A3) @ B2) => (((((bij_betw @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ A3) @ F3)) @ B2) @ A3))))).
% 3.45/1.79 thf(fact_173_inv__into__inv__into__eq, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A),A7:(set @ B),A2:A]: (((((((bij_betw @ A) @ B) @ F3) @ A3) @ A7) => ((((member @ A) @ A2) @ A3) => ((((((hilbert_inv_into @ B) @ A) @ A7) @ ((((hilbert_inv_into @ A) @ B) @ A3) @ F3)) @ A2) = (F3 @ A2))))))).
% 3.45/1.79 thf(fact_174_bij__betw__inv__into__left, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A),A7:(set @ B),A2:A]: (((((((bij_betw @ A) @ B) @ F3) @ A3) @ A7) => ((((member @ A) @ A2) @ A3) => ((((((hilbert_inv_into @ A) @ B) @ A3) @ F3) @ (F3 @ A2)) = A2)))))).
% 3.45/1.79 thf(fact_175_bij__betw__inv__into__right, axiom, (! [A:$tType,B:$tType,F3:(A > B),A3:(set @ A),A7:(set @ B),A8:B]: (((((((bij_betw @ A) @ B) @ F3) @ A3) @ A7) => ((((member @ B) @ A8) @ A7) => ((F3 @ (((((hilbert_inv_into @ A) @ B) @ A3) @ F3) @ A8)) = A8)))))).
% 3.45/1.79 thf(fact_176_inv__into__f__eq, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A),X2:A,Y4:B]: ((((((inj_on @ A) @ B) @ F3) @ A3) => ((((member @ A) @ X2) @ A3) => (((F3 @ X2) = Y4) => ((((((hilbert_inv_into @ A) @ B) @ A3) @ F3) @ Y4) = X2))))))).
% 3.45/1.79 thf(fact_177_f__inv__into__f, axiom, (! [B:$tType,A:$tType,Y4:A,F3:(B > A),A3:(set @ B)]: (((((member @ A) @ Y4) @ ((((image @ B) @ A) @ F3) @ A3)) => ((F3 @ (((((hilbert_inv_into @ B) @ A) @ A3) @ F3) @ Y4)) = Y4))))).
% 3.45/1.79 thf(fact_178_inv__into__into, axiom, (! [A:$tType,B:$tType,X2:A,F3:(B > A),A3:(set @ B)]: (((((member @ A) @ X2) @ ((((image @ B) @ A) @ F3) @ A3)) => (((member @ B) @ (((((hilbert_inv_into @ B) @ A) @ A3) @ F3) @ X2)) @ A3))))).
% 3.45/1.79 thf(fact_179_inv__into__injective, axiom, (! [A:$tType,B:$tType,A3:(set @ A),F3:(A > B),X2:B,Y4:B]: ((((((((hilbert_inv_into @ A) @ B) @ A3) @ F3) @ X2) = (((((hilbert_inv_into @ A) @ B) @ A3) @ F3) @ Y4)) => ((((member @ B) @ X2) @ ((((image @ A) @ B) @ F3) @ A3)) => ((((member @ B) @ Y4) @ ((((image @ A) @ B) @ F3) @ A3)) => (X2 = Y4))))))).
% 3.45/1.79 thf(fact_180_setcompr__eq__image, axiom, (! [A:$tType,B:$tType,F3:(B > A),P:(B > $o)]: ((((collect @ A) @ (^ [Uu:A]: ((? [X4:B]: (((Uu = (F3 @ X4)) & (P @ X4))))))) = ((((image @ B) @ A) @ F3) @ ((collect @ B) @ P)))))).
% 3.45/1.79 thf(fact_181_Setcompr__eq__image, axiom, (! [A:$tType,B:$tType,F3:(B > A),A3:(set @ B)]: ((((collect @ A) @ (^ [Uu:A]: ((? [X4:B]: (((Uu = (F3 @ X4)) & (((member @ B) @ X4) @ A3))))))) = ((((image @ B) @ A) @ F3) @ A3))))).
% 3.45/1.79 thf(fact_182_rangeE, axiom, (! [A:$tType,B:$tType,B3:A,F3:(B > A)]: (((((member @ A) @ B3) @ ((((image @ B) @ A) @ F3) @ (top_top @ (set @ B)))) => (~ ((! [X:B]: ((B3 != (F3 @ X)))))))))).
% 3.45/1.79 thf(fact_183_range__composition, axiom, (! [A:$tType,C:$tType,B:$tType,F3:(C > A),G:(B > C)]: ((((((image @ B) @ A) @ (^ [X4:B]: ((F3 @ (G @ X4))))) @ (top_top @ (set @ B))) = ((((image @ C) @ A) @ F3) @ ((((image @ B) @ C) @ G) @ (top_top @ (set @ B)))))))).
% 3.45/1.79 thf(fact_184_bij__imp__bij__inv, axiom, (! [B:$tType,A:$tType,F3:(A > B)]: (((((((bij_betw @ A) @ B) @ F3) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((((bij_betw @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F3)) @ (top_top @ (set @ B))) @ (top_top @ (set @ A))))))).
% 3.45/1.79 thf(fact_185_bij__inv__eq__iff, axiom, (! [A:$tType,B:$tType,P6:(A > B),X2:A,Y4:B]: (((((((bij_betw @ A) @ B) @ P6) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => ((X2 = (((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ P6) @ Y4)) = ((P6 @ X2) = Y4)))))).
% 3.45/1.79 thf(fact_186_inv__inv__eq, axiom, (! [B:$tType,A:$tType,F3:(A > B)]: (((((((bij_betw @ A) @ B) @ F3) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F3)) = F3))))).
% 3.45/1.79 thf(fact_187_inj__imp__inv__eq, axiom, (! [A:$tType,B:$tType,F3:(A > B),G:(B > A)]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => ((! [X:B]: (((F3 @ (G @ X)) = X))) => (((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F3) = G)))))).
% 3.45/1.79 thf(fact_188_inv__f__eq, axiom, (! [B:$tType,A:$tType,F3:(A > B),X2:A,Y4:B]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => (((F3 @ X2) = Y4) => ((((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F3) @ Y4) = X2)))))).
% 3.45/1.79 thf(fact_189_inv__f__f, axiom, (! [B:$tType,A:$tType,F3:(A > B),X2:A]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => ((((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F3) @ (F3 @ X2)) = X2))))).
% 3.45/1.79 thf(fact_190_bijection_Ointro, axiom, (! [A:$tType,F3:(A > A)]: (((((((bij_betw @ A) @ A) @ F3) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))) => ((hilbert_bijection @ A) @ F3))))).
% 3.45/1.79 thf(fact_191_bijection__def, axiom, (! [A:$tType]: (((hilbert_bijection @ A) = (^ [F:(A > A)]: ((((((bij_betw @ A) @ A) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))))))))).
% 3.45/1.79 thf(fact_192_bijection_Obij, axiom, (! [A:$tType,F3:(A > A)]: ((((hilbert_bijection @ A) @ F3) => (((((bij_betw @ A) @ A) @ F3) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))))))).
% 3.45/1.79 thf(fact_193_surj__imp__inv__eq, axiom, (! [B:$tType,A:$tType,F3:(B > A),G:(A > B)]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((! [X:B]: (((G @ (F3 @ X)) = X))) => (((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F3) = G)))))).
% 3.45/1.79 thf(fact_194_image__f__inv__f, axiom, (! [B:$tType,A:$tType,F3:(B > A),A3:(set @ A)]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((((image @ B) @ A) @ F3) @ ((((image @ A) @ B) @ ((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F3)) @ A3)) = A3))))).
% 3.45/1.79 thf(fact_195_surj__iff__all, axiom, (! [B:$tType,A:$tType,F3:(B > A)]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [X4:A]: (((F3 @ (((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F3) @ X4)) = X4))))))).
% 3.45/1.79 thf(fact_196_surj__f__inv__f, axiom, (! [B:$tType,A:$tType,F3:(B > A),Y4:A]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((F3 @ (((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F3) @ Y4)) = Y4))))).
% 3.45/1.79 thf(fact_197_bijection_Oinj, axiom, (! [A:$tType,F3:(A > A)]: ((((hilbert_bijection @ A) @ F3) => ((((inj_on @ A) @ A) @ F3) @ (top_top @ (set @ A))))))).
% 3.45/1.79 thf(fact_198_bijection_Osurj, axiom, (! [A:$tType,F3:(A > A)]: ((((hilbert_bijection @ A) @ F3) => (((((image @ A) @ A) @ F3) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))).
% 3.45/1.79 thf(fact_199_image__constant, axiom, (! [A:$tType,B:$tType,X2:A,A3:(set @ A),C3:B]: (((((member @ A) @ X2) @ A3) => (((((image @ A) @ B) @ (^ [X4:A]: (C3))) @ A3) = (((insert @ B) @ C3) @ (bot_bot @ (set @ B)))))))).
% 3.45/1.79 thf(fact_200_image__constant__conv, axiom, (! [B:$tType,A:$tType,A3:(set @ B),C3:A]: ((((A3 = (bot_bot @ (set @ B))) => (((((image @ B) @ A) @ (^ [X4:B]: (C3))) @ A3) = (bot_bot @ (set @ A)))) & ((A3 != (bot_bot @ (set @ B))) => (((((image @ B) @ A) @ (^ [X4:B]: (C3))) @ A3) = (((insert @ A) @ C3) @ (bot_bot @ (set @ A))))))))).
% 3.45/1.79 thf(fact_201_full__SetCompr__eq, axiom, (! [A:$tType,B:$tType,F3:(B > A)]: ((((collect @ A) @ (^ [U:A]: ((? [X4:B]: ((U = (F3 @ X4))))))) = ((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))))))).
% 3.45/1.79 thf(fact_202_range__eq__singletonD, axiom, (! [B:$tType,A:$tType,F3:(B > A),A2:A,X2:B]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => ((F3 @ X2) = A2))))).
% 3.45/1.79 thf(fact_203_inj__on__empty, axiom, (! [B:$tType,A:$tType,F3:(A > B)]: (((((inj_on @ A) @ B) @ F3) @ (bot_bot @ (set @ A)))))).
% 3.45/1.79 thf(fact_204_bij__def, axiom, (! [A:$tType,B:$tType,F3:(A > B)]: (((((((bij_betw @ A) @ B) @ F3) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) = (((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) & (((((image @ A) @ B) @ F3) @ (top_top @ (set @ A))) = (top_top @ (set @ B)))))))).
% 3.45/1.79 thf(fact_205_bijI, axiom, (! [A:$tType,B:$tType,F3:(A > B)]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => ((((((image @ A) @ B) @ F3) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => (((((bij_betw @ A) @ B) @ F3) @ (top_top @ (set @ A))) @ (top_top @ (set @ B)))))))).
% 3.45/1.79 thf(fact_206_bij__is__inj, axiom, (! [B:$tType,A:$tType,F3:(A > B)]: (((((((bij_betw @ A) @ B) @ F3) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => ((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))))))).
% 3.45/1.79 thf(fact_207_inj__on__id2, axiom, (! [A:$tType,A3:(set @ A)]: (((((inj_on @ A) @ A) @ (^ [X4:A]: (X4))) @ A3)))).
% 3.45/1.79 thf(fact_208_surjD, axiom, (! [A:$tType,B:$tType,F3:(B > A),Y4:A]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X:B]: ((Y4 = (F3 @ X)))))))).
% 3.45/1.79 thf(fact_209_surjE, axiom, (! [A:$tType,B:$tType,F3:(B > A),Y4:A]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X:B]: ((Y4 != (F3 @ X)))))))))).
% 3.45/1.79 thf(fact_210_surjI, axiom, (! [B:$tType,A:$tType,G:(B > A),F3:(A > B)]: (((! [X:A]: (((G @ (F3 @ X)) = X))) => (((((image @ B) @ A) @ G) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 3.45/1.79 thf(fact_211_surj__def, axiom, (! [B:$tType,A:$tType,F3:(B > A)]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y:A]: ((? [X4:B]: ((Y = (F3 @ X4)))))))))).
% 3.45/1.79 thf(fact_212_injD, axiom, (! [B:$tType,A:$tType,F3:(A > B),X2:A,Y4:A]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => (((F3 @ X2) = (F3 @ Y4)) => (X2 = Y4)))))).
% 3.45/1.79 thf(fact_213_injI, axiom, (! [B:$tType,A:$tType,F3:(A > B)]: (((! [X:A,Y3:A]: ((((F3 @ X) = (F3 @ Y3)) => (X = Y3)))) => ((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))))))).
% 3.45/1.79 thf(fact_214_inj__eq, axiom, (! [B:$tType,A:$tType,F3:(A > B),X2:A,Y4:A]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => (((F3 @ X2) = (F3 @ Y4)) = (X2 = Y4)))))).
% 3.45/1.79 thf(fact_215_inj__def, axiom, (! [B:$tType,A:$tType,F3:(A > B)]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) = (! [X4:A,Y:A]: ((((F3 @ X4) = (F3 @ Y)) => (X4 = Y)))))))).
% 3.45/1.79 thf(fact_216_bij__betw__empty2, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A)]: (((((((bij_betw @ A) @ B) @ F3) @ A3) @ (bot_bot @ (set @ B))) => (A3 = (bot_bot @ (set @ A))))))).
% 3.45/1.79 thf(fact_217_bij__betw__empty1, axiom, (! [A:$tType,B:$tType,F3:(A > B),A3:(set @ B)]: (((((((bij_betw @ A) @ B) @ F3) @ (bot_bot @ (set @ A))) @ A3) => (A3 = (bot_bot @ (set @ B))))))).
% 3.45/1.79 thf(fact_218_bij__pointE, axiom, (! [B:$tType,A:$tType,F3:(A > B),Y4:B]: (((((((bij_betw @ A) @ B) @ F3) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (~ ((! [X:A]: (((Y4 = (F3 @ X)) => (~ ((! [X8:A]: (((Y4 = (F3 @ X8)) => (X8 = X))))))))))))))).
% 3.45/1.79 thf(fact_219_inj__singleton, axiom, (! [A:$tType,A3:(set @ A)]: (((((inj_on @ A) @ (set @ A)) @ (^ [X4:A]: ((((insert @ A) @ X4) @ (bot_bot @ (set @ A)))))) @ A3)))).
% 3.45/1.79 thf(fact_220_inj__fun, axiom, (! [B:$tType,C:$tType,A:$tType,F3:(A > B)]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => ((((inj_on @ A) @ (C > B)) @ (^ [X4:A,Y:C]: ((F3 @ X4)))) @ (top_top @ (set @ A))))))).
% 3.45/1.79 thf(fact_221_inj__img__insertE, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A),X2:B,B2:(set @ B)]: ((((((inj_on @ A) @ B) @ F3) @ A3) => ((~ ((((member @ B) @ X2) @ B2))) => (((((insert @ B) @ X2) @ B2) = ((((image @ A) @ B) @ F3) @ A3)) => (~ ((! [X9:A,A9:(set @ A)]: (((~ ((((member @ A) @ X9) @ A9))) => ((A3 = (((insert @ A) @ X9) @ A9)) => ((X2 = (F3 @ X9)) => (B2 != ((((image @ A) @ B) @ F3) @ A9))))))))))))))).
% 3.45/1.79 thf(fact_222_range__ex1__eq, axiom, (! [B:$tType,A:$tType,F3:(A > B),B3:B]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => ((((member @ B) @ B3) @ ((((image @ A) @ B) @ F3) @ (top_top @ (set @ A)))) = (? [X4:A]: (((B3 = (F3 @ X4)) & (! [Y:A]: (((B3 = (F3 @ Y)) => (Y = X4)))))))))))).
% 3.45/1.79 thf(fact_223_inj__image__eq__iff, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A),B2:(set @ A)]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => ((((((image @ A) @ B) @ F3) @ A3) = ((((image @ A) @ B) @ F3) @ B2)) = (A3 = B2)))))).
% 3.45/1.79 thf(fact_224_inj__image__mem__iff, axiom, (! [B:$tType,A:$tType,F3:(A > B),A2:A,A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => ((((member @ B) @ (F3 @ A2)) @ ((((image @ A) @ B) @ F3) @ A3)) = (((member @ A) @ A2) @ A3)))))).
% 3.45/1.79 thf(fact_225_bij__is__surj, axiom, (! [A:$tType,B:$tType,F3:(A > B)]: (((((((bij_betw @ A) @ B) @ F3) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((((image @ A) @ B) @ F3) @ (top_top @ (set @ A))) = (top_top @ (set @ B))))))).
% 3.45/1.79 thf(fact_226_bij__betw__imp__surj, axiom, (! [A:$tType,B:$tType,F3:(A > B),A3:(set @ A)]: (((((((bij_betw @ A) @ B) @ F3) @ A3) @ (top_top @ (set @ B))) => (((((image @ A) @ B) @ F3) @ (top_top @ (set @ A))) = (top_top @ (set @ B))))))).
% 3.45/1.79 thf(fact_227_inj__on__insert, axiom, (! [B:$tType,A:$tType,F3:(A > B),A2:A,A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F3) @ (((insert @ A) @ A2) @ A3)) = (((((inj_on @ A) @ B) @ F3) @ A3) & (~ ((((member @ B) @ (F3 @ A2)) @ ((((image @ A) @ B) @ F3) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))))))).
% 3.45/1.79 thf(fact_228_top__conj_I2_J, axiom, (! [A:$tType,P:$o,X2:A]: (((P & ((top_top @ (A > $o)) @ X2)) = P)))).
% 3.45/1.79 thf(fact_229_DiffI, axiom, (! [A:$tType,C3:A,A3:(set @ A),B2:(set @ A)]: (((((member @ A) @ C3) @ A3) => ((~ ((((member @ A) @ C3) @ B2))) => (((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A3) @ B2))))))).
% 3.45/1.79 thf(fact_230_Diff__iff, axiom, (! [A:$tType,C3:A,A3:(set @ A),B2:(set @ A)]: (((((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A3) @ B2)) = ((((member @ A) @ C3) @ A3) & (~ ((((member @ A) @ C3) @ B2)))))))).
% 3.45/1.79 thf(fact_231_Diff__idemp, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B2)) @ B2) = (((minus_minus @ (set @ A)) @ A3) @ B2))))).
% 3.45/1.79 thf(fact_232_Diff__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = A3)))).
% 3.45/1.79 thf(fact_233_empty__Diff, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3) = (bot_bot @ (set @ A)))))).
% 3.45/1.79 thf(fact_234_Diff__cancel, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ A3) = (bot_bot @ (set @ A)))))).
% 3.45/1.79 thf(fact_235_insert__Diff1, axiom, (! [A:$tType,X2:A,B2:(set @ A),A3:(set @ A)]: (((((member @ A) @ X2) @ B2) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X2) @ A3)) @ B2) = (((minus_minus @ (set @ A)) @ A3) @ B2)))))).
% 3.45/1.79 thf(fact_236_Diff__insert0, axiom, (! [A:$tType,X2:A,A3:(set @ A),B2:(set @ A)]: (((~ ((((member @ A) @ X2) @ A3))) => ((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X2) @ B2)) = (((minus_minus @ (set @ A)) @ A3) @ B2)))))).
% 3.45/1.79 thf(fact_237_insert__Diff__single, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((insert @ A) @ A2) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = (((insert @ A) @ A2) @ A3))))).
% 3.45/1.79 thf(fact_238_Diff__UNIV, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (top_top @ (set @ A))) = (bot_bot @ (set @ A)))))).
% 3.45/1.79 thf(fact_239_surj__diff__right, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A]: ((((((image @ A) @ A) @ (^ [X4:A]: ((((minus_minus @ A) @ X4) @ A2)))) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))))).
% 3.45/1.79 thf(fact_240_inj__diff__right, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A]: (((((inj_on @ A) @ A) @ (^ [B4:A]: ((((minus_minus @ A) @ B4) @ A2)))) @ (top_top @ (set @ A))))))))).
% 3.45/1.79 thf(fact_241_translation__subtract__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,S3:(set @ A),T:(set @ A)]: ((((((image @ A) @ A) @ (^ [X4:A]: ((((minus_minus @ A) @ X4) @ A2)))) @ (((minus_minus @ (set @ A)) @ S3) @ T)) = (((minus_minus @ (set @ A)) @ ((((image @ A) @ A) @ (^ [X4:A]: ((((minus_minus @ A) @ X4) @ A2)))) @ S3)) @ ((((image @ A) @ A) @ (^ [X4:A]: ((((minus_minus @ A) @ X4) @ A2)))) @ T))))))))).
% 3.45/1.79 thf(fact_242_Diff__insert__absorb, axiom, (! [A:$tType,X2:A,A3:(set @ A)]: (((~ ((((member @ A) @ X2) @ A3))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X2) @ A3)) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) = A3))))).
% 3.45/1.79 thf(fact_243_Diff__insert2, axiom, (! [A:$tType,A3:(set @ A),A2:A,B2:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ B2)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) @ B2))))).
% 3.45/1.79 thf(fact_244_insert__Diff, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => ((((insert @ A) @ A2) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = A3))))).
% 3.45/1.79 thf(fact_245_Diff__insert, axiom, (! [A:$tType,A3:(set @ A),A2:A,B2:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ B2)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B2)) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 3.45/1.79 thf(fact_246_in__image__insert__iff, axiom, (! [A:$tType,B2:(set @ (set @ A)),X2:A,A3:(set @ A)]: (((! [C5:(set @ A)]: (((((member @ (set @ A)) @ C5) @ B2) => (~ ((((member @ A) @ X2) @ C5)))))) => ((((member @ (set @ A)) @ A3) @ ((((image @ (set @ A)) @ (set @ A)) @ ((insert @ A) @ X2)) @ B2)) = ((((member @ A) @ X2) @ A3) & (((member @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))) @ B2))))))).
% 3.45/1.79 thf(fact_247_DiffE, axiom, (! [A:$tType,C3:A,A3:(set @ A),B2:(set @ A)]: (((((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A3) @ B2)) => (~ (((((member @ A) @ C3) @ A3) => (((member @ A) @ C3) @ B2)))))))).
% 3.45/1.79 thf(fact_248_DiffD1, axiom, (! [A:$tType,C3:A,A3:(set @ A),B2:(set @ A)]: (((((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A3) @ B2)) => (((member @ A) @ C3) @ A3))))).
% 3.45/1.79 thf(fact_249_DiffD2, axiom, (! [A:$tType,C3:A,A3:(set @ A),B2:(set @ A)]: (((((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A3) @ B2)) => (~ ((((member @ A) @ C3) @ B2))))))).
% 3.45/1.79 thf(fact_250_set__diff__eq, axiom, (! [A:$tType]: (((minus_minus @ (set @ A)) = (^ [A5:(set @ A),B6:(set @ A)]: (((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ A5) & (~ ((((member @ A) @ X4) @ B6))))))))))))).
% 3.45/1.79 thf(fact_251_insert__Diff__if, axiom, (! [A:$tType,X2:A,B2:(set @ A),A3:(set @ A)]: ((((((member @ A) @ X2) @ B2) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X2) @ A3)) @ B2) = (((minus_minus @ (set @ A)) @ A3) @ B2))) & ((~ ((((member @ A) @ X2) @ B2))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X2) @ A3)) @ B2) = (((insert @ A) @ X2) @ (((minus_minus @ (set @ A)) @ A3) @ B2)))))))).
% 3.45/1.79 thf(fact_252_image__set__diff, axiom, (! [B:$tType,A:$tType,F3:(A > B),A3:(set @ A),B2:(set @ A)]: ((((((inj_on @ A) @ B) @ F3) @ (top_top @ (set @ A))) => (((((image @ A) @ B) @ F3) @ (((minus_minus @ (set @ A)) @ A3) @ B2)) = (((minus_minus @ (set @ B)) @ ((((image @ A) @ B) @ F3) @ A3)) @ ((((image @ A) @ B) @ F3) @ B2))))))).
% 3.45/1.79 thf(fact_253_top__conj_I1_J, axiom, (! [A:$tType,X2:A,P:$o]: (((((top_top @ (A > $o)) @ X2) & P) = P)))).
% 3.45/1.79 thf(fact_254_remove__def, axiom, (! [A:$tType]: (((remove @ A) = (^ [X4:A,A5:(set @ A)]: ((((minus_minus @ (set @ A)) @ A5) @ (((insert @ A) @ X4) @ (bot_bot @ (set @ A)))))))))).
% 3.45/1.79 thf(fact_255_fun__upd__image, axiom, (! [A:$tType,B:$tType,X2:B,A3:(set @ B),F3:(B > A),Y4:A]: ((((((member @ B) @ X2) @ A3) => (((((image @ B) @ A) @ (((((fun_upd @ B) @ A) @ F3) @ X2) @ Y4)) @ A3) = (((insert @ A) @ Y4) @ ((((image @ B) @ A) @ F3) @ (((minus_minus @ (set @ B)) @ A3) @ (((insert @ B) @ X2) @ (bot_bot @ (set @ B)))))))) & ((~ ((((member @ B) @ X2) @ A3))) => (((((image @ B) @ A) @ (((((fun_upd @ B) @ A) @ F3) @ X2) @ Y4)) @ A3) = ((((image @ B) @ A) @ F3) @ A3))))))).
% 3.45/1.79 thf(tcon_fun___Orderings_Otop, axiom, (! [A10:$tType,A11:$tType]: (((top @ A11) => (top @ (A10 > A11)))))).
% 3.45/1.79 thf(tcon_fun___Orderings_Oord, axiom, (! [A10:$tType,A11:$tType]: (((ord @ A11) => (ord @ (A10 > A11)))))).
% 3.45/1.79 thf(tcon_fun___Orderings_Obot, axiom, (! [A10:$tType,A11:$tType]: (((bot @ A11) => (bot @ (A10 > A11)))))).
% 3.45/1.79 thf(tcon_Set_Oset___Orderings_Otop_1, axiom, (! [A10:$tType]: ((top @ (set @ A10))))).
% 3.45/1.79 thf(tcon_Set_Oset___Orderings_Oord_2, axiom, (! [A10:$tType]: ((ord @ (set @ A10))))).
% 3.45/1.79 thf(tcon_Set_Oset___Orderings_Obot_3, axiom, (! [A10:$tType]: ((bot @ (set @ A10))))).
% 3.45/1.79 thf(tcon_HOL_Obool___Orderings_Otop_4, axiom, (top @ $o)).
% 3.45/1.79 thf(tcon_HOL_Obool___Orderings_Oord_5, axiom, (ord @ $o)).
% 3.45/1.79 thf(tcon_HOL_Obool___Orderings_Obot_6, axiom, (bot @ $o)).
% 3.45/1.79 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 3.45/1.79 thf(help_If_2_1_T, axiom, (! [A:$tType,X2:A,Y4:A]: ((((((if @ A) @ $false) @ X2) @ Y4) = Y4)))).
% 3.45/1.79 thf(help_If_1_1_T, axiom, (! [A:$tType,X2:A,Y4:A]: ((((((if @ A) @ $true) @ X2) @ Y4) = X2)))).
% 3.45/1.79 thf(conj_0, hypothesis, (! [X5:a]: (((p @ X5) => (? [X_12:b]: (((q @ X5) @ X_12))))))).
% 3.45/1.79 thf(conj_1, conjecture, (! [X:a]: (((~ ((p @ X))) | ((q @ X) @ ((hilbert_Eps @ b) @ (q @ X))))))).
% 3.45/1.79 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.ttMsP055XI/DTF2DTF17675.p
% 3.45/1.79 ------------------------
% 3.73/1.84 ---- Cleaned THF ---
% 3.73/1.84 thf(ty_t_Product__Type_Oprod,type,
% 3.73/1.84 product_prod: $tType > $tType > $tType ).
% 3.73/1.84
% 3.73/1.84 thf(ty_t_Set_Oset,type,
% 3.73/1.84 set: $tType > $tType ).
% 3.73/1.84
% 3.73/1.84 thf(ty_tf_b,type,
% 3.73/1.84 b: $tType ).
% 3.73/1.84
% 3.73/1.84 thf(ty_tf_a,type,
% 3.73/1.84 a: $tType ).
% 3.73/1.84
% 3.73/1.84 thf(sy_cl_HOL_Otype,type,
% 3.73/1.84 type:
% 3.73/1.84 !>[A: $tType] : $o ).
% 3.73/1.84
% 3.73/1.84 thf(sy_cl_Orderings_Obot,type,
% 3.73/1.84 bot:
% 3.73/1.84 !>[A: $tType] : $o ).
% 3.73/1.84
% 3.73/1.84 thf(sy_cl_Orderings_Oord,type,
% 3.73/1.84 ord:
% 3.73/1.84 !>[A: $tType] : $o ).
% 3.73/1.84
% 3.73/1.84 thf(sy_cl_Orderings_Otop,type,
% 3.73/1.84 top:
% 3.73/1.84 !>[A: $tType] : $o ).
% 3.73/1.84
% 3.73/1.84 thf(sy_cl_Groups_Oab__group__add,type,
% 3.73/1.84 ab_group_add:
% 3.73/1.84 !>[A: $tType] : $o ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_BNF__Def_Opick__middlep,type,
% 3.73/1.84 bNF_pick_middlep:
% 3.73/1.84 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_BNF__Greatest__Fixpoint_OfromCard,type,
% 3.73/1.84 bNF_Gr932048252omCard:
% 3.73/1.84 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > B > A ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2p,type,
% 3.73/1.84 bNF_Greatest_image2p:
% 3.73/1.84 !>[C: $tType,A: $tType,D: $tType,B: $tType] : ( ( C > A ) > ( D > B ) > ( C > D > $o ) > A > B > $o ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_BNF__Greatest__Fixpoint_OtoCard,type,
% 3.73/1.84 bNF_Greatest_toCard:
% 3.73/1.84 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > A > B ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
% 3.73/1.84 bNF_Gr1978856366d_pred:
% 3.73/1.84 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > $o ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_BNF__Greatest__Fixpoint_Ouniv,type,
% 3.73/1.84 bNF_Greatest_univ:
% 3.73/1.84 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Fun_Obij__betw,type,
% 3.73/1.84 bij_betw:
% 3.73/1.84 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Fun_Ofun__upd,type,
% 3.73/1.84 fun_upd:
% 3.73/1.84 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Fun_Oinj__on,type,
% 3.73/1.84 inj_on:
% 3.73/1.84 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 3.73/1.84 minus_minus:
% 3.73/1.84 !>[A: $tType] : ( A > A > A ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Hilbert__Choice_OEps,type,
% 3.73/1.84 hilbert_Eps:
% 3.73/1.84 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Hilbert__Choice_Obijection,type,
% 3.73/1.84 hilbert_bijection:
% 3.73/1.84 !>[A: $tType] : ( ( A > A ) > $o ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Hilbert__Choice_Oinv__into,type,
% 3.73/1.84 hilbert_inv_into:
% 3.73/1.84 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_If,type,
% 3.73/1.84 if:
% 3.73/1.84 !>[A: $tType] : ( $o > A > A > A ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Lattices__Big_Oord__class_Oarg__max,type,
% 3.73/1.84 lattices_ord_arg_max:
% 3.73/1.84 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Lattices__Big_Oord__class_Oarg__max__on,type,
% 3.73/1.84 lattic1534546155max_on:
% 3.73/1.84 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Lattices__Big_Oord__class_Ois__arg__max,type,
% 3.73/1.84 lattic2130478198rg_max:
% 3.73/1.84 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Lifting_Orel__pred__comp,type,
% 3.73/1.84 rel_pred_comp:
% 3.73/1.84 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( B > $o ) > A > $o ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Orderings_Obot__class_Obot,type,
% 3.73/1.84 bot_bot:
% 3.73/1.84 !>[A: $tType] : A ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Orderings_Otop__class_Otop,type,
% 3.73/1.84 top_top:
% 3.73/1.84 !>[A: $tType] : A ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Product__Type_Obool_Ocase__bool,type,
% 3.73/1.84 product_case_bool:
% 3.73/1.84 !>[A: $tType] : ( A > A > $o > A ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Quotient_OQuotient3,type,
% 3.73/1.84 quotient3:
% 3.73/1.84 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Quotient_Oquot__type,type,
% 3.73/1.84 quot_type:
% 3.73/1.84 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > B ) > ( B > ( set @ A ) ) > $o ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Quotient_Oquot__type_Oabs,type,
% 3.73/1.84 quot_abs:
% 3.73/1.84 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > B ) > A > B ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Quotient_Oquot__type_Orep,type,
% 3.73/1.84 quot_rep:
% 3.73/1.84 !>[B: $tType,A: $tType] : ( ( B > ( set @ A ) ) > B > A ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Relation_Oinv__imagep,type,
% 3.73/1.84 inv_imagep:
% 3.73/1.84 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Set_OCollect,type,
% 3.73/1.84 collect:
% 3.73/1.84 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Set_Oimage,type,
% 3.73/1.84 image:
% 3.73/1.84 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Set_Oinsert,type,
% 3.73/1.84 insert:
% 3.73/1.84 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Set_Ois__empty,type,
% 3.73/1.84 is_empty:
% 3.73/1.84 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Set_Ois__singleton,type,
% 3.73/1.84 is_singleton:
% 3.73/1.84 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Set_Oremove,type,
% 3.73/1.84 remove:
% 3.73/1.84 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_Set_Othe__elem,type,
% 3.73/1.84 the_elem:
% 3.73/1.84 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_c_member,type,
% 3.73/1.84 member:
% 3.73/1.84 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.73/1.84
% 3.73/1.84 thf(sy_v_P,type,
% 3.73/1.84 p: a > $o ).
% 3.73/1.84
% 3.73/1.84 thf(sy_v_Q,type,
% 3.73/1.84 q: a > b > $o ).
% 3.73/1.84
% 3.73/1.84 thf(fact_0_some__equality,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o,A2: A] :
% 3.73/1.84 ( ( P @ A2 )
% 3.73/1.84 => ( ! [X: A] :
% 3.73/1.84 ( ( P @ X )
% 3.73/1.84 => ( X = A2 ) )
% 3.73/1.84 => ( ( hilbert_Eps @ A @ P )
% 3.73/1.84 = A2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_1_some__eq__trivial,axiom,
% 3.73/1.84 ! [A: $tType,X2: A] :
% 3.73/1.84 ( ( hilbert_Eps @ A
% 3.73/1.84 @ ^ [Y: A] : ( Y = X2 ) )
% 3.73/1.84 = X2 ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_2_some__sym__eq__trivial,axiom,
% 3.73/1.84 ! [A: $tType,X2: A] :
% 3.73/1.84 ( ( hilbert_Eps @ A
% 3.73/1.84 @ ( ^ [Y2: A,Z: A] : ( Y2 = Z )
% 3.73/1.84 @ X2 ) )
% 3.73/1.84 = X2 ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_3_verit__sko__ex_H,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o,A3: $o] :
% 3.73/1.84 ( ( ( P @ ( hilbert_Eps @ A @ P ) )
% 3.73/1.84 = A3 )
% 3.73/1.84 => ( ( ? [X3: A] : ( P @ X3 ) )
% 3.73/1.84 = A3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_4_verit__sko__forall,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( ^ [P2: A > $o] :
% 3.73/1.84 ! [X3: A] : ( P2 @ X3 ) )
% 3.73/1.84 = ( ^ [P3: A > $o] :
% 3.73/1.84 ( P3
% 3.73/1.84 @ ( hilbert_Eps @ A
% 3.73/1.84 @ ^ [X4: A] :
% 3.73/1.84 ~ ( P3 @ X4 ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_5_someI2,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
% 3.73/1.84 ( ( P @ A2 )
% 3.73/1.84 => ( ! [X: A] :
% 3.73/1.84 ( ( P @ X )
% 3.73/1.84 => ( Q @ X ) )
% 3.73/1.84 => ( Q @ ( hilbert_Eps @ A @ P ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_6_verit__sko__forall_H,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o,A3: $o] :
% 3.73/1.84 ( ( ( P
% 3.73/1.84 @ ( hilbert_Eps @ A
% 3.73/1.84 @ ^ [X4: A] :
% 3.73/1.84 ~ ( P @ X4 ) ) )
% 3.73/1.84 = A3 )
% 3.73/1.84 => ( ( ! [X3: A] : ( P @ X3 ) )
% 3.73/1.84 = A3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_7_verit__sko__forall_H_H,axiom,
% 3.73/1.84 ! [A: $tType,B2: A,A3: A,P: A > $o] :
% 3.73/1.84 ( ( B2 = A3 )
% 3.73/1.84 => ( ( ( hilbert_Eps @ A @ P )
% 3.73/1.84 = A3 )
% 3.73/1.84 = ( ( hilbert_Eps @ A @ P )
% 3.73/1.84 = B2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_8_someI__ex,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o] :
% 3.73/1.84 ( ? [X_1: A] : ( P @ X_1 )
% 3.73/1.84 => ( P @ ( hilbert_Eps @ A @ P ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_9_someI2__ex,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.73/1.84 ( ? [X_1: A] : ( P @ X_1 )
% 3.73/1.84 => ( ! [X: A] :
% 3.73/1.84 ( ( P @ X )
% 3.73/1.84 => ( Q @ X ) )
% 3.73/1.84 => ( Q @ ( hilbert_Eps @ A @ P ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_10_someI2__bex,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] :
% 3.73/1.84 ( ? [X5: A] :
% 3.73/1.84 ( ( member @ A @ X5 @ A3 )
% 3.73/1.84 & ( P @ X5 ) )
% 3.73/1.84 => ( ! [X: A] :
% 3.73/1.84 ( ( ( member @ A @ X @ A3 )
% 3.73/1.84 & ( P @ X ) )
% 3.73/1.84 => ( Q @ X ) )
% 3.73/1.84 => ( Q
% 3.73/1.84 @ ( hilbert_Eps @ A
% 3.73/1.84 @ ^ [X4: A] :
% 3.73/1.84 ( ( member @ A @ X4 @ A3 )
% 3.73/1.84 & ( P @ X4 ) ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_11_some__eq__ex,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o] :
% 3.73/1.84 ( ( P @ ( hilbert_Eps @ A @ P ) )
% 3.73/1.84 = ( ? [X3: A] : ( P @ X3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_12_some__eq__imp,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o,A2: A,B3: A] :
% 3.73/1.84 ( ( ( hilbert_Eps @ A @ P )
% 3.73/1.84 = A2 )
% 3.73/1.84 => ( ( P @ B3 )
% 3.73/1.84 => ( P @ A2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_13_tfl__some,axiom,
% 3.73/1.84 ! [A: $tType,P4: A > $o,X5: A] :
% 3.73/1.84 ( ( P4 @ X5 )
% 3.73/1.84 => ( P4 @ ( hilbert_Eps @ A @ P4 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_14_Eps__cong,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.73/1.84 ( ! [X: A] :
% 3.73/1.84 ( ( P @ X )
% 3.73/1.84 = ( Q @ X ) )
% 3.73/1.84 => ( ( hilbert_Eps @ A @ P )
% 3.73/1.84 = ( hilbert_Eps @ A @ Q ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_15_someI,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o,X2: A] :
% 3.73/1.84 ( ( P @ X2 )
% 3.73/1.84 => ( P @ ( hilbert_Eps @ A @ P ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_16_verit__sko__forall__indirect2,axiom,
% 3.73/1.84 ! [A: $tType,X2: A,P: A > $o,P5: A > $o] :
% 3.73/1.84 ( ( X2
% 3.73/1.84 = ( hilbert_Eps @ A
% 3.73/1.84 @ ^ [X4: A] :
% 3.73/1.84 ~ ( P @ X4 ) ) )
% 3.73/1.84 => ( ! [X: A] :
% 3.73/1.84 ( ( P @ X )
% 3.73/1.84 = ( P5 @ X ) )
% 3.73/1.84 => ( ( ! [X3: A] : ( P5 @ X3 ) )
% 3.73/1.84 = ( P @ X2 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_17_verit__sko__forall__indirect,axiom,
% 3.73/1.84 ! [A: $tType,X2: A,P: A > $o] :
% 3.73/1.84 ( ( X2
% 3.73/1.84 = ( hilbert_Eps @ A
% 3.73/1.84 @ ^ [X4: A] :
% 3.73/1.84 ~ ( P @ X4 ) ) )
% 3.73/1.84 => ( ( ! [X3: A] : ( P @ X3 ) )
% 3.73/1.84 = ( P @ X2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_18_some1__equality,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o,A2: A] :
% 3.73/1.84 ( ? [X5: A] :
% 3.73/1.84 ( ( P @ X5 )
% 3.73/1.84 & ! [Y3: A] :
% 3.73/1.84 ( ( P @ Y3 )
% 3.73/1.84 => ( Y3 = X5 ) ) )
% 3.73/1.84 => ( ( P @ A2 )
% 3.73/1.84 => ( ( hilbert_Eps @ A @ P )
% 3.73/1.84 = A2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_19_verit__sko__ex__indirect2,axiom,
% 3.73/1.84 ! [A: $tType,X2: A,P: A > $o,P5: A > $o] :
% 3.73/1.84 ( ( X2
% 3.73/1.84 = ( hilbert_Eps @ A @ P ) )
% 3.73/1.84 => ( ! [X: A] :
% 3.73/1.84 ( ( P @ X )
% 3.73/1.84 = ( P5 @ X ) )
% 3.73/1.84 => ( ( ? [X3: A] : ( P5 @ X3 ) )
% 3.73/1.84 = ( P @ X2 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_20_verit__sko__ex__indirect,axiom,
% 3.73/1.84 ! [A: $tType,X2: A,P: A > $o] :
% 3.73/1.84 ( ( X2
% 3.73/1.84 = ( hilbert_Eps @ A @ P ) )
% 3.73/1.84 => ( ( ? [X3: A] : ( P @ X3 ) )
% 3.73/1.84 = ( P @ X2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_21_Nitpick_OEps__psimp,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o,X2: A,Y4: A] :
% 3.73/1.84 ( ( P @ X2 )
% 3.73/1.84 => ( ~ ( P @ Y4 )
% 3.73/1.84 => ( ( ( hilbert_Eps @ A @ P )
% 3.73/1.84 = Y4 )
% 3.73/1.84 => ( ( hilbert_Eps @ A @ P )
% 3.73/1.84 = X2 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_22_univ__def,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType] :
% 3.73/1.84 ( ( bNF_Greatest_univ @ B @ A )
% 3.73/1.84 = ( ^ [F: B > A,X6: set @ B] :
% 3.73/1.84 ( F
% 3.73/1.84 @ ( hilbert_Eps @ B
% 3.73/1.84 @ ^ [X4: B] : ( member @ B @ X4 @ X6 ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_23_pick__middlep__def,axiom,
% 3.73/1.84 ! [C: $tType,A: $tType,B: $tType] :
% 3.73/1.84 ( ( bNF_pick_middlep @ B @ A @ C )
% 3.73/1.84 = ( ^ [P3: B > A > $o,Q2: A > C > $o,A4: B,C2: C] :
% 3.73/1.84 ( hilbert_Eps @ A
% 3.73/1.84 @ ^ [B4: A] :
% 3.73/1.84 ( ( P3 @ A4 @ B4 )
% 3.73/1.84 & ( Q2 @ B4 @ C2 ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_24_some__in__eq,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A] :
% 3.73/1.84 ( ( member @ A
% 3.73/1.84 @ ( hilbert_Eps @ A
% 3.73/1.84 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 3.73/1.84 @ A3 )
% 3.73/1.84 = ( A3
% 3.73/1.84 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_25_inv__into__def2,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType] :
% 3.73/1.84 ( ( hilbert_inv_into @ A @ B )
% 3.73/1.84 = ( ^ [A5: set @ A,F: A > B,X4: B] :
% 3.73/1.84 ( hilbert_Eps @ A
% 3.73/1.84 @ ^ [Y: A] :
% 3.73/1.84 ( ( member @ A @ Y @ A5 )
% 3.73/1.84 & ( ( F @ Y )
% 3.73/1.84 = X4 ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_26_inv__into__def,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType] :
% 3.73/1.84 ( ( hilbert_inv_into @ A @ B )
% 3.73/1.84 = ( ^ [A5: set @ A,F: A > B,X4: B] :
% 3.73/1.84 ( hilbert_Eps @ A
% 3.73/1.84 @ ^ [Y: A] :
% 3.73/1.84 ( ( member @ A @ Y @ A5 )
% 3.73/1.84 & ( ( F @ Y )
% 3.73/1.84 = X4 ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_27_quot__type_Osome__collect,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,R: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A ),R2: A] :
% 3.73/1.84 ( ( quot_type @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( ( R @ R2 @ R2 )
% 3.73/1.84 => ( ( R
% 3.73/1.84 @ ( hilbert_Eps @ A
% 3.73/1.84 @ ^ [X4: A] : ( member @ A @ X4 @ ( collect @ A @ ( R @ R2 ) ) ) ) )
% 3.73/1.84 = ( R @ R2 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_28_fromCard__def,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType] :
% 3.73/1.84 ( ( bNF_Gr932048252omCard @ A @ B )
% 3.73/1.84 = ( ^ [A5: set @ A,R3: set @ ( product_prod @ B @ B ),K: B] :
% 3.73/1.84 ( hilbert_Eps @ A
% 3.73/1.84 @ ^ [B4: A] :
% 3.73/1.84 ( ( member @ A @ B4 @ A5 )
% 3.73/1.84 & ( ( bNF_Greatest_toCard @ A @ B @ A5 @ R3 @ B4 )
% 3.73/1.84 = K ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_29_arg__max__def,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType] :
% 3.73/1.84 ( ( ord @ A )
% 3.73/1.84 => ( ( lattices_ord_arg_max @ B @ A )
% 3.73/1.84 = ( ^ [F: B > A,P3: B > $o] : ( hilbert_Eps @ B @ ( lattic2130478198rg_max @ B @ A @ F @ P3 ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_30_case__bool__if,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( product_case_bool @ A )
% 3.73/1.84 = ( ^ [X4: A,Y: A,P3: $o] : ( if @ A @ P3 @ X4 @ Y ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_31_exE__some,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o,C3: A] :
% 3.73/1.84 ( ? [X_1: A] : ( P @ X_1 )
% 3.73/1.84 => ( ( C3
% 3.73/1.84 = ( hilbert_Eps @ A @ P ) )
% 3.73/1.84 => ( P @ C3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_32_quot__type_Orep__inverse,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,R: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A ),X2: B] :
% 3.73/1.84 ( ( quot_type @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( ( Abs @ ( Rep @ X2 ) )
% 3.73/1.84 = X2 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_33_quot__type_Oabs__inverse,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,R: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A ),C3: set @ A] :
% 3.73/1.84 ( ( quot_type @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( ? [X5: A] :
% 3.73/1.84 ( ( R @ X5 @ X5 )
% 3.73/1.84 & ( C3
% 3.73/1.84 = ( collect @ A @ ( R @ X5 ) ) ) )
% 3.73/1.84 => ( ( Rep @ ( Abs @ C3 ) )
% 3.73/1.84 = C3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_34_quot__type_Orep__inject,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,R: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A ),X2: B,Y4: B] :
% 3.73/1.84 ( ( quot_type @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( ( ( Rep @ X2 )
% 3.73/1.84 = ( Rep @ Y4 ) )
% 3.73/1.84 = ( X2 = Y4 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_35_quot__type_Orep__prop,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,R: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A ),Y4: B] :
% 3.73/1.84 ( ( quot_type @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ? [X: A] :
% 3.73/1.84 ( ( R @ X @ X )
% 3.73/1.84 & ( ( Rep @ Y4 )
% 3.73/1.84 = ( collect @ A @ ( R @ X ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_36_arg__max__on__def,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType] :
% 3.73/1.84 ( ( ord @ A )
% 3.73/1.84 => ( ( lattic1534546155max_on @ B @ A )
% 3.73/1.84 = ( ^ [F: B > A,S: set @ B] :
% 3.73/1.84 ( lattices_ord_arg_max @ B @ A @ F
% 3.73/1.84 @ ^ [X4: B] : ( member @ B @ X4 @ S ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_37_empty__Collect__eq,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o] :
% 3.73/1.84 ( ( ( bot_bot @ ( set @ A ) )
% 3.73/1.84 = ( collect @ A @ P ) )
% 3.73/1.84 = ( ! [X4: A] :
% 3.73/1.84 ~ ( P @ X4 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_38_Collect__empty__eq,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o] :
% 3.73/1.84 ( ( ( collect @ A @ P )
% 3.73/1.84 = ( bot_bot @ ( set @ A ) ) )
% 3.73/1.84 = ( ! [X4: A] :
% 3.73/1.84 ~ ( P @ X4 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_39_all__not__in__conv,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A] :
% 3.73/1.84 ( ( ! [X4: A] :
% 3.73/1.84 ~ ( member @ A @ X4 @ A3 ) )
% 3.73/1.84 = ( A3
% 3.73/1.84 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_40_empty__iff,axiom,
% 3.73/1.84 ! [A: $tType,C3: A] :
% 3.73/1.84 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_41_bot__apply,axiom,
% 3.73/1.84 ! [C: $tType,D: $tType] :
% 3.73/1.84 ( ( bot @ C )
% 3.73/1.84 => ( ( bot_bot @ ( D > C ) )
% 3.73/1.84 = ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_42_quot__type_Orep__def,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,R: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A ),A2: B] :
% 3.73/1.84 ( ( quot_type @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( ( quot_rep @ B @ A @ Rep @ A2 )
% 3.73/1.84 = ( hilbert_Eps @ A
% 3.73/1.84 @ ^ [X4: A] : ( member @ A @ X4 @ ( Rep @ A2 ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_43_toCard__def,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType] :
% 3.73/1.84 ( ( bNF_Greatest_toCard @ A @ B )
% 3.73/1.84 = ( ^ [A5: set @ A,R3: set @ ( product_prod @ B @ B )] : ( hilbert_Eps @ ( A > B ) @ ( bNF_Gr1978856366d_pred @ A @ B @ A5 @ R3 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_44_bool_Osplit__sel__asm,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o,F1: A,F2: A,Bool: $o] :
% 3.73/1.84 ( ( P @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 3.73/1.84 = ( ~ ( ( Bool
% 3.73/1.84 & ~ ( P @ F1 ) )
% 3.73/1.84 | ( ~ Bool
% 3.73/1.84 & ~ ( P @ F2 ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_45_mem__Collect__eq,axiom,
% 3.73/1.84 ! [A: $tType,A2: A,P: A > $o] :
% 3.73/1.84 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 3.73/1.84 = ( P @ A2 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_46_Collect__mem__eq,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A] :
% 3.73/1.84 ( ( collect @ A
% 3.73/1.84 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 3.73/1.84 = A3 ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_47_Collect__cong,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.73/1.84 ( ! [X: A] :
% 3.73/1.84 ( ( P @ X )
% 3.73/1.84 = ( Q @ X ) )
% 3.73/1.84 => ( ( collect @ A @ P )
% 3.73/1.84 = ( collect @ A @ Q ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_48_ext,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,G: A > B] :
% 3.73/1.84 ( ! [X: A] :
% 3.73/1.84 ( ( F3 @ X )
% 3.73/1.84 = ( G @ X ) )
% 3.73/1.84 => ( F3 = G ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_49_bot__set__def,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( bot_bot @ ( set @ A ) )
% 3.73/1.84 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_50_quot__type_Orep_Ocong,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType] :
% 3.73/1.84 ( ( quot_rep @ B @ A )
% 3.73/1.84 = ( quot_rep @ B @ A ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_51_bot__fun__def,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType] :
% 3.73/1.84 ( ( bot @ B )
% 3.73/1.84 => ( ( bot_bot @ ( A > B ) )
% 3.73/1.84 = ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_52_emptyE,axiom,
% 3.73/1.84 ! [A: $tType,A2: A] :
% 3.73/1.84 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_53_equals0D,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A,A2: A] :
% 3.73/1.84 ( ( A3
% 3.73/1.84 = ( bot_bot @ ( set @ A ) ) )
% 3.73/1.84 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_54_equals0I,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A] :
% 3.73/1.84 ( ! [Y3: A] :
% 3.73/1.84 ~ ( member @ A @ Y3 @ A3 )
% 3.73/1.84 => ( A3
% 3.73/1.84 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_55_ex__in__conv,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A] :
% 3.73/1.84 ( ( ? [X4: A] : ( member @ A @ X4 @ A3 ) )
% 3.73/1.84 = ( A3
% 3.73/1.84 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_56_empty__def,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( bot_bot @ ( set @ A ) )
% 3.73/1.84 = ( collect @ A
% 3.73/1.84 @ ^ [X4: A] : $false ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_57_old_Obool_Osimps_I4_J,axiom,
% 3.73/1.84 ! [A: $tType,F1: A,F2: A] :
% 3.73/1.84 ( ( product_case_bool @ A @ F1 @ F2 @ $false )
% 3.73/1.84 = F2 ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_58_old_Obool_Osimps_I3_J,axiom,
% 3.73/1.84 ! [A: $tType,F1: A,F2: A] :
% 3.73/1.84 ( ( product_case_bool @ A @ F1 @ F2 @ $true )
% 3.73/1.84 = F1 ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_59_bool_Osplit__sel,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o,F1: A,F2: A,Bool: $o] :
% 3.73/1.84 ( ( P @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 3.73/1.84 = ( ( Bool
% 3.73/1.84 => ( P @ F1 ) )
% 3.73/1.84 & ( ~ Bool
% 3.73/1.84 => ( P @ F2 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_60_bool_Ocase__eq__if,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( product_case_bool @ A )
% 3.73/1.84 = ( ^ [F12: A,F22: A,Bool2: $o] : ( if @ A @ Bool2 @ F12 @ F22 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_61_bool_Ocase__distrib,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,H: A > B,F1: A,F2: A,Bool: $o] :
% 3.73/1.84 ( ( H @ ( product_case_bool @ A @ F1 @ F2 @ Bool ) )
% 3.73/1.84 = ( product_case_bool @ B @ ( H @ F1 ) @ ( H @ F2 ) @ Bool ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_62_Set_Ois__empty__def,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( is_empty @ A )
% 3.73/1.84 = ( ^ [A5: set @ A] :
% 3.73/1.84 ( A5
% 3.73/1.84 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_63_quot__type_Oabs__def,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,R: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A ),X2: A] :
% 3.73/1.84 ( ( quot_type @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( ( quot_abs @ A @ B @ R @ Abs @ X2 )
% 3.73/1.84 = ( Abs @ ( collect @ A @ ( R @ X2 ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_64_image2p__def,axiom,
% 3.73/1.84 ! [D: $tType,B: $tType,A: $tType,C: $tType] :
% 3.73/1.84 ( ( bNF_Greatest_image2p @ C @ A @ D @ B )
% 3.73/1.84 = ( ^ [F: C > A,G2: D > B,R4: C > D > $o,X4: A,Y: B] :
% 3.73/1.84 ? [X7: C,Y5: D] :
% 3.73/1.84 ( ( R4 @ X7 @ Y5 )
% 3.73/1.84 & ( ( F @ X7 )
% 3.73/1.84 = X4 )
% 3.73/1.84 & ( ( G2 @ Y5 )
% 3.73/1.84 = Y ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_65_Collect__empty__eq__bot,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o] :
% 3.73/1.84 ( ( ( collect @ A @ P )
% 3.73/1.84 = ( bot_bot @ ( set @ A ) ) )
% 3.73/1.84 = ( P
% 3.73/1.84 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_66_bot__empty__eq,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( bot_bot @ ( A > $o ) )
% 3.73/1.84 = ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_67_inv__def,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: B > A] :
% 3.73/1.84 ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 )
% 3.73/1.84 = ( ^ [Y: A] :
% 3.73/1.84 ( hilbert_Eps @ B
% 3.73/1.84 @ ^ [X4: B] :
% 3.73/1.84 ( ( F3 @ X4 )
% 3.73/1.84 = Y ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_68_rel__pred__comp__def,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType] :
% 3.73/1.84 ( ( rel_pred_comp @ A @ B )
% 3.73/1.84 = ( ^ [R4: A > B > $o,P3: B > $o,X4: A] :
% 3.73/1.84 ? [Y: B] :
% 3.73/1.84 ( ( R4 @ X4 @ Y )
% 3.73/1.84 & ( P3 @ Y ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_69_is__singletonI_H,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A] :
% 3.73/1.84 ( ( A3
% 3.73/1.84 != ( bot_bot @ ( set @ A ) ) )
% 3.73/1.84 => ( ! [X: A,Y3: A] :
% 3.73/1.84 ( ( member @ A @ X @ A3 )
% 3.73/1.84 => ( ( member @ A @ Y3 @ A3 )
% 3.73/1.84 => ( X = Y3 ) ) )
% 3.73/1.84 => ( is_singleton @ A @ A3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_70_top__apply,axiom,
% 3.73/1.84 ! [C: $tType,D: $tType] :
% 3.73/1.84 ( ( top @ C )
% 3.73/1.84 => ( ( top_top @ ( D > C ) )
% 3.73/1.84 = ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_71_UNIV__I,axiom,
% 3.73/1.84 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_72_Collect__const,axiom,
% 3.73/1.84 ! [A: $tType,P: $o] :
% 3.73/1.84 ( ( P
% 3.73/1.84 => ( ( collect @ A
% 3.73/1.84 @ ^ [S2: A] : P )
% 3.73/1.84 = ( top_top @ ( set @ A ) ) ) )
% 3.73/1.84 & ( ~ P
% 3.73/1.84 => ( ( collect @ A
% 3.73/1.84 @ ^ [S2: A] : P )
% 3.73/1.84 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_73_inv__identity,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) )
% 3.73/1.84 @ ^ [A4: A] : A4 )
% 3.73/1.84 = ( ^ [A4: A] : A4 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_74_UNIV__def,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( top_top @ ( set @ A ) )
% 3.73/1.84 = ( collect @ A
% 3.73/1.84 @ ^ [X4: A] : $true ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_75_UNIV__eq__I,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A] :
% 3.73/1.84 ( ! [X: A] : ( member @ A @ X @ A3 )
% 3.73/1.84 => ( ( top_top @ ( set @ A ) )
% 3.73/1.84 = A3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_76_UNIV__witness,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ? [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_77_image2pE,axiom,
% 3.73/1.84 ! [D: $tType,B: $tType,A: $tType,C: $tType,F3: A > B,G: C > D,R: A > C > $o,Fx: B,Gy: D] :
% 3.73/1.84 ( ( bNF_Greatest_image2p @ A @ B @ C @ D @ F3 @ G @ R @ Fx @ Gy )
% 3.73/1.84 => ~ ! [X: A] :
% 3.73/1.84 ( ( Fx
% 3.73/1.84 = ( F3 @ X ) )
% 3.73/1.84 => ! [Y3: C] :
% 3.73/1.84 ( ( Gy
% 3.73/1.84 = ( G @ Y3 ) )
% 3.73/1.84 => ~ ( R @ X @ Y3 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_78_image2pI,axiom,
% 3.73/1.84 ! [A: $tType,C: $tType,D: $tType,B: $tType,R: A > B > $o,X2: A,Y4: B,F3: A > C,G: B > D] :
% 3.73/1.84 ( ( R @ X2 @ Y4 )
% 3.73/1.84 => ( bNF_Greatest_image2p @ A @ C @ B @ D @ F3 @ G @ R @ ( F3 @ X2 ) @ ( G @ Y4 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_79_quot__type_Oabs_Ocong,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType] :
% 3.73/1.84 ( ( quot_abs @ A @ B )
% 3.73/1.84 = ( quot_abs @ A @ B ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_80_empty__not__UNIV,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( bot_bot @ ( set @ A ) )
% 3.73/1.84 != ( top_top @ ( set @ A ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_81_inv__equality,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,G: B > A,F3: A > B] :
% 3.73/1.84 ( ! [X: A] :
% 3.73/1.84 ( ( G @ ( F3 @ X ) )
% 3.73/1.84 = X )
% 3.73/1.84 => ( ! [Y3: B] :
% 3.73/1.84 ( ( F3 @ ( G @ Y3 ) )
% 3.73/1.84 = Y3 )
% 3.73/1.84 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
% 3.73/1.84 = G ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_82_iso__tuple__UNIV__I,axiom,
% 3.73/1.84 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_83_quot__type_OQuotient,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,R: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A )] :
% 3.73/1.84 ( ( quot_type @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( quotient3 @ A @ B @ R @ ( quot_abs @ A @ B @ R @ Abs ) @ ( quot_rep @ B @ A @ Rep ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_84_in__inv__imagep,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType] :
% 3.73/1.84 ( ( inv_imagep @ A @ B )
% 3.73/1.84 = ( ^ [R3: A > A > $o,F: B > A,X4: B,Y: B] : ( R3 @ ( F @ X4 ) @ ( F @ Y ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_85_is__singletonI,axiom,
% 3.73/1.84 ! [A: $tType,X2: A] : ( is_singleton @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_86_bijection_Oinv__right__eq__iff,axiom,
% 3.73/1.84 ! [A: $tType,F3: A > A,B3: A,A2: A] :
% 3.73/1.84 ( ( hilbert_bijection @ A @ F3 )
% 3.73/1.84 => ( ( B3
% 3.73/1.84 = ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ A2 ) )
% 3.73/1.84 = ( ( F3 @ B3 )
% 3.73/1.84 = A2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_87_bijection_Oinv__left__eq__iff,axiom,
% 3.73/1.84 ! [A: $tType,F3: A > A,A2: A,B3: A] :
% 3.73/1.84 ( ( hilbert_bijection @ A @ F3 )
% 3.73/1.84 => ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ A2 )
% 3.73/1.84 = B3 )
% 3.73/1.84 = ( ( F3 @ B3 )
% 3.73/1.84 = A2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_88_bijection_Oeq__inv__iff,axiom,
% 3.73/1.84 ! [A: $tType,F3: A > A,A2: A,B3: A] :
% 3.73/1.84 ( ( hilbert_bijection @ A @ F3 )
% 3.73/1.84 => ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ A2 )
% 3.73/1.84 = ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ B3 ) )
% 3.73/1.84 = ( A2 = B3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_89_insert__absorb2,axiom,
% 3.73/1.84 ! [A: $tType,X2: A,A3: set @ A] :
% 3.73/1.84 ( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A3 ) )
% 3.73/1.84 = ( insert @ A @ X2 @ A3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_90_insert__iff,axiom,
% 3.73/1.84 ! [A: $tType,A2: A,B3: A,A3: set @ A] :
% 3.73/1.84 ( ( member @ A @ A2 @ ( insert @ A @ B3 @ A3 ) )
% 3.73/1.84 = ( ( A2 = B3 )
% 3.73/1.84 | ( member @ A @ A2 @ A3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_91_insertCI,axiom,
% 3.73/1.84 ! [A: $tType,A2: A,B2: set @ A,B3: A] :
% 3.73/1.84 ( ( ~ ( member @ A @ A2 @ B2 )
% 3.73/1.84 => ( A2 = B3 ) )
% 3.73/1.84 => ( member @ A @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_92_singletonI,axiom,
% 3.73/1.84 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_93_singleton__conv,axiom,
% 3.73/1.84 ! [A: $tType,A2: A] :
% 3.73/1.84 ( ( collect @ A
% 3.73/1.84 @ ^ [X4: A] : ( X4 = A2 ) )
% 3.73/1.84 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_94_singleton__conv2,axiom,
% 3.73/1.84 ! [A: $tType,A2: A] :
% 3.73/1.84 ( ( collect @ A
% 3.73/1.84 @ ( ^ [Y2: A,Z: A] : ( Y2 = Z )
% 3.73/1.84 @ A2 ) )
% 3.73/1.84 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_95_Quotient3__rep__reflp,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A2: B] :
% 3.73/1.84 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( R @ ( Rep @ A2 ) @ ( Rep @ A2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_96_Quotient3__rep__abs,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R2: A] :
% 3.73/1.84 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( ( R @ R2 @ R2 )
% 3.73/1.84 => ( R @ ( Rep @ ( Abs @ R2 ) ) @ R2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_97_Quotient3__rel__rep,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A2: B,B3: B] :
% 3.73/1.84 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( ( R @ ( Rep @ A2 ) @ ( Rep @ B3 ) )
% 3.73/1.84 = ( A2 = B3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_98_Quotient3__rel__abs,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R2: A,S3: A] :
% 3.73/1.84 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( ( R @ R2 @ S3 )
% 3.73/1.84 => ( ( Abs @ R2 )
% 3.73/1.84 = ( Abs @ S3 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_99_Quotient3__abs__rep,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A2: B] :
% 3.73/1.84 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( ( Abs @ ( Rep @ A2 ) )
% 3.73/1.84 = A2 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_100_rep__abs__rsp__left,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
% 3.73/1.84 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( ( R @ X1 @ X22 )
% 3.73/1.84 => ( R @ ( Rep @ ( Abs @ X1 ) ) @ X22 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_101_Quotient3__refl2,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R2: A,S3: A] :
% 3.73/1.84 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( ( R @ R2 @ S3 )
% 3.73/1.84 => ( R @ S3 @ S3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_102_Quotient3__refl1,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R2: A,S3: A] :
% 3.73/1.84 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( ( R @ R2 @ S3 )
% 3.73/1.84 => ( R @ R2 @ R2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_103_Quotient3__rel,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R2: A,S3: A] :
% 3.73/1.84 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( ( ( R @ R2 @ R2 )
% 3.73/1.84 & ( R @ S3 @ S3 )
% 3.73/1.84 & ( ( Abs @ R2 )
% 3.73/1.84 = ( Abs @ S3 ) ) )
% 3.73/1.84 = ( R @ R2 @ S3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_104_Quotient3__def,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType] :
% 3.73/1.84 ( ( quotient3 @ A @ B )
% 3.73/1.84 = ( ^ [R4: A > A > $o,Abs2: A > B,Rep2: B > A] :
% 3.73/1.84 ( ! [A4: B] :
% 3.73/1.84 ( ( Abs2 @ ( Rep2 @ A4 ) )
% 3.73/1.84 = A4 )
% 3.73/1.84 & ! [A4: B] : ( R4 @ ( Rep2 @ A4 ) @ ( Rep2 @ A4 ) )
% 3.73/1.84 & ! [R3: A,S2: A] :
% 3.73/1.84 ( ( R4 @ R3 @ S2 )
% 3.73/1.84 = ( ( R4 @ R3 @ R3 )
% 3.73/1.84 & ( R4 @ S2 @ S2 )
% 3.73/1.84 & ( ( Abs2 @ R3 )
% 3.73/1.84 = ( Abs2 @ S2 ) ) ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_105_rep__abs__rsp,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
% 3.73/1.84 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( ( R @ X1 @ X22 )
% 3.73/1.84 => ( R @ X1 @ ( Rep @ ( Abs @ X22 ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_106_equals__rsp,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,Xa: A,Xb: A,Ya: A,Yb: A] :
% 3.73/1.84 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 3.73/1.84 => ( ( R @ Xa @ Xb )
% 3.73/1.84 => ( ( R @ Ya @ Yb )
% 3.73/1.84 => ( ( R @ Xa @ Ya )
% 3.73/1.84 = ( R @ Xb @ Yb ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_107_Quotient3I,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,Abs: B > A,Rep: A > B,R: B > B > $o] :
% 3.73/1.84 ( ! [A6: A] :
% 3.73/1.84 ( ( Abs @ ( Rep @ A6 ) )
% 3.73/1.84 = A6 )
% 3.73/1.84 => ( ! [A6: A] : ( R @ ( Rep @ A6 ) @ ( Rep @ A6 ) )
% 3.73/1.84 => ( ! [R5: B,S4: B] :
% 3.73/1.84 ( ( R @ R5 @ S4 )
% 3.73/1.84 = ( ( R @ R5 @ R5 )
% 3.73/1.84 & ( R @ S4 @ S4 )
% 3.73/1.84 & ( ( Abs @ R5 )
% 3.73/1.84 = ( Abs @ S4 ) ) ) )
% 3.73/1.84 => ( quotient3 @ B @ A @ R @ Abs @ Rep ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_108_cond__prs,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,R: A > A > $o,Absf: A > B,Repf: B > A,A2: $o,B3: B,C3: B] :
% 3.73/1.84 ( ( quotient3 @ A @ B @ R @ Absf @ Repf )
% 3.73/1.84 => ( ( A2
% 3.73/1.84 => ( ( Absf @ ( if @ A @ A2 @ ( Repf @ B3 ) @ ( Repf @ C3 ) ) )
% 3.73/1.84 = B3 ) )
% 3.73/1.84 & ( ~ A2
% 3.73/1.84 => ( ( Absf @ ( if @ A @ A2 @ ( Repf @ B3 ) @ ( Repf @ C3 ) ) )
% 3.73/1.84 = C3 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_109_bijection_Oeq__iff,axiom,
% 3.73/1.84 ! [A: $tType,F3: A > A,A2: A,B3: A] :
% 3.73/1.84 ( ( hilbert_bijection @ A @ F3 )
% 3.73/1.84 => ( ( ( F3 @ A2 )
% 3.73/1.84 = ( F3 @ B3 ) )
% 3.73/1.84 = ( A2 = B3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_110_bijection_OeqI,axiom,
% 3.73/1.84 ! [A: $tType,F3: A > A,A2: A,B3: A] :
% 3.73/1.84 ( ( hilbert_bijection @ A @ F3 )
% 3.73/1.84 => ( ( ( F3 @ A2 )
% 3.73/1.84 = ( F3 @ B3 ) )
% 3.73/1.84 => ( A2 = B3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_111_mk__disjoint__insert,axiom,
% 3.73/1.84 ! [A: $tType,A2: A,A3: set @ A] :
% 3.73/1.84 ( ( member @ A @ A2 @ A3 )
% 3.73/1.84 => ? [B5: set @ A] :
% 3.73/1.84 ( ( A3
% 3.73/1.84 = ( insert @ A @ A2 @ B5 ) )
% 3.73/1.84 & ~ ( member @ A @ A2 @ B5 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_112_insert__commute,axiom,
% 3.73/1.84 ! [A: $tType,X2: A,Y4: A,A3: set @ A] :
% 3.73/1.84 ( ( insert @ A @ X2 @ ( insert @ A @ Y4 @ A3 ) )
% 3.73/1.84 = ( insert @ A @ Y4 @ ( insert @ A @ X2 @ A3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_113_insert__Collect,axiom,
% 3.73/1.84 ! [A: $tType,A2: A,P: A > $o] :
% 3.73/1.84 ( ( insert @ A @ A2 @ ( collect @ A @ P ) )
% 3.73/1.84 = ( collect @ A
% 3.73/1.84 @ ^ [U: A] :
% 3.73/1.84 ( ( U != A2 )
% 3.73/1.84 => ( P @ U ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_114_insert__eq__iff,axiom,
% 3.73/1.84 ! [A: $tType,A2: A,A3: set @ A,B3: A,B2: set @ A] :
% 3.73/1.84 ( ~ ( member @ A @ A2 @ A3 )
% 3.73/1.84 => ( ~ ( member @ A @ B3 @ B2 )
% 3.73/1.84 => ( ( ( insert @ A @ A2 @ A3 )
% 3.73/1.84 = ( insert @ A @ B3 @ B2 ) )
% 3.73/1.84 = ( ( ( A2 = B3 )
% 3.73/1.84 => ( A3 = B2 ) )
% 3.73/1.84 & ( ( A2 != B3 )
% 3.73/1.84 => ? [C4: set @ A] :
% 3.73/1.84 ( ( A3
% 3.73/1.84 = ( insert @ A @ B3 @ C4 ) )
% 3.73/1.84 & ~ ( member @ A @ B3 @ C4 )
% 3.73/1.84 & ( B2
% 3.73/1.84 = ( insert @ A @ A2 @ C4 ) )
% 3.73/1.84 & ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_115_insert__absorb,axiom,
% 3.73/1.84 ! [A: $tType,A2: A,A3: set @ A] :
% 3.73/1.84 ( ( member @ A @ A2 @ A3 )
% 3.73/1.84 => ( ( insert @ A @ A2 @ A3 )
% 3.73/1.84 = A3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_116_insert__ident,axiom,
% 3.73/1.84 ! [A: $tType,X2: A,A3: set @ A,B2: set @ A] :
% 3.73/1.84 ( ~ ( member @ A @ X2 @ A3 )
% 3.73/1.84 => ( ~ ( member @ A @ X2 @ B2 )
% 3.73/1.84 => ( ( ( insert @ A @ X2 @ A3 )
% 3.73/1.84 = ( insert @ A @ X2 @ B2 ) )
% 3.73/1.84 = ( A3 = B2 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_117_insert__compr,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( insert @ A )
% 3.73/1.84 = ( ^ [A4: A,B6: set @ A] :
% 3.73/1.84 ( collect @ A
% 3.73/1.84 @ ^ [X4: A] :
% 3.73/1.84 ( ( X4 = A4 )
% 3.73/1.84 | ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_118_Set_Oset__insert,axiom,
% 3.73/1.84 ! [A: $tType,X2: A,A3: set @ A] :
% 3.73/1.84 ( ( member @ A @ X2 @ A3 )
% 3.73/1.84 => ~ ! [B5: set @ A] :
% 3.73/1.84 ( ( A3
% 3.73/1.84 = ( insert @ A @ X2 @ B5 ) )
% 3.73/1.84 => ( member @ A @ X2 @ B5 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_119_insertI2,axiom,
% 3.73/1.84 ! [A: $tType,A2: A,B2: set @ A,B3: A] :
% 3.73/1.84 ( ( member @ A @ A2 @ B2 )
% 3.73/1.84 => ( member @ A @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_120_insertI1,axiom,
% 3.73/1.84 ! [A: $tType,A2: A,B2: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B2 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_121_insertE,axiom,
% 3.73/1.84 ! [A: $tType,A2: A,B3: A,A3: set @ A] :
% 3.73/1.84 ( ( member @ A @ A2 @ ( insert @ A @ B3 @ A3 ) )
% 3.73/1.84 => ( ( A2 != B3 )
% 3.73/1.84 => ( member @ A @ A2 @ A3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_122_top__empty__eq,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( top_top @ ( A > $o ) )
% 3.73/1.84 = ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_123_top__set__def,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( top_top @ ( set @ A ) )
% 3.73/1.84 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_124_singleton__inject,axiom,
% 3.73/1.84 ! [A: $tType,A2: A,B3: A] :
% 3.73/1.84 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 3.73/1.84 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.73/1.84 => ( A2 = B3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_125_insert__not__empty,axiom,
% 3.73/1.84 ! [A: $tType,A2: A,A3: set @ A] :
% 3.73/1.84 ( ( insert @ A @ A2 @ A3 )
% 3.73/1.84 != ( bot_bot @ ( set @ A ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_126_doubleton__eq__iff,axiom,
% 3.73/1.84 ! [A: $tType,A2: A,B3: A,C3: A,D2: A] :
% 3.73/1.84 ( ( ( insert @ A @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.73/1.84 = ( insert @ A @ C3 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 3.73/1.84 = ( ( ( A2 = C3 )
% 3.73/1.84 & ( B3 = D2 ) )
% 3.73/1.84 | ( ( A2 = D2 )
% 3.73/1.84 & ( B3 = C3 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_127_singleton__iff,axiom,
% 3.73/1.84 ! [A: $tType,B3: A,A2: A] :
% 3.73/1.84 ( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.73/1.84 = ( B3 = A2 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_128_singletonD,axiom,
% 3.73/1.84 ! [A: $tType,B3: A,A2: A] :
% 3.73/1.84 ( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.73/1.84 => ( B3 = A2 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_129_insert__UNIV,axiom,
% 3.73/1.84 ! [A: $tType,X2: A] :
% 3.73/1.84 ( ( insert @ A @ X2 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 = ( top_top @ ( set @ A ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_130_Collect__conv__if,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o,A2: A] :
% 3.73/1.84 ( ( ( P @ A2 )
% 3.73/1.84 => ( ( collect @ A
% 3.73/1.84 @ ^ [X4: A] :
% 3.73/1.84 ( ( X4 = A2 )
% 3.73/1.84 & ( P @ X4 ) ) )
% 3.73/1.84 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 3.73/1.84 & ( ~ ( P @ A2 )
% 3.73/1.84 => ( ( collect @ A
% 3.73/1.84 @ ^ [X4: A] :
% 3.73/1.84 ( ( X4 = A2 )
% 3.73/1.84 & ( P @ X4 ) ) )
% 3.73/1.84 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_131_Collect__conv__if2,axiom,
% 3.73/1.84 ! [A: $tType,P: A > $o,A2: A] :
% 3.73/1.84 ( ( ( P @ A2 )
% 3.73/1.84 => ( ( collect @ A
% 3.73/1.84 @ ^ [X4: A] :
% 3.73/1.84 ( ( A2 = X4 )
% 3.73/1.84 & ( P @ X4 ) ) )
% 3.73/1.84 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 3.73/1.84 & ( ~ ( P @ A2 )
% 3.73/1.84 => ( ( collect @ A
% 3.73/1.84 @ ^ [X4: A] :
% 3.73/1.84 ( ( A2 = X4 )
% 3.73/1.84 & ( P @ X4 ) ) )
% 3.73/1.84 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_132_inv__imagep__def,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType] :
% 3.73/1.84 ( ( inv_imagep @ B @ A )
% 3.73/1.84 = ( ^ [R3: B > B > $o,F: A > B,X4: A,Y: A] : ( R3 @ ( F @ X4 ) @ ( F @ Y ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_133_is__singleton__def,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( is_singleton @ A )
% 3.73/1.84 = ( ^ [A5: set @ A] :
% 3.73/1.84 ? [X4: A] :
% 3.73/1.84 ( A5
% 3.73/1.84 = ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_134_is__singletonE,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A] :
% 3.73/1.84 ( ( is_singleton @ A @ A3 )
% 3.73/1.84 => ~ ! [X: A] :
% 3.73/1.84 ( A3
% 3.73/1.84 != ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_135_bijection_Oeq__invI,axiom,
% 3.73/1.84 ! [A: $tType,F3: A > A,A2: A,B3: A] :
% 3.73/1.84 ( ( hilbert_bijection @ A @ F3 )
% 3.73/1.84 => ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ A2 )
% 3.73/1.84 = ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ B3 ) )
% 3.73/1.84 => ( A2 = B3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_136_bijection_Oinv__left,axiom,
% 3.73/1.84 ! [A: $tType,F3: A > A,A2: A] :
% 3.73/1.84 ( ( hilbert_bijection @ A @ F3 )
% 3.73/1.84 => ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ ( F3 @ A2 ) )
% 3.73/1.84 = A2 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_137_bijection_Oinv__right,axiom,
% 3.73/1.84 ! [A: $tType,F3: A > A,A2: A] :
% 3.73/1.84 ( ( hilbert_bijection @ A @ F3 )
% 3.73/1.84 => ( ( F3 @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ A2 ) )
% 3.73/1.84 = A2 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_138_is__singleton__the__elem,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( is_singleton @ A )
% 3.73/1.84 = ( ^ [A5: set @ A] :
% 3.73/1.84 ( A5
% 3.73/1.84 = ( insert @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_139_the__elem__eq,axiom,
% 3.73/1.84 ! [A: $tType,X2: A] :
% 3.73/1.84 ( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.73/1.84 = X2 ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_140_range__constant,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,X2: A] :
% 3.73/1.84 ( ( image @ B @ A
% 3.73/1.84 @ ^ [Uu: B] : X2
% 3.73/1.84 @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_141_bijection_Osurj__inv,axiom,
% 3.73/1.84 ! [A: $tType,F3: A > A] :
% 3.73/1.84 ( ( hilbert_bijection @ A @ F3 )
% 3.73/1.84 => ( ( image @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_142_bijection_Oinj__inv,axiom,
% 3.73/1.84 ! [A: $tType,F3: A > A] :
% 3.73/1.84 ( ( hilbert_bijection @ A @ F3 )
% 3.73/1.84 => ( inj_on @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_143_bijection_Obij__inv,axiom,
% 3.73/1.84 ! [A: $tType,F3: A > A] :
% 3.73/1.84 ( ( hilbert_bijection @ A @ F3 )
% 3.73/1.84 => ( bij_betw @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_144_image__eqI,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,B3: A,F3: B > A,X2: B,A3: set @ B] :
% 3.73/1.84 ( ( B3
% 3.73/1.84 = ( F3 @ X2 ) )
% 3.73/1.84 => ( ( member @ B @ X2 @ A3 )
% 3.73/1.84 => ( member @ A @ B3 @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_145_image__ident,axiom,
% 3.73/1.84 ! [A: $tType,Y6: set @ A] :
% 3.73/1.84 ( ( image @ A @ A
% 3.73/1.84 @ ^ [X4: A] : X4
% 3.73/1.84 @ Y6 )
% 3.73/1.84 = Y6 ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_146_top1I,axiom,
% 3.73/1.84 ! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_147_image__is__empty,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
% 3.73/1.84 ( ( ( image @ B @ A @ F3 @ A3 )
% 3.73/1.84 = ( bot_bot @ ( set @ A ) ) )
% 3.73/1.84 = ( A3
% 3.73/1.84 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_148_empty__is__image,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
% 3.73/1.84 ( ( ( bot_bot @ ( set @ A ) )
% 3.73/1.84 = ( image @ B @ A @ F3 @ A3 ) )
% 3.73/1.84 = ( A3
% 3.73/1.84 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_149_image__empty,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: B > A] :
% 3.73/1.84 ( ( image @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) )
% 3.73/1.84 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_150_image__insert,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: B > A,A2: B,B2: set @ B] :
% 3.73/1.84 ( ( image @ B @ A @ F3 @ ( insert @ B @ A2 @ B2 ) )
% 3.73/1.84 = ( insert @ A @ ( F3 @ A2 ) @ ( image @ B @ A @ F3 @ B2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_151_insert__image,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,X2: A,A3: set @ A,F3: A > B] :
% 3.73/1.84 ( ( member @ A @ X2 @ A3 )
% 3.73/1.84 => ( ( insert @ B @ ( F3 @ X2 ) @ ( image @ A @ B @ F3 @ A3 ) )
% 3.73/1.84 = ( image @ A @ B @ F3 @ A3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_152_inv__into__f__f,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X2: A] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ A3 )
% 3.73/1.84 => ( ( member @ A @ X2 @ A3 )
% 3.73/1.84 => ( ( hilbert_inv_into @ A @ B @ A3 @ F3 @ ( F3 @ X2 ) )
% 3.73/1.84 = X2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_153_inj__imp__bij__betw__inv,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,M: set @ A] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( image @ A @ B @ F3 @ M ) @ M ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_154_inj__transfer,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,P: A > $o,X2: A] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( ! [Y3: B] :
% 3.73/1.84 ( ( member @ B @ Y3 @ ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) )
% 3.73/1.84 => ( P @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 @ Y3 ) ) )
% 3.73/1.84 => ( P @ X2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_155_image__inv__f__f,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( image @ A @ B @ F3 @ A3 ) )
% 3.73/1.84 = A3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_156_inj__imp__surj__inv,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_157_surj__imp__inj__inv,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: B > A] :
% 3.73/1.84 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 = ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_158_bijI_H,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: A > B] :
% 3.73/1.84 ( ! [X: A,Y3: A] :
% 3.73/1.84 ( ( ( F3 @ X )
% 3.73/1.84 = ( F3 @ Y3 ) )
% 3.73/1.84 = ( X = Y3 ) )
% 3.73/1.84 => ( ! [Y3: B] :
% 3.73/1.84 ? [X5: A] :
% 3.73/1.84 ( Y3
% 3.73/1.84 = ( F3 @ X5 ) )
% 3.73/1.84 => ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_159_imageE,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,B3: A,F3: B > A,A3: set @ B] :
% 3.73/1.84 ( ( member @ A @ B3 @ ( image @ B @ A @ F3 @ A3 ) )
% 3.73/1.84 => ~ ! [X: B] :
% 3.73/1.84 ( ( B3
% 3.73/1.84 = ( F3 @ X ) )
% 3.73/1.84 => ~ ( member @ B @ X @ A3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_160_imageI,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,X2: A,A3: set @ A,F3: A > B] :
% 3.73/1.84 ( ( member @ A @ X2 @ A3 )
% 3.73/1.84 => ( member @ B @ ( F3 @ X2 ) @ ( image @ A @ B @ F3 @ A3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_161_image__iff,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,Z2: A,F3: B > A,A3: set @ B] :
% 3.73/1.84 ( ( member @ A @ Z2 @ ( image @ B @ A @ F3 @ A3 ) )
% 3.73/1.84 = ( ? [X4: B] :
% 3.73/1.84 ( ( member @ B @ X4 @ A3 )
% 3.73/1.84 & ( Z2
% 3.73/1.84 = ( F3 @ X4 ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_162_bex__imageD,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P: A > $o] :
% 3.73/1.84 ( ? [X5: A] :
% 3.73/1.84 ( ( member @ A @ X5 @ ( image @ B @ A @ F3 @ A3 ) )
% 3.73/1.84 & ( P @ X5 ) )
% 3.73/1.84 => ? [X: B] :
% 3.73/1.84 ( ( member @ B @ X @ A3 )
% 3.73/1.84 & ( P @ ( F3 @ X ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_163_image__cong,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F3: A > B,G: A > B] :
% 3.73/1.84 ( ( M = N )
% 3.73/1.84 => ( ! [X: A] :
% 3.73/1.84 ( ( member @ A @ X @ N )
% 3.73/1.84 => ( ( F3 @ X )
% 3.73/1.84 = ( G @ X ) ) )
% 3.73/1.84 => ( ( image @ A @ B @ F3 @ M )
% 3.73/1.84 = ( image @ A @ B @ G @ N ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_164_ball__imageD,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P: A > $o] :
% 3.73/1.84 ( ! [X: A] :
% 3.73/1.84 ( ( member @ A @ X @ ( image @ B @ A @ F3 @ A3 ) )
% 3.73/1.84 => ( P @ X ) )
% 3.73/1.84 => ! [X5: B] :
% 3.73/1.84 ( ( member @ B @ X5 @ A3 )
% 3.73/1.84 => ( P @ ( F3 @ X5 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_165_image__image,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,C: $tType,F3: B > A,G: C > B,A3: set @ C] :
% 3.73/1.84 ( ( image @ B @ A @ F3 @ ( image @ C @ B @ G @ A3 ) )
% 3.73/1.84 = ( image @ C @ A
% 3.73/1.84 @ ^ [X4: C] : ( F3 @ ( G @ X4 ) )
% 3.73/1.84 @ A3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_166_rev__image__eqI,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,X2: A,A3: set @ A,B3: B,F3: A > B] :
% 3.73/1.84 ( ( member @ A @ X2 @ A3 )
% 3.73/1.84 => ( ( B3
% 3.73/1.84 = ( F3 @ X2 ) )
% 3.73/1.84 => ( member @ B @ B3 @ ( image @ A @ B @ F3 @ A3 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_167_Compr__image__eq,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P: A > $o] :
% 3.73/1.84 ( ( collect @ A
% 3.73/1.84 @ ^ [X4: A] :
% 3.73/1.84 ( ( member @ A @ X4 @ ( image @ B @ A @ F3 @ A3 ) )
% 3.73/1.84 & ( P @ X4 ) ) )
% 3.73/1.84 = ( image @ B @ A @ F3
% 3.73/1.84 @ ( collect @ B
% 3.73/1.84 @ ^ [X4: B] :
% 3.73/1.84 ( ( member @ B @ X4 @ A3 )
% 3.73/1.84 & ( P @ ( F3 @ X4 ) ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_168_the__elem__image__unique,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,A3: set @ A,F3: A > B,X2: A] :
% 3.73/1.84 ( ( A3
% 3.73/1.84 != ( bot_bot @ ( set @ A ) ) )
% 3.73/1.84 => ( ! [Y3: A] :
% 3.73/1.84 ( ( member @ A @ Y3 @ A3 )
% 3.73/1.84 => ( ( F3 @ Y3 )
% 3.73/1.84 = ( F3 @ X2 ) ) )
% 3.73/1.84 => ( ( the_elem @ B @ ( image @ A @ B @ F3 @ A3 ) )
% 3.73/1.84 = ( F3 @ X2 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_169_rangeI,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: B > A,X2: B] : ( member @ A @ ( F3 @ X2 ) @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_170_range__eqI,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,B3: A,F3: B > A,X2: B] :
% 3.73/1.84 ( ( B3
% 3.73/1.84 = ( F3 @ X2 ) )
% 3.73/1.84 => ( member @ A @ B3 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_171_bij__image__Collect__eq,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: A > B,P: A > $o] :
% 3.73/1.84 ( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 => ( ( image @ A @ B @ F3 @ ( collect @ A @ P ) )
% 3.73/1.84 = ( collect @ B
% 3.73/1.84 @ ^ [Y: B] : ( P @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 @ Y ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_172_bij__betw__inv__into,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,B2: set @ B] :
% 3.73/1.84 ( ( bij_betw @ A @ B @ F3 @ A3 @ B2 )
% 3.73/1.84 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A3 @ F3 ) @ B2 @ A3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_173_inv__into__inv__into__eq,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,A7: set @ B,A2: A] :
% 3.73/1.84 ( ( bij_betw @ A @ B @ F3 @ A3 @ A7 )
% 3.73/1.84 => ( ( member @ A @ A2 @ A3 )
% 3.73/1.84 => ( ( hilbert_inv_into @ B @ A @ A7 @ ( hilbert_inv_into @ A @ B @ A3 @ F3 ) @ A2 )
% 3.73/1.84 = ( F3 @ A2 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_174_bij__betw__inv__into__left,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,A7: set @ B,A2: A] :
% 3.73/1.84 ( ( bij_betw @ A @ B @ F3 @ A3 @ A7 )
% 3.73/1.84 => ( ( member @ A @ A2 @ A3 )
% 3.73/1.84 => ( ( hilbert_inv_into @ A @ B @ A3 @ F3 @ ( F3 @ A2 ) )
% 3.73/1.84 = A2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_175_bij__betw__inv__into__right,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: A > B,A3: set @ A,A7: set @ B,A8: B] :
% 3.73/1.84 ( ( bij_betw @ A @ B @ F3 @ A3 @ A7 )
% 3.73/1.84 => ( ( member @ B @ A8 @ A7 )
% 3.73/1.84 => ( ( F3 @ ( hilbert_inv_into @ A @ B @ A3 @ F3 @ A8 ) )
% 3.73/1.84 = A8 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_176_inv__into__f__eq,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X2: A,Y4: B] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ A3 )
% 3.73/1.84 => ( ( member @ A @ X2 @ A3 )
% 3.73/1.84 => ( ( ( F3 @ X2 )
% 3.73/1.84 = Y4 )
% 3.73/1.84 => ( ( hilbert_inv_into @ A @ B @ A3 @ F3 @ Y4 )
% 3.73/1.84 = X2 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_177_f__inv__into__f,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,Y4: A,F3: B > A,A3: set @ B] :
% 3.73/1.84 ( ( member @ A @ Y4 @ ( image @ B @ A @ F3 @ A3 ) )
% 3.73/1.84 => ( ( F3 @ ( hilbert_inv_into @ B @ A @ A3 @ F3 @ Y4 ) )
% 3.73/1.84 = Y4 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_178_inv__into__into,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,X2: A,F3: B > A,A3: set @ B] :
% 3.73/1.84 ( ( member @ A @ X2 @ ( image @ B @ A @ F3 @ A3 ) )
% 3.73/1.84 => ( member @ B @ ( hilbert_inv_into @ B @ A @ A3 @ F3 @ X2 ) @ A3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_179_inv__into__injective,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,A3: set @ A,F3: A > B,X2: B,Y4: B] :
% 3.73/1.84 ( ( ( hilbert_inv_into @ A @ B @ A3 @ F3 @ X2 )
% 3.73/1.84 = ( hilbert_inv_into @ A @ B @ A3 @ F3 @ Y4 ) )
% 3.73/1.84 => ( ( member @ B @ X2 @ ( image @ A @ B @ F3 @ A3 ) )
% 3.73/1.84 => ( ( member @ B @ Y4 @ ( image @ A @ B @ F3 @ A3 ) )
% 3.73/1.84 => ( X2 = Y4 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_180_setcompr__eq__image,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: B > A,P: B > $o] :
% 3.73/1.84 ( ( collect @ A
% 3.73/1.84 @ ^ [Uu: A] :
% 3.73/1.84 ? [X4: B] :
% 3.73/1.84 ( ( Uu
% 3.73/1.84 = ( F3 @ X4 ) )
% 3.73/1.84 & ( P @ X4 ) ) )
% 3.73/1.84 = ( image @ B @ A @ F3 @ ( collect @ B @ P ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_181_Setcompr__eq__image,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
% 3.73/1.84 ( ( collect @ A
% 3.73/1.84 @ ^ [Uu: A] :
% 3.73/1.84 ? [X4: B] :
% 3.73/1.84 ( ( Uu
% 3.73/1.84 = ( F3 @ X4 ) )
% 3.73/1.84 & ( member @ B @ X4 @ A3 ) ) )
% 3.73/1.84 = ( image @ B @ A @ F3 @ A3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_182_rangeE,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,B3: A,F3: B > A] :
% 3.73/1.84 ( ( member @ A @ B3 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
% 3.73/1.84 => ~ ! [X: B] :
% 3.73/1.84 ( B3
% 3.73/1.84 != ( F3 @ X ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_183_range__composition,axiom,
% 3.73/1.84 ! [A: $tType,C: $tType,B: $tType,F3: C > A,G: B > C] :
% 3.73/1.84 ( ( image @ B @ A
% 3.73/1.84 @ ^ [X4: B] : ( F3 @ ( G @ X4 ) )
% 3.73/1.84 @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 = ( image @ C @ A @ F3 @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_184_bij__imp__bij__inv,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B] :
% 3.73/1.84 ( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 => ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_185_bij__inv__eq__iff,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,P6: A > B,X2: A,Y4: B] :
% 3.73/1.84 ( ( bij_betw @ A @ B @ P6 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 => ( ( X2
% 3.73/1.84 = ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P6 @ Y4 ) )
% 3.73/1.84 = ( ( P6 @ X2 )
% 3.73/1.84 = Y4 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_186_inv__inv__eq,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B] :
% 3.73/1.84 ( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 => ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) )
% 3.73/1.84 = F3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_187_inj__imp__inv__eq,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: A > B,G: B > A] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( ! [X: B] :
% 3.73/1.84 ( ( F3 @ ( G @ X ) )
% 3.73/1.84 = X )
% 3.73/1.84 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
% 3.73/1.84 = G ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_188_inv__f__eq,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,X2: A,Y4: B] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( ( ( F3 @ X2 )
% 3.73/1.84 = Y4 )
% 3.73/1.84 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 @ Y4 )
% 3.73/1.84 = X2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_189_inv__f__f,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,X2: A] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 @ ( F3 @ X2 ) )
% 3.73/1.84 = X2 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_190_bijection_Ointro,axiom,
% 3.73/1.84 ! [A: $tType,F3: A > A] :
% 3.73/1.84 ( ( bij_betw @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( hilbert_bijection @ A @ F3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_191_bijection__def,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( hilbert_bijection @ A )
% 3.73/1.84 = ( ^ [F: A > A] : ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_192_bijection_Obij,axiom,
% 3.73/1.84 ! [A: $tType,F3: A > A] :
% 3.73/1.84 ( ( hilbert_bijection @ A @ F3 )
% 3.73/1.84 => ( bij_betw @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_193_surj__imp__inv__eq,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: B > A,G: A > B] :
% 3.73/1.84 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 = ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( ! [X: B] :
% 3.73/1.84 ( ( G @ ( F3 @ X ) )
% 3.73/1.84 = X )
% 3.73/1.84 => ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 )
% 3.73/1.84 = G ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_194_image__f__inv__f,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: B > A,A3: set @ A] :
% 3.73/1.84 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 = ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( ( image @ B @ A @ F3 @ ( image @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 ) @ A3 ) )
% 3.73/1.84 = A3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_195_surj__iff__all,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: B > A] :
% 3.73/1.84 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 = ( top_top @ ( set @ A ) ) )
% 3.73/1.84 = ( ! [X4: A] :
% 3.73/1.84 ( ( F3 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 @ X4 ) )
% 3.73/1.84 = X4 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_196_surj__f__inv__f,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: B > A,Y4: A] :
% 3.73/1.84 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 = ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( ( F3 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 @ Y4 ) )
% 3.73/1.84 = Y4 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_197_bijection_Oinj,axiom,
% 3.73/1.84 ! [A: $tType,F3: A > A] :
% 3.73/1.84 ( ( hilbert_bijection @ A @ F3 )
% 3.73/1.84 => ( inj_on @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_198_bijection_Osurj,axiom,
% 3.73/1.84 ! [A: $tType,F3: A > A] :
% 3.73/1.84 ( ( hilbert_bijection @ A @ F3 )
% 3.73/1.84 => ( ( image @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_199_image__constant,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,X2: A,A3: set @ A,C3: B] :
% 3.73/1.84 ( ( member @ A @ X2 @ A3 )
% 3.73/1.84 => ( ( image @ A @ B
% 3.73/1.84 @ ^ [X4: A] : C3
% 3.73/1.84 @ A3 )
% 3.73/1.84 = ( insert @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_200_image__constant__conv,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,A3: set @ B,C3: A] :
% 3.73/1.84 ( ( ( A3
% 3.73/1.84 = ( bot_bot @ ( set @ B ) ) )
% 3.73/1.84 => ( ( image @ B @ A
% 3.73/1.84 @ ^ [X4: B] : C3
% 3.73/1.84 @ A3 )
% 3.73/1.84 = ( bot_bot @ ( set @ A ) ) ) )
% 3.73/1.84 & ( ( A3
% 3.73/1.84 != ( bot_bot @ ( set @ B ) ) )
% 3.73/1.84 => ( ( image @ B @ A
% 3.73/1.84 @ ^ [X4: B] : C3
% 3.73/1.84 @ A3 )
% 3.73/1.84 = ( insert @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_201_full__SetCompr__eq,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: B > A] :
% 3.73/1.84 ( ( collect @ A
% 3.73/1.84 @ ^ [U: A] :
% 3.73/1.84 ? [X4: B] :
% 3.73/1.84 ( U
% 3.73/1.84 = ( F3 @ X4 ) ) )
% 3.73/1.84 = ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_202_range__eq__singletonD,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: B > A,A2: A,X2: B] :
% 3.73/1.84 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.73/1.84 => ( ( F3 @ X2 )
% 3.73/1.84 = A2 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_203_inj__on__empty,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B] : ( inj_on @ A @ B @ F3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_204_bij__def,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: A > B] :
% 3.73/1.84 ( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 = ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 & ( ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 = ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_205_bijI,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: A > B] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( ( ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 = ( top_top @ ( set @ B ) ) )
% 3.73/1.84 => ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_206_bij__is__inj,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B] :
% 3.73/1.84 ( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 => ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_207_inj__on__id2,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A] :
% 3.73/1.84 ( inj_on @ A @ A
% 3.73/1.84 @ ^ [X4: A] : X4
% 3.73/1.84 @ A3 ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_208_surjD,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: B > A,Y4: A] :
% 3.73/1.84 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 = ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ? [X: B] :
% 3.73/1.84 ( Y4
% 3.73/1.84 = ( F3 @ X ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_209_surjE,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: B > A,Y4: A] :
% 3.73/1.84 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 = ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ~ ! [X: B] :
% 3.73/1.84 ( Y4
% 3.73/1.84 != ( F3 @ X ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_210_surjI,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,G: B > A,F3: A > B] :
% 3.73/1.84 ( ! [X: A] :
% 3.73/1.84 ( ( G @ ( F3 @ X ) )
% 3.73/1.84 = X )
% 3.73/1.84 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_211_surj__def,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: B > A] :
% 3.73/1.84 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 = ( top_top @ ( set @ A ) ) )
% 3.73/1.84 = ( ! [Y: A] :
% 3.73/1.84 ? [X4: B] :
% 3.73/1.84 ( Y
% 3.73/1.84 = ( F3 @ X4 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_212_injD,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,X2: A,Y4: A] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( ( ( F3 @ X2 )
% 3.73/1.84 = ( F3 @ Y4 ) )
% 3.73/1.84 => ( X2 = Y4 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_213_injI,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B] :
% 3.73/1.84 ( ! [X: A,Y3: A] :
% 3.73/1.84 ( ( ( F3 @ X )
% 3.73/1.84 = ( F3 @ Y3 ) )
% 3.73/1.84 => ( X = Y3 ) )
% 3.73/1.84 => ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_214_inj__eq,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,X2: A,Y4: A] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( ( ( F3 @ X2 )
% 3.73/1.84 = ( F3 @ Y4 ) )
% 3.73/1.84 = ( X2 = Y4 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_215_inj__def,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 = ( ! [X4: A,Y: A] :
% 3.73/1.84 ( ( ( F3 @ X4 )
% 3.73/1.84 = ( F3 @ Y ) )
% 3.73/1.84 => ( X4 = Y ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_216_bij__betw__empty2,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A] :
% 3.73/1.84 ( ( bij_betw @ A @ B @ F3 @ A3 @ ( bot_bot @ ( set @ B ) ) )
% 3.73/1.84 => ( A3
% 3.73/1.84 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_217_bij__betw__empty1,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: A > B,A3: set @ B] :
% 3.73/1.84 ( ( bij_betw @ A @ B @ F3 @ ( bot_bot @ ( set @ A ) ) @ A3 )
% 3.73/1.84 => ( A3
% 3.73/1.84 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_218_bij__pointE,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,Y4: B] :
% 3.73/1.84 ( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 => ~ ! [X: A] :
% 3.73/1.84 ( ( Y4
% 3.73/1.84 = ( F3 @ X ) )
% 3.73/1.84 => ~ ! [X8: A] :
% 3.73/1.84 ( ( Y4
% 3.73/1.84 = ( F3 @ X8 ) )
% 3.73/1.84 => ( X8 = X ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_219_inj__singleton,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A] :
% 3.73/1.84 ( inj_on @ A @ ( set @ A )
% 3.73/1.84 @ ^ [X4: A] : ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
% 3.73/1.84 @ A3 ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_220_inj__fun,axiom,
% 3.73/1.84 ! [B: $tType,C: $tType,A: $tType,F3: A > B] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( inj_on @ A @ ( C > B )
% 3.73/1.84 @ ^ [X4: A,Y: C] : ( F3 @ X4 )
% 3.73/1.84 @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_221_inj__img__insertE,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X2: B,B2: set @ B] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ A3 )
% 3.73/1.84 => ( ~ ( member @ B @ X2 @ B2 )
% 3.73/1.84 => ( ( ( insert @ B @ X2 @ B2 )
% 3.73/1.84 = ( image @ A @ B @ F3 @ A3 ) )
% 3.73/1.84 => ~ ! [X9: A,A9: set @ A] :
% 3.73/1.84 ( ~ ( member @ A @ X9 @ A9 )
% 3.73/1.84 => ( ( A3
% 3.73/1.84 = ( insert @ A @ X9 @ A9 ) )
% 3.73/1.84 => ( ( X2
% 3.73/1.84 = ( F3 @ X9 ) )
% 3.73/1.84 => ( B2
% 3.73/1.84 != ( image @ A @ B @ F3 @ A9 ) ) ) ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_222_range__ex1__eq,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,B3: B] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( ( member @ B @ B3 @ ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) )
% 3.73/1.84 = ( ? [X4: A] :
% 3.73/1.84 ( ( B3
% 3.73/1.84 = ( F3 @ X4 ) )
% 3.73/1.84 & ! [Y: A] :
% 3.73/1.84 ( ( B3
% 3.73/1.84 = ( F3 @ Y ) )
% 3.73/1.84 => ( Y = X4 ) ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_223_inj__image__eq__iff,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,B2: set @ A] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( ( ( image @ A @ B @ F3 @ A3 )
% 3.73/1.84 = ( image @ A @ B @ F3 @ B2 ) )
% 3.73/1.84 = ( A3 = B2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_224_inj__image__mem__iff,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,A2: A,A3: set @ A] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( ( member @ B @ ( F3 @ A2 ) @ ( image @ A @ B @ F3 @ A3 ) )
% 3.73/1.84 = ( member @ A @ A2 @ A3 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_225_bij__is__surj,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: A > B] :
% 3.73/1.84 ( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 => ( ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 = ( top_top @ ( set @ B ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_226_bij__betw__imp__surj,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,F3: A > B,A3: set @ A] :
% 3.73/1.84 ( ( bij_betw @ A @ B @ F3 @ A3 @ ( top_top @ ( set @ B ) ) )
% 3.73/1.84 => ( ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 = ( top_top @ ( set @ B ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_227_inj__on__insert,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,A2: A,A3: set @ A] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( insert @ A @ A2 @ A3 ) )
% 3.73/1.84 = ( ( inj_on @ A @ B @ F3 @ A3 )
% 3.73/1.84 & ~ ( member @ B @ ( F3 @ A2 ) @ ( image @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_228_top__conj_I2_J,axiom,
% 3.73/1.84 ! [A: $tType,P: $o,X2: A] :
% 3.73/1.84 ( ( P
% 3.73/1.84 & ( top_top @ ( A > $o ) @ X2 ) )
% 3.73/1.84 = P ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_229_DiffI,axiom,
% 3.73/1.84 ! [A: $tType,C3: A,A3: set @ A,B2: set @ A] :
% 3.73/1.84 ( ( member @ A @ C3 @ A3 )
% 3.73/1.84 => ( ~ ( member @ A @ C3 @ B2 )
% 3.73/1.84 => ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_230_Diff__iff,axiom,
% 3.73/1.84 ! [A: $tType,C3: A,A3: set @ A,B2: set @ A] :
% 3.73/1.84 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
% 3.73/1.84 = ( ( member @ A @ C3 @ A3 )
% 3.73/1.84 & ~ ( member @ A @ C3 @ B2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_231_Diff__idemp,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 3.73/1.84 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ B2 )
% 3.73/1.84 = ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_232_Diff__empty,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A] :
% 3.73/1.84 ( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 3.73/1.84 = A3 ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_233_empty__Diff,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A] :
% 3.73/1.84 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
% 3.73/1.84 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_234_Diff__cancel,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A] :
% 3.73/1.84 ( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
% 3.73/1.84 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_235_insert__Diff1,axiom,
% 3.73/1.84 ! [A: $tType,X2: A,B2: set @ A,A3: set @ A] :
% 3.73/1.84 ( ( member @ A @ X2 @ B2 )
% 3.73/1.84 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B2 )
% 3.73/1.84 = ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_236_Diff__insert0,axiom,
% 3.73/1.84 ! [A: $tType,X2: A,A3: set @ A,B2: set @ A] :
% 3.73/1.84 ( ~ ( member @ A @ X2 @ A3 )
% 3.73/1.84 => ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ B2 ) )
% 3.73/1.84 = ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_237_insert__Diff__single,axiom,
% 3.73/1.84 ! [A: $tType,A2: A,A3: set @ A] :
% 3.73/1.84 ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 3.73/1.84 = ( insert @ A @ A2 @ A3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_238_Diff__UNIV,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A] :
% 3.73/1.84 ( ( minus_minus @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_239_surj__diff__right,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( ab_group_add @ A )
% 3.73/1.84 => ! [A2: A] :
% 3.73/1.84 ( ( image @ A @ A
% 3.73/1.84 @ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
% 3.73/1.84 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_240_inj__diff__right,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( ab_group_add @ A )
% 3.73/1.84 => ! [A2: A] :
% 3.73/1.84 ( inj_on @ A @ A
% 3.73/1.84 @ ^ [B4: A] : ( minus_minus @ A @ B4 @ A2 )
% 3.73/1.84 @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_241_translation__subtract__diff,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( ab_group_add @ A )
% 3.73/1.84 => ! [A2: A,S3: set @ A,T: set @ A] :
% 3.73/1.84 ( ( image @ A @ A
% 3.73/1.84 @ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
% 3.73/1.84 @ ( minus_minus @ ( set @ A ) @ S3 @ T ) )
% 3.73/1.84 = ( minus_minus @ ( set @ A )
% 3.73/1.84 @ ( image @ A @ A
% 3.73/1.84 @ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
% 3.73/1.84 @ S3 )
% 3.73/1.84 @ ( image @ A @ A
% 3.73/1.84 @ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
% 3.73/1.84 @ T ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_242_Diff__insert__absorb,axiom,
% 3.73/1.84 ! [A: $tType,X2: A,A3: set @ A] :
% 3.73/1.84 ( ~ ( member @ A @ X2 @ A3 )
% 3.73/1.84 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.73/1.84 = A3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_243_Diff__insert2,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A,A2: A,B2: set @ A] :
% 3.73/1.84 ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B2 ) )
% 3.73/1.84 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_244_insert__Diff,axiom,
% 3.73/1.84 ! [A: $tType,A2: A,A3: set @ A] :
% 3.73/1.84 ( ( member @ A @ A2 @ A3 )
% 3.73/1.84 => ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 3.73/1.84 = A3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_245_Diff__insert,axiom,
% 3.73/1.84 ! [A: $tType,A3: set @ A,A2: A,B2: set @ A] :
% 3.73/1.84 ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B2 ) )
% 3.73/1.84 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_246_in__image__insert__iff,axiom,
% 3.73/1.84 ! [A: $tType,B2: set @ ( set @ A ),X2: A,A3: set @ A] :
% 3.73/1.84 ( ! [C5: set @ A] :
% 3.73/1.84 ( ( member @ ( set @ A ) @ C5 @ B2 )
% 3.73/1.84 => ~ ( member @ A @ X2 @ C5 ) )
% 3.73/1.84 => ( ( member @ ( set @ A ) @ A3 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X2 ) @ B2 ) )
% 3.73/1.84 = ( ( member @ A @ X2 @ A3 )
% 3.73/1.84 & ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_247_DiffE,axiom,
% 3.73/1.84 ! [A: $tType,C3: A,A3: set @ A,B2: set @ A] :
% 3.73/1.84 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
% 3.73/1.84 => ~ ( ( member @ A @ C3 @ A3 )
% 3.73/1.84 => ( member @ A @ C3 @ B2 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_248_DiffD1,axiom,
% 3.73/1.84 ! [A: $tType,C3: A,A3: set @ A,B2: set @ A] :
% 3.73/1.84 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
% 3.73/1.84 => ( member @ A @ C3 @ A3 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_249_DiffD2,axiom,
% 3.73/1.84 ! [A: $tType,C3: A,A3: set @ A,B2: set @ A] :
% 3.73/1.84 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
% 3.73/1.84 => ~ ( member @ A @ C3 @ B2 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_250_set__diff__eq,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( minus_minus @ ( set @ A ) )
% 3.73/1.84 = ( ^ [A5: set @ A,B6: set @ A] :
% 3.73/1.84 ( collect @ A
% 3.73/1.84 @ ^ [X4: A] :
% 3.73/1.84 ( ( member @ A @ X4 @ A5 )
% 3.73/1.84 & ~ ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_251_insert__Diff__if,axiom,
% 3.73/1.84 ! [A: $tType,X2: A,B2: set @ A,A3: set @ A] :
% 3.73/1.84 ( ( ( member @ A @ X2 @ B2 )
% 3.73/1.84 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B2 )
% 3.73/1.84 = ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) )
% 3.73/1.84 & ( ~ ( member @ A @ X2 @ B2 )
% 3.73/1.84 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B2 )
% 3.73/1.84 = ( insert @ A @ X2 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_252_image__set__diff,axiom,
% 3.73/1.84 ! [B: $tType,A: $tType,F3: A > B,A3: set @ A,B2: set @ A] :
% 3.73/1.84 ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 3.73/1.84 => ( ( image @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
% 3.73/1.84 = ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F3 @ A3 ) @ ( image @ A @ B @ F3 @ B2 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_253_top__conj_I1_J,axiom,
% 3.73/1.84 ! [A: $tType,X2: A,P: $o] :
% 3.73/1.84 ( ( ( top_top @ ( A > $o ) @ X2 )
% 3.73/1.84 & P )
% 3.73/1.84 = P ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_254_remove__def,axiom,
% 3.73/1.84 ! [A: $tType] :
% 3.73/1.84 ( ( remove @ A )
% 3.73/1.84 = ( ^ [X4: A,A5: set @ A] : ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(fact_255_fun__upd__image,axiom,
% 3.73/1.84 ! [A: $tType,B: $tType,X2: B,A3: set @ B,F3: B > A,Y4: A] :
% 3.73/1.84 ( ( ( member @ B @ X2 @ A3 )
% 3.73/1.84 => ( ( image @ B @ A @ ( fun_upd @ B @ A @ F3 @ X2 @ Y4 ) @ A3 )
% 3.73/1.84 = ( insert @ A @ Y4 @ ( image @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
% 3.73/1.84 & ( ~ ( member @ B @ X2 @ A3 )
% 3.73/1.84 => ( ( image @ B @ A @ ( fun_upd @ B @ A @ F3 @ X2 @ Y4 ) @ A3 )
% 3.73/1.84 = ( image @ B @ A @ F3 @ A3 ) ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(tcon_fun___Orderings_Otop,axiom,
% 3.73/1.84 ! [A10: $tType,A11: $tType] :
% 3.73/1.84 ( ( top @ A11 )
% 3.73/1.84 => ( top @ ( A10 > A11 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(tcon_fun___Orderings_Oord,axiom,
% 3.73/1.84 ! [A10: $tType,A11: $tType] :
% 3.73/1.84 ( ( ord @ A11 )
% 3.73/1.84 => ( ord @ ( A10 > A11 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(tcon_fun___Orderings_Obot,axiom,
% 3.73/1.84 ! [A10: $tType,A11: $tType] :
% 3.73/1.84 ( ( bot @ A11 )
% 3.73/1.84 => ( bot @ ( A10 > A11 ) ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(tcon_Set_Oset___Orderings_Otop_1,axiom,
% 3.73/1.84 ! [A10: $tType] : ( top @ ( set @ A10 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(tcon_Set_Oset___Orderings_Oord_2,axiom,
% 3.73/1.84 ! [A10: $tType] : ( ord @ ( set @ A10 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(tcon_Set_Oset___Orderings_Obot_3,axiom,
% 3.73/1.84 ! [A10: $tType] : ( bot @ ( set @ A10 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(tcon_HOL_Obool___Orderings_Otop_4,axiom,
% 3.73/1.84 top @ $o ).
% 3.73/1.84
% 3.73/1.84 thf(tcon_HOL_Obool___Orderings_Oord_5,axiom,
% 3.73/1.84 ord @ $o ).
% 3.73/1.84
% 3.73/1.84 thf(tcon_HOL_Obool___Orderings_Obot_6,axiom,
% 3.73/1.84 bot @ $o ).
% 3.73/1.84
% 3.73/1.84 thf(help_If_3_1_T,axiom,
% 3.73/1.84 ! [P: $o] :
% 3.73/1.84 ( ( P = $true )
% 3.73/1.84 | ( P = $false ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(help_If_2_1_T,axiom,
% 3.73/1.84 ! [A: $tType,X2: A,Y4: A] :
% 3.73/1.84 ( ( if @ A @ $false @ X2 @ Y4 )
% 3.73/1.84 = Y4 ) ).
% 3.73/1.84
% 3.73/1.84 thf(help_If_1_1_T,axiom,
% 3.73/1.84 ! [A: $tType,X2: A,Y4: A] :
% 3.73/1.84 ( ( if @ A @ $true @ X2 @ Y4 )
% 3.73/1.84 = X2 ) ).
% 3.73/1.84
% 3.73/1.84 thf(conj_0,hypothesis,
% 3.73/1.84 ! [X5: a] :
% 3.73/1.84 ( ( p @ X5 )
% 3.73/1.84 => ? [X_12: b] : ( q @ X5 @ X_12 ) ) ).
% 3.73/1.84
% 3.73/1.84 thf(conj_1,conjecture,
% 3.73/1.84 ! [X: a] :
% 3.73/1.84 ( ~ ( p @ X )
% 3.73/1.84 | ( q @ X @ ( hilbert_Eps @ b @ ( q @ X ) ) ) ) ).
% 3.73/1.84 ------------------------
% 3.73/1.88 % (17784)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_17675 for (2999ds/2Mi)
% 3.73/1.88 % (17782)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_17675 for (2999ds/4Mi)
% 3.73/1.88 % (17783)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_17675 for (2999ds/27Mi)
% 3.73/1.88 % (17781)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_17675 for (2999ds/183Mi)
% 3.73/1.88 % (17785)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_17675 for (2999ds/2Mi)
% 3.73/1.88 % (17786)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_17675 for (2999ds/275Mi)
% 3.73/1.88 % (17787)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_17675 for (2999ds/18Mi)
% 3.73/1.88 % (17784)Instruction limit reached!
% 3.73/1.88 % (17784)------------------------------
% 3.73/1.88 % (17784)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.73/1.88 % (17784)Termination reason: Unknown
% 3.73/1.88 % (17784)Termination phase: shuffling
% 3.73/1.88
% 3.73/1.88 % (17784)Memory used [KB]: 1279
% 3.73/1.88 % (17784)Time elapsed: 0.003 s
% 3.73/1.88 % (17784)Instructions burned: 2 (million)
% 3.73/1.88 % (17784)------------------------------
% 3.73/1.88 % (17784)------------------------------
% 3.73/1.88 % (17785)Instruction limit reached!
% 3.73/1.88 % (17785)------------------------------
% 3.73/1.88 % (17785)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.73/1.88 % (17785)Termination reason: Unknown
% 3.73/1.88 % (17785)Termination phase: shuffling
% 3.73/1.88
% 3.73/1.88 % (17785)Memory used [KB]: 1407
% 3.73/1.88 % (17785)Time elapsed: 0.003 s
% 3.73/1.88 % (17785)Instructions burned: 3 (million)
% 3.73/1.88 % (17785)------------------------------
% 3.73/1.88 % (17785)------------------------------
% 3.73/1.88 % (17782)Instruction limit reached!
% 3.73/1.88 % (17782)------------------------------
% 3.73/1.88 % (17782)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.73/1.88 % (17782)Termination reason: Unknown
% 3.73/1.88 % (17782)Termination phase: shuffling
% 3.73/1.88
% 3.73/1.88 % (17782)Memory used [KB]: 1407
% 3.73/1.88 % (17782)Time elapsed: 0.004 s
% 3.73/1.88 % (17782)Instructions burned: 5 (million)
% 3.73/1.88 % (17782)------------------------------
% 3.73/1.88 % (17782)------------------------------
% 3.73/1.89 % (17787)Instruction limit reached!
% 3.73/1.89 % (17787)------------------------------
% 3.73/1.89 % (17787)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.73/1.89 % (17787)Termination reason: Unknown
% 3.73/1.89 % (17787)Termination phase: shuffling
% 3.73/1.89
% 3.73/1.89 % (17787)Memory used [KB]: 1663
% 3.73/1.89 % (17787)Time elapsed: 0.012 s
% 3.73/1.89 % (17787)Instructions burned: 19 (million)
% 3.73/1.89 % (17787)------------------------------
% 3.73/1.89 % (17787)------------------------------
% 3.73/1.89 % (17783)Instruction limit reached!
% 3.73/1.89 % (17783)------------------------------
% 3.73/1.89 % (17783)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.73/1.89 % (17783)Termination reason: Unknown
% 3.73/1.89 % (17783)Termination phase: Property scanning
% 3.73/1.89
% 3.73/1.89 % (17783)Memory used [KB]: 1791
% 3.73/1.89 % (17783)Time elapsed: 0.017 s
% 3.73/1.89 % (17783)Instructions burned: 28 (million)
% 3.73/1.89 % (17783)------------------------------
% 3.73/1.89 % (17783)------------------------------
% 3.79/1.90 % (17788)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_17675 for (2999ds/3Mi)
% 3.79/1.90 % (17789)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_17675 for (2999ds/37Mi)
% 3.79/1.90 % (17790)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_17675 for (2999ds/15Mi)
% 3.79/1.90 % (17788)Instruction limit reached!
% 3.79/1.90 % (17788)------------------------------
% 3.79/1.90 % (17788)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.79/1.90 % (17788)Termination reason: Unknown
% 3.79/1.90 % (17788)Termination phase: shuffling
% 3.79/1.90
% 3.79/1.90 % (17788)Memory used [KB]: 1279
% 3.79/1.90 % (17788)Time elapsed: 0.004 s
% 3.79/1.90 % (17788)Instructions burned: 3 (million)
% 3.79/1.90 % (17788)------------------------------
% 3.79/1.90 % (17788)------------------------------
% 3.79/1.91 % (17791)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_17675 for (2999ds/3Mi)
% 3.79/1.91 % (17790)Instruction limit reached!
% 3.79/1.91 % (17790)------------------------------
% 3.79/1.91 % (17790)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.79/1.91 % (17790)Termination reason: Unknown
% 3.79/1.91 % (17790)Termination phase: shuffling
% 3.79/1.91 % (17786)First to succeed.
% 3.79/1.91
% 3.79/1.91 % (17790)Memory used [KB]: 1663
% 3.79/1.91 % (17790)Time elapsed: 0.011 s
% 3.79/1.91 % (17790)Instructions burned: 16 (million)
% 3.79/1.91 % (17790)------------------------------
% 3.79/1.91 % (17790)------------------------------
% 3.79/1.91 % (17791)Instruction limit reached!
% 3.79/1.91 % (17791)------------------------------
% 3.79/1.91 % (17791)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.79/1.91 % (17791)Termination reason: Unknown
% 3.79/1.91 % (17791)Termination phase: shuffling
% 3.79/1.91
% 3.79/1.91 % (17791)Memory used [KB]: 1407
% 3.79/1.91 % (17791)Time elapsed: 0.004 s
% 3.79/1.91 % (17791)Instructions burned: 3 (million)
% 3.79/1.91 % (17791)------------------------------
% 3.79/1.91 % (17791)------------------------------
% 3.79/1.91 % (17786)Refutation found. Thanks to Tanya!
% 3.79/1.91 % SZS status Theorem for DTF2THF_17675
% 3.79/1.91 % SZS output start Proof for DTF2THF_17675
% 3.79/1.91 thf(type_def_6, type, set: ($tType) > $tType).
% 3.79/1.91 thf(type_def_7, type, product_prod: ($tType * $tType) > $tType).
% 3.79/1.91 thf(type_def_8, type, a: $tType).
% 3.79/1.91 thf(type_def_9, type, b: $tType).
% 3.79/1.91 thf(func_def_0, type, product_prod: ($tType * $tType) > $tType).
% 3.79/1.91 thf(func_def_1, type, set: ($tType) > $tType).
% 3.79/1.91 thf(func_def_2, type, b: $tType).
% 3.79/1.91 thf(func_def_3, type, a: $tType).
% 3.79/1.91 thf(func_def_4, type, type: !>[X0: $tType]:($o)).
% 3.79/1.91 thf(func_def_5, type, bot: !>[X0: $tType]:($o)).
% 3.79/1.91 thf(func_def_6, type, ord: !>[X0: $tType]:($o)).
% 3.79/1.91 thf(func_def_7, type, top: !>[X0: $tType]:($o)).
% 3.79/1.91 thf(func_def_8, type, ab_group_add: !>[X0: $tType]:($o)).
% 3.79/1.91 thf(func_def_9, type, bNF_pick_middlep: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > X0 > X2 > X1)).
% 3.79/1.91 thf(func_def_10, type, bNF_Gr932048252omCard: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ product_prod @ X1 @ X1 > X1 > X0)).
% 3.79/1.91 thf(func_def_11, type, bNF_Greatest_image2p: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > (X0 > X2 > $o) > X1 > X3 > $o)).
% 3.79/1.91 thf(func_def_12, type, bNF_Greatest_toCard: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ product_prod @ X1 @ X1 > X0 > X1)).
% 3.79/1.91 thf(func_def_13, type, bNF_Gr1978856366d_pred: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ product_prod @ X1 @ X1 > (X0 > X1) > $o)).
% 3.79/1.91 thf(func_def_14, type, bNF_Greatest_univ: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X1)).
% 3.79/1.91 thf(func_def_15, type, bij_betw: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1 > $o)).
% 3.79/1.91 thf(func_def_16, type, fun_upd: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0 > X1 > X0 > X1)).
% 3.79/1.91 thf(func_def_17, type, inj_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 3.79/1.91 thf(func_def_18, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.79/1.91 thf(func_def_19, type, hilbert_Eps: !>[X0: $tType]:((X0 > $o) > X0)).
% 3.79/1.91 thf(func_def_20, type, hilbert_bijection: !>[X0: $tType]:((X0 > X0) > $o)).
% 3.79/1.91 thf(func_def_21, type, hilbert_inv_into: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X1 > X0)).
% 3.79/1.91 thf(func_def_22, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.79/1.91 thf(func_def_23, type, lattices_ord_arg_max: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > X0)).
% 3.79/1.91 thf(func_def_24, type, lattic1534546155max_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X0)).
% 3.79/1.91 thf(func_def_25, type, lattic2130478198rg_max: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > X0 > $o)).
% 3.79/1.91 thf(func_def_26, type, rel_pred_comp: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X1 > $o) > X0 > $o)).
% 3.79/1.91 thf(func_def_27, type, bot_bot: !>[X0: $tType]:(X0)).
% 3.79/1.91 thf(func_def_28, type, top_top: !>[X0: $tType]:(X0)).
% 3.79/1.91 thf(func_def_29, type, product_case_bool: !>[X0: $tType]:(X0 > X0 > $o > X0)).
% 3.79/1.91 thf(func_def_30, type, quotient3: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > (X1 > X0) > $o)).
% 3.79/1.91 thf(func_def_31, type, quot_type: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (set @ X0 > X1) > (X1 > set @ X0) > $o)).
% 3.79/1.91 thf(func_def_32, type, quot_abs: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (set @ X0 > X1) > X0 > X1)).
% 3.79/1.91 thf(func_def_33, type, quot_rep: !>[X0: $tType, X1: $tType]:((X0 > set @ X1) > X0 > X1)).
% 3.79/1.91 thf(func_def_34, type, inv_imagep: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > X1 > X1 > $o)).
% 3.79/1.91 thf(func_def_35, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.79/1.91 thf(func_def_36, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 3.79/1.91 thf(func_def_37, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 3.79/1.91 thf(func_def_38, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 3.79/1.91 thf(func_def_39, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 3.79/1.91 thf(func_def_40, type, remove: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 3.79/1.91 thf(func_def_41, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 3.79/1.91 thf(func_def_42, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.79/1.91 thf(func_def_43, type, p: a > $o).
% 3.79/1.91 thf(func_def_44, type, q: a > b > $o).
% 3.79/1.91 thf(func_def_63, type, sK0: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 3.79/1.91 thf(func_def_64, type, sK1: !>[X0: $tType]:((X0 > $o) > X0)).
% 3.79/1.91 thf(func_def_65, type, sK2: !>[X0: $tType]:(set @ X0 > X0)).
% 3.79/1.91 thf(func_def_66, type, sK3: !>[X0: $tType]:((X0 > $o) > X0)).
% 3.79/1.91 thf(func_def_67, type, sK4: !>[X0: $tType]:(set @ X0 > X0)).
% 3.79/1.91 thf(func_def_68, type, sK5: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.79/1.91 thf(func_def_69, type, sK6: !>[X0: $tType]:(set @ X0 > X0)).
% 3.79/1.91 thf(func_def_70, type, sK7: !>[X0: $tType]:((X0 > $o) > X0)).
% 3.79/1.91 thf(func_def_71, type, sK8: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.79/1.91 thf(func_def_72, type, sK9: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.79/1.91 thf(func_def_73, type, sK10: a).
% 3.79/1.91 thf(func_def_74, type, sK11: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.79/1.91 thf(func_def_75, type, sK12: !>[X0: $tType]:((X0 > $o) > X0)).
% 3.79/1.91 thf(func_def_76, type, sK13: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.79/1.91 thf(func_def_77, type, sK14: a > b).
% 3.79/1.91 thf(func_def_78, type, sK15: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 3.79/1.91 thf(func_def_80, type, ph17: !>[X0: $tType]:(X0)).
% 3.79/1.91 thf(f975,plain,(
% 3.79/1.91 $false),
% 3.79/1.91 inference(subsumption_resolution,[status(thm)],[f952,f865])).
% 3.79/1.91 thf(f865,plain,(
% 3.79/1.91 ((p @ sK10) = $true)),
% 3.79/1.91 inference(cnf_transformation,[status(thm)],[f817])).
% 3.79/1.91 thf(f817,plain,(
% 3.79/1.91 ((p @ sK10) = $true) & ((q @ sK10 @ (hilbert_Eps @ b @ (q @ sK10))) != $true)),
% 3.79/1.91 inference(skolemisation,[status(esa),new_symbols(skolem,[sK10])],[f780,f816])).
% 3.79/1.91 thf(f816,plain,(
% 3.79/1.91 ? [X0 : a] : (((p @ X0) = $true) & ((q @ X0 @ (hilbert_Eps @ b @ (q @ X0))) != $true)) => (((p @ sK10) = $true) & ((q @ sK10 @ (hilbert_Eps @ b @ (q @ sK10))) != $true))),
% 3.79/1.91 introduced(definition,[],[choice_axiom])).
% 3.79/1.91 thf(f780,plain,(
% 3.79/1.91 ? [X0 : a] : (((p @ X0) = $true) & ((q @ X0 @ (hilbert_Eps @ b @ (q @ X0))) != $true))),
% 3.79/1.91 inference(ennf_transformation,[status(thm)],[f745])).
% 3.79/1.91 thf(f745,plain,(
% 3.79/1.91 ~! [X0 : a] : (((q @ X0 @ (hilbert_Eps @ b @ (q @ X0))) = $true) | ((p @ X0) != $true))),
% 3.79/1.91 inference(flattening,[status(thm)],[f320])).
% 3.79/1.91 thf(f320,plain,(
% 3.79/1.91 ~! [X0 : a] : (((q @ X0 @ (hilbert_Eps @ b @ (q @ X0))) = $true) | ~((p @ X0) = $true))),
% 3.79/1.91 inference(fool_elimination,[status(thm)],[f319])).
% 3.79/1.91 thf(f319,plain,(
% 3.79/1.91 ~! [X0 : a] : ((q @ X0 @ (hilbert_Eps @ b @ (q @ X0))) | ~(p @ X0))),
% 3.79/1.91 inference(rectify,[status(thm)],[f271])).
% 3.79/1.91 thf(f271,negated_conjecture,(
% 3.79/1.91 ~! [X6 : a] : ((q @ X6 @ (hilbert_Eps @ b @ (q @ X6))) | ~(p @ X6))),
% 3.79/1.91 inference(negated_conjecture,[status(cth)],[f270])).
% 3.79/1.91 thf(f270,conjecture,(
% 3.79/1.91 ! [X6 : a] : ((q @ X6 @ (hilbert_Eps @ b @ (q @ X6))) | ~(p @ X6))),
% 3.79/1.91 file('/export/starexec/sandbox/tmp/tmp.ttMsP055XI/DTF2THF_17675.p',conj_1)).
% 3.79/1.91 thf(f952,plain,(
% 3.79/1.91 ((p @ sK10) != $true)),
% 3.79/1.91 inference(trivial_inequality_removal,[status(thm)],[f932])).
% 3.79/1.91 thf(f932,plain,(
% 3.79/1.91 ((p @ sK10) != $true) | ($true != $true)),
% 3.79/1.91 inference(superposition,[status(thm)],[f929,f876])).
% 3.79/1.91 thf(f876,plain,(
% 3.79/1.91 ( ! [X0 : a] : (((q @ X0 @ (sK14 @ X0)) = $true) | ((p @ X0) != $true)) )),
% 3.79/1.91 inference(cnf_transformation,[status(thm)],[f829])).
% 3.79/1.91 thf(f829,plain,(
% 3.79/1.91 ! [X0 : a] : (((p @ X0) != $true) | ((q @ X0 @ (sK14 @ X0)) = $true))),
% 3.79/1.91 inference(skolemisation,[status(esa),new_symbols(skolem,[sK14])],[f768,f828])).
% 3.79/1.91 thf(f828,plain,(
% 3.79/1.91 ! [X0 : a] : (? [X1 : b] : ((q @ X0 @ X1) = $true) => ((q @ X0 @ (sK14 @ X0)) = $true))),
% 3.79/1.91 introduced(definition,[],[choice_axiom])).
% 3.79/1.91 thf(f768,plain,(
% 3.79/1.91 ! [X0 : a] : (((p @ X0) != $true) | ? [X1 : b] : ((q @ X0 @ X1) = $true))),
% 3.79/1.91 inference(ennf_transformation,[status(thm)],[f552])).
% 3.79/1.91 thf(f552,plain,(
% 3.79/1.91 ! [X0 : a] : (((p @ X0) = $true) => ? [X1 : b] : ((q @ X0 @ X1) = $true))),
% 3.79/1.91 inference(fool_elimination,[status(thm)],[f551])).
% 3.79/1.91 thf(f551,plain,(
% 3.79/1.91 ! [X0 : a] : ((p @ X0) => ? [X1 : b] : (q @ X0 @ X1))),
% 3.79/1.91 inference(rectify,[status(thm)],[f269])).
% 3.79/1.91 thf(f269,axiom,(
% 3.79/1.91 ! [X19 : a] : ((p @ X19) => ? [X90 : b] : (q @ X19 @ X90))),
% 3.79/1.91 file('/export/starexec/sandbox/tmp/tmp.ttMsP055XI/DTF2THF_17675.p',conj_0)).
% 3.79/1.91 thf(f929,plain,(
% 3.79/1.91 ( ! [X0 : b] : (($true != (q @ sK10 @ X0))) )),
% 3.79/1.91 inference(trivial_inequality_removal,[status(thm)],[f908])).
% 3.79/1.91 thf(f908,plain,(
% 3.79/1.91 ( ! [X0 : b] : (($true != (q @ sK10 @ X0)) | ($true != $true)) )),
% 3.79/1.91 inference(superposition,[status(thm)],[f864,f882])).
% 3.79/1.91 thf(f882,plain,(
% 3.79/1.91 ( ! [X0 : $tType,X2 : X0,X1 : X0 > $o] : (((X1 @ (hilbert_Eps @ X0 @ X1)) = $true) | ((X1 @ X2) != $true)) )),
% 3.79/1.91 inference(cnf_transformation,[status(thm)],[f757])).
% 3.79/1.91 thf(f757,plain,(
% 3.79/1.91 ! [X0 : $tType,X1 : X0 > $o] : (! [X2 : X0] : ((X1 @ X2) != $true) | ((X1 @ (hilbert_Eps @ X0 @ X1)) = $true))),
% 3.79/1.91 inference(ennf_transformation,[status(thm)],[f574])).
% 3.79/1.91 thf(f574,plain,(
% 3.79/1.91 ! [X0 : $tType,X1 : X0 > $o] : (? [X2 : X0] : ((X1 @ X2) = $true) => ((X1 @ (hilbert_Eps @ X0 @ X1)) = $true))),
% 3.79/1.91 inference(fool_elimination,[status(thm)],[f573])).
% 3.79/1.91 thf(f573,plain,(
% 3.79/1.91 ! [X0 : $tType,X1 : X0 > $o] : (? [X2 : X0] : (X1 @ X2) => (X1 @ (hilbert_Eps @ X0 @ X1)))),
% 3.79/1.91 inference(rectify,[status(thm)],[f9])).
% 3.79/1.91 thf(f9,axiom,(
% 3.79/1.91 ! [X0 : $tType,X4 : X0 > $o] : (? [X18 : X0] : (X4 @ X18) => (X4 @ (hilbert_Eps @ X0 @ X4)))),
% 3.79/1.91 file('/export/starexec/sandbox/tmp/tmp.ttMsP055XI/DTF2THF_17675.p',fact_8_someI__ex)).
% 3.79/1.91 thf(f864,plain,(
% 3.79/1.91 ((q @ sK10 @ (hilbert_Eps @ b @ (q @ sK10))) != $true)),
% 3.79/1.91 inference(cnf_transformation,[status(thm)],[f817])).
% 3.79/1.91 % SZS output end Proof for DTF2THF_17675
% 3.79/1.91 % (17786)------------------------------
% 3.79/1.91 % (17786)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.79/1.91 % (17786)Termination reason: Refutation
% 3.79/1.91
% 3.79/1.91 % (17786)Memory used [KB]: 6524
% 3.79/1.91 % (17786)Time elapsed: 0.032 s
% 3.79/1.91 % (17786)Instructions burned: 51 (million)
% 3.79/1.91 % (17786)------------------------------
% 3.79/1.91 % (17786)------------------------------
% 3.79/1.91 % (17780)Success in time 0.062 s
%------------------------------------------------------------------------------