%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP028^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n022.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 08:54:58 AM UTC 2026
% Result : Theorem 2.83s 1.66s
% Output : Refutation 2.83s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP028^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.14 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.35 % Computer : n022.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Tue Feb 24 08:23:51 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.26/0.48 ---- Original DTF file ---
% 0.26/0.48 thf(spec,logic,$$dhol).
% 0.26/0.48 %------------------------------------------------------------------------------
% 0.26/0.48 % File : ITP028^2 : TPTP v9.2.1. Released v7.5.0.
% 0.26/0.48 % Domain : Interactive Theorem Proving
% 0.26/0.48 % Problem : Sledgehammer Algebra8 problem prob_962__6421504_1
% 0.26/0.48 % Version : Especial.
% 0.26/0.48 % English :
% 0.26/0.48
% 0.26/0.48 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.26/0.48 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.26/0.48 % Source : [Des21]
% 0.26/0.48 % Names : Algebra8/prob_962__6421504_1 [Des21]
% 0.26/0.48
% 0.26/0.48 % Status : Theorem
% 0.26/0.48 % Rating : 0.00 v7.5.0
% 0.26/0.48 % Syntax : Number of formulae : 322 ( 122 unt; 50 typ; 0 def)
% 0.26/0.48 % Number of atoms : 645 ( 220 equ; 0 cnn)
% 0.26/0.48 % Maximal formula atoms : 13 ( 2 avg)
% 0.26/0.48 % Number of connectives : 4431 ( 65 ~; 9 |; 40 &;4008 @)
% 0.26/0.48 % ( 0 <=>; 309 =>; 0 <=; 0 <~>)
% 0.26/0.48 % Maximal formula depth : 28 ( 9 avg)
% 0.26/0.48 % Number of types : 3 ( 2 usr)
% 0.26/0.48 % Number of type conns : 421 ( 421 >; 0 *; 0 +; 0 <<)
% 0.26/0.48 % Number of symbols : 51 ( 48 usr; 7 con; 0-9 aty)
% 0.26/0.48 % Number of variables : 1350 ( 77 ^;1168 !; 30 ?;1350 :)
% 0.26/0.48 % ( 75 !>; 0 ?*; 0 @-; 0 @+)
% 0.26/0.48 % SPC : TH1_THM_EQU_NAR
% 0.26/0.48
% 0.26/0.48 % Comments : This file was generated by Sledgehammer 2021-02-23 16:30:50.699
% 0.26/0.48 %------------------------------------------------------------------------------
% 0.26/0.48 % Could-be-implicit typings (4)
% 0.26/0.48 thf(ty_t_Product__Type_Oprod,type,
% 0.26/0.48 product_prod: $tType > $tType > $tType ).
% 0.26/0.48
% 0.26/0.48 thf(ty_t_Set_Oset,type,
% 0.26/0.48 set: $tType > $tType ).
% 0.26/0.48
% 0.26/0.48 thf(ty_tf_e,type,
% 0.26/0.48 e: $tType ).
% 0.26/0.48
% 0.26/0.48 thf(ty_tf_a,type,
% 0.26/0.48 a: $tType ).
% 0.26/0.48
% 0.26/0.48 % Explicit typings (46)
% 0.26/0.48 thf(sy_cl_HOL_Otype,type,
% 0.26/0.48 type:
% 0.26/0.48 !>[A: $tType] : $o ).
% 0.26/0.48
% 0.26/0.48 thf(sy_cl_Orderings_Oord,type,
% 0.26/0.48 ord:
% 0.26/0.48 !>[A: $tType] : $o ).
% 0.26/0.48
% 0.26/0.48 thf(sy_cl_Orderings_Otop,type,
% 0.26/0.48 top:
% 0.26/0.48 !>[A: $tType] : $o ).
% 0.26/0.48
% 0.26/0.48 thf(sy_cl_Orderings_Oorder,type,
% 0.26/0.48 order:
% 0.26/0.48 !>[A: $tType] : $o ).
% 0.26/0.48
% 0.26/0.48 thf(sy_cl_Orderings_Olinorder,type,
% 0.26/0.48 linorder:
% 0.26/0.48 !>[A: $tType] : $o ).
% 0.26/0.48
% 0.26/0.48 thf(sy_cl_Orderings_Opreorder,type,
% 0.26/0.48 preorder:
% 0.26/0.48 !>[A: $tType] : $o ).
% 0.26/0.48
% 0.26/0.48 thf(sy_cl_Orderings_Oorder__top,type,
% 0.26/0.48 order_top:
% 0.26/0.48 !>[A: $tType] : $o ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.26/0.48 bNF_Ca1785829860lChain:
% 0.26/0.48 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_BNF__Def_Ocsquare,type,
% 0.26/0.48 bNF_csquare:
% 0.26/0.48 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( set @ A ) > ( B > C ) > ( D > C ) > ( A > B ) > ( A > D ) > $o ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_BNF__Def_OfstOp,type,
% 0.26/0.48 bNF_fstOp:
% 0.26/0.48 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_BNF__Def_Opick__middlep,type,
% 0.26/0.48 bNF_pick_middlep:
% 0.26/0.48 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_BNF__Def_OsndOp,type,
% 0.26/0.48 bNF_sndOp:
% 0.26/0.48 !>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 0.26/0.48 bNF_Greatest_image2:
% 0.26/0.48 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Finite__Set_OFpow,type,
% 0.26/0.48 finite_Fpow:
% 0.26/0.48 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Fun__Def_Oin__rel,type,
% 0.26/0.48 fun_in_rel:
% 0.26/0.48 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > A > B > $o ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.26/0.48 ord_less_eq:
% 0.26/0.48 !>[A: $tType] : ( A > A > $o ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 0.26/0.48 order_Greatest:
% 0.26/0.48 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 0.26/0.48 order_strict_mono:
% 0.26/0.48 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.26/0.48 top_top:
% 0.26/0.48 !>[A: $tType] : A ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Product__Type_OPair,type,
% 0.26/0.48 product_Pair:
% 0.26/0.48 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Product__Type_Ocurry,type,
% 0.26/0.48 product_curry:
% 0.26/0.48 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.26/0.48 produc2004651681e_prod:
% 0.26/0.48 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Product__Type_Omap__prod,type,
% 0.26/0.48 product_map_prod:
% 0.26/0.48 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.26/0.48 product_rec_prod:
% 0.26/0.48 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.26/0.48 product_case_prod:
% 0.26/0.48 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.26/0.48 product_fst:
% 0.26/0.48 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.26/0.48 product_snd:
% 0.26/0.48 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.26/0.48 product_swap:
% 0.26/0.48 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Relation_ODomain,type,
% 0.26/0.48 domain:
% 0.26/0.48 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Relation_ORange,type,
% 0.26/0.48 range:
% 0.26/0.48 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Relation_Oconverse,type,
% 0.26/0.48 converse:
% 0.26/0.48 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Relation_Oinv__image,type,
% 0.26/0.48 inv_image:
% 0.26/0.48 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Relation_Oinv__imagep,type,
% 0.26/0.48 inv_imagep:
% 0.26/0.48 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Relation_Orelcompp,type,
% 0.26/0.48 relcompp:
% 0.26/0.48 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Set_OCollect,type,
% 0.26/0.48 collect:
% 0.26/0.48 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Set_OPow,type,
% 0.26/0.48 pow:
% 0.26/0.48 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_Set_Oimage,type,
% 0.26/0.48 image:
% 0.26/0.48 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_c_member,type,
% 0.26/0.48 member:
% 0.26/0.48 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_v_C,type,
% 0.26/0.48 c: set @ ( product_prod @ ( set @ a ) @ ( a > e ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sy_v_a1,type,
% 0.26/0.48 a1: set @ a ).
% 0.26/0.48
% 0.26/0.48 thf(sy_v_a2,type,
% 0.26/0.48 a2: set @ a ).
% 0.26/0.48
% 0.26/0.48 thf(sy_v_a3,type,
% 0.26/0.48 a3: set @ a ).
% 0.26/0.48
% 0.26/0.48 thf(sy_v_b1,type,
% 0.26/0.48 b1: a > e ).
% 0.26/0.48
% 0.26/0.48 thf(sy_v_b2,type,
% 0.26/0.48 b2: a > e ).
% 0.26/0.48
% 0.26/0.48 thf(sy_v_b3,type,
% 0.26/0.48 b3: a > e ).
% 0.26/0.48
% 0.26/0.48 thf(sy_v_thesis____,type,
% 0.26/0.48 thesis: $o ).
% 0.26/0.48
% 0.26/0.48 % Relevant facts (254)
% 0.26/0.48 thf(fact_0__092_060open_062_092_060forall_062a_092_060in_062C_O_A_092_060forall_062b_092_060in_062C_O_Afst_Aa_A_092_060subseteq_062_Afst_Ab_A_092_060or_062_Afst_Ab_A_092_060subseteq_062_Afst_Aa_092_060close_062,axiom,
% 0.26/0.48 ! [X: product_prod @ ( set @ a ) @ ( a > e )] :
% 0.26/0.48 ( ( member @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ X @ c )
% 0.26/0.48 => ! [Xa: product_prod @ ( set @ a ) @ ( a > e )] :
% 0.26/0.48 ( ( member @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ Xa @ c )
% 0.26/0.48 => ( ( ord_less_eq @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) @ X ) @ ( product_fst @ ( set @ a ) @ ( a > e ) @ Xa ) )
% 0.26/0.48 | ( ord_less_eq @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) @ Xa ) @ ( product_fst @ ( set @ a ) @ ( a > e ) @ X ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % \<open>\<forall>a\<in>C. \<forall>b\<in>C. fst a \<subseteq> fst b \<or> fst b \<subseteq> fst a\<close>
% 0.26/0.48 thf(fact_1__092_060open_062_092_060exists_062a_092_060in_062fst_A_096_AC_O_Aa1_A_092_060subseteq_062_Aa_A_092_060and_062_Aa2_A_092_060subseteq_062_Aa_A_092_060and_062_Aa3_A_092_060subseteq_062_Aa_092_060close_062,axiom,
% 0.26/0.48 ? [X2: set @ a] :
% 0.26/0.48 ( ( member @ ( set @ a ) @ X2 @ ( image @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) ) @ c ) )
% 0.26/0.48 & ( ord_less_eq @ ( set @ a ) @ a1 @ X2 )
% 0.26/0.48 & ( ord_less_eq @ ( set @ a ) @ a2 @ X2 )
% 0.26/0.48 & ( ord_less_eq @ ( set @ a ) @ a3 @ X2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % \<open>\<exists>a\<in>fst ` C. a1 \<subseteq> a \<and> a2 \<subseteq> a \<and> a3 \<subseteq> a\<close>
% 0.26/0.48 thf(fact_2__092_060open_062a1_A_092_060subseteq_062_Aa2_A_092_060or_062_Aa2_A_092_060subseteq_062_Aa1_092_060close_062,axiom,
% 0.26/0.48 ( ( ord_less_eq @ ( set @ a ) @ a1 @ a2 )
% 0.26/0.48 | ( ord_less_eq @ ( set @ a ) @ a2 @ a1 ) ) ).
% 0.26/0.48
% 0.26/0.48 % \<open>a1 \<subseteq> a2 \<or> a2 \<subseteq> a1\<close>
% 0.26/0.48 thf(fact_3__092_060open_062a1_A_092_060subseteq_062_Aa3_A_092_060or_062_Aa3_A_092_060subseteq_062_Aa1_092_060close_062,axiom,
% 0.26/0.48 ( ( ord_less_eq @ ( set @ a ) @ a1 @ a3 )
% 0.26/0.48 | ( ord_less_eq @ ( set @ a ) @ a3 @ a1 ) ) ).
% 0.26/0.48
% 0.26/0.48 % \<open>a1 \<subseteq> a3 \<or> a3 \<subseteq> a1\<close>
% 0.26/0.48 thf(fact_4__092_060open_062a2_A_092_060subseteq_062_Aa3_A_092_060or_062_Aa3_A_092_060subseteq_062_Aa2_092_060close_062,axiom,
% 0.26/0.48 ( ( ord_less_eq @ ( set @ a ) @ a2 @ a3 )
% 0.26/0.48 | ( ord_less_eq @ ( set @ a ) @ a3 @ a2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % \<open>a2 \<subseteq> a3 \<or> a3 \<subseteq> a2\<close>
% 0.26/0.48 thf(fact_5_A_I3_J,axiom,
% 0.26/0.48 member @ ( set @ a ) @ a3 @ ( image @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) ) @ c ) ).
% 0.26/0.48
% 0.26/0.48 % A(3)
% 0.26/0.48 thf(fact_6_A_I2_J,axiom,
% 0.26/0.48 member @ ( set @ a ) @ a2 @ ( image @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) ) @ c ) ).
% 0.26/0.48
% 0.26/0.48 % A(2)
% 0.26/0.48 thf(fact_7_A_I1_J,axiom,
% 0.26/0.48 member @ ( set @ a ) @ a1 @ ( image @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) ) @ c ) ).
% 0.26/0.48
% 0.26/0.48 % A(1)
% 0.26/0.48 thf(fact_8__092_060open_062_Ia1_M_Ab1_J_A_092_060in_062_AC_092_060close_062,axiom,
% 0.26/0.48 member @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( product_Pair @ ( set @ a ) @ ( a > e ) @ a1 @ b1 ) @ c ).
% 0.26/0.48
% 0.26/0.48 % \<open>(a1, b1) \<in> C\<close>
% 0.26/0.48 thf(fact_9__092_060open_062_Ia2_M_Ab2_J_A_092_060in_062_AC_092_060close_062,axiom,
% 0.26/0.48 member @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( product_Pair @ ( set @ a ) @ ( a > e ) @ a2 @ b2 ) @ c ).
% 0.26/0.48
% 0.26/0.48 % \<open>(a2, b2) \<in> C\<close>
% 0.26/0.48 thf(fact_10__092_060open_062_Ia3_M_Ab3_J_A_092_060in_062_AC_092_060close_062,axiom,
% 0.26/0.48 member @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( product_Pair @ ( set @ a ) @ ( a > e ) @ a3 @ b3 ) @ c ).
% 0.26/0.48
% 0.26/0.48 % \<open>(a3, b3) \<in> C\<close>
% 0.26/0.48 thf(fact_11_subsetI,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.48 ( ! [X2: A] :
% 0.26/0.48 ( ( member @ A @ X2 @ A2 )
% 0.26/0.48 => ( member @ A @ X2 @ B2 ) )
% 0.26/0.48 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % subsetI
% 0.26/0.48 thf(fact_12_subset__antisym,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.26/0.48 => ( A2 = B2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % subset_antisym
% 0.26/0.48 thf(fact_13_image__eqI,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,B3: A,F: B > A,X3: B,A2: set @ B] :
% 0.26/0.48 ( ( B3
% 0.26/0.48 = ( F @ X3 ) )
% 0.26/0.48 => ( ( member @ B @ X3 @ A2 )
% 0.26/0.48 => ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % image_eqI
% 0.26/0.48 thf(fact_14_order__refl,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( preorder @ A )
% 0.26/0.48 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % order_refl
% 0.26/0.48 thf(fact_15_image__mono,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > B] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.48 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % image_mono
% 0.26/0.48 thf(fact_16_image__subsetI,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,A2: set @ A,F: A > B,B2: set @ B] :
% 0.26/0.48 ( ! [X2: A] :
% 0.26/0.48 ( ( member @ A @ X2 @ A2 )
% 0.26/0.48 => ( member @ B @ ( F @ X2 ) @ B2 ) )
% 0.26/0.48 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % image_subsetI
% 0.26/0.48 thf(fact_17_subset__imageE,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 0.26/0.48 => ~ ! [C2: set @ B] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ B ) @ C2 @ A2 )
% 0.26/0.48 => ( B2
% 0.26/0.48 != ( image @ B @ A @ F @ C2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % subset_imageE
% 0.26/0.48 thf(fact_18_image__subset__iff,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 0.26/0.48 = ( ! [X4: B] :
% 0.26/0.48 ( ( member @ B @ X4 @ A2 )
% 0.26/0.48 => ( member @ A @ ( F @ X4 ) @ B2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % image_subset_iff
% 0.26/0.48 thf(fact_19_subset__image__iff,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 0.26/0.48 = ( ? [AA: set @ B] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
% 0.26/0.48 & ( B2
% 0.26/0.48 = ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % subset_image_iff
% 0.26/0.48 thf(fact_20_PairE__lemma,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 0.26/0.48 ? [X2: A,Y: B] :
% 0.26/0.48 ( P
% 0.26/0.48 = ( product_Pair @ A @ B @ X2 @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 % PairE_lemma
% 0.26/0.48 thf(fact_21_dual__order_Oantisym,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( order @ A )
% 0.26/0.48 => ! [B3: A,A3: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.26/0.48 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.26/0.48 => ( A3 = B3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % dual_order.antisym
% 0.26/0.48 thf(fact_22_dual__order_Oeq__iff,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( order @ A )
% 0.26/0.48 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.26/0.48 = ( ^ [A4: A,B4: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.26/0.48 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % dual_order.eq_iff
% 0.26/0.48 thf(fact_23_dual__order_Otrans,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( order @ A )
% 0.26/0.48 => ! [B3: A,A3: A,C3: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.26/0.48 => ( ( ord_less_eq @ A @ C3 @ B3 )
% 0.26/0.48 => ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % dual_order.trans
% 0.26/0.48 thf(fact_24_linorder__wlog,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( linorder @ A )
% 0.26/0.48 => ! [P2: A > A > $o,A3: A,B3: A] :
% 0.26/0.48 ( ! [A5: A,B5: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.26/0.48 => ( P2 @ A5 @ B5 ) )
% 0.26/0.48 => ( ! [A5: A,B5: A] :
% 0.26/0.48 ( ( P2 @ B5 @ A5 )
% 0.26/0.48 => ( P2 @ A5 @ B5 ) )
% 0.26/0.48 => ( P2 @ A3 @ B3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % linorder_wlog
% 0.26/0.48 thf(fact_25_dual__order_Orefl,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( order @ A )
% 0.26/0.48 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % dual_order.refl
% 0.26/0.48 thf(fact_26_order__trans,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( preorder @ A )
% 0.26/0.48 => ! [X3: A,Y3: A,Z2: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.26/0.48 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.26/0.48 => ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % order_trans
% 0.26/0.48 thf(fact_27_order__class_Oorder_Oantisym,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( order @ A )
% 0.26/0.48 => ! [A3: A,B3: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.26/0.48 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.26/0.48 => ( A3 = B3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % order_class.order.antisym
% 0.26/0.48 thf(fact_28_ord__le__eq__trans,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( ord @ A )
% 0.26/0.48 => ! [A3: A,B3: A,C3: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.26/0.48 => ( ( B3 = C3 )
% 0.26/0.48 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % ord_le_eq_trans
% 0.26/0.48 thf(fact_29_ord__eq__le__trans,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( ord @ A )
% 0.26/0.48 => ! [A3: A,B3: A,C3: A] :
% 0.26/0.48 ( ( A3 = B3 )
% 0.26/0.48 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 0.26/0.48 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % ord_eq_le_trans
% 0.26/0.48 thf(fact_30_order__class_Oorder_Oeq__iff,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( order @ A )
% 0.26/0.48 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.26/0.48 = ( ^ [A4: A,B4: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.26/0.48 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % order_class.order.eq_iff
% 0.26/0.48 thf(fact_31_antisym__conv,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( order @ A )
% 0.26/0.48 => ! [Y3: A,X3: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.26/0.48 => ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.26/0.48 = ( X3 = Y3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % antisym_conv
% 0.26/0.48 thf(fact_32_le__cases3,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( linorder @ A )
% 0.26/0.48 => ! [X3: A,Y3: A,Z2: A] :
% 0.26/0.48 ( ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.26/0.48 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 0.26/0.48 => ( ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.26/0.48 => ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
% 0.26/0.48 => ( ( ( ord_less_eq @ A @ X3 @ Z2 )
% 0.26/0.48 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 0.26/0.48 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 0.26/0.48 => ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
% 0.26/0.48 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.26/0.48 => ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
% 0.26/0.48 => ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
% 0.26/0.48 => ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_cases3
% 0.26/0.48 thf(fact_33_order_Otrans,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( order @ A )
% 0.26/0.48 => ! [A3: A,B3: A,C3: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.26/0.48 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 0.26/0.48 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % order.trans
% 0.26/0.48 thf(fact_34_le__cases,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( linorder @ A )
% 0.26/0.48 => ! [X3: A,Y3: A] :
% 0.26/0.48 ( ~ ( ord_less_eq @ A @ X3 @ Y3 )
% 0.26/0.48 => ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_cases
% 0.26/0.48 thf(fact_35_eq__refl,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( preorder @ A )
% 0.26/0.48 => ! [X3: A,Y3: A] :
% 0.26/0.48 ( ( X3 = Y3 )
% 0.26/0.48 => ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % eq_refl
% 0.26/0.48 thf(fact_36_linear,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( linorder @ A )
% 0.26/0.48 => ! [X3: A,Y3: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.26/0.48 | ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % linear
% 0.26/0.48 thf(fact_37_antisym,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( order @ A )
% 0.26/0.48 => ! [X3: A,Y3: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.26/0.48 => ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.26/0.48 => ( X3 = Y3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % antisym
% 0.26/0.48 thf(fact_38_eq__iff,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( order @ A )
% 0.26/0.48 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.26/0.48 = ( ^ [X4: A,Y4: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.26/0.48 & ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % eq_iff
% 0.26/0.48 thf(fact_39_ord__le__eq__subst,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType] :
% 0.26/0.48 ( ( ( ord @ B )
% 0.26/0.48 & ( ord @ A ) )
% 0.26/0.48 => ! [A3: A,B3: A,F: A > B,C3: B] :
% 0.26/0.48 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.26/0.48 => ( ( ( F @ B3 )
% 0.26/0.48 = C3 )
% 0.26/0.48 => ( ! [X2: A,Y: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.26/0.48 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.26/0.48 => ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % ord_le_eq_subst
% 0.26/0.48 thf(fact_40_ord__eq__le__subst,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType] :
% 0.26/0.48 ( ( ( ord @ B )
% 0.26/0.48 & ( ord @ A ) )
% 0.26/0.48 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 0.26/0.48 ( ( A3
% 0.26/0.48 = ( F @ B3 ) )
% 0.26/0.48 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 0.26/0.48 => ( ! [X2: B,Y: B] :
% 0.26/0.48 ( ( ord_less_eq @ B @ X2 @ Y )
% 0.26/0.48 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.26/0.48 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % ord_eq_le_subst
% 0.26/0.48 thf(fact_41_order__subst2,axiom,
% 0.26/0.48 ! [A: $tType,C: $tType] :
% 0.26/0.48 ( ( ( order @ C )
% 0.26/0.48 & ( order @ A ) )
% 0.26/0.48 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 0.26/0.48 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.26/0.48 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
% 0.26/0.48 => ( ! [X2: A,Y: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.26/0.48 => ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.26/0.48 => ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % order_subst2
% 0.26/0.48 thf(fact_42_order__subst1,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType] :
% 0.26/0.48 ( ( ( order @ B )
% 0.26/0.48 & ( order @ A ) )
% 0.26/0.48 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 0.26/0.48 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.26/0.48 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 0.26/0.48 => ( ! [X2: B,Y: B] :
% 0.26/0.48 ( ( ord_less_eq @ B @ X2 @ Y )
% 0.26/0.48 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.26/0.48 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % order_subst1
% 0.26/0.48 thf(fact_43_le__fun__def,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType] :
% 0.26/0.48 ( ( ord @ B )
% 0.26/0.48 => ( ( ord_less_eq @ ( A > B ) )
% 0.26/0.48 = ( ^ [F2: A > B,G: A > B] :
% 0.26/0.48 ! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_fun_def
% 0.26/0.48 thf(fact_44_le__funI,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType] :
% 0.26/0.48 ( ( ord @ B )
% 0.26/0.48 => ! [F: A > B,G2: A > B] :
% 0.26/0.48 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G2 @ X2 ) )
% 0.26/0.48 => ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_funI
% 0.26/0.48 thf(fact_45_mem__Collect__eq,axiom,
% 0.26/0.48 ! [A: $tType,A3: A,P2: A > $o] :
% 0.26/0.48 ( ( member @ A @ A3 @ ( collect @ A @ P2 ) )
% 0.26/0.48 = ( P2 @ A3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % mem_Collect_eq
% 0.26/0.48 thf(fact_46_Collect__mem__eq,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A] :
% 0.26/0.48 ( ( collect @ A
% 0.26/0.48 @ ^ [X4: A] : ( member @ A @ X4 @ A2 ) )
% 0.26/0.48 = A2 ) ).
% 0.26/0.48
% 0.26/0.48 % Collect_mem_eq
% 0.26/0.48 thf(fact_47_Collect__cong,axiom,
% 0.26/0.48 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.26/0.48 ( ! [X2: A] :
% 0.26/0.48 ( ( P2 @ X2 )
% 0.26/0.48 = ( Q @ X2 ) )
% 0.26/0.48 => ( ( collect @ A @ P2 )
% 0.26/0.48 = ( collect @ A @ Q ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Collect_cong
% 0.26/0.48 thf(fact_48_ext,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,F: A > B,G2: A > B] :
% 0.26/0.48 ( ! [X2: A] :
% 0.26/0.48 ( ( F @ X2 )
% 0.26/0.48 = ( G2 @ X2 ) )
% 0.26/0.48 => ( F = G2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % ext
% 0.26/0.48 thf(fact_49_le__funE,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType] :
% 0.26/0.48 ( ( ord @ B )
% 0.26/0.48 => ! [F: A > B,G2: A > B,X3: A] :
% 0.26/0.48 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 0.26/0.48 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_funE
% 0.26/0.48 thf(fact_50_le__funD,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType] :
% 0.26/0.48 ( ( ord @ B )
% 0.26/0.48 => ! [F: A > B,G2: A > B,X3: A] :
% 0.26/0.48 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 0.26/0.48 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_funD
% 0.26/0.48 thf(fact_51_rev__image__eqI,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,X3: A,A2: set @ A,B3: B,F: A > B] :
% 0.26/0.48 ( ( member @ A @ X3 @ A2 )
% 0.26/0.48 => ( ( B3
% 0.26/0.48 = ( F @ X3 ) )
% 0.26/0.48 => ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % rev_image_eqI
% 0.26/0.48 thf(fact_52_ball__imageD,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
% 0.26/0.48 ( ! [X2: A] :
% 0.26/0.48 ( ( member @ A @ X2 @ ( image @ B @ A @ F @ A2 ) )
% 0.26/0.48 => ( P2 @ X2 ) )
% 0.26/0.48 => ! [X: B] :
% 0.26/0.48 ( ( member @ B @ X @ A2 )
% 0.26/0.48 => ( P2 @ ( F @ X ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % ball_imageD
% 0.26/0.48 thf(fact_53_image__cong,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G2: A > B] :
% 0.26/0.48 ( ( M = N )
% 0.26/0.48 => ( ! [X2: A] :
% 0.26/0.48 ( ( member @ A @ X2 @ N )
% 0.26/0.48 => ( ( F @ X2 )
% 0.26/0.48 = ( G2 @ X2 ) ) )
% 0.26/0.48 => ( ( image @ A @ B @ F @ M )
% 0.26/0.48 = ( image @ A @ B @ G2 @ N ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % image_cong
% 0.26/0.48 thf(fact_54_bex__imageD,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
% 0.26/0.48 ( ? [X: A] :
% 0.26/0.48 ( ( member @ A @ X @ ( image @ B @ A @ F @ A2 ) )
% 0.26/0.48 & ( P2 @ X ) )
% 0.26/0.48 => ? [X2: B] :
% 0.26/0.48 ( ( member @ B @ X2 @ A2 )
% 0.26/0.48 & ( P2 @ ( F @ X2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % bex_imageD
% 0.26/0.48 thf(fact_55_image__iff,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,Z2: A,F: B > A,A2: set @ B] :
% 0.26/0.48 ( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A2 ) )
% 0.26/0.48 = ( ? [X4: B] :
% 0.26/0.48 ( ( member @ B @ X4 @ A2 )
% 0.26/0.48 & ( Z2
% 0.26/0.48 = ( F @ X4 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % image_iff
% 0.26/0.48 thf(fact_56_imageI,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,X3: A,A2: set @ A,F: A > B] :
% 0.26/0.48 ( ( member @ A @ X3 @ A2 )
% 0.26/0.48 => ( member @ B @ ( F @ X3 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % imageI
% 0.26/0.48 thf(fact_57_Collect__mono__iff,axiom,
% 0.26/0.48 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
% 0.26/0.48 = ( ! [X4: A] :
% 0.26/0.48 ( ( P2 @ X4 )
% 0.26/0.48 => ( Q @ X4 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Collect_mono_iff
% 0.26/0.48 thf(fact_58_set__eq__subset,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( ^ [Y2: set @ A,Z: set @ A] : ( Y2 = Z ) )
% 0.26/0.48 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.26/0.48 & ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % set_eq_subset
% 0.26/0.48 thf(fact_59_subset__trans,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C4 )
% 0.26/0.48 => ( ord_less_eq @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % subset_trans
% 0.26/0.48 thf(fact_60_Collect__mono,axiom,
% 0.26/0.48 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.26/0.48 ( ! [X2: A] :
% 0.26/0.48 ( ( P2 @ X2 )
% 0.26/0.48 => ( Q @ X2 ) )
% 0.26/0.48 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Collect_mono
% 0.26/0.48 thf(fact_61_subset__iff,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) )
% 0.26/0.48 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.26/0.48 ! [T2: A] :
% 0.26/0.48 ( ( member @ A @ T2 @ A6 )
% 0.26/0.48 => ( member @ A @ T2 @ B6 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % subset_iff
% 0.26/0.48 thf(fact_62_equalityD2,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.48 ( ( A2 = B2 )
% 0.26/0.48 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % equalityD2
% 0.26/0.48 thf(fact_63_subset__eq,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) )
% 0.26/0.48 = ( ^ [A6: set @ A,B6: set @ A] :
% 0.26/0.48 ! [X4: A] :
% 0.26/0.48 ( ( member @ A @ X4 @ A6 )
% 0.26/0.48 => ( member @ A @ X4 @ B6 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % subset_eq
% 0.26/0.48 thf(fact_64_equalityE,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.48 ( ( A2 = B2 )
% 0.26/0.48 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.48 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % equalityE
% 0.26/0.48 thf(fact_65_subsetD,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.48 => ( ( member @ A @ C3 @ A2 )
% 0.26/0.48 => ( member @ A @ C3 @ B2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % subsetD
% 0.26/0.48 thf(fact_66_in__mono,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.48 => ( ( member @ A @ X3 @ A2 )
% 0.26/0.48 => ( member @ A @ X3 @ B2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % in_mono
% 0.26/0.48 thf(fact_67_prod_Oinject,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 0.26/0.48 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.26/0.48 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.26/0.48 = ( ( X1 = Y1 )
% 0.26/0.48 & ( X22 = Y22 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod.inject
% 0.26/0.48 thf(fact_68_old_Oprod_Oinject,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,A3: A,B3: B,A7: A,B7: B] :
% 0.26/0.48 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.26/0.48 = ( product_Pair @ A @ B @ A7 @ B7 ) )
% 0.26/0.48 = ( ( A3 = A7 )
% 0.26/0.48 & ( B3 = B7 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % old.prod.inject
% 0.26/0.48 thf(fact_69_eq__fst__iff,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,A3: A,P: product_prod @ A @ B] :
% 0.26/0.48 ( ( A3
% 0.26/0.48 = ( product_fst @ A @ B @ P ) )
% 0.26/0.48 = ( ? [B4: B] :
% 0.26/0.48 ( P
% 0.26/0.48 = ( product_Pair @ A @ B @ A3 @ B4 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % eq_fst_iff
% 0.26/0.48 thf(fact_70_fst__conv,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 0.26/0.48 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.26/0.48 = X1 ) ).
% 0.26/0.48
% 0.26/0.48 % fst_conv
% 0.26/0.48 thf(fact_71_fst__eqD,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,X3: A,Y3: B,A3: A] :
% 0.26/0.48 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 0.26/0.48 = A3 )
% 0.26/0.48 => ( X3 = A3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % fst_eqD
% 0.26/0.48 thf(fact_72_fstI,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,X3: product_prod @ A @ B,Y3: A,Z2: B] :
% 0.26/0.48 ( ( X3
% 0.26/0.48 = ( product_Pair @ A @ B @ Y3 @ Z2 ) )
% 0.26/0.48 => ( ( product_fst @ A @ B @ X3 )
% 0.26/0.48 = Y3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % fstI
% 0.26/0.48 thf(fact_73_all__subset__image,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: ( set @ A ) > $o] :
% 0.26/0.48 ( ( ! [B6: set @ A] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F @ A2 ) )
% 0.26/0.48 => ( P2 @ B6 ) ) )
% 0.26/0.48 = ( ! [B6: set @ B] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ B ) @ B6 @ A2 )
% 0.26/0.48 => ( P2 @ ( image @ B @ A @ F @ B6 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % all_subset_image
% 0.26/0.48 thf(fact_74_relChain__def,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType] :
% 0.26/0.48 ( ( ord @ B )
% 0.26/0.48 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.26/0.48 = ( ^ [R: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.26/0.48 ! [I: A,J: A] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
% 0.26/0.48 => ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % relChain_def
% 0.26/0.48 thf(fact_75_Greatest__equality,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( order @ A )
% 0.26/0.48 => ! [P2: A > $o,X3: A] :
% 0.26/0.48 ( ( P2 @ X3 )
% 0.26/0.48 => ( ! [Y: A] :
% 0.26/0.48 ( ( P2 @ Y )
% 0.26/0.48 => ( ord_less_eq @ A @ Y @ X3 ) )
% 0.26/0.48 => ( ( order_Greatest @ A @ P2 )
% 0.26/0.48 = X3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Greatest_equality
% 0.26/0.48 thf(fact_76_GreatestI2__order,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( order @ A )
% 0.26/0.48 => ! [P2: A > $o,X3: A,Q: A > $o] :
% 0.26/0.48 ( ( P2 @ X3 )
% 0.26/0.48 => ( ! [Y: A] :
% 0.26/0.48 ( ( P2 @ Y )
% 0.26/0.48 => ( ord_less_eq @ A @ Y @ X3 ) )
% 0.26/0.48 => ( ! [X2: A] :
% 0.26/0.48 ( ( P2 @ X2 )
% 0.26/0.48 => ( ! [Y5: A] :
% 0.26/0.48 ( ( P2 @ Y5 )
% 0.26/0.48 => ( ord_less_eq @ A @ Y5 @ X2 ) )
% 0.26/0.48 => ( Q @ X2 ) ) )
% 0.26/0.48 => ( Q @ ( order_Greatest @ A @ P2 ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % GreatestI2_order
% 0.26/0.48 thf(fact_77_old_Oprod_Oinducts,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.26/0.48 ( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.26/0.48 => ( P2 @ Prod ) ) ).
% 0.26/0.48
% 0.26/0.48 % old.prod.inducts
% 0.26/0.48 thf(fact_78_old_Oprod_Oexhaust,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 0.26/0.48 ~ ! [A5: A,B5: B] :
% 0.26/0.48 ( Y3
% 0.26/0.48 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% 0.26/0.48
% 0.26/0.48 % old.prod.exhaust
% 0.26/0.48 thf(fact_79_prod__induct7,axiom,
% 0.26/0.48 ! [G3: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 0.26/0.48 ( ! [A5: A,B5: B,C5: C,D2: D,E2: E,F4: F3,G4: G3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
% 0.26/0.48 => ( P2 @ X3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod_induct7
% 0.26/0.48 thf(fact_80_prod__induct6,axiom,
% 0.26/0.48 ! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 0.26/0.48 ( ! [A5: A,B5: B,C5: C,D2: D,E2: E,F4: F3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
% 0.26/0.48 => ( P2 @ X3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod_induct6
% 0.26/0.48 thf(fact_81_prod__induct5,axiom,
% 0.26/0.48 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.26/0.48 ( ! [A5: A,B5: B,C5: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C5 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.26/0.48 => ( P2 @ X3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod_induct5
% 0.26/0.48 thf(fact_82_prod__induct4,axiom,
% 0.26/0.48 ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.26/0.48 ( ! [A5: A,B5: B,C5: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C5 @ D2 ) ) ) )
% 0.26/0.48 => ( P2 @ X3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod_induct4
% 0.26/0.48 thf(fact_83_prod__induct3,axiom,
% 0.26/0.48 ! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.26/0.48 ( ! [A5: A,B5: B,C5: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C5 ) ) )
% 0.26/0.48 => ( P2 @ X3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod_induct3
% 0.26/0.48 thf(fact_84_prod__cases7,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 0.26/0.48 ~ ! [A5: A,B5: B,C5: C,D2: D,E2: E,F4: F3,G4: G3] :
% 0.26/0.48 ( Y3
% 0.26/0.48 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod_cases7
% 0.26/0.48 thf(fact_85_prod__cases6,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 0.26/0.48 ~ ! [A5: A,B5: B,C5: C,D2: D,E2: E,F4: F3] :
% 0.26/0.48 ( Y3
% 0.26/0.48 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod_cases6
% 0.26/0.48 thf(fact_86_prod__cases5,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.26/0.48 ~ ! [A5: A,B5: B,C5: C,D2: D,E2: E] :
% 0.26/0.48 ( Y3
% 0.26/0.48 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C5 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod_cases5
% 0.26/0.48 thf(fact_87_prod__cases4,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.26/0.48 ~ ! [A5: A,B5: B,C5: C,D2: D] :
% 0.26/0.48 ( Y3
% 0.26/0.48 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C5 @ D2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod_cases4
% 0.26/0.48 thf(fact_88_prod__cases3,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.26/0.48 ~ ! [A5: A,B5: B,C5: C] :
% 0.26/0.48 ( Y3
% 0.26/0.48 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C5 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod_cases3
% 0.26/0.48 thf(fact_89_Pair__inject,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,A3: A,B3: B,A7: A,B7: B] :
% 0.26/0.48 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.26/0.48 = ( product_Pair @ A @ B @ A7 @ B7 ) )
% 0.26/0.48 => ~ ( ( A3 = A7 )
% 0.26/0.48 => ( B3 != B7 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Pair_inject
% 0.26/0.48 thf(fact_90_prod__cases,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 0.26/0.48 ( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.26/0.48 => ( P2 @ P ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod_cases
% 0.26/0.48 thf(fact_91_old_Oprod_Orec,axiom,
% 0.26/0.48 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
% 0.26/0.48 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.26/0.48 = ( F1 @ A3 @ B3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % old.prod.rec
% 0.26/0.48 thf(fact_92_internal__case__prod__conv,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A3: B,B3: C] :
% 0.26/0.48 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 0.26/0.48 = ( C3 @ A3 @ B3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % internal_case_prod_conv
% 0.26/0.48 thf(fact_93_image__Fpow__mono,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 0.26/0.48 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( finite_Fpow @ B @ A2 ) ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % image_Fpow_mono
% 0.26/0.48 thf(fact_94_not__sub,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.48 ( ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.48 => ? [A5: A] :
% 0.26/0.48 ( ( member @ A @ A5 @ A2 )
% 0.26/0.48 & ~ ( member @ A @ A5 @ B2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % not_sub
% 0.26/0.48 thf(fact_95_eqsets__sub,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.48 ( ( A2 = B2 )
% 0.26/0.48 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % eqsets_sub
% 0.26/0.48 thf(fact_96_sub__which1,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
% 0.26/0.48 ( ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.48 | ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) )
% 0.26/0.48 => ( ( member @ A @ X3 @ A2 )
% 0.26/0.48 => ( ~ ( member @ A @ X3 @ B2 )
% 0.26/0.48 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sub_which1
% 0.26/0.48 thf(fact_97_Fpow__mono,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.48 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A2 ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Fpow_mono
% 0.26/0.48 thf(fact_98_le__convert,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( ord @ A )
% 0.26/0.48 => ! [A3: A,B3: A,C3: A] :
% 0.26/0.48 ( ( A3 = B3 )
% 0.26/0.48 => ( ( ord_less_eq @ A @ A3 @ C3 )
% 0.26/0.48 => ( ord_less_eq @ A @ B3 @ C3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_convert
% 0.26/0.48 thf(fact_99_ge__convert,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( ord @ A )
% 0.26/0.48 => ! [A3: A,B3: A,C3: A] :
% 0.26/0.48 ( ( A3 = B3 )
% 0.26/0.48 => ( ( ord_less_eq @ A @ C3 @ A3 )
% 0.26/0.48 => ( ord_less_eq @ A @ C3 @ B3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % ge_convert
% 0.26/0.48 thf(fact_100_mem__in__image3,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,B3: A,F: B > A,A2: set @ B] :
% 0.26/0.48 ( ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) )
% 0.26/0.48 => ? [X2: B] :
% 0.26/0.48 ( ( member @ B @ X2 @ A2 )
% 0.26/0.48 & ( B3
% 0.26/0.48 = ( F @ X2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % mem_in_image3
% 0.26/0.48 thf(fact_101_mem__in__image2,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,A3: A,A2: set @ A,F: A > B] :
% 0.26/0.48 ( ( member @ A @ A3 @ A2 )
% 0.26/0.48 => ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % mem_in_image2
% 0.26/0.48 thf(fact_102_mem__in__image1,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,B2: set @ B,A3: A] :
% 0.26/0.48 ( ! [X2: A] :
% 0.26/0.48 ( ( member @ A @ X2 @ A2 )
% 0.26/0.48 => ( member @ B @ ( F @ X2 ) @ B2 ) )
% 0.26/0.48 => ( ( member @ A @ A3 @ A2 )
% 0.26/0.48 => ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % mem_in_image1
% 0.26/0.48 thf(fact_103_proper__subset,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.48 => ( ~ ( member @ A @ X3 @ A2 )
% 0.26/0.48 => ( ( member @ A @ X3 @ B2 )
% 0.26/0.48 => ( A2 != B2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % proper_subset
% 0.26/0.48 thf(fact_104_not__subseteq,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.48 ( ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.48 => ? [X2: A] :
% 0.26/0.48 ( ( member @ A @ X2 @ A2 )
% 0.26/0.48 & ~ ( member @ A @ X2 @ B2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % not_subseteq
% 0.26/0.48 thf(fact_105_subset__self,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.26/0.48
% 0.26/0.48 % subset_self
% 0.26/0.48 thf(fact_106_sets__not__eq,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.48 ( ( A2 != B2 )
% 0.26/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.26/0.48 => ? [X2: A] :
% 0.26/0.48 ( ( member @ A @ X2 @ A2 )
% 0.26/0.48 & ~ ( member @ A @ X2 @ B2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sets_not_eq
% 0.26/0.48 thf(fact_107_sub__which2,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
% 0.26/0.48 ( ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.48 | ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) )
% 0.26/0.48 => ( ~ ( member @ A @ X3 @ A2 )
% 0.26/0.48 => ( ( member @ A @ X3 @ B2 )
% 0.26/0.48 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sub_which2
% 0.26/0.48 thf(fact_108_image__Pow__mono,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 0.26/0.48 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) ) @ ( pow @ A @ B2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % image_Pow_mono
% 0.26/0.48 thf(fact_109_Sup_OSUP__cong,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C4: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 0.26/0.48 ( ( A2 = B2 )
% 0.26/0.48 => ( ! [X2: B] :
% 0.26/0.48 ( ( member @ B @ X2 @ B2 )
% 0.26/0.48 => ( ( C4 @ X2 )
% 0.26/0.48 = ( D3 @ X2 ) ) )
% 0.26/0.48 => ( ( Sup @ ( image @ B @ A @ C4 @ A2 ) )
% 0.26/0.48 = ( Sup @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Sup.SUP_cong
% 0.26/0.48 thf(fact_110_Inf_OINF__cong,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C4: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 0.26/0.48 ( ( A2 = B2 )
% 0.26/0.48 => ( ! [X2: B] :
% 0.26/0.48 ( ( member @ B @ X2 @ B2 )
% 0.26/0.48 => ( ( C4 @ X2 )
% 0.26/0.48 = ( D3 @ X2 ) ) )
% 0.26/0.48 => ( ( Inf @ ( image @ B @ A @ C4 @ A2 ) )
% 0.26/0.48 = ( Inf @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Inf.INF_cong
% 0.26/0.48 thf(fact_111_subrelI,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ! [X2: A,Y: B] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R2 )
% 0.26/0.48 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S ) )
% 0.26/0.48 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ).
% 0.26/0.48
% 0.26/0.48 % subrelI
% 0.26/0.48 thf(fact_112_ssubst__Pair__rhs,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,R2: A,S: B,R3: set @ ( product_prod @ A @ B ),S2: B] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S ) @ R3 )
% 0.26/0.48 => ( ( S2 = S )
% 0.26/0.48 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % ssubst_Pair_rhs
% 0.26/0.48 thf(fact_113_PowI,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.48 => ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % PowI
% 0.26/0.48 thf(fact_114_Pow__iff,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.48 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 0.26/0.48 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % Pow_iff
% 0.26/0.48 thf(fact_115_Pow__top,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A] : ( member @ ( set @ A ) @ A2 @ ( pow @ A @ A2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % Pow_top
% 0.26/0.48 thf(fact_116_PowD,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.48 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 0.26/0.48 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % PowD
% 0.26/0.48 thf(fact_117_Pow__mono,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.26/0.48 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A2 ) @ ( pow @ A @ B2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Pow_mono
% 0.26/0.48 thf(fact_118_image__Pow__surj,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 0.26/0.48 ( ( ( image @ B @ A @ F @ A2 )
% 0.26/0.48 = B2 )
% 0.26/0.48 => ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) )
% 0.26/0.48 = ( pow @ A @ B2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % image_Pow_surj
% 0.26/0.48 thf(fact_119_Fpow__subset__Pow,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A2 ) @ ( pow @ A @ A2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % Fpow_subset_Pow
% 0.26/0.48 thf(fact_120_Cantors__paradox,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A] :
% 0.26/0.48 ~ ? [F5: A > ( set @ A )] :
% 0.26/0.48 ( ( image @ A @ ( set @ A ) @ F5 @ A2 )
% 0.26/0.48 = ( pow @ A @ A2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % Cantors_paradox
% 0.26/0.48 thf(fact_121_curry__conv,axiom,
% 0.26/0.48 ! [A: $tType,C: $tType,B: $tType] :
% 0.26/0.48 ( ( product_curry @ B @ C @ A )
% 0.26/0.48 = ( ^ [F2: ( product_prod @ B @ C ) > A,A4: B,B4: C] : ( F2 @ ( product_Pair @ B @ C @ A4 @ B4 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % curry_conv
% 0.26/0.48 thf(fact_122_in__inv__imagep,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType] :
% 0.26/0.48 ( ( inv_imagep @ A @ B )
% 0.26/0.48 = ( ^ [R: A > A > $o,F2: B > A,X4: B,Y4: B] : ( R @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % in_inv_imagep
% 0.26/0.48 thf(fact_123_verit__la__disequality,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( linorder @ A )
% 0.26/0.48 => ! [A3: A,B3: A] :
% 0.26/0.48 ( ( A3 = B3 )
% 0.26/0.48 | ~ ( ord_less_eq @ A @ A3 @ B3 )
% 0.26/0.48 | ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % verit_la_disequality
% 0.26/0.48 thf(fact_124_le__rel__bool__arg__iff,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( ord @ A )
% 0.26/0.48 => ( ( ord_less_eq @ ( $o > A ) )
% 0.26/0.48 = ( ^ [X5: $o > A,Y6: $o > A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ ( X5 @ $false ) @ ( Y6 @ $false ) )
% 0.26/0.48 & ( ord_less_eq @ A @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % le_rel_bool_arg_iff
% 0.26/0.48 thf(fact_125_curryI,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 0.26/0.48 ( ( F @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.26/0.48 => ( product_curry @ A @ B @ $o @ F @ A3 @ B3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % curryI
% 0.26/0.48 thf(fact_126_curryD,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 0.26/0.48 ( ( product_curry @ A @ B @ $o @ F @ A3 @ B3 )
% 0.26/0.48 => ( F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % curryD
% 0.26/0.48 thf(fact_127_curryE,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 0.26/0.48 ( ( product_curry @ A @ B @ $o @ F @ A3 @ B3 )
% 0.26/0.48 => ( F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % curryE
% 0.26/0.48 thf(fact_128_in__inv__image,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,X3: A,Y3: A,R2: set @ ( product_prod @ B @ B ),F: A > B] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( inv_image @ B @ A @ R2 @ F ) )
% 0.26/0.48 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) @ R2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % in_inv_image
% 0.26/0.48 thf(fact_129_fst__eq__Domain,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ R3 )
% 0.26/0.48 = ( domain @ A @ B @ R3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % fst_eq_Domain
% 0.26/0.48 thf(fact_130_Domain__fst,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType] :
% 0.26/0.48 ( ( domain @ A @ B )
% 0.26/0.48 = ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Domain_fst
% 0.26/0.48 thf(fact_131_Range__mono,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S )
% 0.26/0.48 => ( ord_less_eq @ ( set @ B ) @ ( range @ A @ B @ R2 ) @ ( range @ A @ B @ S ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Range_mono
% 0.26/0.48 thf(fact_132_RangeE,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,B3: A,R2: set @ ( product_prod @ B @ A )] :
% 0.26/0.48 ( ( member @ A @ B3 @ ( range @ B @ A @ R2 ) )
% 0.26/0.48 => ~ ! [A5: B] :
% 0.26/0.48 ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A5 @ B3 ) @ R2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % RangeE
% 0.26/0.48 thf(fact_133_Range__iff,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,A3: A,R2: set @ ( product_prod @ B @ A )] :
% 0.26/0.48 ( ( member @ A @ A3 @ ( range @ B @ A @ R2 ) )
% 0.26/0.48 = ( ? [Y4: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y4 @ A3 ) @ R2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Range_iff
% 0.26/0.48 thf(fact_134_Range_Ocases,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,A3: B,R2: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( member @ B @ A3 @ ( range @ A @ B @ R2 ) )
% 0.26/0.48 => ~ ! [A5: A] :
% 0.26/0.48 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ A3 ) @ R2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % Range.cases
% 0.26/0.48 thf(fact_135_Range_Osimps,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,A3: B,R2: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( member @ B @ A3 @ ( range @ A @ B @ R2 ) )
% 0.26/0.48 = ( ? [A4: A,B4: B] :
% 0.26/0.48 ( ( A3 = B4 )
% 0.26/0.48 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Range.simps
% 0.26/0.48 thf(fact_136_Range_Ointros,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,A3: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R2 )
% 0.26/0.48 => ( member @ B @ B3 @ ( range @ A @ B @ R2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Range.intros
% 0.26/0.48 thf(fact_137_Range_Oinducts,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,X3: B,R2: set @ ( product_prod @ A @ B ),P2: B > $o] :
% 0.26/0.48 ( ( member @ B @ X3 @ ( range @ A @ B @ R2 ) )
% 0.26/0.48 => ( ! [A5: A,B5: B] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R2 )
% 0.26/0.48 => ( P2 @ B5 ) )
% 0.26/0.48 => ( P2 @ X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Range.inducts
% 0.26/0.48 thf(fact_138_DomainE,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
% 0.26/0.48 => ~ ! [B5: B] :
% 0.26/0.48 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % DomainE
% 0.26/0.48 thf(fact_139_Domain__iff,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
% 0.26/0.48 = ( ? [Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y4 ) @ R2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Domain_iff
% 0.26/0.48 thf(fact_140_Domain_Ocases,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
% 0.26/0.48 => ~ ! [B5: B] :
% 0.26/0.48 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % Domain.cases
% 0.26/0.48 thf(fact_141_Domain_Osimps,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
% 0.26/0.48 = ( ? [A4: A,B4: B] :
% 0.26/0.48 ( ( A3 = A4 )
% 0.26/0.48 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Domain.simps
% 0.26/0.48 thf(fact_142_Domain_ODomainI,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,A3: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R2 )
% 0.26/0.48 => ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Domain.DomainI
% 0.26/0.48 thf(fact_143_Domain_Oinducts,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,X3: A,R2: set @ ( product_prod @ A @ B ),P2: A > $o] :
% 0.26/0.48 ( ( member @ A @ X3 @ ( domain @ A @ B @ R2 ) )
% 0.26/0.48 => ( ! [A5: A,B5: B] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R2 )
% 0.26/0.48 => ( P2 @ A5 ) )
% 0.26/0.48 => ( P2 @ X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Domain.inducts
% 0.26/0.48 thf(fact_144_Domain__mono,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S )
% 0.26/0.48 => ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R2 ) @ ( domain @ A @ B @ S ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Domain_mono
% 0.26/0.48 thf(fact_145_Range__converse,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( range @ B @ A @ ( converse @ A @ B @ R2 ) )
% 0.26/0.48 = ( domain @ A @ B @ R2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % Range_converse
% 0.26/0.48 thf(fact_146_Domain__converse,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A )] :
% 0.26/0.48 ( ( domain @ A @ B @ ( converse @ B @ A @ R2 ) )
% 0.26/0.48 = ( range @ B @ A @ R2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % Domain_converse
% 0.26/0.48 thf(fact_147_pair__in__swap__image,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,Y3: A,X3: B,A2: set @ ( product_prod @ B @ A )] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ X3 ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A2 ) )
% 0.26/0.48 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ A2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % pair_in_swap_image
% 0.26/0.48 thf(fact_148_swap__simp,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,X3: B,Y3: A] :
% 0.26/0.48 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y3 ) )
% 0.26/0.48 = ( product_Pair @ A @ B @ Y3 @ X3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % swap_simp
% 0.26/0.48 thf(fact_149_converse__converse,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( converse @ B @ A @ ( converse @ A @ B @ R2 ) )
% 0.26/0.48 = R2 ) ).
% 0.26/0.48
% 0.26/0.48 % converse_converse
% 0.26/0.48 thf(fact_150_converse__inject,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A )] :
% 0.26/0.48 ( ( ( converse @ B @ A @ R2 )
% 0.26/0.48 = ( converse @ B @ A @ S ) )
% 0.26/0.48 = ( R2 = S ) ) ).
% 0.26/0.48
% 0.26/0.48 % converse_inject
% 0.26/0.48 thf(fact_151_swap__swap,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 0.26/0.48 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
% 0.26/0.48 = P ) ).
% 0.26/0.48
% 0.26/0.48 % swap_swap
% 0.26/0.48 thf(fact_152_converse__iff,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,A3: A,B3: B,R2: set @ ( product_prod @ B @ A )] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( converse @ B @ A @ R2 ) )
% 0.26/0.48 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A3 ) @ R2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % converse_iff
% 0.26/0.48 thf(fact_153_converse__mono,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A )] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S ) )
% 0.26/0.48 = ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S ) ) ).
% 0.26/0.48
% 0.26/0.48 % converse_mono
% 0.26/0.48 thf(fact_154_converse__inv__image,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ B @ B ),F: A > B] :
% 0.26/0.48 ( ( converse @ A @ A @ ( inv_image @ B @ A @ R3 @ F ) )
% 0.26/0.48 = ( inv_image @ B @ A @ ( converse @ B @ B @ R3 ) @ F ) ) ).
% 0.26/0.48
% 0.26/0.48 % converse_inv_image
% 0.26/0.48 thf(fact_155_converse_Oinducts,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,X1: B,X22: A,R2: set @ ( product_prod @ A @ B ),P2: B > A > $o] :
% 0.26/0.48 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X1 @ X22 ) @ ( converse @ A @ B @ R2 ) )
% 0.26/0.48 => ( ! [A5: A,B5: B] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R2 )
% 0.26/0.48 => ( P2 @ B5 @ A5 ) )
% 0.26/0.48 => ( P2 @ X1 @ X22 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % converse.inducts
% 0.26/0.48 thf(fact_156_converse_Ointros,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,A3: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R2 )
% 0.26/0.48 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A3 ) @ ( converse @ A @ B @ R2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % converse.intros
% 0.26/0.48 thf(fact_157_converse_Osimps,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,A1: B,A22: A,R2: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R2 ) )
% 0.26/0.48 = ( ? [A4: A,B4: B] :
% 0.26/0.48 ( ( A1 = B4 )
% 0.26/0.48 & ( A22 = A4 )
% 0.26/0.48 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % converse.simps
% 0.26/0.48 thf(fact_158_converse_Ocases,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,A1: B,A22: A,R2: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R2 ) )
% 0.26/0.48 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A22 @ A1 ) @ R2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % converse.cases
% 0.26/0.48 thf(fact_159_converseE,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R2: set @ ( product_prod @ B @ A )] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R2 ) )
% 0.26/0.48 => ~ ! [X2: B,Y: A] :
% 0.26/0.48 ( ( Yx
% 0.26/0.48 = ( product_Pair @ A @ B @ Y @ X2 ) )
% 0.26/0.48 => ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y ) @ R2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % converseE
% 0.26/0.48 thf(fact_160_converseD,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,A3: A,B3: B,R2: set @ ( product_prod @ B @ A )] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( converse @ B @ A @ R2 ) )
% 0.26/0.48 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A3 ) @ R2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % converseD
% 0.26/0.48 thf(fact_161_converse__subset__swap,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ A )] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ ( converse @ B @ A @ S ) )
% 0.26/0.48 = ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( converse @ A @ B @ R2 ) @ S ) ) ).
% 0.26/0.48
% 0.26/0.48 % converse_subset_swap
% 0.26/0.48 thf(fact_162_surj__swap,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType] :
% 0.26/0.48 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.26/0.48 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % surj_swap
% 0.26/0.48 thf(fact_163_prod_Oswap__def,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType] :
% 0.26/0.48 ( ( product_swap @ A @ B )
% 0.26/0.48 = ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod.swap_def
% 0.26/0.48 thf(fact_164_map__prod__imageI,axiom,
% 0.26/0.48 ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B ),F: A > C,G2: B > D] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 )
% 0.26/0.48 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A3 ) @ ( G2 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G2 ) @ R3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % map_prod_imageI
% 0.26/0.48 thf(fact_165_top__apply,axiom,
% 0.26/0.48 ! [C: $tType,D: $tType] :
% 0.26/0.48 ( ( top @ C )
% 0.26/0.48 => ( ( top_top @ ( D > C ) )
% 0.26/0.48 = ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % top_apply
% 0.26/0.48 thf(fact_166_UNIV__I,axiom,
% 0.26/0.48 ! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % UNIV_I
% 0.26/0.48 thf(fact_167_map__prod__simp,axiom,
% 0.26/0.48 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G2: D > B,A3: C,B3: D] :
% 0.26/0.48 ( ( product_map_prod @ C @ A @ D @ B @ F @ G2 @ ( product_Pair @ C @ D @ A3 @ B3 ) )
% 0.26/0.48 = ( product_Pair @ A @ B @ ( F @ A3 ) @ ( G2 @ B3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % map_prod_simp
% 0.26/0.48 thf(fact_168_Pow__UNIV,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( pow @ A @ ( top_top @ ( set @ A ) ) )
% 0.26/0.48 = ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Pow_UNIV
% 0.26/0.48 thf(fact_169_fst__map__prod,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G2: D > B,X3: product_prod @ C @ D] :
% 0.26/0.48 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G2 @ X3 ) )
% 0.26/0.48 = ( F @ ( product_fst @ C @ D @ X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % fst_map_prod
% 0.26/0.48 thf(fact_170_snd__map__prod,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G2: D > A,X3: product_prod @ C @ D] :
% 0.26/0.48 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G2 @ X3 ) )
% 0.26/0.48 = ( G2 @ ( product_snd @ C @ D @ X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % snd_map_prod
% 0.26/0.48 thf(fact_171_converse__UNIV,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType] :
% 0.26/0.48 ( ( converse @ B @ A @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.26/0.48 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % converse_UNIV
% 0.26/0.48 thf(fact_172_prod_Ocollapse,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.26/0.48 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.26/0.48 = Prod ) ).
% 0.26/0.48
% 0.26/0.48 % prod.collapse
% 0.26/0.48 thf(fact_173_fst__swap,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,X3: product_prod @ B @ A] :
% 0.26/0.48 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X3 ) )
% 0.26/0.48 = ( product_snd @ B @ A @ X3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % fst_swap
% 0.26/0.48 thf(fact_174_snd__swap,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,X3: product_prod @ A @ B] :
% 0.26/0.48 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X3 ) )
% 0.26/0.48 = ( product_fst @ A @ B @ X3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % snd_swap
% 0.26/0.48 thf(fact_175_range__fst,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType] :
% 0.26/0.48 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.26/0.48 = ( top_top @ ( set @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % range_fst
% 0.26/0.48 thf(fact_176_range__snd,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType] :
% 0.26/0.48 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.26/0.48 = ( top_top @ ( set @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % range_snd
% 0.26/0.48 thf(fact_177_rangeI,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,F: B > A,X3: B] : ( member @ A @ ( F @ X3 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % rangeI
% 0.26/0.48 thf(fact_178_range__eqI,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,B3: A,F: B > A,X3: B] :
% 0.26/0.48 ( ( B3
% 0.26/0.48 = ( F @ X3 ) )
% 0.26/0.48 => ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % range_eqI
% 0.26/0.48 thf(fact_179_surj__def,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,F: B > A] :
% 0.26/0.48 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.26/0.48 = ( top_top @ ( set @ A ) ) )
% 0.26/0.48 = ( ! [Y4: A] :
% 0.26/0.48 ? [X4: B] :
% 0.26/0.48 ( Y4
% 0.26/0.48 = ( F @ X4 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % surj_def
% 0.26/0.48 thf(fact_180_surjI,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,G2: B > A,F: A > B] :
% 0.26/0.48 ( ! [X2: A] :
% 0.26/0.48 ( ( G2 @ ( F @ X2 ) )
% 0.26/0.48 = X2 )
% 0.26/0.48 => ( ( image @ B @ A @ G2 @ ( top_top @ ( set @ B ) ) )
% 0.26/0.48 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % surjI
% 0.26/0.48 thf(fact_181_surjE,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,F: B > A,Y3: A] :
% 0.26/0.48 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.26/0.48 = ( top_top @ ( set @ A ) ) )
% 0.26/0.48 => ~ ! [X2: B] :
% 0.26/0.48 ( Y3
% 0.26/0.48 != ( F @ X2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % surjE
% 0.26/0.48 thf(fact_182_surjD,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,F: B > A,Y3: A] :
% 0.26/0.48 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.26/0.48 = ( top_top @ ( set @ A ) ) )
% 0.26/0.48 => ? [X2: B] :
% 0.26/0.48 ( Y3
% 0.26/0.48 = ( F @ X2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % surjD
% 0.26/0.48 thf(fact_183_map__prod__surj,axiom,
% 0.26/0.48 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G2: C > D] :
% 0.26/0.48 ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.26/0.48 = ( top_top @ ( set @ B ) ) )
% 0.26/0.48 => ( ( ( image @ C @ D @ G2 @ ( top_top @ ( set @ C ) ) )
% 0.26/0.48 = ( top_top @ ( set @ D ) ) )
% 0.26/0.48 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G2 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 0.26/0.48 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % map_prod_surj
% 0.26/0.48 thf(fact_184_top_Oextremum__uniqueI,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( order_top @ A )
% 0.26/0.48 => ! [A3: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.26/0.48 => ( A3
% 0.26/0.48 = ( top_top @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % top.extremum_uniqueI
% 0.26/0.48 thf(fact_185_top_Oextremum__unique,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( order_top @ A )
% 0.26/0.48 => ! [A3: A] :
% 0.26/0.48 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.26/0.48 = ( A3
% 0.26/0.48 = ( top_top @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % top.extremum_unique
% 0.26/0.48 thf(fact_186_top__greatest,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( order_top @ A )
% 0.26/0.48 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % top_greatest
% 0.26/0.48 thf(fact_187_subset__UNIV,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % subset_UNIV
% 0.26/0.48 thf(fact_188_sndI,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,X3: product_prod @ A @ B,Y3: A,Z2: B] :
% 0.26/0.48 ( ( X3
% 0.26/0.48 = ( product_Pair @ A @ B @ Y3 @ Z2 ) )
% 0.26/0.48 => ( ( product_snd @ A @ B @ X3 )
% 0.26/0.48 = Z2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % sndI
% 0.26/0.48 thf(fact_189_snd__eqD,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,X3: B,Y3: A,A3: A] :
% 0.26/0.48 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y3 ) )
% 0.26/0.48 = A3 )
% 0.26/0.48 => ( Y3 = A3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % snd_eqD
% 0.26/0.48 thf(fact_190_snd__conv,axiom,
% 0.26/0.48 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 0.26/0.48 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 0.26/0.48 = X22 ) ).
% 0.26/0.48
% 0.26/0.48 % snd_conv
% 0.26/0.48 thf(fact_191_eq__snd__iff,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,B3: A,P: product_prod @ B @ A] :
% 0.26/0.48 ( ( B3
% 0.26/0.48 = ( product_snd @ B @ A @ P ) )
% 0.26/0.48 = ( ? [A4: B] :
% 0.26/0.48 ( P
% 0.26/0.48 = ( product_Pair @ B @ A @ A4 @ B3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % eq_snd_iff
% 0.26/0.48 thf(fact_192_prod__eq__iff,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType] :
% 0.26/0.48 ( ( ^ [Y2: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y2 = Z ) )
% 0.26/0.48 = ( ^ [S3: product_prod @ A @ B,T2: product_prod @ A @ B] :
% 0.26/0.48 ( ( ( product_fst @ A @ B @ S3 )
% 0.26/0.48 = ( product_fst @ A @ B @ T2 ) )
% 0.26/0.48 & ( ( product_snd @ A @ B @ S3 )
% 0.26/0.48 = ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod_eq_iff
% 0.26/0.48 thf(fact_193_prod_Oexpand,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 0.26/0.48 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.26/0.48 = ( product_fst @ A @ B @ Prod2 ) )
% 0.26/0.48 & ( ( product_snd @ A @ B @ Prod )
% 0.26/0.48 = ( product_snd @ A @ B @ Prod2 ) ) )
% 0.26/0.48 => ( Prod = Prod2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod.expand
% 0.26/0.48 thf(fact_194_prod__eqI,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,P: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 0.26/0.48 ( ( ( product_fst @ A @ B @ P )
% 0.26/0.48 = ( product_fst @ A @ B @ Q2 ) )
% 0.26/0.48 => ( ( ( product_snd @ A @ B @ P )
% 0.26/0.48 = ( product_snd @ A @ B @ Q2 ) )
% 0.26/0.48 => ( P = Q2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod_eqI
% 0.26/0.48 thf(fact_195_UNIV__eq__I,axiom,
% 0.26/0.48 ! [A: $tType,A2: set @ A] :
% 0.26/0.48 ( ! [X2: A] : ( member @ A @ X2 @ A2 )
% 0.26/0.48 => ( ( top_top @ ( set @ A ) )
% 0.26/0.48 = A2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % UNIV_eq_I
% 0.26/0.48 thf(fact_196_UNIV__witness,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % UNIV_witness
% 0.26/0.48 thf(fact_197_surjective__pairing,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,T3: product_prod @ A @ B] :
% 0.26/0.48 ( T3
% 0.26/0.48 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T3 ) @ ( product_snd @ A @ B @ T3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % surjective_pairing
% 0.26/0.48 thf(fact_198_prod_Oexhaust__sel,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.26/0.48 ( Prod
% 0.26/0.48 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod.exhaust_sel
% 0.26/0.48 thf(fact_199_snd__eq__Range,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A )] :
% 0.26/0.48 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ R3 )
% 0.26/0.48 = ( range @ B @ A @ R3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % snd_eq_Range
% 0.26/0.48 thf(fact_200_Range__snd,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType] :
% 0.26/0.48 ( ( range @ B @ A )
% 0.26/0.48 = ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Range_snd
% 0.26/0.48 thf(fact_201_prod__fun__imageE,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F: C > A,G2: D > B,R3: set @ ( product_prod @ C @ D )] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G2 ) @ R3 ) )
% 0.26/0.48 => ~ ! [X2: C,Y: D] :
% 0.26/0.48 ( ( C3
% 0.26/0.48 = ( product_Pair @ A @ B @ ( F @ X2 ) @ ( G2 @ Y ) ) )
% 0.26/0.48 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X2 @ Y ) @ R3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod_fun_imageE
% 0.26/0.48 thf(fact_202_exI__realizer,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,P2: A > B > $o,Y3: A,X3: B] :
% 0.26/0.48 ( ( P2 @ Y3 @ X3 )
% 0.26/0.48 => ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y3 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % exI_realizer
% 0.26/0.48 thf(fact_203_conjI__realizer,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,P2: A > $o,P: A,Q: B > $o,Q2: B] :
% 0.26/0.48 ( ( P2 @ P )
% 0.26/0.48 => ( ( Q @ Q2 )
% 0.26/0.48 => ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) )
% 0.26/0.48 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % conjI_realizer
% 0.26/0.48 thf(fact_204_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,P2: A > B > $o,X3: A,Y3: B,A3: product_prod @ A @ B] :
% 0.26/0.48 ( ( P2 @ X3 @ Y3 )
% 0.26/0.48 => ( ( A3
% 0.26/0.48 = ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 0.26/0.48 => ( P2 @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % BNF_Greatest_Fixpoint.subst_Pair
% 0.26/0.48 thf(fact_205_top__set__def,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( top_top @ ( set @ A ) )
% 0.26/0.48 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % top_set_def
% 0.26/0.48 thf(fact_206_top__empty__eq,axiom,
% 0.26/0.48 ! [A: $tType] :
% 0.26/0.48 ( ( top_top @ ( A > $o ) )
% 0.26/0.48 = ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % top_empty_eq
% 0.26/0.48 thf(fact_207_exE__realizer_H,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A] :
% 0.26/0.48 ( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
% 0.26/0.48 => ~ ! [X2: B,Y: A] :
% 0.26/0.48 ~ ( P2 @ Y @ X2 ) ) ).
% 0.26/0.48
% 0.26/0.48 % exE_realizer'
% 0.26/0.48 thf(fact_208_Collect__split__mono__strong,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,X6: set @ A,A2: set @ ( product_prod @ A @ B ),Y7: set @ B,P2: A > B > $o,Q: A > B > $o] :
% 0.26/0.48 ( ( X6
% 0.26/0.48 = ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A2 ) )
% 0.26/0.48 => ( ( Y7
% 0.26/0.48 = ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A2 ) )
% 0.26/0.48 => ( ! [X2: A] :
% 0.26/0.48 ( ( member @ A @ X2 @ X6 )
% 0.26/0.48 => ! [Xa2: B] :
% 0.26/0.48 ( ( member @ B @ Xa2 @ Y7 )
% 0.26/0.48 => ( ( P2 @ X2 @ Xa2 )
% 0.26/0.48 => ( Q @ X2 @ Xa2 ) ) ) )
% 0.26/0.48 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) ) )
% 0.26/0.48 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Collect_split_mono_strong
% 0.26/0.48 thf(fact_209_sndOp__def,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,C: $tType] :
% 0.26/0.48 ( ( bNF_sndOp @ C @ A @ B )
% 0.26/0.48 = ( ^ [P4: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sndOp_def
% 0.26/0.48 thf(fact_210_top1I,axiom,
% 0.26/0.48 ! [A: $tType,X3: A] : ( top_top @ ( A > $o ) @ X3 ) ).
% 0.26/0.48
% 0.26/0.48 % top1I
% 0.26/0.48 thf(fact_211_curry__case__prod,axiom,
% 0.26/0.48 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 0.26/0.48 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
% 0.26/0.48 = F ) ).
% 0.26/0.48
% 0.26/0.48 % curry_case_prod
% 0.26/0.48 thf(fact_212_case__prod__curry,axiom,
% 0.26/0.48 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.26/0.48 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 0.26/0.48 = F ) ).
% 0.26/0.48
% 0.26/0.48 % case_prod_curry
% 0.26/0.48 thf(fact_213_Product__Type_OCollect__case__prodD,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,X3: product_prod @ A @ B,A2: A > B > $o] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ X3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A2 ) ) )
% 0.26/0.48 => ( A2 @ ( product_fst @ A @ B @ X3 ) @ ( product_snd @ A @ B @ X3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Product_Type.Collect_case_prodD
% 0.26/0.48 thf(fact_214_prod_Ocase__eq__if,axiom,
% 0.26/0.48 ! [C: $tType,B: $tType,A: $tType] :
% 0.26/0.48 ( ( product_case_prod @ A @ B @ C )
% 0.26/0.48 = ( ^ [F2: A > B > C,Prod3: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod.case_eq_if
% 0.26/0.48 thf(fact_215_case__prod__beta,axiom,
% 0.26/0.48 ! [A: $tType,C: $tType,B: $tType] :
% 0.26/0.48 ( ( product_case_prod @ B @ C @ A )
% 0.26/0.48 = ( ^ [F2: B > C > A,P3: product_prod @ B @ C] : ( F2 @ ( product_fst @ B @ C @ P3 ) @ ( product_snd @ B @ C @ P3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % case_prod_beta
% 0.26/0.48 thf(fact_216_internal__case__prod__def,axiom,
% 0.26/0.48 ! [C: $tType,B: $tType,A: $tType] :
% 0.26/0.48 ( ( produc2004651681e_prod @ A @ B @ C )
% 0.26/0.48 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.26/0.48
% 0.26/0.48 % internal_case_prod_def
% 0.26/0.48 thf(fact_217_Collect__case__prod__mono,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,A2: A > B > $o,B2: A > B > $o] :
% 0.26/0.48 ( ( ord_less_eq @ ( A > B > $o ) @ A2 @ B2 )
% 0.26/0.48 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A2 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Collect_case_prod_mono
% 0.26/0.48 thf(fact_218_old_Oprod_Ocase,axiom,
% 0.26/0.48 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
% 0.26/0.48 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.26/0.48 = ( F @ X1 @ X22 ) ) ).
% 0.26/0.48
% 0.26/0.48 % old.prod.case
% 0.26/0.48 thf(fact_219_prod_Osplit__sel,axiom,
% 0.26/0.48 ! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.26/0.48 ( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.26/0.48 = ( ( Prod
% 0.26/0.48 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.26/0.48 => ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod.split_sel
% 0.26/0.48 thf(fact_220_prod_Osplit__sel__asm,axiom,
% 0.26/0.48 ! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.26/0.48 ( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.26/0.48 = ( ~ ( ( Prod
% 0.26/0.48 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.26/0.48 & ~ ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % prod.split_sel_asm
% 0.26/0.48 thf(fact_221_fstOp__def,axiom,
% 0.26/0.48 ! [C: $tType,B: $tType,A: $tType] :
% 0.26/0.48 ( ( bNF_fstOp @ A @ B @ C )
% 0.26/0.48 = ( ^ [P4: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % fstOp_def
% 0.26/0.48 thf(fact_222_split__cong,axiom,
% 0.26/0.48 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F: A > B > C,G2: A > B > C,P: product_prod @ A @ B] :
% 0.26/0.48 ( ! [X2: A,Y: B] :
% 0.26/0.48 ( ( ( product_Pair @ A @ B @ X2 @ Y )
% 0.26/0.48 = Q2 )
% 0.26/0.48 => ( ( F @ X2 @ Y )
% 0.26/0.48 = ( G2 @ X2 @ Y ) ) )
% 0.26/0.48 => ( ( P = Q2 )
% 0.26/0.48 => ( ( product_case_prod @ A @ B @ C @ F @ P )
% 0.26/0.48 = ( product_case_prod @ A @ B @ C @ G2 @ Q2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % split_cong
% 0.26/0.48 thf(fact_223_predicate2I,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,P2: A > B > $o,Q: A > B > $o] :
% 0.26/0.48 ( ! [X2: A,Y: B] :
% 0.26/0.48 ( ( P2 @ X2 @ Y )
% 0.26/0.48 => ( Q @ X2 @ Y ) )
% 0.26/0.48 => ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q ) ) ).
% 0.26/0.48
% 0.26/0.48 % predicate2I
% 0.26/0.48 thf(fact_224_mem__case__prodE,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,C: $tType,Z2: A,C3: B > C > ( set @ A ),P: product_prod @ B @ C] :
% 0.26/0.48 ( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P ) )
% 0.26/0.48 => ~ ! [X2: B,Y: C] :
% 0.26/0.48 ( ( P
% 0.26/0.48 = ( product_Pair @ B @ C @ X2 @ Y ) )
% 0.26/0.48 => ~ ( member @ A @ Z2 @ ( C3 @ X2 @ Y ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % mem_case_prodE
% 0.26/0.48 thf(fact_225_rev__predicate2D,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,P2: A > B > $o,X3: A,Y3: B,Q: A > B > $o] :
% 0.26/0.48 ( ( P2 @ X3 @ Y3 )
% 0.26/0.48 => ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
% 0.26/0.48 => ( Q @ X3 @ Y3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % rev_predicate2D
% 0.26/0.48 thf(fact_226_predicate2D,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,P2: A > B > $o,Q: A > B > $o,X3: A,Y3: B] :
% 0.26/0.48 ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
% 0.26/0.48 => ( ( P2 @ X3 @ Y3 )
% 0.26/0.48 => ( Q @ X3 @ Y3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % predicate2D
% 0.26/0.48 thf(fact_227_refl__ge__eq,axiom,
% 0.26/0.48 ! [A: $tType,R3: A > A > $o] :
% 0.26/0.48 ( ! [X2: A] : ( R3 @ X2 @ X2 )
% 0.26/0.48 => ( ord_less_eq @ ( A > A > $o )
% 0.26/0.48 @ ^ [Y2: A,Z: A] : ( Y2 = Z )
% 0.26/0.48 @ R3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % refl_ge_eq
% 0.26/0.48 thf(fact_228_ge__eq__refl,axiom,
% 0.26/0.48 ! [A: $tType,R3: A > A > $o,X3: A] :
% 0.26/0.48 ( ( ord_less_eq @ ( A > A > $o )
% 0.26/0.48 @ ^ [Y2: A,Z: A] : ( Y2 = Z )
% 0.26/0.48 @ R3 )
% 0.26/0.48 => ( R3 @ X3 @ X3 ) ) ).
% 0.26/0.48
% 0.26/0.48 % ge_eq_refl
% 0.26/0.48 thf(fact_229_Collect__case__prod__in__rel__leE,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,X6: set @ ( product_prod @ A @ B ),Y7: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( fun_in_rel @ A @ B @ Y7 ) ) ) )
% 0.26/0.48 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ Y7 ) ) ).
% 0.26/0.48
% 0.26/0.48 % Collect_case_prod_in_rel_leE
% 0.26/0.48 thf(fact_230_Collect__case__prod__in__rel__leI,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,X6: set @ ( product_prod @ A @ B ),Y7: set @ ( product_prod @ A @ B )] :
% 0.26/0.48 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ Y7 )
% 0.26/0.48 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( fun_in_rel @ A @ B @ Y7 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Collect_case_prod_in_rel_leI
% 0.26/0.48 thf(fact_231_in__rel__def,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType] :
% 0.26/0.48 ( ( fun_in_rel @ A @ B )
% 0.26/0.48 = ( ^ [R4: set @ ( product_prod @ A @ B ),X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % in_rel_def
% 0.26/0.48 thf(fact_232_csquare__fstOp__sndOp,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,C: $tType,F: ( A > B > $o ) > ( product_prod @ A @ B ) > $o,P2: A > C > $o,Q: C > B > $o] : ( bNF_csquare @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ C @ ( product_prod @ C @ B ) @ ( collect @ ( product_prod @ A @ B ) @ ( F @ ( relcompp @ A @ C @ B @ P2 @ Q ) ) ) @ ( product_snd @ A @ C ) @ ( product_fst @ C @ B ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q ) ) ).
% 0.26/0.48
% 0.26/0.48 % csquare_fstOp_sndOp
% 0.26/0.48 thf(fact_233_predicate2D__conj,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,P2: A > B > $o,Q: A > B > $o,R3: $o,X3: A,Y3: B] :
% 0.26/0.48 ( ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
% 0.26/0.48 & R3 )
% 0.26/0.48 => ( R3
% 0.26/0.48 & ( ( P2 @ X3 @ Y3 )
% 0.26/0.48 => ( Q @ X3 @ Y3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % predicate2D_conj
% 0.26/0.48 thf(fact_234_fstOp__in,axiom,
% 0.26/0.48 ! [B: $tType,C: $tType,A: $tType,Ac2: product_prod @ A @ B,P2: A > C > $o,Q: C > B > $o] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P2 @ Q ) ) ) )
% 0.26/0.48 => ( member @ ( product_prod @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q @ Ac2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ P2 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % fstOp_in
% 0.26/0.48 thf(fact_235_sndOp__in,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,C: $tType,Ac2: product_prod @ A @ B,P2: A > C > $o,Q: C > B > $o] :
% 0.26/0.48 ( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P2 @ Q ) ) ) )
% 0.26/0.48 => ( member @ ( product_prod @ C @ B ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q @ Ac2 ) @ ( collect @ ( product_prod @ C @ B ) @ ( product_case_prod @ C @ B @ $o @ Q ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % sndOp_in
% 0.26/0.48 thf(fact_236_pick__middlep,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,C: $tType,P2: A > B > $o,Q: B > C > $o,A3: A,C3: C] :
% 0.26/0.48 ( ( relcompp @ A @ B @ C @ P2 @ Q @ A3 @ C3 )
% 0.26/0.48 => ( ( P2 @ A3 @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q @ A3 @ C3 ) )
% 0.26/0.48 & ( Q @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q @ A3 @ C3 ) @ C3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % pick_middlep
% 0.26/0.48 thf(fact_237_nchotomy__relcomppE,axiom,
% 0.26/0.48 ! [C: $tType,B: $tType,A: $tType,D: $tType,F: B > A,R2: C > A > $o,S: A > D > $o,A3: C,C3: D] :
% 0.26/0.48 ( ! [Y: A] :
% 0.26/0.48 ? [X: B] :
% 0.26/0.48 ( Y
% 0.26/0.48 = ( F @ X ) )
% 0.26/0.48 => ( ( relcompp @ C @ A @ D @ R2 @ S @ A3 @ C3 )
% 0.26/0.48 => ~ ! [B5: B] :
% 0.26/0.48 ( ( R2 @ A3 @ ( F @ B5 ) )
% 0.26/0.48 => ~ ( S @ ( F @ B5 ) @ C3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % nchotomy_relcomppE
% 0.26/0.48 thf(fact_238_relcompp_OrelcompI,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,C: $tType,R2: A > B > $o,A3: A,B3: B,S: B > C > $o,C3: C] :
% 0.26/0.48 ( ( R2 @ A3 @ B3 )
% 0.26/0.48 => ( ( S @ B3 @ C3 )
% 0.26/0.48 => ( relcompp @ A @ B @ C @ R2 @ S @ A3 @ C3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % relcompp.relcompI
% 0.26/0.48 thf(fact_239_relcompp_Oinducts,axiom,
% 0.26/0.48 ! [B: $tType,A: $tType,C: $tType,R2: A > B > $o,S: B > C > $o,X1: A,X22: C,P2: A > C > $o] :
% 0.26/0.48 ( ( relcompp @ A @ B @ C @ R2 @ S @ X1 @ X22 )
% 0.26/0.48 => ( ! [A5: A,B5: B,C5: C] :
% 0.26/0.48 ( ( R2 @ A5 @ B5 )
% 0.26/0.48 => ( ( S @ B5 @ C5 )
% 0.26/0.48 => ( P2 @ A5 @ C5 ) ) )
% 0.26/0.48 => ( P2 @ X1 @ X22 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % relcompp.inducts
% 0.26/0.48 thf(fact_240_relcompp__assoc,axiom,
% 0.26/0.48 ! [A: $tType,D: $tType,B: $tType,C: $tType,R2: A > D > $o,S: D > C > $o,T3: C > B > $o] :
% 0.26/0.48 ( ( relcompp @ A @ C @ B @ ( relcompp @ A @ D @ C @ R2 @ S ) @ T3 )
% 0.26/0.48 = ( relcompp @ A @ D @ B @ R2 @ ( relcompp @ D @ C @ B @ S @ T3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % relcompp_assoc
% 0.26/0.48 thf(fact_241_relcompp__apply,axiom,
% 0.26/0.48 ! [C: $tType,B: $tType,A: $tType] :
% 0.26/0.48 ( ( relcompp @ A @ B @ C )
% 0.26/0.48 = ( ^ [R4: A > B > $o,S4: B > C > $o,A4: A,C6: C] :
% 0.26/0.48 ? [B4: B] :
% 0.26/0.48 ( ( R4 @ A4 @ B4 )
% 0.26/0.48 & ( S4 @ B4 @ C6 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % relcompp_apply
% 0.26/0.48 thf(fact_242_relcompp_Osimps,axiom,
% 0.26/0.48 ! [C: $tType,B: $tType,A: $tType] :
% 0.26/0.48 ( ( relcompp @ A @ B @ C )
% 0.26/0.48 = ( ^ [R: A > B > $o,S3: B > C > $o,A12: A,A23: C] :
% 0.26/0.48 ? [A4: A,B4: B,C6: C] :
% 0.26/0.48 ( ( A12 = A4 )
% 0.26/0.48 & ( A23 = C6 )
% 0.26/0.48 & ( R @ A4 @ B4 )
% 0.26/0.48 & ( S3 @ B4 @ C6 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % relcompp.simps
% 0.26/0.48 thf(fact_243_relcompp_Ocases,axiom,
% 0.26/0.48 ! [A: $tType,B: $tType,C: $tType,R2: A > B > $o,S: B > C > $o,A1: A,A22: C] :
% 0.26/0.48 ( ( relcompp @ A @ B @ C @ R2 @ S @ A1 @ A22 )
% 0.26/0.49 => ~ ! [B5: B] :
% 0.26/0.49 ( ( R2 @ A1 @ B5 )
% 0.26/0.49 => ~ ( S @ B5 @ A22 ) ) ) ).
% 0.26/0.49
% 0.26/0.49 % relcompp.cases
% 0.26/0.49 thf(fact_244_relcomppE,axiom,
% 0.26/0.49 ! [A: $tType,B: $tType,C: $tType,R2: A > B > $o,S: B > C > $o,A3: A,C3: C] :
% 0.26/0.49 ( ( relcompp @ A @ B @ C @ R2 @ S @ A3 @ C3 )
% 0.26/0.49 => ~ ! [B5: B] :
% 0.26/0.49 ( ( R2 @ A3 @ B5 )
% 0.26/0.49 => ~ ( S @ B5 @ C3 ) ) ) ).
% 0.26/0.49
% 0.26/0.49 % relcomppE
% 0.26/0.49 thf(fact_245_eq__OO,axiom,
% 0.26/0.49 ! [B: $tType,A: $tType,R3: A > B > $o] :
% 0.26/0.49 ( ( relcompp @ A @ A @ B
% 0.26/0.49 @ ^ [Y2: A,Z: A] : ( Y2 = Z )
% 0.26/0.49 @ R3 )
% 0.26/0.49 = R3 ) ).
% 0.26/0.49
% 0.26/0.49 % eq_OO
% 0.26/0.49 thf(fact_246_OO__eq,axiom,
% 0.26/0.49 ! [B: $tType,A: $tType,R3: A > B > $o] :
% 0.26/0.49 ( ( relcompp @ A @ B @ B @ R3
% 0.26/0.49 @ ^ [Y2: B,Z: B] : ( Y2 = Z ) )
% 0.26/0.49 = R3 ) ).
% 0.26/0.49
% 0.26/0.49 % OO_eq
% 0.26/0.49 thf(fact_247_csquare__def,axiom,
% 0.26/0.49 ! [B: $tType,C: $tType,D: $tType,A: $tType] :
% 0.26/0.49 ( ( bNF_csquare @ A @ B @ C @ D )
% 0.26/0.49 = ( ^ [A6: set @ A,F12: B > C,F22: D > C,P1: A > B,P22: A > D] :
% 0.26/0.49 ! [X4: A] :
% 0.26/0.49 ( ( member @ A @ X4 @ A6 )
% 0.26/0.49 => ( ( F12 @ ( P1 @ X4 ) )
% 0.26/0.49 = ( F22 @ ( P22 @ X4 ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 % csquare_def
% 0.26/0.49 thf(fact_248_leq__OOI,axiom,
% 0.26/0.49 ! [A: $tType,R3: A > A > $o] :
% 0.26/0.49 ( ( R3
% 0.26/0.49 = ( ^ [Y2: A,Z: A] : ( Y2 = Z ) ) )
% 0.26/0.49 => ( ord_less_eq @ ( A > A > $o ) @ R3 @ ( relcompp @ A @ A @ A @ R3 @ R3 ) ) ) ).
% 0.26/0.49
% 0.26/0.49 % leq_OOI
% 0.26/0.49 thf(fact_249_relcompp__mono,axiom,
% 0.26/0.49 ! [A: $tType,C: $tType,B: $tType,R5: A > B > $o,R2: A > B > $o,S2: B > C > $o,S: B > C > $o] :
% 0.26/0.49 ( ( ord_less_eq @ ( A > B > $o ) @ R5 @ R2 )
% 0.26/0.49 => ( ( ord_less_eq @ ( B > C > $o ) @ S2 @ S )
% 0.26/0.49 => ( ord_less_eq @ ( A > C > $o ) @ ( relcompp @ A @ B @ C @ R5 @ S2 ) @ ( relcompp @ A @ B @ C @ R2 @ S ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 % relcompp_mono
% 0.26/0.49 thf(fact_250_image2__eqI,axiom,
% 0.26/0.49 ! [A: $tType,C: $tType,B: $tType,B3: A,F: B > A,X3: B,C3: C,G2: B > C,A2: set @ B] :
% 0.26/0.49 ( ( B3
% 0.26/0.49 = ( F @ X3 ) )
% 0.26/0.49 => ( ( C3
% 0.26/0.49 = ( G2 @ X3 ) )
% 0.26/0.49 => ( ( member @ B @ X3 @ A2 )
% 0.26/0.49 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B3 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A2 @ F @ G2 ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 % image2_eqI
% 0.26/0.49 thf(fact_251_strict__mono__inv,axiom,
% 0.26/0.49 ! [A: $tType,B: $tType] :
% 0.26/0.49 ( ( ( linorder @ B )
% 0.26/0.49 & ( linorder @ A ) )
% 0.26/0.49 => ! [F: A > B,G2: B > A] :
% 0.26/0.49 ( ( order_strict_mono @ A @ B @ F )
% 0.26/0.49 => ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.26/0.49 = ( top_top @ ( set @ B ) ) )
% 0.26/0.49 => ( ! [X2: A] :
% 0.26/0.49 ( ( G2 @ ( F @ X2 ) )
% 0.26/0.49 = X2 )
% 0.26/0.49 => ( order_strict_mono @ B @ A @ G2 ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 % strict_mono_inv
% 0.26/0.49 thf(fact_252_strict__mono__less__eq,axiom,
% 0.26/0.49 ! [B: $tType,A: $tType] :
% 0.26/0.49 ( ( ( linorder @ A )
% 0.26/0.49 & ( order @ B ) )
% 0.26/0.49 => ! [F: A > B,X3: A,Y3: A] :
% 0.26/0.49 ( ( order_strict_mono @ A @ B @ F )
% 0.26/0.49 => ( ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y3 ) )
% 0.26/0.49 = ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 % strict_mono_less_eq
% 0.26/0.49 thf(fact_253_strict__mono__eq,axiom,
% 0.26/0.49 ! [B: $tType,A: $tType] :
% 0.26/0.49 ( ( ( linorder @ A )
% 0.26/0.49 & ( order @ B ) )
% 0.26/0.49 => ! [F: A > B,X3: A,Y3: A] :
% 0.26/0.49 ( ( order_strict_mono @ A @ B @ F )
% 0.26/0.49 => ( ( ( F @ X3 )
% 0.26/0.49 = ( F @ Y3 ) )
% 0.26/0.49 = ( X3 = Y3 ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 % strict_mono_eq
% 0.26/0.49
% 0.26/0.49 % Type constructors (16)
% 0.26/0.49 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.26/0.49 ! [A8: $tType,A9: $tType] :
% 0.26/0.49 ( ( order_top @ A9 )
% 0.26/0.49 => ( order_top @ ( A8 > A9 ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.26/0.49 ! [A8: $tType,A9: $tType] :
% 0.26/0.49 ( ( preorder @ A9 )
% 0.26/0.49 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.26/0.49 ! [A8: $tType,A9: $tType] :
% 0.26/0.49 ( ( order @ A9 )
% 0.26/0.49 => ( order @ ( A8 > A9 ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(tcon_fun___Orderings_Otop,axiom,
% 0.26/0.49 ! [A8: $tType,A9: $tType] :
% 0.26/0.49 ( ( top @ A9 )
% 0.26/0.49 => ( top @ ( A8 > A9 ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(tcon_fun___Orderings_Oord,axiom,
% 0.26/0.49 ! [A8: $tType,A9: $tType] :
% 0.26/0.49 ( ( ord @ A9 )
% 0.26/0.49 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(tcon_Set_Oset___Orderings_Oorder__top_1,axiom,
% 0.26/0.49 ! [A8: $tType] : ( order_top @ ( set @ A8 ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
% 0.26/0.49 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
% 0.26/0.49 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(tcon_Set_Oset___Orderings_Otop_4,axiom,
% 0.26/0.49 ! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
% 0.26/0.49 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(tcon_HOL_Obool___Orderings_Oorder__top_6,axiom,
% 0.26/0.49 order_top @ $o ).
% 0.26/0.49
% 0.26/0.49 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 0.26/0.49 preorder @ $o ).
% 0.26/0.49
% 0.26/0.49 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.26/0.49 linorder @ $o ).
% 0.26/0.49
% 0.26/0.49 thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
% 0.26/0.49 order @ $o ).
% 0.26/0.49
% 0.26/0.49 thf(tcon_HOL_Obool___Orderings_Otop_9,axiom,
% 0.26/0.49 top @ $o ).
% 0.26/0.49
% 0.26/0.49 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 0.26/0.49 ord @ $o ).
% 0.26/0.49
% 0.26/0.49 % Conjectures (2)
% 0.26/0.49 thf(conj_0,hypothesis,
% 0.26/0.49 ! [A10: set @ a] :
% 0.26/0.49 ( ( member @ ( set @ a ) @ A10 @ ( image @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) ) @ c ) )
% 0.26/0.49 => ( ( ord_less_eq @ ( set @ a ) @ a1 @ A10 )
% 0.26/0.49 => ( ( ord_less_eq @ ( set @ a ) @ a2 @ A10 )
% 0.26/0.49 => ( ( ord_less_eq @ ( set @ a ) @ a3 @ A10 )
% 0.26/0.49 => thesis ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(conj_1,conjecture,
% 0.26/0.49 thesis ).
% 0.26/0.49
% 0.26/0.49 %------------------------------------------------------------------------------
% 0.26/0.49 ------------------------
% 2.77/1.52 ---- Embedded in THF ---
% 2.77/1.52 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.77/1.52 %%% Generated on Tue Feb 24 08:23:52 EST 2026
% 2.77/1.52 %%% using '$$dhol' embedding, version 1.3.0.
% 2.77/1.52 %%% Logic specification used:
% 2.77/1.52 %%% thf(spec, logic, $$dhol).
% 2.77/1.52
% 2.77/1.52 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.HLLIQP6qlU/DTF2DTF11799.p
% 2.77/1.52 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.77/1.52 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.77/1.52 thf(ty_tf_e, type, e: $tType).
% 2.77/1.52 thf(ty_tf_a, type, a: $tType).
% 2.77/1.52 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.77/1.52 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.77/1.52 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 2.77/1.52 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.77/1.52 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.77/1.52 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.77/1.52 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: ($o))).
% 2.77/1.52 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca1785829860lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 2.77/1.52 thf(sy_c_BNF__Def_Ocsquare, type, bNF_csquare: (!> [A:$tType,B:$tType,C:$tType,D:$tType]: (((set @ A) > ((B > C) > ((D > C) > ((A > B) > ((A > D) > $o)))))))).
% 2.77/1.52 thf(sy_c_BNF__Def_OfstOp, type, bNF_fstOp: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((B > (C > $o)) > (((product_prod @ A) @ C) > ((product_prod @ A) @ B))))))).
% 2.77/1.52 thf(sy_c_BNF__Def_Opick__middlep, type, bNF_pick_middlep: (!> [B:$tType,A:$tType,C:$tType]: (((B > (A > $o)) > ((A > (C > $o)) > (B > (C > A))))))).
% 2.77/1.52 thf(sy_c_BNF__Def_OsndOp, type, bNF_sndOp: (!> [C:$tType,A:$tType,B:$tType]: (((C > (A > $o)) > ((A > (B > $o)) > (((product_prod @ C) @ B) > ((product_prod @ A) @ B))))))).
% 2.77/1.52 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2, type, bNF_Greatest_image2: (!> [C:$tType,A:$tType,B:$tType]: (((set @ C) > ((C > A) > ((C > B) > (set @ ((product_prod @ A) @ B)))))))).
% 2.77/1.52 thf(sy_c_Finite__Set_OFpow, type, finite_Fpow: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))).
% 2.77/1.52 thf(sy_c_Fun__Def_Oin__rel, type, fun_in_rel: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (A > (B > $o)))))).
% 2.77/1.52 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.77/1.52 thf(sy_c_Orderings_Oorder__class_OGreatest, type, order_Greatest: (!> [A:$tType]: (((A > $o) > A)))).
% 2.77/1.52 thf(sy_c_Orderings_Oorder__class_Ostrict__mono, type, order_strict_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.77/1.52 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.77/1.52 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.77/1.52 thf(sy_c_Product__Type_Ocurry, type, product_curry: (!> [A:$tType,B:$tType,C:$tType]: (((((product_prod @ A) @ B) > C) > (A > (B > C)))))).
% 2.77/1.52 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.77/1.52 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 2.77/1.52 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.77/1.52 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.77/1.52 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 2.77/1.52 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 2.77/1.52 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 2.77/1.52 thf(sy_c_Relation_ODomain, type, domain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ A))))).
% 2.77/1.52 thf(sy_c_Relation_ORange, type, range: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ B))))).
% 2.77/1.52 thf(sy_c_Relation_Oconverse, type, converse: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ B) @ A)))))).
% 2.77/1.52 thf(sy_c_Relation_Oinv__image, type, inv_image: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ B)) > ((A > B) > (set @ ((product_prod @ A) @ A))))))).
% 2.77/1.52 thf(sy_c_Relation_Oinv__imagep, type, inv_imagep: (!> [B:$tType,A:$tType]: (((B > (B > $o)) > ((A > B) > (A > (A > $o))))))).
% 2.77/1.52 thf(sy_c_Relation_Orelcompp, type, relcompp: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((B > (C > $o)) > (A > (C > $o))))))).
% 2.77/1.52 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.77/1.52 thf(sy_c_Set_OPow, type, pow: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))).
% 2.77/1.52 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.77/1.52 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.77/1.52 thf(sy_v_C, type, c: (set @ ((product_prod @ (set @ a)) @ (a > e)))).
% 2.77/1.52 thf(sy_v_a1, type, a1: (set @ a)).
% 2.77/1.52 thf(sy_v_a2, type, a2: (set @ a)).
% 2.77/1.52 thf(sy_v_a3, type, a3: (set @ a)).
% 2.77/1.52 thf(sy_v_b1, type, b1: (a > e)).
% 2.77/1.52 thf(sy_v_b2, type, b2: (a > e)).
% 2.77/1.52 thf(sy_v_b3, type, b3: (a > e)).
% 2.77/1.52 thf(sy_v_thesis____, type, thesis: $o).
% 2.77/1.52 thf(fact_0__092_060open_062_092_060forall_062a_092_060in_062C_O_A_092_060forall_062b_092_060in_062C_O_Afst_Aa_A_092_060subseteq_062_Afst_Ab_A_092_060or_062_Afst_Ab_A_092_060subseteq_062_Afst_Aa_092_060close_062, axiom, (! [X:((product_prod @ (set @ a)) @ (a > e))]: (((((member @ ((product_prod @ (set @ a)) @ (a > e))) @ X) @ c) => (! [Xa:((product_prod @ (set @ a)) @ (a > e))]: (((((member @ ((product_prod @ (set @ a)) @ (a > e))) @ Xa) @ c) => ((((ord_less_eq @ (set @ a)) @ (((product_fst @ (set @ a)) @ (a > e)) @ X)) @ (((product_fst @ (set @ a)) @ (a > e)) @ Xa)) | (((ord_less_eq @ (set @ a)) @ (((product_fst @ (set @ a)) @ (a > e)) @ Xa)) @ (((product_fst @ (set @ a)) @ (a > e)) @ X)))))))))).
% 2.77/1.52 thf(fact_1__092_060open_062_092_060exists_062a_092_060in_062fst_A_096_AC_O_Aa1_A_092_060subseteq_062_Aa_A_092_060and_062_Aa2_A_092_060subseteq_062_Aa_A_092_060and_062_Aa3_A_092_060subseteq_062_Aa_092_060close_062, axiom, (? [X2:(set @ a)]: (((((member @ (set @ a)) @ X2) @ ((((image @ ((product_prod @ (set @ a)) @ (a > e))) @ (set @ a)) @ ((product_fst @ (set @ a)) @ (a > e))) @ c)) & ((((ord_less_eq @ (set @ a)) @ a1) @ X2) & ((((ord_less_eq @ (set @ a)) @ a2) @ X2) & (((ord_less_eq @ (set @ a)) @ a3) @ X2))))))).
% 2.77/1.52 thf(fact_2__092_060open_062a1_A_092_060subseteq_062_Aa2_A_092_060or_062_Aa2_A_092_060subseteq_062_Aa1_092_060close_062, axiom, ((((ord_less_eq @ (set @ a)) @ a1) @ a2) | (((ord_less_eq @ (set @ a)) @ a2) @ a1))).
% 2.77/1.52 thf(fact_3__092_060open_062a1_A_092_060subseteq_062_Aa3_A_092_060or_062_Aa3_A_092_060subseteq_062_Aa1_092_060close_062, axiom, ((((ord_less_eq @ (set @ a)) @ a1) @ a3) | (((ord_less_eq @ (set @ a)) @ a3) @ a1))).
% 2.77/1.52 thf(fact_4__092_060open_062a2_A_092_060subseteq_062_Aa3_A_092_060or_062_Aa3_A_092_060subseteq_062_Aa2_092_060close_062, axiom, ((((ord_less_eq @ (set @ a)) @ a2) @ a3) | (((ord_less_eq @ (set @ a)) @ a3) @ a2))).
% 2.77/1.52 thf(fact_5_A_I3_J, axiom, (((member @ (set @ a)) @ a3) @ ((((image @ ((product_prod @ (set @ a)) @ (a > e))) @ (set @ a)) @ ((product_fst @ (set @ a)) @ (a > e))) @ c))).
% 2.77/1.52 thf(fact_6_A_I2_J, axiom, (((member @ (set @ a)) @ a2) @ ((((image @ ((product_prod @ (set @ a)) @ (a > e))) @ (set @ a)) @ ((product_fst @ (set @ a)) @ (a > e))) @ c))).
% 2.77/1.52 thf(fact_7_A_I1_J, axiom, (((member @ (set @ a)) @ a1) @ ((((image @ ((product_prod @ (set @ a)) @ (a > e))) @ (set @ a)) @ ((product_fst @ (set @ a)) @ (a > e))) @ c))).
% 2.77/1.52 thf(fact_8__092_060open_062_Ia1_M_Ab1_J_A_092_060in_062_AC_092_060close_062, axiom, (((member @ ((product_prod @ (set @ a)) @ (a > e))) @ ((((product_Pair @ (set @ a)) @ (a > e)) @ a1) @ b1)) @ c)).
% 2.77/1.52 thf(fact_9__092_060open_062_Ia2_M_Ab2_J_A_092_060in_062_AC_092_060close_062, axiom, (((member @ ((product_prod @ (set @ a)) @ (a > e))) @ ((((product_Pair @ (set @ a)) @ (a > e)) @ a2) @ b2)) @ c)).
% 2.77/1.52 thf(fact_10__092_060open_062_Ia3_M_Ab3_J_A_092_060in_062_AC_092_060close_062, axiom, (((member @ ((product_prod @ (set @ a)) @ (a > e))) @ ((((product_Pair @ (set @ a)) @ (a > e)) @ a3) @ b3)) @ c)).
% 2.77/1.52 thf(fact_11_subsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ A2) => (((member @ A) @ X2) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.77/1.52 thf(fact_12_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => (A2 = B2)))))).
% 2.77/1.52 thf(fact_13_image__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),X3:B,A2:(set @ B)]: (((B3 = (F @ X3)) => ((((member @ B) @ X3) @ A2) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ A2))))))).
% 2.77/1.52 thf(fact_14_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ X3) @ X3))))))).
% 2.77/1.52 thf(fact_15_image__mono, axiom, (! [B:$tType,A:$tType,A2:(set @ A),B2:(set @ A),F:(A > B)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ ((((image @ A) @ B) @ F) @ B2)))))).
% 2.77/1.52 thf(fact_16_image__subsetI, axiom, (! [A:$tType,B:$tType,A2:(set @ A),F:(A > B),B2:(set @ B)]: (((! [X2:A]: (((((member @ A) @ X2) @ A2) => (((member @ B) @ (F @ X2)) @ B2)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ B2))))).
% 2.77/1.52 thf(fact_17_subset__imageE, axiom, (! [A:$tType,B:$tType,B2:(set @ A),F:(B > A),A2:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B2) @ ((((image @ B) @ A) @ F) @ A2)) => (~ ((! [C2:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C2) @ A2) => (B2 != ((((image @ B) @ A) @ F) @ C2))))))))))).
% 2.77/1.52 thf(fact_18_image__subset__iff, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A2)) @ B2) = (! [X4:B]: (((((member @ B) @ X4) @ A2) => (((member @ A) @ (F @ X4)) @ B2)))))))).
% 2.77/1.52 thf(fact_19_subset__image__iff, axiom, (! [A:$tType,B:$tType,B2:(set @ A),F:(B > A),A2:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B2) @ ((((image @ B) @ A) @ F) @ A2)) = (? [AA:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ AA) @ A2) & (B2 = ((((image @ B) @ A) @ F) @ AA))))))))).
% 2.77/1.52 thf(fact_20_PairE__lemma, axiom, (! [A:$tType,B:$tType,P:((product_prod @ A) @ B)]: ((? [X2:A,Y:B]: ((P = ((((product_Pair @ A) @ B) @ X2) @ Y))))))).
% 2.77/1.52 thf(fact_21_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ A3) @ B3) => (A3 = B3))))))))).
% 2.77/1.52 thf(fact_22_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (((ord_less_eq @ A) @ A4) @ B4))))))))).
% 2.77/1.52 thf(fact_23_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C3) @ B3) => (((ord_less_eq @ A) @ C3) @ A3))))))))).
% 2.77/1.52 thf(fact_24_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P2:(A > (A > $o)),A3:A,B3:A]: (((! [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((P2 @ A5) @ B5)))) => ((! [A5:A,B5:A]: ((((P2 @ B5) @ A5) => ((P2 @ A5) @ B5)))) => ((P2 @ A3) @ B3))))))))).
% 2.77/1.52 thf(fact_25_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 2.77/1.52 thf(fact_26_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y3:A,Z2:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z2) => (((ord_less_eq @ A) @ X3) @ Z2))))))))).
% 2.77/1.52 thf(fact_27_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ A3) => (A3 = B3))))))))).
% 2.77/1.52 thf(fact_28_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((B3 = C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 2.77/1.52 thf(fact_29_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C3:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ B3) @ C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 2.77/1.52 thf(fact_30_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) & (((ord_less_eq @ A) @ B4) @ A4))))))))).
% 2.77/1.52 thf(fact_31_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y3:A,X3:A]: (((((ord_less_eq @ A) @ Y3) @ X3) => ((((ord_less_eq @ A) @ X3) @ Y3) = (X3 = Y3))))))))).
% 2.77/1.52 thf(fact_32_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A,Z2:A]: ((((((ord_less_eq @ A) @ X3) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ Z2)))) => (((((ord_less_eq @ A) @ Y3) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Z2)))) => (((((ord_less_eq @ A) @ X3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y3)))) => (((((ord_less_eq @ A) @ Z2) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ X3)))) => (((((ord_less_eq @ A) @ Y3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X3)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Y3))))))))))))))))).
% 2.77/1.52 thf(fact_33_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 2.77/1.52 thf(fact_34_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 2.77/1.52 thf(fact_35_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y3:A]: (((X3 = Y3) => (((ord_less_eq @ A) @ X3) @ Y3)))))))).
% 2.77/1.52 thf(fact_36_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) | (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 2.77/1.52 thf(fact_37_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ X3) => (X3 = Y3))))))))).
% 2.77/1.52 thf(fact_38_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) & (((ord_less_eq @ A) @ Y4) @ X4))))))))).
% 2.77/1.52 thf(fact_39_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B3:A,F:(A > B),C3:B]: (((((ord_less_eq @ A) @ A3) @ B3) => (((F @ B3) = C3) => ((! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => (((ord_less_eq @ B) @ (F @ X2)) @ (F @ Y))))) => (((ord_less_eq @ B) @ (F @ A3)) @ C3)))))))))).
% 2.77/1.52 thf(fact_40_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B3:B,C3:B]: (((A3 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C3) => ((! [X2:B,Y:B]: (((((ord_less_eq @ B) @ X2) @ Y) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less_eq @ A) @ A3) @ (F @ C3))))))))))).
% 2.77/1.52 thf(fact_41_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B3:A,F:(A > C),C3:C]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C3) => ((! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => (((ord_less_eq @ C) @ (F @ X2)) @ (F @ Y))))) => (((ord_less_eq @ C) @ (F @ A3)) @ C3)))))))))).
% 2.77/1.52 thf(fact_42_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B3:B,C3:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C3) => ((! [X2:B,Y:B]: (((((ord_less_eq @ B) @ X2) @ Y) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less_eq @ A) @ A3) @ (F @ C3))))))))))).
% 2.77/1.52 thf(fact_43_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G:(A > B)]: ((! [X4:A]: ((((ord_less_eq @ B) @ (F2 @ X4)) @ (G @ X4))))))))))).
% 2.77/1.52 thf(fact_44_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G2:(A > B)]: (((! [X2:A]: ((((ord_less_eq @ B) @ (F @ X2)) @ (G2 @ X2)))) => (((ord_less_eq @ (A > B)) @ F) @ G2)))))))).
% 2.77/1.52 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P2:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P2)) = (P2 @ A3))))).
% 2.77/1.52 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A2)))) = A2)))).
% 2.77/1.52 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P2 @ X2) = (Q @ X2)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q)))))).
% 2.77/1.52 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G2:(A > B)]: (((! [X2:A]: (((F @ X2) = (G2 @ X2)))) => (F = G2))))).
% 2.77/1.52 thf(fact_49_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G2:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F) @ G2) => (((ord_less_eq @ B) @ (F @ X3)) @ (G2 @ X3))))))))).
% 2.77/1.52 thf(fact_50_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G2:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F) @ G2) => (((ord_less_eq @ B) @ (F @ X3)) @ (G2 @ X3))))))))).
% 2.77/1.52 thf(fact_51_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X3:A,A2:(set @ A),B3:B,F:(A > B)]: (((((member @ A) @ X3) @ A2) => ((B3 = (F @ X3)) => (((member @ B) @ B3) @ ((((image @ A) @ B) @ F) @ A2))))))).
% 2.77/1.52 thf(fact_52_ball__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),P2:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ ((((image @ B) @ A) @ F) @ A2)) => (P2 @ X2)))) => (! [X:B]: (((((member @ B) @ X) @ A2) => (P2 @ (F @ X))))))))).
% 2.77/1.52 thf(fact_53_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N:(set @ A),F:(A > B),G2:(A > B)]: (((M = N) => ((! [X2:A]: (((((member @ A) @ X2) @ N) => ((F @ X2) = (G2 @ X2))))) => (((((image @ A) @ B) @ F) @ M) = ((((image @ A) @ B) @ G2) @ N))))))).
% 2.77/1.52 thf(fact_54_bex__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),P2:(A > $o)]: (((? [X:A]: (((((member @ A) @ X) @ ((((image @ B) @ A) @ F) @ A2)) & (P2 @ X)))) => (? [X2:B]: (((((member @ B) @ X2) @ A2) & (P2 @ (F @ X2))))))))).
% 2.77/1.52 thf(fact_55_image__iff, axiom, (! [A:$tType,B:$tType,Z2:A,F:(B > A),A2:(set @ B)]: (((((member @ A) @ Z2) @ ((((image @ B) @ A) @ F) @ A2)) = (? [X4:B]: (((((member @ B) @ X4) @ A2) & (Z2 = (F @ X4))))))))).
% 2.77/1.52 thf(fact_56_imageI, axiom, (! [B:$tType,A:$tType,X3:A,A2:(set @ A),F:(A > B)]: (((((member @ A) @ X3) @ A2) => (((member @ B) @ (F @ X3)) @ ((((image @ A) @ B) @ F) @ A2)))))).
% 2.77/1.52 thf(fact_57_Collect__mono__iff, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P2)) @ ((collect @ A) @ Q)) = (! [X4:A]: (((P2 @ X4) => (Q @ X4)))))))).
% 2.77/1.52 thf(fact_58_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y2:(set @ A),Z:(set @ A)]: ((Y2 = Z))) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A6)))))))).
% 2.77/1.52 thf(fact_59_subset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C4) => (((ord_less_eq @ (set @ A)) @ A2) @ C4)))))).
% 2.77/1.52 thf(fact_60_Collect__mono, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P2 @ X2) => (Q @ X2)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P2)) @ ((collect @ A) @ Q)))))).
% 2.77/1.52 thf(fact_61_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A6) => (((member @ A) @ T2) @ B6)))))))))).
% 2.77/1.52 thf(fact_62_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 2.77/1.52 thf(fact_63_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((! [X4:A]: (((((member @ A) @ X4) @ A6) => (((member @ A) @ X4) @ B6)))))))))).
% 2.77/1.52 thf(fact_64_equalityE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A2)))))))))).
% 2.77/1.52 thf(fact_65_subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C3) @ A2) => (((member @ A) @ C3) @ B2)))))).
% 2.77/1.52 thf(fact_66_in__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X3) @ A2) => (((member @ A) @ X3) @ B2)))))).
% 2.77/1.52 thf(fact_67_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y22:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y22)) = ((X1 = Y1) & (X22 = Y22)))))).
% 2.77/1.52 thf(fact_68_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A7:A,B7:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = ((((product_Pair @ A) @ B) @ A7) @ B7)) = ((A3 = A7) & (B3 = B7)))))).
% 2.77/1.52 thf(fact_69_eq__fst__iff, axiom, (! [A:$tType,B:$tType,A3:A,P:((product_prod @ A) @ B)]: (((A3 = (((product_fst @ A) @ B) @ P)) = (? [B4:B]: ((P = ((((product_Pair @ A) @ B) @ A3) @ B4)))))))).
% 2.77/1.52 thf(fact_70_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X22:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = X1)))).
% 2.77/1.52 thf(fact_71_fst__eqD, axiom, (! [B:$tType,A:$tType,X3:A,Y3:B,A3:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) = A3) => (X3 = A3))))).
% 2.77/1.52 thf(fact_72_fstI, axiom, (! [B:$tType,A:$tType,X3:((product_prod @ A) @ B),Y3:A,Z2:B]: (((X3 = ((((product_Pair @ A) @ B) @ Y3) @ Z2)) => ((((product_fst @ A) @ B) @ X3) = Y3))))).
% 2.77/1.52 thf(fact_73_all__subset__image, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),P2:((set @ A) > $o)]: (((! [B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B6) @ ((((image @ B) @ A) @ F) @ A2)) => (P2 @ B6)))) = (! [B6:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ B6) @ A2) => (P2 @ ((((image @ B) @ A) @ F) @ B6))))))))).
% 2.77/1.52 thf(fact_74_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca1785829860lChain @ A) @ B) = (^ [R:(set @ ((product_prod @ A) @ A)),As:(A > B)]: ((! [I:A,J:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R) => (((ord_less_eq @ B) @ (As @ I)) @ (As @ J)))))))))))).
% 2.77/1.52 thf(fact_75_Greatest__equality, axiom, (! [A:$tType]: (((order @ A) => (! [P2:(A > $o),X3:A]: (((P2 @ X3) => ((! [Y:A]: (((P2 @ Y) => (((ord_less_eq @ A) @ Y) @ X3)))) => (((order_Greatest @ A) @ P2) = X3))))))))).
% 2.77/1.52 thf(fact_76_GreatestI2__order, axiom, (! [A:$tType]: (((order @ A) => (! [P2:(A > $o),X3:A,Q:(A > $o)]: (((P2 @ X3) => ((! [Y:A]: (((P2 @ Y) => (((ord_less_eq @ A) @ Y) @ X3)))) => ((! [X2:A]: (((P2 @ X2) => ((! [Y5:A]: (((P2 @ Y5) => (((ord_less_eq @ A) @ Y5) @ X2)))) => (Q @ X2))))) => (Q @ ((order_Greatest @ A) @ P2))))))))))).
% 2.77/1.52 thf(fact_77_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P2 @ Prod))))).
% 2.77/1.52 thf(fact_78_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y3:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B5:B]: ((Y3 != ((((product_Pair @ A) @ B) @ A5) @ B5))))))))).
% 2.77/1.52 thf(fact_79_prod__induct7, axiom, (! [G3:$tType,F3:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))))))]: (((! [A5:A,B5:B,C5:C,D2:D,E2:E,F4:F3,G4:G3]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))) @ C5) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F3) @ G3)) @ E2) @ ((((product_Pair @ F3) @ G3) @ F4) @ G4))))))))) => (P2 @ X3))))).
% 2.77/1.52 thf(fact_80_prod__induct6, axiom, (! [F3:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))))]: (((! [A5:A,B5:B,C5:C,D2:D,E2:E,F4:F3]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))) @ C5) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F3)) @ D2) @ ((((product_Pair @ E) @ F3) @ E2) @ F4)))))))) => (P2 @ X3))))).
% 2.77/1.52 thf(fact_81_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A5:A,B5:B,C5:C,D2:D,E2:E]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C5) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P2 @ X3))))).
% 2.77/1.52 thf(fact_82_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A5:A,B5:B,C5:C,D2:D]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C5) @ D2)))))) => (P2 @ X3))))).
% 2.77/1.52 thf(fact_83_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B5:B,C5:C]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B5) @ C5))))) => (P2 @ X3))))).
% 2.77/1.52 thf(fact_84_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F3:$tType,G3:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))))))]: ((~ ((! [A5:A,B5:B,C5:C,D2:D,E2:E,F4:F3,G4:G3]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))) @ C5) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F3) @ G3)) @ E2) @ ((((product_Pair @ F3) @ G3) @ F4) @ G4)))))))))))))).
% 2.77/1.52 thf(fact_85_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F3:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))))]: ((~ ((! [A5:A,B5:B,C5:C,D2:D,E2:E,F4:F3]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))) @ C5) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F3)) @ D2) @ ((((product_Pair @ E) @ F3) @ E2) @ F4))))))))))))).
% 2.77/1.52 thf(fact_86_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A5:A,B5:B,C5:C,D2:D,E2:E]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C5) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 2.77/1.52 thf(fact_87_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A5:A,B5:B,C5:C,D2:D]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C5) @ D2))))))))))).
% 2.77/1.52 thf(fact_88_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B5:B,C5:C]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B5) @ C5)))))))))).
% 2.77/1.52 thf(fact_89_Pair__inject, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A7:A,B7:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = ((((product_Pair @ A) @ B) @ A7) @ B7)) => (~ (((A3 = A7) => (B3 != B7)))))))).
% 2.77/1.52 thf(fact_90_prod__cases, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),P:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P2 @ P))))).
% 2.77/1.52 thf(fact_91_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A3:A,B3:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((F1 @ A3) @ B3))))).
% 2.77/1.52 thf(fact_92_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(B > (C > A)),A3:B,B3:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C3) @ ((((product_Pair @ B) @ C) @ A3) @ B3)) = ((C3 @ A3) @ B3))))).
% 2.77/1.52 thf(fact_93_image__Fpow__mono, axiom, (! [B:$tType,A:$tType,F:(B > A),A2:(set @ B),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A2)) @ B2) => (((ord_less_eq @ (set @ (set @ A))) @ ((((image @ (set @ B)) @ (set @ A)) @ (((image @ B) @ A) @ F)) @ ((finite_Fpow @ B) @ A2))) @ ((finite_Fpow @ A) @ B2)))))).
% 2.77/1.52 thf(fact_94_not__sub, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((~ ((((ord_less_eq @ (set @ A)) @ A2) @ B2))) => (? [A5:A]: (((((member @ A) @ A5) @ A2) & (~ ((((member @ A) @ A5) @ B2)))))))))).
% 2.77/1.52 thf(fact_95_eqsets__sub, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.77/1.52 thf(fact_96_sub__which1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X3:A]: ((((((ord_less_eq @ (set @ A)) @ A2) @ B2) | (((ord_less_eq @ (set @ A)) @ B2) @ A2)) => ((((member @ A) @ X3) @ A2) => ((~ ((((member @ A) @ X3) @ B2))) => (((ord_less_eq @ (set @ A)) @ B2) @ A2))))))).
% 2.77/1.52 thf(fact_97_Fpow__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ (set @ A))) @ ((finite_Fpow @ A) @ A2)) @ ((finite_Fpow @ A) @ B2)))))).
% 2.77/1.52 thf(fact_98_le__convert, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C3:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ A3) @ C3) => (((ord_less_eq @ A) @ B3) @ C3))))))))).
% 2.77/1.52 thf(fact_99_ge__convert, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C3:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ C3) @ A3) => (((ord_less_eq @ A) @ C3) @ B3))))))))).
% 2.77/1.52 thf(fact_100_mem__in__image3, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),A2:(set @ B)]: (((((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ A2)) => (? [X2:B]: (((((member @ B) @ X2) @ A2) & (B3 = (F @ X2))))))))).
% 2.77/1.52 thf(fact_101_mem__in__image2, axiom, (! [B:$tType,A:$tType,A3:A,A2:(set @ A),F:(A > B)]: (((((member @ A) @ A3) @ A2) => (((member @ B) @ (F @ A3)) @ ((((image @ A) @ B) @ F) @ A2)))))).
% 2.77/1.52 thf(fact_102_mem__in__image1, axiom, (! [B:$tType,A:$tType,A2:(set @ A),F:(A > B),B2:(set @ B),A3:A]: (((! [X2:A]: (((((member @ A) @ X2) @ A2) => (((member @ B) @ (F @ X2)) @ B2)))) => ((((member @ A) @ A3) @ A2) => (((member @ B) @ (F @ A3)) @ ((((image @ A) @ B) @ F) @ A2))))))).
% 2.77/1.52 thf(fact_103_proper__subset, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((~ ((((member @ A) @ X3) @ A2))) => ((((member @ A) @ X3) @ B2) => (A2 != B2))))))).
% 2.77/1.52 thf(fact_104_not__subseteq, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((~ ((((ord_less_eq @ (set @ A)) @ A2) @ B2))) => (? [X2:A]: (((((member @ A) @ X2) @ A2) & (~ ((((member @ A) @ X2) @ B2)))))))))).
% 2.77/1.52 thf(fact_105_subset__self, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 2.77/1.52 thf(fact_106_sets__not__eq, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 != B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => (? [X2:A]: (((((member @ A) @ X2) @ A2) & (~ ((((member @ A) @ X2) @ B2))))))))))).
% 2.77/1.52 thf(fact_107_sub__which2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X3:A]: ((((((ord_less_eq @ (set @ A)) @ A2) @ B2) | (((ord_less_eq @ (set @ A)) @ B2) @ A2)) => ((~ ((((member @ A) @ X3) @ A2))) => ((((member @ A) @ X3) @ B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))))).
% 2.77/1.52 thf(fact_108_image__Pow__mono, axiom, (! [B:$tType,A:$tType,F:(B > A),A2:(set @ B),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A2)) @ B2) => (((ord_less_eq @ (set @ (set @ A))) @ ((((image @ (set @ B)) @ (set @ A)) @ (((image @ B) @ A) @ F)) @ ((pow @ B) @ A2))) @ ((pow @ A) @ B2)))))).
% 2.77/1.52 thf(fact_109_Sup_OSUP__cong, axiom, (! [A:$tType,B:$tType,A2:(set @ B),B2:(set @ B),C4:(B > A),D3:(B > A),Sup:((set @ A) > A)]: (((A2 = B2) => ((! [X2:B]: (((((member @ B) @ X2) @ B2) => ((C4 @ X2) = (D3 @ X2))))) => ((Sup @ ((((image @ B) @ A) @ C4) @ A2)) = (Sup @ ((((image @ B) @ A) @ D3) @ B2)))))))).
% 2.77/1.52 thf(fact_110_Inf_OINF__cong, axiom, (! [A:$tType,B:$tType,A2:(set @ B),B2:(set @ B),C4:(B > A),D3:(B > A),Inf:((set @ A) > A)]: (((A2 = B2) => ((! [X2:B]: (((((member @ B) @ X2) @ B2) => ((C4 @ X2) = (D3 @ X2))))) => ((Inf @ ((((image @ B) @ A) @ C4) @ A2)) = (Inf @ ((((image @ B) @ A) @ D3) @ B2)))))))).
% 2.77/1.52 thf(fact_111_subrelI, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((! [X2:A,Y:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ R2) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ S)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R2) @ S))))).
% 2.77/1.52 thf(fact_112_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R2:A,S:B,R3:(set @ ((product_prod @ A) @ B)),S2:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R2) @ S)) @ R3) => ((S2 = S) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R2) @ S2)) @ R3)))))).
% 2.77/1.52 thf(fact_113_PowI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((member @ (set @ A)) @ A2) @ ((pow @ A) @ B2)))))).
% 2.77/1.52 thf(fact_114_Pow__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((member @ (set @ A)) @ A2) @ ((pow @ A) @ B2)) = (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.77/1.52 thf(fact_115_Pow__top, axiom, (! [A:$tType,A2:(set @ A)]: ((((member @ (set @ A)) @ A2) @ ((pow @ A) @ A2))))).
% 2.77/1.52 thf(fact_116_PowD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((member @ (set @ A)) @ A2) @ ((pow @ A) @ B2)) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.77/1.52 thf(fact_117_Pow__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ (set @ A))) @ ((pow @ A) @ A2)) @ ((pow @ A) @ B2)))))).
% 2.77/1.52 thf(fact_118_image__Pow__surj, axiom, (! [B:$tType,A:$tType,F:(B > A),A2:(set @ B),B2:(set @ A)]: (((((((image @ B) @ A) @ F) @ A2) = B2) => (((((image @ (set @ B)) @ (set @ A)) @ (((image @ B) @ A) @ F)) @ ((pow @ B) @ A2)) = ((pow @ A) @ B2)))))).
% 2.77/1.52 thf(fact_119_Fpow__subset__Pow, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ (set @ A))) @ ((finite_Fpow @ A) @ A2)) @ ((pow @ A) @ A2))))).
% 2.77/1.52 thf(fact_120_Cantors__paradox, axiom, (! [A:$tType,A2:(set @ A)]: ((~ ((? [F5:(A > (set @ A))]: ((((((image @ A) @ (set @ A)) @ F5) @ A2) = ((pow @ A) @ A2))))))))).
% 2.77/1.52 thf(fact_121_curry__conv, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_curry @ B) @ C) @ A) = (^ [F2:(((product_prod @ B) @ C) > A),A4:B,B4:C]: ((F2 @ ((((product_Pair @ B) @ C) @ A4) @ B4)))))))).
% 2.77/1.52 thf(fact_122_in__inv__imagep, axiom, (! [B:$tType,A:$tType]: ((((inv_imagep @ A) @ B) = (^ [R:(A > (A > $o)),F2:(B > A),X4:B,Y4:B]: (((R @ (F2 @ X4)) @ (F2 @ Y4)))))))).
% 2.77/1.52 thf(fact_123_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B3:A]: (((A3 = B3) | ((~ ((((ord_less_eq @ A) @ A3) @ B3))) | (~ ((((ord_less_eq @ A) @ B3) @ A3))))))))))).
% 2.77/1.52 thf(fact_124_le__rel__bool__arg__iff, axiom, (! [A:$tType]: (((ord @ A) => ((ord_less_eq @ ($o > A)) = (^ [X5:($o > A),Y6:($o > A)]: (((((ord_less_eq @ A) @ (X5 @ $false)) @ (Y6 @ $false)) & (((ord_less_eq @ A) @ (X5 @ $true)) @ (Y6 @ $true)))))))))).
% 2.77/1.52 thf(fact_125_curryI, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A3:A,B3:B]: (((F @ ((((product_Pair @ A) @ B) @ A3) @ B3)) => ((((((product_curry @ A) @ B) @ $o) @ F) @ A3) @ B3))))).
% 2.77/1.52 thf(fact_126_curryD, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A3:A,B3:B]: ((((((((product_curry @ A) @ B) @ $o) @ F) @ A3) @ B3) => (F @ ((((product_Pair @ A) @ B) @ A3) @ B3)))))).
% 2.77/1.52 thf(fact_127_curryE, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A3:A,B3:B]: ((((((((product_curry @ A) @ B) @ $o) @ F) @ A3) @ B3) => (F @ ((((product_Pair @ A) @ B) @ A3) @ B3)))))).
% 2.77/1.52 thf(fact_128_in__inv__image, axiom, (! [A:$tType,B:$tType,X3:A,Y3:A,R2:(set @ ((product_prod @ B) @ B)),F:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ ((((inv_image @ B) @ A) @ R2) @ F)) = (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F @ X3)) @ (F @ Y3))) @ R2))))).
% 2.77/1.52 thf(fact_129_fst__eq__Domain, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B))]: ((((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ R3) = (((domain @ A) @ B) @ R3))))).
% 2.77/1.52 thf(fact_130_Domain__fst, axiom, (! [B:$tType,A:$tType]: ((((domain @ A) @ B) = (((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)))))).
% 2.77/1.52 thf(fact_131_Range__mono, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R2) @ S) => (((ord_less_eq @ (set @ B)) @ (((range @ A) @ B) @ R2)) @ (((range @ A) @ B) @ S)))))).
% 2.77/1.52 thf(fact_132_RangeE, axiom, (! [A:$tType,B:$tType,B3:A,R2:(set @ ((product_prod @ B) @ A))]: (((((member @ A) @ B3) @ (((range @ B) @ A) @ R2)) => (~ ((! [A5:B]: ((~ ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A5) @ B3)) @ R2))))))))))).
% 2.77/1.52 thf(fact_133_Range__iff, axiom, (! [A:$tType,B:$tType,A3:A,R2:(set @ ((product_prod @ B) @ A))]: (((((member @ A) @ A3) @ (((range @ B) @ A) @ R2)) = (? [Y4:B]: ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ Y4) @ A3)) @ R2))))))).
% 2.77/1.52 thf(fact_134_Range_Ocases, axiom, (! [B:$tType,A:$tType,A3:B,R2:(set @ ((product_prod @ A) @ B))]: (((((member @ B) @ A3) @ (((range @ A) @ B) @ R2)) => (~ ((! [A5:A]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ A3)) @ R2))))))))))).
% 2.77/1.52 thf(fact_135_Range_Osimps, axiom, (! [B:$tType,A:$tType,A3:B,R2:(set @ ((product_prod @ A) @ B))]: (((((member @ B) @ A3) @ (((range @ A) @ B) @ R2)) = (? [A4:A,B4:B]: (((A3 = B4) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ R2)))))))).
% 2.77/1.52 thf(fact_136_Range_Ointros, axiom, (! [B:$tType,A:$tType,A3:A,B3:B,R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R2) => (((member @ B) @ B3) @ (((range @ A) @ B) @ R2)))))).
% 2.77/1.52 thf(fact_137_Range_Oinducts, axiom, (! [A:$tType,B:$tType,X3:B,R2:(set @ ((product_prod @ A) @ B)),P2:(B > $o)]: (((((member @ B) @ X3) @ (((range @ A) @ B) @ R2)) => ((! [A5:A,B5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ R2) => (P2 @ B5)))) => (P2 @ X3)))))).
% 2.77/1.52 thf(fact_138_DomainE, axiom, (! [B:$tType,A:$tType,A3:A,R2:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ (((domain @ A) @ B) @ R2)) => (~ ((! [B5:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B5)) @ R2))))))))))).
% 2.77/1.52 thf(fact_139_Domain__iff, axiom, (! [A:$tType,B:$tType,A3:A,R2:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ (((domain @ A) @ B) @ R2)) = (? [Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ Y4)) @ R2))))))).
% 2.77/1.52 thf(fact_140_Domain_Ocases, axiom, (! [B:$tType,A:$tType,A3:A,R2:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ (((domain @ A) @ B) @ R2)) => (~ ((! [B5:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B5)) @ R2))))))))))).
% 2.77/1.52 thf(fact_141_Domain_Osimps, axiom, (! [B:$tType,A:$tType,A3:A,R2:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ (((domain @ A) @ B) @ R2)) = (? [A4:A,B4:B]: (((A3 = A4) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ R2)))))))).
% 2.77/1.52 thf(fact_142_Domain_ODomainI, axiom, (! [B:$tType,A:$tType,A3:A,B3:B,R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R2) => (((member @ A) @ A3) @ (((domain @ A) @ B) @ R2)))))).
% 2.77/1.52 thf(fact_143_Domain_Oinducts, axiom, (! [B:$tType,A:$tType,X3:A,R2:(set @ ((product_prod @ A) @ B)),P2:(A > $o)]: (((((member @ A) @ X3) @ (((domain @ A) @ B) @ R2)) => ((! [A5:A,B5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ R2) => (P2 @ A5)))) => (P2 @ X3)))))).
% 2.77/1.52 thf(fact_144_Domain__mono, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R2) @ S) => (((ord_less_eq @ (set @ A)) @ (((domain @ A) @ B) @ R2)) @ (((domain @ A) @ B) @ S)))))).
% 2.77/1.52 thf(fact_145_Range__converse, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B))]: (((((range @ B) @ A) @ (((converse @ A) @ B) @ R2)) = (((domain @ A) @ B) @ R2))))).
% 2.77/1.52 thf(fact_146_Domain__converse, axiom, (! [A:$tType,B:$tType,R2:(set @ ((product_prod @ B) @ A))]: (((((domain @ A) @ B) @ (((converse @ B) @ A) @ R2)) = (((range @ B) @ A) @ R2))))).
% 2.77/1.52 thf(fact_147_pair__in__swap__image, axiom, (! [A:$tType,B:$tType,Y3:A,X3:B,A2:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y3) @ X3)) @ ((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ A2)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X3) @ Y3)) @ A2))))).
% 2.77/1.52 thf(fact_148_swap__simp, axiom, (! [A:$tType,B:$tType,X3:B,Y3:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X3) @ Y3)) = ((((product_Pair @ A) @ B) @ Y3) @ X3))))).
% 2.77/1.52 thf(fact_149_converse__converse, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B))]: (((((converse @ B) @ A) @ (((converse @ A) @ B) @ R2)) = R2)))).
% 2.77/1.52 thf(fact_150_converse__inject, axiom, (! [A:$tType,B:$tType,R2:(set @ ((product_prod @ B) @ A)),S:(set @ ((product_prod @ B) @ A))]: ((((((converse @ B) @ A) @ R2) = (((converse @ B) @ A) @ S)) = (R2 = S))))).
% 2.77/1.52 thf(fact_151_swap__swap, axiom, (! [B:$tType,A:$tType,P:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P)) = P)))).
% 2.77/1.52 thf(fact_152_converse__iff, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,R2:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ (((converse @ B) @ A) @ R2)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ B3) @ A3)) @ R2))))).
% 2.77/1.52 thf(fact_153_converse__mono, axiom, (! [A:$tType,B:$tType,R2:(set @ ((product_prod @ B) @ A)),S:(set @ ((product_prod @ B) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ (((converse @ B) @ A) @ R2)) @ (((converse @ B) @ A) @ S)) = (((ord_less_eq @ (set @ ((product_prod @ B) @ A))) @ R2) @ S))))).
% 2.77/1.52 thf(fact_154_converse__inv__image, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ B) @ B)),F:(A > B)]: (((((converse @ A) @ A) @ ((((inv_image @ B) @ A) @ R3) @ F)) = ((((inv_image @ B) @ A) @ (((converse @ B) @ B) @ R3)) @ F))))).
% 2.77/1.52 thf(fact_155_converse_Oinducts, axiom, (! [B:$tType,A:$tType,X1:B,X22:A,R2:(set @ ((product_prod @ A) @ B)),P2:(B > (A > $o))]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X1) @ X22)) @ (((converse @ A) @ B) @ R2)) => ((! [A5:A,B5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ R2) => ((P2 @ B5) @ A5)))) => ((P2 @ X1) @ X22)))))).
% 2.77/1.52 thf(fact_156_converse_Ointros, axiom, (! [B:$tType,A:$tType,A3:A,B3:B,R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R2) => (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ B3) @ A3)) @ (((converse @ A) @ B) @ R2)))))).
% 2.77/1.52 thf(fact_157_converse_Osimps, axiom, (! [B:$tType,A:$tType,A1:B,A22:A,R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A1) @ A22)) @ (((converse @ A) @ B) @ R2)) = (? [A4:A,B4:B]: (((A1 = B4) & ((A22 = A4) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ R2))))))))).
% 2.77/1.52 thf(fact_158_converse_Ocases, axiom, (! [B:$tType,A:$tType,A1:B,A22:A,R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A1) @ A22)) @ (((converse @ A) @ B) @ R2)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A22) @ A1)) @ R2))))).
% 2.77/1.52 thf(fact_159_converseE, axiom, (! [A:$tType,B:$tType,Yx:((product_prod @ A) @ B),R2:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ Yx) @ (((converse @ B) @ A) @ R2)) => (~ ((! [X2:B,Y:A]: (((Yx = ((((product_Pair @ A) @ B) @ Y) @ X2)) => (~ ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X2) @ Y)) @ R2)))))))))))).
% 2.77/1.52 thf(fact_160_converseD, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,R2:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ (((converse @ B) @ A) @ R2)) => (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ B3) @ A3)) @ R2))))).
% 2.77/1.52 thf(fact_161_converse__subset__swap, axiom, (! [A:$tType,B:$tType,R2:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ B) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R2) @ (((converse @ B) @ A) @ S)) = (((ord_less_eq @ (set @ ((product_prod @ B) @ A))) @ (((converse @ A) @ B) @ R2)) @ S))))).
% 2.77/1.52 thf(fact_162_surj__swap, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ ((product_prod @ A) @ B))))))).
% 2.77/1.52 thf(fact_163_prod_Oswap__def, axiom, (! [B:$tType,A:$tType]: ((((product_swap @ A) @ B) = (^ [P3:((product_prod @ A) @ B)]: (((((product_Pair @ B) @ A) @ (((product_snd @ A) @ B) @ P3)) @ (((product_fst @ A) @ B) @ P3)))))))).
% 2.77/1.52 thf(fact_164_map__prod__imageI, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,A3:A,B3:B,R3:(set @ ((product_prod @ A) @ B)),F:(A > C),G2:(B > D)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R3) => (((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (F @ A3)) @ (G2 @ B3))) @ ((((image @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F) @ G2)) @ R3)))))).
% 2.77/1.52 thf(fact_165_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X4:D]: ((top_top @ C)))))))).
% 2.77/1.52 thf(fact_166_UNIV__I, axiom, (! [A:$tType,X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))).
% 2.77/1.52 thf(fact_167_map__prod__simp, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F:(C > A),G2:(D > B),A3:C,B3:D]: (((((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G2) @ ((((product_Pair @ C) @ D) @ A3) @ B3)) = ((((product_Pair @ A) @ B) @ (F @ A3)) @ (G2 @ B3)))))).
% 2.77/1.52 thf(fact_168_Pow__UNIV, axiom, (! [A:$tType]: ((((pow @ A) @ (top_top @ (set @ A))) = (top_top @ (set @ (set @ A))))))).
% 2.77/1.52 thf(fact_169_fst__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(C > A),G2:(D > B),X3:((product_prod @ C) @ D)]: (((((product_fst @ A) @ B) @ (((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G2) @ X3)) = (F @ (((product_fst @ C) @ D) @ X3)))))).
% 2.77/1.52 thf(fact_170_snd__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(C > B),G2:(D > A),X3:((product_prod @ C) @ D)]: (((((product_snd @ B) @ A) @ (((((((product_map_prod @ C) @ B) @ D) @ A) @ F) @ G2) @ X3)) = (G2 @ (((product_snd @ C) @ D) @ X3)))))).
% 2.77/1.52 thf(fact_171_converse__UNIV, axiom, (! [B:$tType,A:$tType]: (((((converse @ B) @ A) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ ((product_prod @ A) @ B))))))).
% 2.77/1.52 thf(fact_172_prod_Ocollapse, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)) = Prod)))).
% 2.77/1.52 thf(fact_173_fst__swap, axiom, (! [A:$tType,B:$tType,X3:((product_prod @ B) @ A)]: (((((product_fst @ A) @ B) @ (((product_swap @ B) @ A) @ X3)) = (((product_snd @ B) @ A) @ X3))))).
% 2.77/1.52 thf(fact_174_snd__swap, axiom, (! [B:$tType,A:$tType,X3:((product_prod @ A) @ B)]: (((((product_snd @ B) @ A) @ (((product_swap @ A) @ B) @ X3)) = (((product_fst @ A) @ B) @ X3))))).
% 2.77/1.52 thf(fact_175_range__fst, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = (top_top @ (set @ A)))))).
% 2.77/1.52 thf(fact_176_range__snd, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ A)))))).
% 2.77/1.52 thf(fact_177_rangeI, axiom, (! [A:$tType,B:$tType,F:(B > A),X3:B]: ((((member @ A) @ (F @ X3)) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B))))))).
% 2.77/1.52 thf(fact_178_range__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),X3:B]: (((B3 = (F @ X3)) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))))))).
% 2.77/1.52 thf(fact_179_surj__def, axiom, (! [B:$tType,A:$tType,F:(B > A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y4:A]: ((? [X4:B]: ((Y4 = (F @ X4)))))))))).
% 2.77/1.52 thf(fact_180_surjI, axiom, (! [B:$tType,A:$tType,G2:(B > A),F:(A > B)]: (((! [X2:A]: (((G2 @ (F @ X2)) = X2))) => (((((image @ B) @ A) @ G2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.77/1.52 thf(fact_181_surjE, axiom, (! [A:$tType,B:$tType,F:(B > A),Y3:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X2:B]: ((Y3 != (F @ X2)))))))))).
% 2.77/1.52 thf(fact_182_surjD, axiom, (! [A:$tType,B:$tType,F:(B > A),Y3:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X2:B]: ((Y3 = (F @ X2)))))))).
% 2.77/1.52 thf(fact_183_map__prod__surj, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F:(A > B),G2:(C > D)]: (((((((image @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((((((image @ C) @ D) @ G2) @ (top_top @ (set @ C))) = (top_top @ (set @ D))) => (((((image @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)) @ ((((((product_map_prod @ A) @ B) @ C) @ D) @ F) @ G2)) @ (top_top @ (set @ ((product_prod @ A) @ C)))) = (top_top @ (set @ ((product_prod @ B) @ D))))))))).
% 2.77/1.52 thf(fact_184_top_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) => (A3 = (top_top @ A))))))))).
% 2.77/1.52 thf(fact_185_top_Oextremum__unique, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) = (A3 = (top_top @ A))))))))).
% 2.77/1.52 thf(fact_186_top__greatest, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ (top_top @ A)))))))).
% 2.77/1.52 thf(fact_187_subset__UNIV, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ (top_top @ (set @ A)))))).
% 2.77/1.52 thf(fact_188_sndI, axiom, (! [A:$tType,B:$tType,X3:((product_prod @ A) @ B),Y3:A,Z2:B]: (((X3 = ((((product_Pair @ A) @ B) @ Y3) @ Z2)) => ((((product_snd @ A) @ B) @ X3) = Z2))))).
% 2.77/1.52 thf(fact_189_snd__eqD, axiom, (! [B:$tType,A:$tType,X3:B,Y3:A,A3:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X3) @ Y3)) = A3) => (Y3 = A3))))).
% 2.77/1.52 thf(fact_190_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X22:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X22)) = X22)))).
% 2.77/1.52 thf(fact_191_eq__snd__iff, axiom, (! [A:$tType,B:$tType,B3:A,P:((product_prod @ B) @ A)]: (((B3 = (((product_snd @ B) @ A) @ P)) = (? [A4:B]: ((P = ((((product_Pair @ B) @ A) @ A4) @ B3)))))))).
% 2.77/1.52 thf(fact_192_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y2:((product_prod @ A) @ B),Z:((product_prod @ A) @ B)]: ((Y2 = Z))) = (^ [S3:((product_prod @ A) @ B),T2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S3) = (((product_fst @ A) @ B) @ T2)) & ((((product_snd @ A) @ B) @ S3) = (((product_snd @ A) @ B) @ T2))))))))).
% 2.77/1.52 thf(fact_193_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod2:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod2)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod2))) => (Prod = Prod2))))).
% 2.77/1.52 thf(fact_194_prod__eqI, axiom, (! [B:$tType,A:$tType,P:((product_prod @ A) @ B),Q2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P) = (((product_fst @ A) @ B) @ Q2)) => (((((product_snd @ A) @ B) @ P) = (((product_snd @ A) @ B) @ Q2)) => (P = Q2)))))).
% 2.77/1.52 thf(fact_195_UNIV__eq__I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X2:A]: ((((member @ A) @ X2) @ A2))) => ((top_top @ (set @ A)) = A2))))).
% 2.77/1.52 thf(fact_196_UNIV__witness, axiom, (! [A:$tType]: ((? [X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))))).
% 2.77/1.52 thf(fact_197_surjective__pairing, axiom, (! [B:$tType,A:$tType,T3:((product_prod @ A) @ B)]: ((T3 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T3)) @ (((product_snd @ A) @ B) @ T3)))))).
% 2.77/1.52 thf(fact_198_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))).
% 2.77/1.52 thf(fact_199_snd__eq__Range, axiom, (! [A:$tType,B:$tType,R3:(set @ ((product_prod @ B) @ A))]: ((((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ R3) = (((range @ B) @ A) @ R3))))).
% 2.77/1.52 thf(fact_200_Range__snd, axiom, (! [A:$tType,B:$tType]: ((((range @ B) @ A) = (((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)))))).
% 2.77/1.52 thf(fact_201_prod__fun__imageE, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,C3:((product_prod @ A) @ B),F:(C > A),G2:(D > B),R3:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ A) @ B)) @ C3) @ ((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G2)) @ R3)) => (~ ((! [X2:C,Y:D]: (((C3 = ((((product_Pair @ A) @ B) @ (F @ X2)) @ (G2 @ Y))) => (~ ((((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ X2) @ Y)) @ R3)))))))))))).
% 2.77/1.52 thf(fact_202_exI__realizer, axiom, (! [B:$tType,A:$tType,P2:(A > (B > $o)),Y3:A,X3:B]: ((((P2 @ Y3) @ X3) => ((P2 @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X3) @ Y3))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X3) @ Y3))))))).
% 2.77/1.52 thf(fact_203_conjI__realizer, axiom, (! [A:$tType,B:$tType,P2:(A > $o),P:A,Q:(B > $o),Q2:B]: (((P2 @ P) => ((Q @ Q2) => ((P2 @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P) @ Q2))) & (Q @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P) @ Q2))))))))).
% 2.77/1.52 thf(fact_204_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P2:(A > (B > $o)),X3:A,Y3:B,A3:((product_prod @ A) @ B)]: ((((P2 @ X3) @ Y3) => ((A3 = ((((product_Pair @ A) @ B) @ X3) @ Y3)) => ((P2 @ (((product_fst @ A) @ B) @ A3)) @ (((product_snd @ A) @ B) @ A3))))))).
% 2.77/1.52 thf(fact_205_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.77/1.52 thf(fact_206_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X4:A]: ((((member @ A) @ X4) @ (top_top @ (set @ A))))))))).
% 2.77/1.52 thf(fact_207_exE__realizer_H, axiom, (! [A:$tType,B:$tType,P2:(A > (B > $o)),P:((product_prod @ B) @ A)]: ((((P2 @ (((product_snd @ B) @ A) @ P)) @ (((product_fst @ B) @ A) @ P)) => (~ ((! [X2:B,Y:A]: ((~ (((P2 @ Y) @ X2))))))))))).
% 2.77/1.52 thf(fact_208_Collect__split__mono__strong, axiom, (! [B:$tType,A:$tType,X6:(set @ A),A2:(set @ ((product_prod @ A) @ B)),Y7:(set @ B),P2:(A > (B > $o)),Q:(A > (B > $o))]: (((X6 = ((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ A2)) => ((Y7 = ((((image @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ A2)) => ((! [X2:A]: (((((member @ A) @ X2) @ X6) => (! [Xa2:B]: (((((member @ B) @ Xa2) @ Y7) => (((P2 @ X2) @ Xa2) => ((Q @ X2) @ Xa2)))))))) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A2) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ P2))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A2) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ Q)))))))))).
% 2.77/1.52 thf(fact_209_sndOp__def, axiom, (! [B:$tType,A:$tType,C:$tType]: (((((bNF_sndOp @ C) @ A) @ B) = (^ [P4:(C > (A > $o)),Q3:(A > (B > $o)),Ac:((product_prod @ C) @ B)]: (((((product_Pair @ A) @ B) @ (((((((bNF_pick_middlep @ C) @ A) @ B) @ P4) @ Q3) @ (((product_fst @ C) @ B) @ Ac)) @ (((product_snd @ C) @ B) @ Ac))) @ (((product_snd @ C) @ B) @ Ac)))))))).
% 2.77/1.52 thf(fact_210_top1I, axiom, (! [A:$tType,X3:A]: (((top_top @ (A > $o)) @ X3)))).
% 2.77/1.52 thf(fact_211_curry__case__prod, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > C))]: ((((((product_curry @ A) @ B) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F)) = F)))).
% 2.77/1.52 thf(fact_212_case__prod__curry, axiom, (! [C:$tType,B:$tType,A:$tType,F:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ ((((product_curry @ A) @ B) @ C) @ F)) = F)))).
% 2.77/1.52 thf(fact_213_Product__Type_OCollect__case__prodD, axiom, (! [B:$tType,A:$tType,X3:((product_prod @ A) @ B),A2:(A > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ X3) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A2))) => ((A2 @ (((product_fst @ A) @ B) @ X3)) @ (((product_snd @ A) @ B) @ X3)))))).
% 2.77/1.52 thf(fact_214_prod_Ocase__eq__if, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [F2:(A > (B > C)),Prod3:((product_prod @ A) @ B)]: (((F2 @ (((product_fst @ A) @ B) @ Prod3)) @ (((product_snd @ A) @ B) @ Prod3)))))))).
% 2.77/1.52 thf(fact_215_case__prod__beta, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ A) = (^ [F2:(B > (C > A)),P3:((product_prod @ B) @ C)]: (((F2 @ (((product_fst @ B) @ C) @ P3)) @ (((product_snd @ B) @ C) @ P3)))))))).
% 2.77/1.52 thf(fact_216_internal__case__prod__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((produc2004651681e_prod @ A) @ B) @ C) = (((product_case_prod @ A) @ B) @ C))))).
% 2.77/1.52 thf(fact_217_Collect__case__prod__mono, axiom, (! [B:$tType,A:$tType,A2:(A > (B > $o)),B2:(A > (B > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ A2) @ B2) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A2))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ B2))))))).
% 2.77/1.52 thf(fact_218_old_Oprod_Ocase, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > (B > C)),X1:A,X22:B]: (((((((product_case_prod @ A) @ B) @ C) @ F) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = ((F @ X1) @ X22))))).
% 2.77/1.52 thf(fact_219_prod_Osplit__sel, axiom, (! [C:$tType,B:$tType,A:$tType,P2:(C > $o),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P2 @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) => (P2 @ ((F @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))).
% 2.77/1.52 thf(fact_220_prod_Osplit__sel__asm, axiom, (! [C:$tType,B:$tType,A:$tType,P2:(C > $o),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P2 @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = (~ (((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) & (~ ((P2 @ ((F @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))))))).
% 2.77/1.52 thf(fact_221_fstOp__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((bNF_fstOp @ A) @ B) @ C) = (^ [P4:(A > (B > $o)),Q3:(B > (C > $o)),Ac:((product_prod @ A) @ C)]: (((((product_Pair @ A) @ B) @ (((product_fst @ A) @ C) @ Ac)) @ (((((((bNF_pick_middlep @ A) @ B) @ C) @ P4) @ Q3) @ (((product_fst @ A) @ C) @ Ac)) @ (((product_snd @ A) @ C) @ Ac))))))))).
% 2.77/1.52 thf(fact_222_split__cong, axiom, (! [C:$tType,B:$tType,A:$tType,Q2:((product_prod @ A) @ B),F:(A > (B > C)),G2:(A > (B > C)),P:((product_prod @ A) @ B)]: (((! [X2:A,Y:B]: (((((((product_Pair @ A) @ B) @ X2) @ Y) = Q2) => (((F @ X2) @ Y) = ((G2 @ X2) @ Y))))) => ((P = Q2) => ((((((product_case_prod @ A) @ B) @ C) @ F) @ P) = (((((product_case_prod @ A) @ B) @ C) @ G2) @ Q2))))))).
% 2.77/1.52 thf(fact_223_predicate2I, axiom, (! [B:$tType,A:$tType,P2:(A > (B > $o)),Q:(A > (B > $o))]: (((! [X2:A,Y:B]: ((((P2 @ X2) @ Y) => ((Q @ X2) @ Y)))) => (((ord_less_eq @ (A > (B > $o))) @ P2) @ Q))))).
% 2.77/1.52 thf(fact_224_mem__case__prodE, axiom, (! [B:$tType,A:$tType,C:$tType,Z2:A,C3:(B > (C > (set @ A))),P:((product_prod @ B) @ C)]: (((((member @ A) @ Z2) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C3) @ P)) => (~ ((! [X2:B,Y:C]: (((P = ((((product_Pair @ B) @ C) @ X2) @ Y)) => (~ ((((member @ A) @ Z2) @ ((C3 @ X2) @ Y))))))))))))).
% 2.77/1.52 thf(fact_225_rev__predicate2D, axiom, (! [A:$tType,B:$tType,P2:(A > (B > $o)),X3:A,Y3:B,Q:(A > (B > $o))]: ((((P2 @ X3) @ Y3) => ((((ord_less_eq @ (A > (B > $o))) @ P2) @ Q) => ((Q @ X3) @ Y3)))))).
% 2.77/1.52 thf(fact_226_predicate2D, axiom, (! [A:$tType,B:$tType,P2:(A > (B > $o)),Q:(A > (B > $o)),X3:A,Y3:B]: (((((ord_less_eq @ (A > (B > $o))) @ P2) @ Q) => (((P2 @ X3) @ Y3) => ((Q @ X3) @ Y3)))))).
% 2.77/1.52 thf(fact_227_refl__ge__eq, axiom, (! [A:$tType,R3:(A > (A > $o))]: (((! [X2:A]: (((R3 @ X2) @ X2))) => (((ord_less_eq @ (A > (A > $o))) @ (^ [Y2:A,Z:A]: ((Y2 = Z)))) @ R3))))).
% 2.77/1.52 thf(fact_228_ge__eq__refl, axiom, (! [A:$tType,R3:(A > (A > $o)),X3:A]: (((((ord_less_eq @ (A > (A > $o))) @ (^ [Y2:A,Z:A]: ((Y2 = Z)))) @ R3) => ((R3 @ X3) @ X3))))).
% 2.77/1.52 thf(fact_229_Collect__case__prod__in__rel__leE, axiom, (! [B:$tType,A:$tType,X6:(set @ ((product_prod @ A) @ B)),Y7:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ X6) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (((fun_in_rel @ A) @ B) @ Y7)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ X6) @ Y7))))).
% 2.77/1.52 thf(fact_230_Collect__case__prod__in__rel__leI, axiom, (! [B:$tType,A:$tType,X6:(set @ ((product_prod @ A) @ B)),Y7:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ X6) @ Y7) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ X6) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (((fun_in_rel @ A) @ B) @ Y7)))))))).
% 2.77/1.52 thf(fact_231_in__rel__def, axiom, (! [B:$tType,A:$tType]: ((((fun_in_rel @ A) @ B) = (^ [R4:(set @ ((product_prod @ A) @ B)),X4:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y4)) @ R4))))))).
% 2.77/1.52 thf(fact_232_csquare__fstOp__sndOp, axiom, (! [A:$tType,B:$tType,C:$tType,F:((A > (B > $o)) > (((product_prod @ A) @ B) > $o)),P2:(A > (C > $o)),Q:(C > (B > $o))]: ((((((((((bNF_csquare @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C)) @ C) @ ((product_prod @ C) @ B)) @ ((collect @ ((product_prod @ A) @ B)) @ (F @ (((((relcompp @ A) @ C) @ B) @ P2) @ Q)))) @ ((product_snd @ A) @ C)) @ ((product_fst @ C) @ B)) @ (((((bNF_fstOp @ A) @ C) @ B) @ P2) @ Q)) @ (((((bNF_sndOp @ A) @ C) @ B) @ P2) @ Q))))).
% 2.77/1.52 thf(fact_233_predicate2D__conj, axiom, (! [A:$tType,B:$tType,P2:(A > (B > $o)),Q:(A > (B > $o)),R3:$o,X3:A,Y3:B]: ((((((ord_less_eq @ (A > (B > $o))) @ P2) @ Q) & R3) => (R3 & (((P2 @ X3) @ Y3) => ((Q @ X3) @ Y3))))))).
% 2.77/1.52 thf(fact_234_fstOp__in, axiom, (! [B:$tType,C:$tType,A:$tType,Ac2:((product_prod @ A) @ B),P2:(A > (C > $o)),Q:(C > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ Ac2) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (((((relcompp @ A) @ C) @ B) @ P2) @ Q)))) => (((member @ ((product_prod @ A) @ C)) @ ((((((bNF_fstOp @ A) @ C) @ B) @ P2) @ Q) @ Ac2)) @ ((collect @ ((product_prod @ A) @ C)) @ ((((product_case_prod @ A) @ C) @ $o) @ P2))))))).
% 2.77/1.52 thf(fact_235_sndOp__in, axiom, (! [A:$tType,B:$tType,C:$tType,Ac2:((product_prod @ A) @ B),P2:(A > (C > $o)),Q:(C > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ Ac2) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (((((relcompp @ A) @ C) @ B) @ P2) @ Q)))) => (((member @ ((product_prod @ C) @ B)) @ ((((((bNF_sndOp @ A) @ C) @ B) @ P2) @ Q) @ Ac2)) @ ((collect @ ((product_prod @ C) @ B)) @ ((((product_case_prod @ C) @ B) @ $o) @ Q))))))).
% 2.77/1.52 thf(fact_236_pick__middlep, axiom, (! [B:$tType,A:$tType,C:$tType,P2:(A > (B > $o)),Q:(B > (C > $o)),A3:A,C3:C]: (((((((((relcompp @ A) @ B) @ C) @ P2) @ Q) @ A3) @ C3) => (((P2 @ A3) @ (((((((bNF_pick_middlep @ A) @ B) @ C) @ P2) @ Q) @ A3) @ C3)) & ((Q @ (((((((bNF_pick_middlep @ A) @ B) @ C) @ P2) @ Q) @ A3) @ C3)) @ C3)))))).
% 2.77/1.52 thf(fact_237_nchotomy__relcomppE, axiom, (! [C:$tType,B:$tType,A:$tType,D:$tType,F:(B > A),R2:(C > (A > $o)),S:(A > (D > $o)),A3:C,C3:D]: (((! [Y:A]: ((? [X:B]: ((Y = (F @ X)))))) => ((((((((relcompp @ C) @ A) @ D) @ R2) @ S) @ A3) @ C3) => (~ ((! [B5:B]: ((((R2 @ A3) @ (F @ B5)) => (~ (((S @ (F @ B5)) @ C3))))))))))))).
% 2.77/1.52 thf(fact_238_relcompp_OrelcompI, axiom, (! [A:$tType,B:$tType,C:$tType,R2:(A > (B > $o)),A3:A,B3:B,S:(B > (C > $o)),C3:C]: ((((R2 @ A3) @ B3) => (((S @ B3) @ C3) => (((((((relcompp @ A) @ B) @ C) @ R2) @ S) @ A3) @ C3)))))).
% 2.77/1.52 thf(fact_239_relcompp_Oinducts, axiom, (! [B:$tType,A:$tType,C:$tType,R2:(A > (B > $o)),S:(B > (C > $o)),X1:A,X22:C,P2:(A > (C > $o))]: (((((((((relcompp @ A) @ B) @ C) @ R2) @ S) @ X1) @ X22) => ((! [A5:A,B5:B,C5:C]: ((((R2 @ A5) @ B5) => (((S @ B5) @ C5) => ((P2 @ A5) @ C5))))) => ((P2 @ X1) @ X22)))))).
% 2.77/1.52 thf(fact_240_relcompp__assoc, axiom, (! [A:$tType,D:$tType,B:$tType,C:$tType,R2:(A > (D > $o)),S:(D > (C > $o)),T3:(C > (B > $o))]: (((((((relcompp @ A) @ C) @ B) @ (((((relcompp @ A) @ D) @ C) @ R2) @ S)) @ T3) = (((((relcompp @ A) @ D) @ B) @ R2) @ (((((relcompp @ D) @ C) @ B) @ S) @ T3)))))).
% 2.77/1.52 thf(fact_241_relcompp__apply, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((relcompp @ A) @ B) @ C) = (^ [R4:(A > (B > $o)),S4:(B > (C > $o)),A4:A,C6:C]: ((? [B4:B]: ((((R4 @ A4) @ B4) & ((S4 @ B4) @ C6)))))))))).
% 2.77/1.52 thf(fact_242_relcompp_Osimps, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((relcompp @ A) @ B) @ C) = (^ [R:(A > (B > $o)),S3:(B > (C > $o)),A12:A,A23:C]: ((? [A4:A,B4:B,C6:C]: (((A12 = A4) & ((A23 = C6) & (((R @ A4) @ B4) & ((S3 @ B4) @ C6)))))))))))).
% 2.77/1.52 thf(fact_243_relcompp_Ocases, axiom, (! [A:$tType,B:$tType,C:$tType,R2:(A > (B > $o)),S:(B > (C > $o)),A1:A,A22:C]: (((((((((relcompp @ A) @ B) @ C) @ R2) @ S) @ A1) @ A22) => (~ ((! [B5:B]: ((((R2 @ A1) @ B5) => (~ (((S @ B5) @ A22)))))))))))).
% 2.77/1.52 thf(fact_244_relcomppE, axiom, (! [A:$tType,B:$tType,C:$tType,R2:(A > (B > $o)),S:(B > (C > $o)),A3:A,C3:C]: (((((((((relcompp @ A) @ B) @ C) @ R2) @ S) @ A3) @ C3) => (~ ((! [B5:B]: ((((R2 @ A3) @ B5) => (~ (((S @ B5) @ C3)))))))))))).
% 2.77/1.52 thf(fact_245_eq__OO, axiom, (! [B:$tType,A:$tType,R3:(A > (B > $o))]: (((((((relcompp @ A) @ A) @ B) @ (^ [Y2:A,Z:A]: ((Y2 = Z)))) @ R3) = R3)))).
% 2.77/1.52 thf(fact_246_OO__eq, axiom, (! [B:$tType,A:$tType,R3:(A > (B > $o))]: (((((((relcompp @ A) @ B) @ B) @ R3) @ (^ [Y2:B,Z:B]: ((Y2 = Z)))) = R3)))).
% 2.77/1.52 thf(fact_247_csquare__def, axiom, (! [B:$tType,C:$tType,D:$tType,A:$tType]: ((((((bNF_csquare @ A) @ B) @ C) @ D) = (^ [A6:(set @ A),F12:(B > C),F22:(D > C),P1:(A > B),P22:(A > D)]: ((! [X4:A]: (((((member @ A) @ X4) @ A6) => ((F12 @ (P1 @ X4)) = (F22 @ (P22 @ X4)))))))))))).
% 2.77/1.52 thf(fact_248_leq__OOI, axiom, (! [A:$tType,R3:(A > (A > $o))]: (((R3 = (^ [Y2:A,Z:A]: ((Y2 = Z)))) => (((ord_less_eq @ (A > (A > $o))) @ R3) @ (((((relcompp @ A) @ A) @ A) @ R3) @ R3)))))).
% 2.77/1.52 thf(fact_249_relcompp__mono, axiom, (! [A:$tType,C:$tType,B:$tType,R5:(A > (B > $o)),R2:(A > (B > $o)),S2:(B > (C > $o)),S:(B > (C > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ R5) @ R2) => ((((ord_less_eq @ (B > (C > $o))) @ S2) @ S) => (((ord_less_eq @ (A > (C > $o))) @ (((((relcompp @ A) @ B) @ C) @ R5) @ S2)) @ (((((relcompp @ A) @ B) @ C) @ R2) @ S))))))).
% 2.77/1.52 thf(fact_250_image2__eqI, axiom, (! [A:$tType,C:$tType,B:$tType,B3:A,F:(B > A),X3:B,C3:C,G2:(B > C),A2:(set @ B)]: (((B3 = (F @ X3)) => ((C3 = (G2 @ X3)) => ((((member @ B) @ X3) @ A2) => (((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ B3) @ C3)) @ ((((((bNF_Greatest_image2 @ B) @ A) @ C) @ A2) @ F) @ G2)))))))).
% 2.77/1.52 thf(fact_251_strict__mono__inv, axiom, (! [A:$tType,B:$tType]: ((((linorder @ B) & (linorder @ A)) => (! [F:(A > B),G2:(B > A)]: (((((order_strict_mono @ A) @ B) @ F) => ((((((image @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((! [X2:A]: (((G2 @ (F @ X2)) = X2))) => (((order_strict_mono @ B) @ A) @ G2)))))))))).
% 2.77/1.52 thf(fact_252_strict__mono__less__eq, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F:(A > B),X3:A,Y3:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y3)) = (((ord_less_eq @ A) @ X3) @ Y3))))))))).
% 2.77/1.52 thf(fact_253_strict__mono__eq, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F:(A > B),X3:A,Y3:A]: (((((order_strict_mono @ A) @ B) @ F) => (((F @ X3) = (F @ Y3)) = (X3 = Y3))))))))).
% 2.77/1.52 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A8:$tType,A9:$tType]: (((order_top @ A9) => (order_top @ (A8 > A9)))))).
% 2.77/1.52 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 2.77/1.52 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 2.77/1.52 thf(tcon_fun___Orderings_Otop, axiom, (! [A8:$tType,A9:$tType]: (((top @ A9) => (top @ (A8 > A9)))))).
% 2.77/1.52 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 2.77/1.52 thf(tcon_Set_Oset___Orderings_Oorder__top_1, axiom, (! [A8:$tType]: ((order_top @ (set @ A8))))).
% 2.77/1.52 thf(tcon_Set_Oset___Orderings_Opreorder_2, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 2.77/1.52 thf(tcon_Set_Oset___Orderings_Oorder_3, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 2.77/1.52 thf(tcon_Set_Oset___Orderings_Otop_4, axiom, (! [A8:$tType]: ((top @ (set @ A8))))).
% 2.77/1.52 thf(tcon_Set_Oset___Orderings_Oord_5, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 2.77/1.52 thf(tcon_HOL_Obool___Orderings_Oorder__top_6, axiom, (order_top @ $o)).
% 2.77/1.52 thf(tcon_HOL_Obool___Orderings_Opreorder_7, axiom, (preorder @ $o)).
% 2.77/1.52 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 2.77/1.52 thf(tcon_HOL_Obool___Orderings_Oorder_8, axiom, (order @ $o)).
% 2.77/1.52 thf(tcon_HOL_Obool___Orderings_Otop_9, axiom, (top @ $o)).
% 2.77/1.52 thf(tcon_HOL_Obool___Orderings_Oord_10, axiom, (ord @ $o)).
% 2.77/1.52 thf(conj_0, hypothesis, (! [A10:(set @ a)]: (((((member @ (set @ a)) @ A10) @ ((((image @ ((product_prod @ (set @ a)) @ (a > e))) @ (set @ a)) @ ((product_fst @ (set @ a)) @ (a > e))) @ c)) => ((((ord_less_eq @ (set @ a)) @ a1) @ A10) => ((((ord_less_eq @ (set @ a)) @ a2) @ A10) => ((((ord_less_eq @ (set @ a)) @ a3) @ A10) => thesis))))))).
% 2.77/1.52 thf(conj_1, conjecture, thesis).
% 2.77/1.52 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.HLLIQP6qlU/DTF2DTF11799.p
% 2.77/1.52 ------------------------
% 2.77/1.55 ---- Cleaned THF ---
% 2.77/1.55 thf(ty_t_Product__Type_Oprod,type,
% 2.77/1.55 product_prod: $tType > $tType > $tType ).
% 2.77/1.55
% 2.77/1.55 thf(ty_t_Set_Oset,type,
% 2.77/1.55 set: $tType > $tType ).
% 2.77/1.55
% 2.77/1.55 thf(ty_tf_e,type,
% 2.77/1.55 e: $tType ).
% 2.77/1.55
% 2.77/1.55 thf(ty_tf_a,type,
% 2.77/1.55 a: $tType ).
% 2.77/1.55
% 2.77/1.55 thf(sy_cl_HOL_Otype,type,
% 2.77/1.55 type:
% 2.77/1.55 !>[A: $tType] : $o ).
% 2.77/1.55
% 2.77/1.55 thf(sy_cl_Orderings_Oord,type,
% 2.77/1.55 ord:
% 2.77/1.55 !>[A: $tType] : $o ).
% 2.77/1.55
% 2.77/1.55 thf(sy_cl_Orderings_Otop,type,
% 2.77/1.55 top:
% 2.77/1.55 !>[A: $tType] : $o ).
% 2.77/1.55
% 2.77/1.55 thf(sy_cl_Orderings_Oorder,type,
% 2.77/1.55 order:
% 2.77/1.55 !>[A: $tType] : $o ).
% 2.77/1.55
% 2.77/1.55 thf(sy_cl_Orderings_Olinorder,type,
% 2.77/1.55 linorder:
% 2.77/1.55 !>[A: $tType] : $o ).
% 2.77/1.55
% 2.77/1.55 thf(sy_cl_Orderings_Opreorder,type,
% 2.77/1.55 preorder:
% 2.77/1.55 !>[A: $tType] : $o ).
% 2.77/1.55
% 2.77/1.55 thf(sy_cl_Orderings_Oorder__top,type,
% 2.77/1.55 order_top:
% 2.77/1.55 !>[A: $tType] : $o ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 2.77/1.55 bNF_Ca1785829860lChain:
% 2.77/1.55 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_BNF__Def_Ocsquare,type,
% 2.77/1.55 bNF_csquare:
% 2.77/1.55 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( set @ A ) > ( B > C ) > ( D > C ) > ( A > B ) > ( A > D ) > $o ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_BNF__Def_OfstOp,type,
% 2.77/1.55 bNF_fstOp:
% 2.77/1.55 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_BNF__Def_Opick__middlep,type,
% 2.77/1.55 bNF_pick_middlep:
% 2.77/1.55 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_BNF__Def_OsndOp,type,
% 2.77/1.55 bNF_sndOp:
% 2.77/1.55 !>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 2.77/1.55 bNF_Greatest_image2:
% 2.77/1.55 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Finite__Set_OFpow,type,
% 2.77/1.55 finite_Fpow:
% 2.77/1.55 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Fun__Def_Oin__rel,type,
% 2.77/1.55 fun_in_rel:
% 2.77/1.55 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > A > B > $o ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.77/1.55 ord_less_eq:
% 2.77/1.55 !>[A: $tType] : ( A > A > $o ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 2.77/1.55 order_Greatest:
% 2.77/1.55 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 2.77/1.55 order_strict_mono:
% 2.77/1.55 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.77/1.55 top_top:
% 2.77/1.55 !>[A: $tType] : A ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Product__Type_OPair,type,
% 2.77/1.55 product_Pair:
% 2.77/1.55 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Product__Type_Ocurry,type,
% 2.77/1.55 product_curry:
% 2.77/1.55 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 2.77/1.55 produc2004651681e_prod:
% 2.77/1.55 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Product__Type_Omap__prod,type,
% 2.77/1.55 product_map_prod:
% 2.77/1.55 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.77/1.55 product_rec_prod:
% 2.77/1.55 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 2.77/1.55 product_case_prod:
% 2.77/1.55 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 2.77/1.55 product_fst:
% 2.77/1.55 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 2.77/1.55 product_snd:
% 2.77/1.55 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 2.77/1.55 product_swap:
% 2.77/1.55 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Relation_ODomain,type,
% 2.77/1.55 domain:
% 2.77/1.55 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Relation_ORange,type,
% 2.77/1.55 range:
% 2.77/1.55 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Relation_Oconverse,type,
% 2.77/1.55 converse:
% 2.77/1.55 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ A ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Relation_Oinv__image,type,
% 2.77/1.55 inv_image:
% 2.77/1.55 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Relation_Oinv__imagep,type,
% 2.77/1.55 inv_imagep:
% 2.77/1.55 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Relation_Orelcompp,type,
% 2.77/1.55 relcompp:
% 2.77/1.55 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Set_OCollect,type,
% 2.77/1.55 collect:
% 2.77/1.55 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Set_OPow,type,
% 2.77/1.55 pow:
% 2.77/1.55 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_Set_Oimage,type,
% 2.77/1.55 image:
% 2.77/1.55 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_c_member,type,
% 2.77/1.55 member:
% 2.77/1.55 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_v_C,type,
% 2.77/1.55 c: set @ ( product_prod @ ( set @ a ) @ ( a > e ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(sy_v_a1,type,
% 2.77/1.55 a1: set @ a ).
% 2.77/1.55
% 2.77/1.55 thf(sy_v_a2,type,
% 2.77/1.55 a2: set @ a ).
% 2.77/1.55
% 2.77/1.55 thf(sy_v_a3,type,
% 2.77/1.55 a3: set @ a ).
% 2.77/1.55
% 2.77/1.55 thf(sy_v_b1,type,
% 2.77/1.55 b1: a > e ).
% 2.77/1.55
% 2.77/1.55 thf(sy_v_b2,type,
% 2.77/1.55 b2: a > e ).
% 2.77/1.55
% 2.77/1.55 thf(sy_v_b3,type,
% 2.77/1.55 b3: a > e ).
% 2.77/1.55
% 2.77/1.55 thf(sy_v_thesis____,type,
% 2.77/1.55 thesis: $o ).
% 2.77/1.55
% 2.77/1.55 thf(fact_0__092_060open_062_092_060forall_062a_092_060in_062C_O_A_092_060forall_062b_092_060in_062C_O_Afst_Aa_A_092_060subseteq_062_Afst_Ab_A_092_060or_062_Afst_Ab_A_092_060subseteq_062_Afst_Aa_092_060close_062,axiom,
% 2.77/1.55 ! [X: product_prod @ ( set @ a ) @ ( a > e )] :
% 2.77/1.55 ( ( member @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ X @ c )
% 2.77/1.55 => ! [Xa: product_prod @ ( set @ a ) @ ( a > e )] :
% 2.77/1.55 ( ( member @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ Xa @ c )
% 2.77/1.55 => ( ( ord_less_eq @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) @ X ) @ ( product_fst @ ( set @ a ) @ ( a > e ) @ Xa ) )
% 2.77/1.55 | ( ord_less_eq @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) @ Xa ) @ ( product_fst @ ( set @ a ) @ ( a > e ) @ X ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_1__092_060open_062_092_060exists_062a_092_060in_062fst_A_096_AC_O_Aa1_A_092_060subseteq_062_Aa_A_092_060and_062_Aa2_A_092_060subseteq_062_Aa_A_092_060and_062_Aa3_A_092_060subseteq_062_Aa_092_060close_062,axiom,
% 2.77/1.55 ? [X2: set @ a] :
% 2.77/1.55 ( ( member @ ( set @ a ) @ X2 @ ( image @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) ) @ c ) )
% 2.77/1.55 & ( ord_less_eq @ ( set @ a ) @ a1 @ X2 )
% 2.77/1.55 & ( ord_less_eq @ ( set @ a ) @ a2 @ X2 )
% 2.77/1.55 & ( ord_less_eq @ ( set @ a ) @ a3 @ X2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_2__092_060open_062a1_A_092_060subseteq_062_Aa2_A_092_060or_062_Aa2_A_092_060subseteq_062_Aa1_092_060close_062,axiom,
% 2.77/1.55 ( ( ord_less_eq @ ( set @ a ) @ a1 @ a2 )
% 2.77/1.55 | ( ord_less_eq @ ( set @ a ) @ a2 @ a1 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_3__092_060open_062a1_A_092_060subseteq_062_Aa3_A_092_060or_062_Aa3_A_092_060subseteq_062_Aa1_092_060close_062,axiom,
% 2.77/1.55 ( ( ord_less_eq @ ( set @ a ) @ a1 @ a3 )
% 2.77/1.55 | ( ord_less_eq @ ( set @ a ) @ a3 @ a1 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_4__092_060open_062a2_A_092_060subseteq_062_Aa3_A_092_060or_062_Aa3_A_092_060subseteq_062_Aa2_092_060close_062,axiom,
% 2.77/1.55 ( ( ord_less_eq @ ( set @ a ) @ a2 @ a3 )
% 2.77/1.55 | ( ord_less_eq @ ( set @ a ) @ a3 @ a2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_5_A_I3_J,axiom,
% 2.77/1.55 member @ ( set @ a ) @ a3 @ ( image @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) ) @ c ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_6_A_I2_J,axiom,
% 2.77/1.55 member @ ( set @ a ) @ a2 @ ( image @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) ) @ c ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_7_A_I1_J,axiom,
% 2.77/1.55 member @ ( set @ a ) @ a1 @ ( image @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) ) @ c ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_8__092_060open_062_Ia1_M_Ab1_J_A_092_060in_062_AC_092_060close_062,axiom,
% 2.77/1.55 member @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( product_Pair @ ( set @ a ) @ ( a > e ) @ a1 @ b1 ) @ c ).
% 2.77/1.55
% 2.77/1.55 thf(fact_9__092_060open_062_Ia2_M_Ab2_J_A_092_060in_062_AC_092_060close_062,axiom,
% 2.77/1.55 member @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( product_Pair @ ( set @ a ) @ ( a > e ) @ a2 @ b2 ) @ c ).
% 2.77/1.55
% 2.77/1.55 thf(fact_10__092_060open_062_Ia3_M_Ab3_J_A_092_060in_062_AC_092_060close_062,axiom,
% 2.77/1.55 member @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( product_Pair @ ( set @ a ) @ ( a > e ) @ a3 @ b3 ) @ c ).
% 2.77/1.55
% 2.77/1.55 thf(fact_11_subsetI,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.77/1.55 ( ! [X2: A] :
% 2.77/1.55 ( ( member @ A @ X2 @ A2 )
% 2.77/1.55 => ( member @ A @ X2 @ B2 ) )
% 2.77/1.55 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_12_subset__antisym,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.77/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.77/1.55 => ( A2 = B2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_13_image__eqI,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,B3: A,F: B > A,X3: B,A2: set @ B] :
% 2.77/1.55 ( ( B3
% 2.77/1.55 = ( F @ X3 ) )
% 2.77/1.55 => ( ( member @ B @ X3 @ A2 )
% 2.77/1.55 => ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_14_order__refl,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( preorder @ A )
% 2.77/1.55 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_15_image__mono,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > B] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.77/1.55 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_16_image__subsetI,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,A2: set @ A,F: A > B,B2: set @ B] :
% 2.77/1.55 ( ! [X2: A] :
% 2.77/1.55 ( ( member @ A @ X2 @ A2 )
% 2.77/1.55 => ( member @ B @ ( F @ X2 ) @ B2 ) )
% 2.77/1.55 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_17_subset__imageE,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 2.77/1.55 => ~ ! [C2: set @ B] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ B ) @ C2 @ A2 )
% 2.77/1.55 => ( B2
% 2.77/1.55 != ( image @ B @ A @ F @ C2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_18_image__subset__iff,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 2.77/1.55 = ( ! [X4: B] :
% 2.77/1.55 ( ( member @ B @ X4 @ A2 )
% 2.77/1.55 => ( member @ A @ ( F @ X4 ) @ B2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_19_subset__image__iff,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 2.77/1.55 = ( ? [AA: set @ B] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
% 2.77/1.55 & ( B2
% 2.77/1.55 = ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_20_PairE__lemma,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 2.77/1.55 ? [X2: A,Y: B] :
% 2.77/1.55 ( P
% 2.77/1.55 = ( product_Pair @ A @ B @ X2 @ Y ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_21_dual__order_Oantisym,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( order @ A )
% 2.77/1.55 => ! [B3: A,A3: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.77/1.55 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.77/1.55 => ( A3 = B3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_22_dual__order_Oeq__iff,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( order @ A )
% 2.77/1.55 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.77/1.55 = ( ^ [A4: A,B4: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.77/1.55 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_23_dual__order_Otrans,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( order @ A )
% 2.77/1.55 => ! [B3: A,A3: A,C3: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.77/1.55 => ( ( ord_less_eq @ A @ C3 @ B3 )
% 2.77/1.55 => ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_24_linorder__wlog,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( linorder @ A )
% 2.77/1.55 => ! [P2: A > A > $o,A3: A,B3: A] :
% 2.77/1.55 ( ! [A5: A,B5: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.77/1.55 => ( P2 @ A5 @ B5 ) )
% 2.77/1.55 => ( ! [A5: A,B5: A] :
% 2.77/1.55 ( ( P2 @ B5 @ A5 )
% 2.77/1.55 => ( P2 @ A5 @ B5 ) )
% 2.77/1.55 => ( P2 @ A3 @ B3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_25_dual__order_Orefl,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( order @ A )
% 2.77/1.55 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_26_order__trans,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( preorder @ A )
% 2.77/1.55 => ! [X3: A,Y3: A,Z2: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.77/1.55 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 2.77/1.55 => ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_27_order__class_Oorder_Oantisym,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( order @ A )
% 2.77/1.55 => ! [A3: A,B3: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.77/1.55 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.77/1.55 => ( A3 = B3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_28_ord__le__eq__trans,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( ord @ A )
% 2.77/1.55 => ! [A3: A,B3: A,C3: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.77/1.55 => ( ( B3 = C3 )
% 2.77/1.55 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_29_ord__eq__le__trans,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( ord @ A )
% 2.77/1.55 => ! [A3: A,B3: A,C3: A] :
% 2.77/1.55 ( ( A3 = B3 )
% 2.77/1.55 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 2.77/1.55 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_30_order__class_Oorder_Oeq__iff,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( order @ A )
% 2.77/1.55 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.77/1.55 = ( ^ [A4: A,B4: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.77/1.55 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_31_antisym__conv,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( order @ A )
% 2.77/1.55 => ! [Y3: A,X3: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ Y3 @ X3 )
% 2.77/1.55 => ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.77/1.55 = ( X3 = Y3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_32_le__cases3,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( linorder @ A )
% 2.77/1.55 => ! [X3: A,Y3: A,Z2: A] :
% 2.77/1.55 ( ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.77/1.55 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 2.77/1.55 => ( ( ( ord_less_eq @ A @ Y3 @ X3 )
% 2.77/1.55 => ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
% 2.77/1.55 => ( ( ( ord_less_eq @ A @ X3 @ Z2 )
% 2.77/1.55 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 2.77/1.55 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 2.77/1.55 => ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
% 2.77/1.55 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 2.77/1.55 => ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
% 2.77/1.55 => ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
% 2.77/1.55 => ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_33_order_Otrans,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( order @ A )
% 2.77/1.55 => ! [A3: A,B3: A,C3: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.77/1.55 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 2.77/1.55 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_34_le__cases,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( linorder @ A )
% 2.77/1.55 => ! [X3: A,Y3: A] :
% 2.77/1.55 ( ~ ( ord_less_eq @ A @ X3 @ Y3 )
% 2.77/1.55 => ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_35_eq__refl,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( preorder @ A )
% 2.77/1.55 => ! [X3: A,Y3: A] :
% 2.77/1.55 ( ( X3 = Y3 )
% 2.77/1.55 => ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_36_linear,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( linorder @ A )
% 2.77/1.55 => ! [X3: A,Y3: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.77/1.55 | ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_37_antisym,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( order @ A )
% 2.77/1.55 => ! [X3: A,Y3: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.77/1.55 => ( ( ord_less_eq @ A @ Y3 @ X3 )
% 2.77/1.55 => ( X3 = Y3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_38_eq__iff,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( order @ A )
% 2.77/1.55 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.77/1.55 = ( ^ [X4: A,Y4: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 2.77/1.55 & ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_39_ord__le__eq__subst,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType] :
% 2.77/1.55 ( ( ( ord @ B )
% 2.77/1.55 & ( ord @ A ) )
% 2.77/1.55 => ! [A3: A,B3: A,F: A > B,C3: B] :
% 2.77/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.77/1.55 => ( ( ( F @ B3 )
% 2.77/1.55 = C3 )
% 2.77/1.55 => ( ! [X2: A,Y: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.77/1.55 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.77/1.55 => ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_40_ord__eq__le__subst,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType] :
% 2.77/1.55 ( ( ( ord @ B )
% 2.77/1.55 & ( ord @ A ) )
% 2.77/1.55 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 2.77/1.55 ( ( A3
% 2.77/1.55 = ( F @ B3 ) )
% 2.77/1.55 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 2.77/1.55 => ( ! [X2: B,Y: B] :
% 2.77/1.55 ( ( ord_less_eq @ B @ X2 @ Y )
% 2.77/1.55 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.77/1.55 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_41_order__subst2,axiom,
% 2.77/1.55 ! [A: $tType,C: $tType] :
% 2.77/1.55 ( ( ( order @ C )
% 2.77/1.55 & ( order @ A ) )
% 2.77/1.55 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 2.77/1.55 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.77/1.55 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
% 2.77/1.55 => ( ! [X2: A,Y: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.77/1.55 => ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.77/1.55 => ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_42_order__subst1,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType] :
% 2.77/1.55 ( ( ( order @ B )
% 2.77/1.55 & ( order @ A ) )
% 2.77/1.55 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 2.77/1.55 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 2.77/1.55 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 2.77/1.55 => ( ! [X2: B,Y: B] :
% 2.77/1.55 ( ( ord_less_eq @ B @ X2 @ Y )
% 2.77/1.55 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.77/1.55 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_43_le__fun__def,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType] :
% 2.77/1.55 ( ( ord @ B )
% 2.77/1.55 => ( ( ord_less_eq @ ( A > B ) )
% 2.77/1.55 = ( ^ [F2: A > B,G: A > B] :
% 2.77/1.55 ! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_44_le__funI,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType] :
% 2.77/1.55 ( ( ord @ B )
% 2.77/1.55 => ! [F: A > B,G2: A > B] :
% 2.77/1.55 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G2 @ X2 ) )
% 2.77/1.55 => ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_45_mem__Collect__eq,axiom,
% 2.77/1.55 ! [A: $tType,A3: A,P2: A > $o] :
% 2.77/1.55 ( ( member @ A @ A3 @ ( collect @ A @ P2 ) )
% 2.77/1.55 = ( P2 @ A3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_46_Collect__mem__eq,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A] :
% 2.77/1.55 ( ( collect @ A
% 2.77/1.55 @ ^ [X4: A] : ( member @ A @ X4 @ A2 ) )
% 2.77/1.55 = A2 ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_47_Collect__cong,axiom,
% 2.77/1.55 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.77/1.55 ( ! [X2: A] :
% 2.77/1.55 ( ( P2 @ X2 )
% 2.77/1.55 = ( Q @ X2 ) )
% 2.77/1.55 => ( ( collect @ A @ P2 )
% 2.77/1.55 = ( collect @ A @ Q ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_48_ext,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,F: A > B,G2: A > B] :
% 2.77/1.55 ( ! [X2: A] :
% 2.77/1.55 ( ( F @ X2 )
% 2.77/1.55 = ( G2 @ X2 ) )
% 2.77/1.55 => ( F = G2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_49_le__funE,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType] :
% 2.77/1.55 ( ( ord @ B )
% 2.77/1.55 => ! [F: A > B,G2: A > B,X3: A] :
% 2.77/1.55 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 2.77/1.55 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_50_le__funD,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType] :
% 2.77/1.55 ( ( ord @ B )
% 2.77/1.55 => ! [F: A > B,G2: A > B,X3: A] :
% 2.77/1.55 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 2.77/1.55 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_51_rev__image__eqI,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,X3: A,A2: set @ A,B3: B,F: A > B] :
% 2.77/1.55 ( ( member @ A @ X3 @ A2 )
% 2.77/1.55 => ( ( B3
% 2.77/1.55 = ( F @ X3 ) )
% 2.77/1.55 => ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_52_ball__imageD,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
% 2.77/1.55 ( ! [X2: A] :
% 2.77/1.55 ( ( member @ A @ X2 @ ( image @ B @ A @ F @ A2 ) )
% 2.77/1.55 => ( P2 @ X2 ) )
% 2.77/1.55 => ! [X: B] :
% 2.77/1.55 ( ( member @ B @ X @ A2 )
% 2.77/1.55 => ( P2 @ ( F @ X ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_53_image__cong,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G2: A > B] :
% 2.77/1.55 ( ( M = N )
% 2.77/1.55 => ( ! [X2: A] :
% 2.77/1.55 ( ( member @ A @ X2 @ N )
% 2.77/1.55 => ( ( F @ X2 )
% 2.77/1.55 = ( G2 @ X2 ) ) )
% 2.77/1.55 => ( ( image @ A @ B @ F @ M )
% 2.77/1.55 = ( image @ A @ B @ G2 @ N ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_54_bex__imageD,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
% 2.77/1.55 ( ? [X: A] :
% 2.77/1.55 ( ( member @ A @ X @ ( image @ B @ A @ F @ A2 ) )
% 2.77/1.55 & ( P2 @ X ) )
% 2.77/1.55 => ? [X2: B] :
% 2.77/1.55 ( ( member @ B @ X2 @ A2 )
% 2.77/1.55 & ( P2 @ ( F @ X2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_55_image__iff,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,Z2: A,F: B > A,A2: set @ B] :
% 2.77/1.55 ( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A2 ) )
% 2.77/1.55 = ( ? [X4: B] :
% 2.77/1.55 ( ( member @ B @ X4 @ A2 )
% 2.77/1.55 & ( Z2
% 2.77/1.55 = ( F @ X4 ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_56_imageI,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,X3: A,A2: set @ A,F: A > B] :
% 2.77/1.55 ( ( member @ A @ X3 @ A2 )
% 2.77/1.55 => ( member @ B @ ( F @ X3 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_57_Collect__mono__iff,axiom,
% 2.77/1.55 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
% 2.77/1.55 = ( ! [X4: A] :
% 2.77/1.55 ( ( P2 @ X4 )
% 2.77/1.55 => ( Q @ X4 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_58_set__eq__subset,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( ^ [Y2: set @ A,Z: set @ A] : ( Y2 = Z ) )
% 2.77/1.55 = ( ^ [A6: set @ A,B6: set @ A] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.77/1.55 & ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_59_subset__trans,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.77/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C4 )
% 2.77/1.55 => ( ord_less_eq @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_60_Collect__mono,axiom,
% 2.77/1.55 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.77/1.55 ( ! [X2: A] :
% 2.77/1.55 ( ( P2 @ X2 )
% 2.77/1.55 => ( Q @ X2 ) )
% 2.77/1.55 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_61_subset__iff,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 2.77/1.55 = ( ^ [A6: set @ A,B6: set @ A] :
% 2.77/1.55 ! [T2: A] :
% 2.77/1.55 ( ( member @ A @ T2 @ A6 )
% 2.77/1.55 => ( member @ A @ T2 @ B6 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_62_equalityD2,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.77/1.55 ( ( A2 = B2 )
% 2.77/1.55 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_63_subset__eq,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) )
% 2.77/1.55 = ( ^ [A6: set @ A,B6: set @ A] :
% 2.77/1.55 ! [X4: A] :
% 2.77/1.55 ( ( member @ A @ X4 @ A6 )
% 2.77/1.55 => ( member @ A @ X4 @ B6 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_64_equalityE,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.77/1.55 ( ( A2 = B2 )
% 2.77/1.55 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.77/1.55 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_65_subsetD,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.77/1.55 => ( ( member @ A @ C3 @ A2 )
% 2.77/1.55 => ( member @ A @ C3 @ B2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_66_in__mono,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.77/1.55 => ( ( member @ A @ X3 @ A2 )
% 2.77/1.55 => ( member @ A @ X3 @ B2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_67_prod_Oinject,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 2.77/1.55 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 2.77/1.55 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 2.77/1.55 = ( ( X1 = Y1 )
% 2.77/1.55 & ( X22 = Y22 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_68_old_Oprod_Oinject,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,A3: A,B3: B,A7: A,B7: B] :
% 2.77/1.55 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 2.77/1.55 = ( product_Pair @ A @ B @ A7 @ B7 ) )
% 2.77/1.55 = ( ( A3 = A7 )
% 2.77/1.55 & ( B3 = B7 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_69_eq__fst__iff,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,A3: A,P: product_prod @ A @ B] :
% 2.77/1.55 ( ( A3
% 2.77/1.55 = ( product_fst @ A @ B @ P ) )
% 2.77/1.55 = ( ? [B4: B] :
% 2.77/1.55 ( P
% 2.77/1.55 = ( product_Pair @ A @ B @ A3 @ B4 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_70_fst__conv,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 2.77/1.55 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 2.77/1.55 = X1 ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_71_fst__eqD,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,X3: A,Y3: B,A3: A] :
% 2.77/1.55 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 2.77/1.55 = A3 )
% 2.77/1.55 => ( X3 = A3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_72_fstI,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,X3: product_prod @ A @ B,Y3: A,Z2: B] :
% 2.77/1.55 ( ( X3
% 2.77/1.55 = ( product_Pair @ A @ B @ Y3 @ Z2 ) )
% 2.77/1.55 => ( ( product_fst @ A @ B @ X3 )
% 2.77/1.55 = Y3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_73_all__subset__image,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: ( set @ A ) > $o] :
% 2.77/1.55 ( ( ! [B6: set @ A] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F @ A2 ) )
% 2.77/1.55 => ( P2 @ B6 ) ) )
% 2.77/1.55 = ( ! [B6: set @ B] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ B ) @ B6 @ A2 )
% 2.77/1.55 => ( P2 @ ( image @ B @ A @ F @ B6 ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_74_relChain__def,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType] :
% 2.77/1.55 ( ( ord @ B )
% 2.77/1.55 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 2.77/1.55 = ( ^ [R: set @ ( product_prod @ A @ A ),As: A > B] :
% 2.77/1.55 ! [I: A,J: A] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
% 2.77/1.55 => ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_75_Greatest__equality,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( order @ A )
% 2.77/1.55 => ! [P2: A > $o,X3: A] :
% 2.77/1.55 ( ( P2 @ X3 )
% 2.77/1.55 => ( ! [Y: A] :
% 2.77/1.55 ( ( P2 @ Y )
% 2.77/1.55 => ( ord_less_eq @ A @ Y @ X3 ) )
% 2.77/1.55 => ( ( order_Greatest @ A @ P2 )
% 2.77/1.55 = X3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_76_GreatestI2__order,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( order @ A )
% 2.77/1.55 => ! [P2: A > $o,X3: A,Q: A > $o] :
% 2.77/1.55 ( ( P2 @ X3 )
% 2.77/1.55 => ( ! [Y: A] :
% 2.77/1.55 ( ( P2 @ Y )
% 2.77/1.55 => ( ord_less_eq @ A @ Y @ X3 ) )
% 2.77/1.55 => ( ! [X2: A] :
% 2.77/1.55 ( ( P2 @ X2 )
% 2.77/1.55 => ( ! [Y5: A] :
% 2.77/1.55 ( ( P2 @ Y5 )
% 2.77/1.55 => ( ord_less_eq @ A @ Y5 @ X2 ) )
% 2.77/1.55 => ( Q @ X2 ) ) )
% 2.77/1.55 => ( Q @ ( order_Greatest @ A @ P2 ) ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_77_old_Oprod_Oinducts,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.77/1.55 ( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.77/1.55 => ( P2 @ Prod ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_78_old_Oprod_Oexhaust,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 2.77/1.55 ~ ! [A5: A,B5: B] :
% 2.77/1.55 ( Y3
% 2.77/1.55 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_79_prod__induct7,axiom,
% 2.77/1.55 ! [G3: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 2.77/1.55 ( ! [A5: A,B5: B,C5: C,D2: D,E2: E,F4: F3,G4: G3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
% 2.77/1.55 => ( P2 @ X3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_80_prod__induct6,axiom,
% 2.77/1.55 ! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 2.77/1.55 ( ! [A5: A,B5: B,C5: C,D2: D,E2: E,F4: F3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
% 2.77/1.55 => ( P2 @ X3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_81_prod__induct5,axiom,
% 2.77/1.55 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.77/1.55 ( ! [A5: A,B5: B,C5: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C5 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 2.77/1.55 => ( P2 @ X3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_82_prod__induct4,axiom,
% 2.77/1.55 ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.77/1.55 ( ! [A5: A,B5: B,C5: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C5 @ D2 ) ) ) )
% 2.77/1.55 => ( P2 @ X3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_83_prod__induct3,axiom,
% 2.77/1.55 ! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.77/1.55 ( ! [A5: A,B5: B,C5: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C5 ) ) )
% 2.77/1.55 => ( P2 @ X3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_84_prod__cases7,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 2.77/1.55 ~ ! [A5: A,B5: B,C5: C,D2: D,E2: E,F4: F3,G4: G3] :
% 2.77/1.55 ( Y3
% 2.77/1.55 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_85_prod__cases6,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 2.77/1.55 ~ ! [A5: A,B5: B,C5: C,D2: D,E2: E,F4: F3] :
% 2.77/1.55 ( Y3
% 2.77/1.55 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_86_prod__cases5,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.77/1.55 ~ ! [A5: A,B5: B,C5: C,D2: D,E2: E] :
% 2.77/1.55 ( Y3
% 2.77/1.55 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C5 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_87_prod__cases4,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.77/1.55 ~ ! [A5: A,B5: B,C5: C,D2: D] :
% 2.77/1.55 ( Y3
% 2.77/1.55 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C5 @ D2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_88_prod__cases3,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.77/1.55 ~ ! [A5: A,B5: B,C5: C] :
% 2.77/1.55 ( Y3
% 2.77/1.55 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C5 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_89_Pair__inject,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,A3: A,B3: B,A7: A,B7: B] :
% 2.77/1.55 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 2.77/1.55 = ( product_Pair @ A @ B @ A7 @ B7 ) )
% 2.77/1.55 => ~ ( ( A3 = A7 )
% 2.77/1.55 => ( B3 != B7 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_90_prod__cases,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 2.77/1.55 ( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.77/1.55 => ( P2 @ P ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_91_old_Oprod_Orec,axiom,
% 2.77/1.55 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
% 2.77/1.55 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.77/1.55 = ( F1 @ A3 @ B3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_92_internal__case__prod__conv,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A3: B,B3: C] :
% 2.77/1.55 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 2.77/1.55 = ( C3 @ A3 @ B3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_93_image__Fpow__mono,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 2.77/1.55 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( finite_Fpow @ B @ A2 ) ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_94_not__sub,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.77/1.55 ( ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.77/1.55 => ? [A5: A] :
% 2.77/1.55 ( ( member @ A @ A5 @ A2 )
% 2.77/1.55 & ~ ( member @ A @ A5 @ B2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_95_eqsets__sub,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.77/1.55 ( ( A2 = B2 )
% 2.77/1.55 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_96_sub__which1,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
% 2.77/1.55 ( ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.77/1.55 | ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) )
% 2.77/1.55 => ( ( member @ A @ X3 @ A2 )
% 2.77/1.55 => ( ~ ( member @ A @ X3 @ B2 )
% 2.77/1.55 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_97_Fpow__mono,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.77/1.55 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A2 ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_98_le__convert,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( ord @ A )
% 2.77/1.55 => ! [A3: A,B3: A,C3: A] :
% 2.77/1.55 ( ( A3 = B3 )
% 2.77/1.55 => ( ( ord_less_eq @ A @ A3 @ C3 )
% 2.77/1.55 => ( ord_less_eq @ A @ B3 @ C3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_99_ge__convert,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( ord @ A )
% 2.77/1.55 => ! [A3: A,B3: A,C3: A] :
% 2.77/1.55 ( ( A3 = B3 )
% 2.77/1.55 => ( ( ord_less_eq @ A @ C3 @ A3 )
% 2.77/1.55 => ( ord_less_eq @ A @ C3 @ B3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_100_mem__in__image3,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,B3: A,F: B > A,A2: set @ B] :
% 2.77/1.55 ( ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) )
% 2.77/1.55 => ? [X2: B] :
% 2.77/1.55 ( ( member @ B @ X2 @ A2 )
% 2.77/1.55 & ( B3
% 2.77/1.55 = ( F @ X2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_101_mem__in__image2,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,A3: A,A2: set @ A,F: A > B] :
% 2.77/1.55 ( ( member @ A @ A3 @ A2 )
% 2.77/1.55 => ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_102_mem__in__image1,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,B2: set @ B,A3: A] :
% 2.77/1.55 ( ! [X2: A] :
% 2.77/1.55 ( ( member @ A @ X2 @ A2 )
% 2.77/1.55 => ( member @ B @ ( F @ X2 ) @ B2 ) )
% 2.77/1.55 => ( ( member @ A @ A3 @ A2 )
% 2.77/1.55 => ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_103_proper__subset,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.77/1.55 => ( ~ ( member @ A @ X3 @ A2 )
% 2.77/1.55 => ( ( member @ A @ X3 @ B2 )
% 2.77/1.55 => ( A2 != B2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_104_not__subseteq,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.77/1.55 ( ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.77/1.55 => ? [X2: A] :
% 2.77/1.55 ( ( member @ A @ X2 @ A2 )
% 2.77/1.55 & ~ ( member @ A @ X2 @ B2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_105_subset__self,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_106_sets__not__eq,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.77/1.55 ( ( A2 != B2 )
% 2.77/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.77/1.55 => ? [X2: A] :
% 2.77/1.55 ( ( member @ A @ X2 @ A2 )
% 2.77/1.55 & ~ ( member @ A @ X2 @ B2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_107_sub__which2,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
% 2.77/1.55 ( ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.77/1.55 | ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) )
% 2.77/1.55 => ( ~ ( member @ A @ X3 @ A2 )
% 2.77/1.55 => ( ( member @ A @ X3 @ B2 )
% 2.77/1.55 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_108_image__Pow__mono,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 2.77/1.55 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) ) @ ( pow @ A @ B2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_109_Sup_OSUP__cong,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C4: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 2.77/1.55 ( ( A2 = B2 )
% 2.77/1.55 => ( ! [X2: B] :
% 2.77/1.55 ( ( member @ B @ X2 @ B2 )
% 2.77/1.55 => ( ( C4 @ X2 )
% 2.77/1.55 = ( D3 @ X2 ) ) )
% 2.77/1.55 => ( ( Sup @ ( image @ B @ A @ C4 @ A2 ) )
% 2.77/1.55 = ( Sup @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_110_Inf_OINF__cong,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C4: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 2.77/1.55 ( ( A2 = B2 )
% 2.77/1.55 => ( ! [X2: B] :
% 2.77/1.55 ( ( member @ B @ X2 @ B2 )
% 2.77/1.55 => ( ( C4 @ X2 )
% 2.77/1.55 = ( D3 @ X2 ) ) )
% 2.77/1.55 => ( ( Inf @ ( image @ B @ A @ C4 @ A2 ) )
% 2.77/1.55 = ( Inf @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_111_subrelI,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ! [X2: A,Y: B] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R2 )
% 2.77/1.55 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S ) )
% 2.77/1.55 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_112_ssubst__Pair__rhs,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,R2: A,S: B,R3: set @ ( product_prod @ A @ B ),S2: B] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S ) @ R3 )
% 2.77/1.55 => ( ( S2 = S )
% 2.77/1.55 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_113_PowI,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.77/1.55 => ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_114_Pow__iff,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.77/1.55 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 2.77/1.55 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_115_Pow__top,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A] : ( member @ ( set @ A ) @ A2 @ ( pow @ A @ A2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_116_PowD,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.77/1.55 ( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
% 2.77/1.55 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_117_Pow__mono,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.77/1.55 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A2 ) @ ( pow @ A @ B2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_118_image__Pow__surj,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 2.77/1.55 ( ( ( image @ B @ A @ F @ A2 )
% 2.77/1.55 = B2 )
% 2.77/1.55 => ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) )
% 2.77/1.55 = ( pow @ A @ B2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_119_Fpow__subset__Pow,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A2 ) @ ( pow @ A @ A2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_120_Cantors__paradox,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A] :
% 2.77/1.55 ~ ? [F5: A > ( set @ A )] :
% 2.77/1.55 ( ( image @ A @ ( set @ A ) @ F5 @ A2 )
% 2.77/1.55 = ( pow @ A @ A2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_121_curry__conv,axiom,
% 2.77/1.55 ! [A: $tType,C: $tType,B: $tType] :
% 2.77/1.55 ( ( product_curry @ B @ C @ A )
% 2.77/1.55 = ( ^ [F2: ( product_prod @ B @ C ) > A,A4: B,B4: C] : ( F2 @ ( product_Pair @ B @ C @ A4 @ B4 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_122_in__inv__imagep,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType] :
% 2.77/1.55 ( ( inv_imagep @ A @ B )
% 2.77/1.55 = ( ^ [R: A > A > $o,F2: B > A,X4: B,Y4: B] : ( R @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_123_verit__la__disequality,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( linorder @ A )
% 2.77/1.55 => ! [A3: A,B3: A] :
% 2.77/1.55 ( ( A3 = B3 )
% 2.77/1.55 | ~ ( ord_less_eq @ A @ A3 @ B3 )
% 2.77/1.55 | ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_124_le__rel__bool__arg__iff,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( ord @ A )
% 2.77/1.55 => ( ( ord_less_eq @ ( $o > A ) )
% 2.77/1.55 = ( ^ [X5: $o > A,Y6: $o > A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ ( X5 @ $false ) @ ( Y6 @ $false ) )
% 2.77/1.55 & ( ord_less_eq @ A @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_125_curryI,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 2.77/1.55 ( ( F @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.77/1.55 => ( product_curry @ A @ B @ $o @ F @ A3 @ B3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_126_curryD,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 2.77/1.55 ( ( product_curry @ A @ B @ $o @ F @ A3 @ B3 )
% 2.77/1.55 => ( F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_127_curryE,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 2.77/1.55 ( ( product_curry @ A @ B @ $o @ F @ A3 @ B3 )
% 2.77/1.55 => ( F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_128_in__inv__image,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,X3: A,Y3: A,R2: set @ ( product_prod @ B @ B ),F: A > B] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( inv_image @ B @ A @ R2 @ F ) )
% 2.77/1.55 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) @ R2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_129_fst__eq__Domain,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ R3 )
% 2.77/1.55 = ( domain @ A @ B @ R3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_130_Domain__fst,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType] :
% 2.77/1.55 ( ( domain @ A @ B )
% 2.77/1.55 = ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_131_Range__mono,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S )
% 2.77/1.55 => ( ord_less_eq @ ( set @ B ) @ ( range @ A @ B @ R2 ) @ ( range @ A @ B @ S ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_132_RangeE,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,B3: A,R2: set @ ( product_prod @ B @ A )] :
% 2.77/1.55 ( ( member @ A @ B3 @ ( range @ B @ A @ R2 ) )
% 2.77/1.55 => ~ ! [A5: B] :
% 2.77/1.55 ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A5 @ B3 ) @ R2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_133_Range__iff,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,A3: A,R2: set @ ( product_prod @ B @ A )] :
% 2.77/1.55 ( ( member @ A @ A3 @ ( range @ B @ A @ R2 ) )
% 2.77/1.55 = ( ? [Y4: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y4 @ A3 ) @ R2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_134_Range_Ocases,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,A3: B,R2: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( member @ B @ A3 @ ( range @ A @ B @ R2 ) )
% 2.77/1.55 => ~ ! [A5: A] :
% 2.77/1.55 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ A3 ) @ R2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_135_Range_Osimps,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,A3: B,R2: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( member @ B @ A3 @ ( range @ A @ B @ R2 ) )
% 2.77/1.55 = ( ? [A4: A,B4: B] :
% 2.77/1.55 ( ( A3 = B4 )
% 2.77/1.55 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_136_Range_Ointros,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,A3: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R2 )
% 2.77/1.55 => ( member @ B @ B3 @ ( range @ A @ B @ R2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_137_Range_Oinducts,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,X3: B,R2: set @ ( product_prod @ A @ B ),P2: B > $o] :
% 2.77/1.55 ( ( member @ B @ X3 @ ( range @ A @ B @ R2 ) )
% 2.77/1.55 => ( ! [A5: A,B5: B] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R2 )
% 2.77/1.55 => ( P2 @ B5 ) )
% 2.77/1.55 => ( P2 @ X3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_138_DomainE,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
% 2.77/1.55 => ~ ! [B5: B] :
% 2.77/1.55 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_139_Domain__iff,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
% 2.77/1.55 = ( ? [Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y4 ) @ R2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_140_Domain_Ocases,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
% 2.77/1.55 => ~ ! [B5: B] :
% 2.77/1.55 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_141_Domain_Osimps,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
% 2.77/1.55 = ( ? [A4: A,B4: B] :
% 2.77/1.55 ( ( A3 = A4 )
% 2.77/1.55 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_142_Domain_ODomainI,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,A3: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R2 )
% 2.77/1.55 => ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_143_Domain_Oinducts,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,X3: A,R2: set @ ( product_prod @ A @ B ),P2: A > $o] :
% 2.77/1.55 ( ( member @ A @ X3 @ ( domain @ A @ B @ R2 ) )
% 2.77/1.55 => ( ! [A5: A,B5: B] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R2 )
% 2.77/1.55 => ( P2 @ A5 ) )
% 2.77/1.55 => ( P2 @ X3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_144_Domain__mono,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S )
% 2.77/1.55 => ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R2 ) @ ( domain @ A @ B @ S ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_145_Range__converse,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( range @ B @ A @ ( converse @ A @ B @ R2 ) )
% 2.77/1.55 = ( domain @ A @ B @ R2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_146_Domain__converse,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A )] :
% 2.77/1.55 ( ( domain @ A @ B @ ( converse @ B @ A @ R2 ) )
% 2.77/1.55 = ( range @ B @ A @ R2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_147_pair__in__swap__image,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,Y3: A,X3: B,A2: set @ ( product_prod @ B @ A )] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ X3 ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A2 ) )
% 2.77/1.55 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ A2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_148_swap__simp,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,X3: B,Y3: A] :
% 2.77/1.55 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y3 ) )
% 2.77/1.55 = ( product_Pair @ A @ B @ Y3 @ X3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_149_converse__converse,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( converse @ B @ A @ ( converse @ A @ B @ R2 ) )
% 2.77/1.55 = R2 ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_150_converse__inject,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A )] :
% 2.77/1.55 ( ( ( converse @ B @ A @ R2 )
% 2.77/1.55 = ( converse @ B @ A @ S ) )
% 2.77/1.55 = ( R2 = S ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_151_swap__swap,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 2.77/1.55 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
% 2.77/1.55 = P ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_152_converse__iff,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,A3: A,B3: B,R2: set @ ( product_prod @ B @ A )] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( converse @ B @ A @ R2 ) )
% 2.77/1.55 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A3 ) @ R2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_153_converse__mono,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A )] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S ) )
% 2.77/1.55 = ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_154_converse__inv__image,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ B @ B ),F: A > B] :
% 2.77/1.55 ( ( converse @ A @ A @ ( inv_image @ B @ A @ R3 @ F ) )
% 2.77/1.55 = ( inv_image @ B @ A @ ( converse @ B @ B @ R3 ) @ F ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_155_converse_Oinducts,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,X1: B,X22: A,R2: set @ ( product_prod @ A @ B ),P2: B > A > $o] :
% 2.77/1.55 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X1 @ X22 ) @ ( converse @ A @ B @ R2 ) )
% 2.77/1.55 => ( ! [A5: A,B5: B] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R2 )
% 2.77/1.55 => ( P2 @ B5 @ A5 ) )
% 2.77/1.55 => ( P2 @ X1 @ X22 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_156_converse_Ointros,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,A3: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R2 )
% 2.77/1.55 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A3 ) @ ( converse @ A @ B @ R2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_157_converse_Osimps,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,A1: B,A22: A,R2: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R2 ) )
% 2.77/1.55 = ( ? [A4: A,B4: B] :
% 2.77/1.55 ( ( A1 = B4 )
% 2.77/1.55 & ( A22 = A4 )
% 2.77/1.55 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_158_converse_Ocases,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,A1: B,A22: A,R2: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R2 ) )
% 2.77/1.55 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A22 @ A1 ) @ R2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_159_converseE,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R2: set @ ( product_prod @ B @ A )] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R2 ) )
% 2.77/1.55 => ~ ! [X2: B,Y: A] :
% 2.77/1.55 ( ( Yx
% 2.77/1.55 = ( product_Pair @ A @ B @ Y @ X2 ) )
% 2.77/1.55 => ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y ) @ R2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_160_converseD,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,A3: A,B3: B,R2: set @ ( product_prod @ B @ A )] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( converse @ B @ A @ R2 ) )
% 2.77/1.55 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A3 ) @ R2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_161_converse__subset__swap,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ A )] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ ( converse @ B @ A @ S ) )
% 2.77/1.55 = ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( converse @ A @ B @ R2 ) @ S ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_162_surj__swap,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType] :
% 2.77/1.55 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.77/1.55 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_163_prod_Oswap__def,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType] :
% 2.77/1.55 ( ( product_swap @ A @ B )
% 2.77/1.55 = ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_164_map__prod__imageI,axiom,
% 2.77/1.55 ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B ),F: A > C,G2: B > D] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 )
% 2.77/1.55 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A3 ) @ ( G2 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G2 ) @ R3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_165_top__apply,axiom,
% 2.77/1.55 ! [C: $tType,D: $tType] :
% 2.77/1.55 ( ( top @ C )
% 2.77/1.55 => ( ( top_top @ ( D > C ) )
% 2.77/1.55 = ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_166_UNIV__I,axiom,
% 2.77/1.55 ! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_167_map__prod__simp,axiom,
% 2.77/1.55 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G2: D > B,A3: C,B3: D] :
% 2.77/1.55 ( ( product_map_prod @ C @ A @ D @ B @ F @ G2 @ ( product_Pair @ C @ D @ A3 @ B3 ) )
% 2.77/1.55 = ( product_Pair @ A @ B @ ( F @ A3 ) @ ( G2 @ B3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_168_Pow__UNIV,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( pow @ A @ ( top_top @ ( set @ A ) ) )
% 2.77/1.55 = ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_169_fst__map__prod,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G2: D > B,X3: product_prod @ C @ D] :
% 2.77/1.55 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G2 @ X3 ) )
% 2.77/1.55 = ( F @ ( product_fst @ C @ D @ X3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_170_snd__map__prod,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G2: D > A,X3: product_prod @ C @ D] :
% 2.77/1.55 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G2 @ X3 ) )
% 2.77/1.55 = ( G2 @ ( product_snd @ C @ D @ X3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_171_converse__UNIV,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType] :
% 2.77/1.55 ( ( converse @ B @ A @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.77/1.55 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_172_prod_Ocollapse,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.77/1.55 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 2.77/1.55 = Prod ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_173_fst__swap,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,X3: product_prod @ B @ A] :
% 2.77/1.55 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X3 ) )
% 2.77/1.55 = ( product_snd @ B @ A @ X3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_174_snd__swap,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,X3: product_prod @ A @ B] :
% 2.77/1.55 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X3 ) )
% 2.77/1.55 = ( product_fst @ A @ B @ X3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_175_range__fst,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType] :
% 2.77/1.55 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.77/1.55 = ( top_top @ ( set @ A ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_176_range__snd,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType] :
% 2.77/1.55 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.77/1.55 = ( top_top @ ( set @ A ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_177_rangeI,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,F: B > A,X3: B] : ( member @ A @ ( F @ X3 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_178_range__eqI,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,B3: A,F: B > A,X3: B] :
% 2.77/1.55 ( ( B3
% 2.77/1.55 = ( F @ X3 ) )
% 2.77/1.55 => ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_179_surj__def,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,F: B > A] :
% 2.77/1.55 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.77/1.55 = ( top_top @ ( set @ A ) ) )
% 2.77/1.55 = ( ! [Y4: A] :
% 2.77/1.55 ? [X4: B] :
% 2.77/1.55 ( Y4
% 2.77/1.55 = ( F @ X4 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_180_surjI,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,G2: B > A,F: A > B] :
% 2.77/1.55 ( ! [X2: A] :
% 2.77/1.55 ( ( G2 @ ( F @ X2 ) )
% 2.77/1.55 = X2 )
% 2.77/1.55 => ( ( image @ B @ A @ G2 @ ( top_top @ ( set @ B ) ) )
% 2.77/1.55 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_181_surjE,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,F: B > A,Y3: A] :
% 2.77/1.55 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.77/1.55 = ( top_top @ ( set @ A ) ) )
% 2.77/1.55 => ~ ! [X2: B] :
% 2.77/1.55 ( Y3
% 2.77/1.55 != ( F @ X2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_182_surjD,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,F: B > A,Y3: A] :
% 2.77/1.55 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.77/1.55 = ( top_top @ ( set @ A ) ) )
% 2.77/1.55 => ? [X2: B] :
% 2.77/1.55 ( Y3
% 2.77/1.55 = ( F @ X2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_183_map__prod__surj,axiom,
% 2.77/1.55 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G2: C > D] :
% 2.77/1.55 ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.77/1.55 = ( top_top @ ( set @ B ) ) )
% 2.77/1.55 => ( ( ( image @ C @ D @ G2 @ ( top_top @ ( set @ C ) ) )
% 2.77/1.55 = ( top_top @ ( set @ D ) ) )
% 2.77/1.55 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G2 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 2.77/1.55 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_184_top_Oextremum__uniqueI,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( order_top @ A )
% 2.77/1.55 => ! [A3: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 2.77/1.55 => ( A3
% 2.77/1.55 = ( top_top @ A ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_185_top_Oextremum__unique,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( order_top @ A )
% 2.77/1.55 => ! [A3: A] :
% 2.77/1.55 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 2.77/1.55 = ( A3
% 2.77/1.55 = ( top_top @ A ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_186_top__greatest,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( order_top @ A )
% 2.77/1.55 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_187_subset__UNIV,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_188_sndI,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,X3: product_prod @ A @ B,Y3: A,Z2: B] :
% 2.77/1.55 ( ( X3
% 2.77/1.55 = ( product_Pair @ A @ B @ Y3 @ Z2 ) )
% 2.77/1.55 => ( ( product_snd @ A @ B @ X3 )
% 2.77/1.55 = Z2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_189_snd__eqD,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,X3: B,Y3: A,A3: A] :
% 2.77/1.55 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y3 ) )
% 2.77/1.55 = A3 )
% 2.77/1.55 => ( Y3 = A3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_190_snd__conv,axiom,
% 2.77/1.55 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 2.77/1.55 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 2.77/1.55 = X22 ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_191_eq__snd__iff,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,B3: A,P: product_prod @ B @ A] :
% 2.77/1.55 ( ( B3
% 2.77/1.55 = ( product_snd @ B @ A @ P ) )
% 2.77/1.55 = ( ? [A4: B] :
% 2.77/1.55 ( P
% 2.77/1.55 = ( product_Pair @ B @ A @ A4 @ B3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_192_prod__eq__iff,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType] :
% 2.77/1.55 ( ( ^ [Y2: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y2 = Z ) )
% 2.77/1.55 = ( ^ [S3: product_prod @ A @ B,T2: product_prod @ A @ B] :
% 2.77/1.55 ( ( ( product_fst @ A @ B @ S3 )
% 2.77/1.55 = ( product_fst @ A @ B @ T2 ) )
% 2.77/1.55 & ( ( product_snd @ A @ B @ S3 )
% 2.77/1.55 = ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_193_prod_Oexpand,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 2.77/1.55 ( ( ( ( product_fst @ A @ B @ Prod )
% 2.77/1.55 = ( product_fst @ A @ B @ Prod2 ) )
% 2.77/1.55 & ( ( product_snd @ A @ B @ Prod )
% 2.77/1.55 = ( product_snd @ A @ B @ Prod2 ) ) )
% 2.77/1.55 => ( Prod = Prod2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_194_prod__eqI,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,P: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 2.77/1.55 ( ( ( product_fst @ A @ B @ P )
% 2.77/1.55 = ( product_fst @ A @ B @ Q2 ) )
% 2.77/1.55 => ( ( ( product_snd @ A @ B @ P )
% 2.77/1.55 = ( product_snd @ A @ B @ Q2 ) )
% 2.77/1.55 => ( P = Q2 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_195_UNIV__eq__I,axiom,
% 2.77/1.55 ! [A: $tType,A2: set @ A] :
% 2.77/1.55 ( ! [X2: A] : ( member @ A @ X2 @ A2 )
% 2.77/1.55 => ( ( top_top @ ( set @ A ) )
% 2.77/1.55 = A2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_196_UNIV__witness,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_197_surjective__pairing,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,T3: product_prod @ A @ B] :
% 2.77/1.55 ( T3
% 2.77/1.55 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T3 ) @ ( product_snd @ A @ B @ T3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_198_prod_Oexhaust__sel,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.77/1.55 ( Prod
% 2.77/1.55 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_199_snd__eq__Range,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A )] :
% 2.77/1.55 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ R3 )
% 2.77/1.55 = ( range @ B @ A @ R3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_200_Range__snd,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType] :
% 2.77/1.55 ( ( range @ B @ A )
% 2.77/1.55 = ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_201_prod__fun__imageE,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F: C > A,G2: D > B,R3: set @ ( product_prod @ C @ D )] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G2 ) @ R3 ) )
% 2.77/1.55 => ~ ! [X2: C,Y: D] :
% 2.77/1.55 ( ( C3
% 2.77/1.55 = ( product_Pair @ A @ B @ ( F @ X2 ) @ ( G2 @ Y ) ) )
% 2.77/1.55 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X2 @ Y ) @ R3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_202_exI__realizer,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,P2: A > B > $o,Y3: A,X3: B] :
% 2.77/1.55 ( ( P2 @ Y3 @ X3 )
% 2.77/1.55 => ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y3 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_203_conjI__realizer,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,P2: A > $o,P: A,Q: B > $o,Q2: B] :
% 2.77/1.55 ( ( P2 @ P )
% 2.77/1.55 => ( ( Q @ Q2 )
% 2.77/1.55 => ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) )
% 2.77/1.55 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_204_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,P2: A > B > $o,X3: A,Y3: B,A3: product_prod @ A @ B] :
% 2.77/1.55 ( ( P2 @ X3 @ Y3 )
% 2.77/1.55 => ( ( A3
% 2.77/1.55 = ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 2.77/1.55 => ( P2 @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_205_top__set__def,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( top_top @ ( set @ A ) )
% 2.77/1.55 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_206_top__empty__eq,axiom,
% 2.77/1.55 ! [A: $tType] :
% 2.77/1.55 ( ( top_top @ ( A > $o ) )
% 2.77/1.55 = ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_207_exE__realizer_H,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A] :
% 2.77/1.55 ( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
% 2.77/1.55 => ~ ! [X2: B,Y: A] :
% 2.77/1.55 ~ ( P2 @ Y @ X2 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_208_Collect__split__mono__strong,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,X6: set @ A,A2: set @ ( product_prod @ A @ B ),Y7: set @ B,P2: A > B > $o,Q: A > B > $o] :
% 2.77/1.55 ( ( X6
% 2.77/1.55 = ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A2 ) )
% 2.77/1.55 => ( ( Y7
% 2.77/1.55 = ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A2 ) )
% 2.77/1.55 => ( ! [X2: A] :
% 2.77/1.55 ( ( member @ A @ X2 @ X6 )
% 2.77/1.55 => ! [Xa2: B] :
% 2.77/1.55 ( ( member @ B @ Xa2 @ Y7 )
% 2.77/1.55 => ( ( P2 @ X2 @ Xa2 )
% 2.77/1.55 => ( Q @ X2 @ Xa2 ) ) ) )
% 2.77/1.55 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) ) )
% 2.77/1.55 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_209_sndOp__def,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,C: $tType] :
% 2.77/1.55 ( ( bNF_sndOp @ C @ A @ B )
% 2.77/1.55 = ( ^ [P4: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_210_top1I,axiom,
% 2.77/1.55 ! [A: $tType,X3: A] : ( top_top @ ( A > $o ) @ X3 ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_211_curry__case__prod,axiom,
% 2.77/1.55 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 2.77/1.55 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
% 2.77/1.55 = F ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_212_case__prod__curry,axiom,
% 2.77/1.55 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 2.77/1.55 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 2.77/1.55 = F ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_213_Product__Type_OCollect__case__prodD,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,X3: product_prod @ A @ B,A2: A > B > $o] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ X3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A2 ) ) )
% 2.77/1.55 => ( A2 @ ( product_fst @ A @ B @ X3 ) @ ( product_snd @ A @ B @ X3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_214_prod_Ocase__eq__if,axiom,
% 2.77/1.55 ! [C: $tType,B: $tType,A: $tType] :
% 2.77/1.55 ( ( product_case_prod @ A @ B @ C )
% 2.77/1.55 = ( ^ [F2: A > B > C,Prod3: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_215_case__prod__beta,axiom,
% 2.77/1.55 ! [A: $tType,C: $tType,B: $tType] :
% 2.77/1.55 ( ( product_case_prod @ B @ C @ A )
% 2.77/1.55 = ( ^ [F2: B > C > A,P3: product_prod @ B @ C] : ( F2 @ ( product_fst @ B @ C @ P3 ) @ ( product_snd @ B @ C @ P3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_216_internal__case__prod__def,axiom,
% 2.77/1.55 ! [C: $tType,B: $tType,A: $tType] :
% 2.77/1.55 ( ( produc2004651681e_prod @ A @ B @ C )
% 2.77/1.55 = ( product_case_prod @ A @ B @ C ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_217_Collect__case__prod__mono,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,A2: A > B > $o,B2: A > B > $o] :
% 2.77/1.55 ( ( ord_less_eq @ ( A > B > $o ) @ A2 @ B2 )
% 2.77/1.55 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A2 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_218_old_Oprod_Ocase,axiom,
% 2.77/1.55 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
% 2.77/1.55 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 2.77/1.55 = ( F @ X1 @ X22 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_219_prod_Osplit__sel,axiom,
% 2.77/1.55 ! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 2.77/1.55 ( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 2.77/1.55 = ( ( Prod
% 2.77/1.55 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 2.77/1.55 => ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_220_prod_Osplit__sel__asm,axiom,
% 2.77/1.55 ! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 2.77/1.55 ( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 2.77/1.55 = ( ~ ( ( Prod
% 2.77/1.55 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 2.77/1.55 & ~ ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_221_fstOp__def,axiom,
% 2.77/1.55 ! [C: $tType,B: $tType,A: $tType] :
% 2.77/1.55 ( ( bNF_fstOp @ A @ B @ C )
% 2.77/1.55 = ( ^ [P4: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_222_split__cong,axiom,
% 2.77/1.55 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F: A > B > C,G2: A > B > C,P: product_prod @ A @ B] :
% 2.77/1.55 ( ! [X2: A,Y: B] :
% 2.77/1.55 ( ( ( product_Pair @ A @ B @ X2 @ Y )
% 2.77/1.55 = Q2 )
% 2.77/1.55 => ( ( F @ X2 @ Y )
% 2.77/1.55 = ( G2 @ X2 @ Y ) ) )
% 2.77/1.55 => ( ( P = Q2 )
% 2.77/1.55 => ( ( product_case_prod @ A @ B @ C @ F @ P )
% 2.77/1.55 = ( product_case_prod @ A @ B @ C @ G2 @ Q2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_223_predicate2I,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,P2: A > B > $o,Q: A > B > $o] :
% 2.77/1.55 ( ! [X2: A,Y: B] :
% 2.77/1.55 ( ( P2 @ X2 @ Y )
% 2.77/1.55 => ( Q @ X2 @ Y ) )
% 2.77/1.55 => ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_224_mem__case__prodE,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,C: $tType,Z2: A,C3: B > C > ( set @ A ),P: product_prod @ B @ C] :
% 2.77/1.55 ( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P ) )
% 2.77/1.55 => ~ ! [X2: B,Y: C] :
% 2.77/1.55 ( ( P
% 2.77/1.55 = ( product_Pair @ B @ C @ X2 @ Y ) )
% 2.77/1.55 => ~ ( member @ A @ Z2 @ ( C3 @ X2 @ Y ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_225_rev__predicate2D,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,P2: A > B > $o,X3: A,Y3: B,Q: A > B > $o] :
% 2.77/1.55 ( ( P2 @ X3 @ Y3 )
% 2.77/1.55 => ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
% 2.77/1.55 => ( Q @ X3 @ Y3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_226_predicate2D,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,P2: A > B > $o,Q: A > B > $o,X3: A,Y3: B] :
% 2.77/1.55 ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
% 2.77/1.55 => ( ( P2 @ X3 @ Y3 )
% 2.77/1.55 => ( Q @ X3 @ Y3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_227_refl__ge__eq,axiom,
% 2.77/1.55 ! [A: $tType,R3: A > A > $o] :
% 2.77/1.55 ( ! [X2: A] : ( R3 @ X2 @ X2 )
% 2.77/1.55 => ( ord_less_eq @ ( A > A > $o )
% 2.77/1.55 @ ^ [Y2: A,Z: A] : ( Y2 = Z )
% 2.77/1.55 @ R3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_228_ge__eq__refl,axiom,
% 2.77/1.55 ! [A: $tType,R3: A > A > $o,X3: A] :
% 2.77/1.55 ( ( ord_less_eq @ ( A > A > $o )
% 2.77/1.55 @ ^ [Y2: A,Z: A] : ( Y2 = Z )
% 2.77/1.55 @ R3 )
% 2.77/1.55 => ( R3 @ X3 @ X3 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_229_Collect__case__prod__in__rel__leE,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,X6: set @ ( product_prod @ A @ B ),Y7: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( fun_in_rel @ A @ B @ Y7 ) ) ) )
% 2.77/1.55 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ Y7 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_230_Collect__case__prod__in__rel__leI,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,X6: set @ ( product_prod @ A @ B ),Y7: set @ ( product_prod @ A @ B )] :
% 2.77/1.55 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ Y7 )
% 2.77/1.55 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( fun_in_rel @ A @ B @ Y7 ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_231_in__rel__def,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType] :
% 2.77/1.55 ( ( fun_in_rel @ A @ B )
% 2.77/1.55 = ( ^ [R4: set @ ( product_prod @ A @ B ),X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R4 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_232_csquare__fstOp__sndOp,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,C: $tType,F: ( A > B > $o ) > ( product_prod @ A @ B ) > $o,P2: A > C > $o,Q: C > B > $o] : ( bNF_csquare @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ C @ ( product_prod @ C @ B ) @ ( collect @ ( product_prod @ A @ B ) @ ( F @ ( relcompp @ A @ C @ B @ P2 @ Q ) ) ) @ ( product_snd @ A @ C ) @ ( product_fst @ C @ B ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_233_predicate2D__conj,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,P2: A > B > $o,Q: A > B > $o,R3: $o,X3: A,Y3: B] :
% 2.77/1.55 ( ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
% 2.77/1.55 & R3 )
% 2.77/1.55 => ( R3
% 2.77/1.55 & ( ( P2 @ X3 @ Y3 )
% 2.77/1.55 => ( Q @ X3 @ Y3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_234_fstOp__in,axiom,
% 2.77/1.55 ! [B: $tType,C: $tType,A: $tType,Ac2: product_prod @ A @ B,P2: A > C > $o,Q: C > B > $o] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P2 @ Q ) ) ) )
% 2.77/1.55 => ( member @ ( product_prod @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q @ Ac2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ P2 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_235_sndOp__in,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,C: $tType,Ac2: product_prod @ A @ B,P2: A > C > $o,Q: C > B > $o] :
% 2.77/1.55 ( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P2 @ Q ) ) ) )
% 2.77/1.55 => ( member @ ( product_prod @ C @ B ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q @ Ac2 ) @ ( collect @ ( product_prod @ C @ B ) @ ( product_case_prod @ C @ B @ $o @ Q ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_236_pick__middlep,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,C: $tType,P2: A > B > $o,Q: B > C > $o,A3: A,C3: C] :
% 2.77/1.55 ( ( relcompp @ A @ B @ C @ P2 @ Q @ A3 @ C3 )
% 2.77/1.55 => ( ( P2 @ A3 @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q @ A3 @ C3 ) )
% 2.77/1.55 & ( Q @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q @ A3 @ C3 ) @ C3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_237_nchotomy__relcomppE,axiom,
% 2.77/1.55 ! [C: $tType,B: $tType,A: $tType,D: $tType,F: B > A,R2: C > A > $o,S: A > D > $o,A3: C,C3: D] :
% 2.77/1.55 ( ! [Y: A] :
% 2.77/1.55 ? [X: B] :
% 2.77/1.55 ( Y
% 2.77/1.55 = ( F @ X ) )
% 2.77/1.55 => ( ( relcompp @ C @ A @ D @ R2 @ S @ A3 @ C3 )
% 2.77/1.55 => ~ ! [B5: B] :
% 2.77/1.55 ( ( R2 @ A3 @ ( F @ B5 ) )
% 2.77/1.55 => ~ ( S @ ( F @ B5 ) @ C3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_238_relcompp_OrelcompI,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,C: $tType,R2: A > B > $o,A3: A,B3: B,S: B > C > $o,C3: C] :
% 2.77/1.55 ( ( R2 @ A3 @ B3 )
% 2.77/1.55 => ( ( S @ B3 @ C3 )
% 2.77/1.55 => ( relcompp @ A @ B @ C @ R2 @ S @ A3 @ C3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_239_relcompp_Oinducts,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,C: $tType,R2: A > B > $o,S: B > C > $o,X1: A,X22: C,P2: A > C > $o] :
% 2.77/1.55 ( ( relcompp @ A @ B @ C @ R2 @ S @ X1 @ X22 )
% 2.77/1.55 => ( ! [A5: A,B5: B,C5: C] :
% 2.77/1.55 ( ( R2 @ A5 @ B5 )
% 2.77/1.55 => ( ( S @ B5 @ C5 )
% 2.77/1.55 => ( P2 @ A5 @ C5 ) ) )
% 2.77/1.55 => ( P2 @ X1 @ X22 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_240_relcompp__assoc,axiom,
% 2.77/1.55 ! [A: $tType,D: $tType,B: $tType,C: $tType,R2: A > D > $o,S: D > C > $o,T3: C > B > $o] :
% 2.77/1.55 ( ( relcompp @ A @ C @ B @ ( relcompp @ A @ D @ C @ R2 @ S ) @ T3 )
% 2.77/1.55 = ( relcompp @ A @ D @ B @ R2 @ ( relcompp @ D @ C @ B @ S @ T3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_241_relcompp__apply,axiom,
% 2.77/1.55 ! [C: $tType,B: $tType,A: $tType] :
% 2.77/1.55 ( ( relcompp @ A @ B @ C )
% 2.77/1.55 = ( ^ [R4: A > B > $o,S4: B > C > $o,A4: A,C6: C] :
% 2.77/1.55 ? [B4: B] :
% 2.77/1.55 ( ( R4 @ A4 @ B4 )
% 2.77/1.55 & ( S4 @ B4 @ C6 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_242_relcompp_Osimps,axiom,
% 2.77/1.55 ! [C: $tType,B: $tType,A: $tType] :
% 2.77/1.55 ( ( relcompp @ A @ B @ C )
% 2.77/1.55 = ( ^ [R: A > B > $o,S3: B > C > $o,A12: A,A23: C] :
% 2.77/1.55 ? [A4: A,B4: B,C6: C] :
% 2.77/1.55 ( ( A12 = A4 )
% 2.77/1.55 & ( A23 = C6 )
% 2.77/1.55 & ( R @ A4 @ B4 )
% 2.77/1.55 & ( S3 @ B4 @ C6 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_243_relcompp_Ocases,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,C: $tType,R2: A > B > $o,S: B > C > $o,A1: A,A22: C] :
% 2.77/1.55 ( ( relcompp @ A @ B @ C @ R2 @ S @ A1 @ A22 )
% 2.77/1.55 => ~ ! [B5: B] :
% 2.77/1.55 ( ( R2 @ A1 @ B5 )
% 2.77/1.55 => ~ ( S @ B5 @ A22 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_244_relcomppE,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType,C: $tType,R2: A > B > $o,S: B > C > $o,A3: A,C3: C] :
% 2.77/1.55 ( ( relcompp @ A @ B @ C @ R2 @ S @ A3 @ C3 )
% 2.77/1.55 => ~ ! [B5: B] :
% 2.77/1.55 ( ( R2 @ A3 @ B5 )
% 2.77/1.55 => ~ ( S @ B5 @ C3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_245_eq__OO,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,R3: A > B > $o] :
% 2.77/1.55 ( ( relcompp @ A @ A @ B
% 2.77/1.55 @ ^ [Y2: A,Z: A] : ( Y2 = Z )
% 2.77/1.55 @ R3 )
% 2.77/1.55 = R3 ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_246_OO__eq,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType,R3: A > B > $o] :
% 2.77/1.55 ( ( relcompp @ A @ B @ B @ R3
% 2.77/1.55 @ ^ [Y2: B,Z: B] : ( Y2 = Z ) )
% 2.77/1.55 = R3 ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_247_csquare__def,axiom,
% 2.77/1.55 ! [B: $tType,C: $tType,D: $tType,A: $tType] :
% 2.77/1.55 ( ( bNF_csquare @ A @ B @ C @ D )
% 2.77/1.55 = ( ^ [A6: set @ A,F12: B > C,F22: D > C,P1: A > B,P22: A > D] :
% 2.77/1.55 ! [X4: A] :
% 2.77/1.55 ( ( member @ A @ X4 @ A6 )
% 2.77/1.55 => ( ( F12 @ ( P1 @ X4 ) )
% 2.77/1.55 = ( F22 @ ( P22 @ X4 ) ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_248_leq__OOI,axiom,
% 2.77/1.55 ! [A: $tType,R3: A > A > $o] :
% 2.77/1.55 ( ( R3
% 2.77/1.55 = ( ^ [Y2: A,Z: A] : ( Y2 = Z ) ) )
% 2.77/1.55 => ( ord_less_eq @ ( A > A > $o ) @ R3 @ ( relcompp @ A @ A @ A @ R3 @ R3 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_249_relcompp__mono,axiom,
% 2.77/1.55 ! [A: $tType,C: $tType,B: $tType,R5: A > B > $o,R2: A > B > $o,S2: B > C > $o,S: B > C > $o] :
% 2.77/1.55 ( ( ord_less_eq @ ( A > B > $o ) @ R5 @ R2 )
% 2.77/1.55 => ( ( ord_less_eq @ ( B > C > $o ) @ S2 @ S )
% 2.77/1.55 => ( ord_less_eq @ ( A > C > $o ) @ ( relcompp @ A @ B @ C @ R5 @ S2 ) @ ( relcompp @ A @ B @ C @ R2 @ S ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_250_image2__eqI,axiom,
% 2.77/1.55 ! [A: $tType,C: $tType,B: $tType,B3: A,F: B > A,X3: B,C3: C,G2: B > C,A2: set @ B] :
% 2.77/1.55 ( ( B3
% 2.77/1.55 = ( F @ X3 ) )
% 2.77/1.55 => ( ( C3
% 2.77/1.55 = ( G2 @ X3 ) )
% 2.77/1.55 => ( ( member @ B @ X3 @ A2 )
% 2.77/1.55 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B3 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A2 @ F @ G2 ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_251_strict__mono__inv,axiom,
% 2.77/1.55 ! [A: $tType,B: $tType] :
% 2.77/1.55 ( ( ( linorder @ B )
% 2.77/1.55 & ( linorder @ A ) )
% 2.77/1.55 => ! [F: A > B,G2: B > A] :
% 2.77/1.55 ( ( order_strict_mono @ A @ B @ F )
% 2.77/1.55 => ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.77/1.55 = ( top_top @ ( set @ B ) ) )
% 2.77/1.55 => ( ! [X2: A] :
% 2.77/1.55 ( ( G2 @ ( F @ X2 ) )
% 2.77/1.55 = X2 )
% 2.77/1.55 => ( order_strict_mono @ B @ A @ G2 ) ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_252_strict__mono__less__eq,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType] :
% 2.77/1.55 ( ( ( linorder @ A )
% 2.77/1.55 & ( order @ B ) )
% 2.77/1.55 => ! [F: A > B,X3: A,Y3: A] :
% 2.77/1.55 ( ( order_strict_mono @ A @ B @ F )
% 2.77/1.55 => ( ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y3 ) )
% 2.77/1.55 = ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(fact_253_strict__mono__eq,axiom,
% 2.77/1.55 ! [B: $tType,A: $tType] :
% 2.77/1.55 ( ( ( linorder @ A )
% 2.77/1.55 & ( order @ B ) )
% 2.77/1.55 => ! [F: A > B,X3: A,Y3: A] :
% 2.77/1.55 ( ( order_strict_mono @ A @ B @ F )
% 2.77/1.55 => ( ( ( F @ X3 )
% 2.77/1.55 = ( F @ Y3 ) )
% 2.77/1.55 = ( X3 = Y3 ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 2.77/1.55 ! [A8: $tType,A9: $tType] :
% 2.77/1.55 ( ( order_top @ A9 )
% 2.77/1.55 => ( order_top @ ( A8 > A9 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.77/1.55 ! [A8: $tType,A9: $tType] :
% 2.77/1.55 ( ( preorder @ A9 )
% 2.77/1.55 => ( preorder @ ( A8 > A9 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.77/1.55 ! [A8: $tType,A9: $tType] :
% 2.77/1.55 ( ( order @ A9 )
% 2.77/1.55 => ( order @ ( A8 > A9 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(tcon_fun___Orderings_Otop,axiom,
% 2.77/1.55 ! [A8: $tType,A9: $tType] :
% 2.77/1.55 ( ( top @ A9 )
% 2.77/1.55 => ( top @ ( A8 > A9 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(tcon_fun___Orderings_Oord,axiom,
% 2.77/1.55 ! [A8: $tType,A9: $tType] :
% 2.77/1.55 ( ( ord @ A9 )
% 2.77/1.55 => ( ord @ ( A8 > A9 ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(tcon_Set_Oset___Orderings_Oorder__top_1,axiom,
% 2.77/1.55 ! [A8: $tType] : ( order_top @ ( set @ A8 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
% 2.77/1.55 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
% 2.77/1.55 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(tcon_Set_Oset___Orderings_Otop_4,axiom,
% 2.77/1.55 ! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
% 2.77/1.55 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(tcon_HOL_Obool___Orderings_Oorder__top_6,axiom,
% 2.77/1.55 order_top @ $o ).
% 2.77/1.55
% 2.77/1.55 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 2.77/1.55 preorder @ $o ).
% 2.77/1.55
% 2.77/1.55 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.77/1.55 linorder @ $o ).
% 2.77/1.55
% 2.77/1.55 thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
% 2.77/1.55 order @ $o ).
% 2.77/1.55
% 2.77/1.55 thf(tcon_HOL_Obool___Orderings_Otop_9,axiom,
% 2.77/1.55 top @ $o ).
% 2.77/1.55
% 2.77/1.55 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 2.77/1.55 ord @ $o ).
% 2.77/1.55
% 2.77/1.55 thf(conj_0,hypothesis,
% 2.77/1.55 ! [A10: set @ a] :
% 2.77/1.55 ( ( member @ ( set @ a ) @ A10 @ ( image @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) ) @ c ) )
% 2.77/1.55 => ( ( ord_less_eq @ ( set @ a ) @ a1 @ A10 )
% 2.77/1.55 => ( ( ord_less_eq @ ( set @ a ) @ a2 @ A10 )
% 2.77/1.55 => ( ( ord_less_eq @ ( set @ a ) @ a3 @ A10 )
% 2.77/1.55 => thesis ) ) ) ) ).
% 2.77/1.55
% 2.77/1.55 thf(conj_1,conjecture,
% 2.77/1.55 thesis ).
% 2.77/1.55 ------------------------
% 2.77/1.58 % (12484)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_11799 for (2999ds/4Mi)
% 2.77/1.58 % (12483)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/183Mi)
% 2.77/1.58 % (12486)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/2Mi)
% 2.77/1.58 % (12484)Instruction limit reached!
% 2.77/1.58 % (12484)------------------------------
% 2.77/1.58 % (12484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.58 % (12484)Termination reason: Unknown
% 2.77/1.58 % (12484)Termination phase: shuffling
% 2.77/1.58
% 2.77/1.58 % (12484)Memory used [KB]: 1407
% 2.77/1.58 % (12484)Time elapsed: 0.003 s
% 2.77/1.58 % (12484)Instructions burned: 4 (million)
% 2.77/1.58 % (12484)------------------------------
% 2.77/1.58 % (12484)------------------------------
% 2.77/1.59 % (12486)Instruction limit reached!
% 2.77/1.59 % (12486)------------------------------
% 2.77/1.59 % (12486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.59 % (12486)Termination reason: Unknown
% 2.77/1.59 % (12486)Termination phase: shuffling
% 2.77/1.59
% 2.77/1.59 % (12486)Memory used [KB]: 1407
% 2.77/1.59 % (12486)Time elapsed: 0.002 s
% 2.77/1.59 % (12486)Instructions burned: 3 (million)
% 2.77/1.59 % (12486)------------------------------
% 2.77/1.59 % (12486)------------------------------
% 2.77/1.59 % (12485)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_11799 for (2999ds/27Mi)
% 2.77/1.59 % (12487)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_11799 for (2999ds/2Mi)
% 2.77/1.59 % (12488)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_11799 for (2999ds/275Mi)
% 2.77/1.59 % (12489)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/18Mi)
% 2.77/1.59 % (12487)Instruction limit reached!
% 2.77/1.59 % (12487)------------------------------
% 2.77/1.59 % (12487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.59 % (12487)Termination reason: Unknown
% 2.77/1.59 % (12487)Termination phase: shuffling
% 2.77/1.59
% 2.77/1.59 % (12487)Memory used [KB]: 1407
% 2.77/1.59 % (12487)Time elapsed: 0.004 s
% 2.77/1.59 % (12487)Instructions burned: 3 (million)
% 2.77/1.59 % (12487)------------------------------
% 2.77/1.59 % (12487)------------------------------
% 2.77/1.59 % (12495)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/3Mi)
% 2.77/1.60 % (12496)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_11799 for (2999ds/37Mi)
% 2.83/1.60 % (12495)Instruction limit reached!
% 2.83/1.60 % (12495)------------------------------
% 2.83/1.60 % (12495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.60 % (12495)Termination reason: Unknown
% 2.83/1.60 % (12495)Termination phase: shuffling
% 2.83/1.60
% 2.83/1.60 % (12495)Memory used [KB]: 1407
% 2.83/1.60 % (12495)Time elapsed: 0.003 s
% 2.83/1.60 % (12495)Instructions burned: 5 (million)
% 2.83/1.60 % (12495)------------------------------
% 2.83/1.60 % (12495)------------------------------
% 2.83/1.61 % (12489)Instruction limit reached!
% 2.83/1.61 % (12489)------------------------------
% 2.83/1.61 % (12489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.61 % (12489)Termination reason: Unknown
% 2.83/1.61 % (12489)Termination phase: shuffling
% 2.83/1.61
% 2.83/1.61 % (12489)Memory used [KB]: 1791
% 2.83/1.61 % (12489)Time elapsed: 0.015 s
% 2.83/1.61 % (12489)Instructions burned: 18 (million)
% 2.83/1.61 % (12489)------------------------------
% 2.83/1.61 % (12489)------------------------------
% 2.83/1.61 % (12503)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_11799 for (2999ds/3Mi)
% 2.83/1.61 % (12503)Instruction limit reached!
% 2.83/1.61 % (12503)------------------------------
% 2.83/1.61 % (12503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.61 % (12503)Termination reason: Unknown
% 2.83/1.61 % (12503)Termination phase: shuffling
% 2.83/1.61
% 2.83/1.61 % (12503)Memory used [KB]: 1407
% 2.83/1.61 % (12503)Time elapsed: 0.002 s
% 2.83/1.61 % (12503)Instructions burned: 3 (million)
% 2.83/1.61 % (12503)------------------------------
% 2.83/1.61 % (12503)------------------------------
% 2.83/1.61 % (12496)Instruction limit reached!
% 2.83/1.61 % (12496)------------------------------
% 2.83/1.61 % (12496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.61 % (12496)Termination reason: Unknown
% 2.83/1.61 % (12496)Termination phase: shuffling
% 2.83/1.61
% 2.83/1.61 % (12496)Memory used [KB]: 2174
% 2.83/1.61 % (12496)Time elapsed: 0.014 s
% 2.83/1.61 % (12496)Instructions burned: 39 (million)
% 2.83/1.61 % (12496)------------------------------
% 2.83/1.61 % (12496)------------------------------
% 2.83/1.61 % (12485)Instruction limit reached!
% 2.83/1.61 % (12485)------------------------------
% 2.83/1.61 % (12485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.61 % (12485)Termination reason: Unknown
% 2.83/1.61 % (12485)Termination phase: Property scanning
% 2.83/1.61
% 2.83/1.61 % (12485)Memory used [KB]: 1918
% 2.83/1.61 % (12485)Time elapsed: 0.020 s
% 2.83/1.61 % (12485)Instructions burned: 28 (million)
% 2.83/1.61 % (12485)------------------------------
% 2.83/1.61 % (12485)------------------------------
% 2.83/1.61 % (12500)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_11799 for (2999ds/15Mi)
% 2.83/1.62 % (12510)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/7Mi)
% 2.83/1.62 % (12511)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/16Mi)
% 2.83/1.62 % (12510)Instruction limit reached!
% 2.83/1.62 % (12510)------------------------------
% 2.83/1.62 % (12510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.62 % (12510)Termination reason: Unknown
% 2.83/1.62 % (12510)Termination phase: shuffling
% 2.83/1.62
% 2.83/1.62 % (12510)Memory used [KB]: 1535
% 2.83/1.62 % (12510)Time elapsed: 0.004 s
% 2.83/1.62 % (12510)Instructions burned: 9 (million)
% 2.83/1.62 % (12510)------------------------------
% 2.83/1.62 % (12510)------------------------------
% 2.83/1.62 % (12500)Instruction limit reached!
% 2.83/1.62 % (12500)------------------------------
% 2.83/1.62 % (12500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.62 % (12500)Termination reason: Unknown
% 2.83/1.62 % (12500)Termination phase: shuffling
% 2.83/1.62
% 2.83/1.62 % (12500)Memory used [KB]: 1663
% 2.83/1.62 % (12500)Time elapsed: 0.012 s
% 2.83/1.62 % (12500)Instructions burned: 16 (million)
% 2.83/1.62 % (12500)------------------------------
% 2.83/1.62 % (12500)------------------------------
% 2.83/1.62 % (12511)Instruction limit reached!
% 2.83/1.62 % (12511)------------------------------
% 2.83/1.62 % (12511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.62 % (12511)Termination reason: Unknown
% 2.83/1.62 % (12511)Termination phase: shuffling
% 2.83/1.62
% 2.83/1.62 % (12511)Memory used [KB]: 1791
% 2.83/1.62 % (12511)Time elapsed: 0.008 s
% 2.83/1.62 % (12511)Instructions burned: 17 (million)
% 2.83/1.62 % (12511)------------------------------
% 2.83/1.62 % (12511)------------------------------
% 2.83/1.63 % (12508)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/1041Mi)
% 2.83/1.63 % (12513)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/3Mi)
% 2.83/1.63 % (12514)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/3Mi)
% 2.83/1.63 % (12513)Instruction limit reached!
% 2.83/1.63 % (12513)------------------------------
% 2.83/1.63 % (12513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.63 % (12513)Termination reason: Unknown
% 2.83/1.63 % (12513)Termination phase: shuffling
% 2.83/1.63
% 2.83/1.63 % (12513)Memory used [KB]: 1407
% 2.83/1.63 % (12513)Time elapsed: 0.004 s
% 2.83/1.63 % (12513)Instructions burned: 3 (million)
% 2.83/1.63 % (12513)------------------------------
% 2.83/1.63 % (12513)------------------------------
% 2.83/1.63 % (12514)Instruction limit reached!
% 2.83/1.63 % (12514)------------------------------
% 2.83/1.63 % (12514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.63 % (12514)Termination reason: Unknown
% 2.83/1.63 % (12514)Termination phase: shuffling
% 2.83/1.63
% 2.83/1.63 % (12514)Memory used [KB]: 1407
% 2.83/1.63 % (12514)Time elapsed: 0.002 s
% 2.83/1.63 % (12514)Instructions burned: 3 (million)
% 2.83/1.63 % (12514)------------------------------
% 2.83/1.63 % (12514)------------------------------
% 2.83/1.63 % (12519)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/3Mi)
% 2.83/1.64 % (12519)Instruction limit reached!
% 2.83/1.64 % (12519)------------------------------
% 2.83/1.64 % (12519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.64 % (12519)Termination reason: Unknown
% 2.83/1.64 % (12519)Termination phase: shuffling
% 2.83/1.64
% 2.83/1.64 % (12519)Memory used [KB]: 1407
% 2.83/1.64 % (12519)Time elapsed: 0.003 s
% 2.83/1.64 % (12519)Instructions burned: 4 (million)
% 2.83/1.64 % (12519)------------------------------
% 2.83/1.64 % (12519)------------------------------
% 2.83/1.64 % (12516)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/7Mi)
% 2.83/1.64 % (12523)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/18Mi)
% 2.83/1.64 % (12516)Instruction limit reached!
% 2.83/1.64 % (12516)------------------------------
% 2.83/1.64 % (12516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.64 % (12516)Termination reason: Unknown
% 2.83/1.64 % (12516)Termination phase: shuffling
% 2.83/1.64
% 2.83/1.64 % (12516)Memory used [KB]: 1535
% 2.83/1.64 % (12516)Time elapsed: 0.006 s
% 2.83/1.64 % (12516)Instructions burned: 8 (million)
% 2.83/1.64 % (12516)------------------------------
% 2.83/1.64 % (12516)------------------------------
% 2.83/1.64 % (12523)Instruction limit reached!
% 2.83/1.64 % (12523)------------------------------
% 2.83/1.64 % (12523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.64 % (12523)Termination reason: Unknown
% 2.83/1.64 % (12523)Termination phase: shuffling
% 2.83/1.64
% 2.83/1.64 % (12523)Memory used [KB]: 1791
% 2.83/1.64 % (12523)Time elapsed: 0.007 s
% 2.83/1.64 % (12523)Instructions burned: 18 (million)
% 2.83/1.64 % (12523)------------------------------
% 2.83/1.64 % (12523)------------------------------
% 2.83/1.64 % (12483)Instruction limit reached!
% 2.83/1.64 % (12483)------------------------------
% 2.83/1.64 % (12483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.64 % (12483)Termination reason: Unknown
% 2.83/1.64 % (12483)Termination phase: Saturation
% 2.83/1.64
% 2.83/1.64 % (12483)Memory used [KB]: 7419
% 2.83/1.64 % (12483)Time elapsed: 0.062 s
% 2.83/1.64 % (12483)Instructions burned: 184 (million)
% 2.83/1.64 % (12483)------------------------------
% 2.83/1.64 % (12483)------------------------------
% 2.83/1.65 % (12527)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/710Mi)
% 2.83/1.65 % (12522)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/4Mi)
% 2.83/1.65 % (12522)Instruction limit reached!
% 2.83/1.65 % (12522)------------------------------
% 2.83/1.65 % (12522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.65 % (12522)Termination reason: Unknown
% 2.83/1.65 % (12522)Termination phase: shuffling
% 2.83/1.65
% 2.83/1.65 % (12522)Memory used [KB]: 1407
% 2.83/1.65 % (12522)Time elapsed: 0.004 s
% 2.83/1.65 % (12522)Instructions burned: 4 (million)
% 2.83/1.65 % (12522)------------------------------
% 2.83/1.65 % (12522)------------------------------
% 2.83/1.65 % (12530)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/6Mi)
% 2.83/1.65 % (12488)First to succeed.
% 2.83/1.65 % (12533)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/902Mi)
% 2.83/1.65 % (12530)Instruction limit reached!
% 2.83/1.65 % (12530)------------------------------
% 2.83/1.65 % (12530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.65 % (12530)Termination reason: Unknown
% 2.83/1.65 % (12530)Termination phase: shuffling
% 2.83/1.65
% 2.83/1.65 % (12530)Memory used [KB]: 1407
% 2.83/1.65 % (12530)Time elapsed: 0.003 s
% 2.83/1.65 % (12530)Instructions burned: 6 (million)
% 2.83/1.65 % (12530)------------------------------
% 2.83/1.65 % (12530)------------------------------
% 2.83/1.66 % (12534)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_11799 for (2999ds/21Mi)
% 2.83/1.66 % (12488)Refutation found. Thanks to Tanya!
% 2.83/1.66 % SZS status Theorem for DTF2THF_11799
% 2.83/1.66 % SZS output start Proof for DTF2THF_11799
% 2.83/1.66 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 2.83/1.66 thf(type_def_6, type, set: ($tType) > $tType).
% 2.83/1.66 thf(type_def_8, type, a: $tType).
% 2.83/1.66 thf(type_def_9, type, e: $tType).
% 2.83/1.66 thf(func_def_0, type, product_prod: ($tType * $tType) > $tType).
% 2.83/1.66 thf(func_def_1, type, set: ($tType) > $tType).
% 2.83/1.66 thf(func_def_2, type, e: $tType).
% 2.83/1.66 thf(func_def_3, type, a: $tType).
% 2.83/1.66 thf(func_def_4, type, type: !>[X0: $tType]:($o)).
% 2.83/1.66 thf(func_def_5, type, ord: !>[X0: $tType]:($o)).
% 2.83/1.66 thf(func_def_6, type, top: !>[X0: $tType]:($o)).
% 2.83/1.66 thf(func_def_7, type, order: !>[X0: $tType]:($o)).
% 2.83/1.66 thf(func_def_8, type, linorder: !>[X0: $tType]:($o)).
% 2.83/1.66 thf(func_def_9, type, preorder: !>[X0: $tType]:($o)).
% 2.83/1.66 thf(func_def_10, type, order_top: !>[X0: $tType]:($o)).
% 2.83/1.66 thf(func_def_11, type, bNF_Ca1785829860lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 2.83/1.66 thf(func_def_12, type, bNF_csquare: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(set @ X0 > (X1 > X2) > (X3 > X2) > (X0 > X1) > (X0 > X3) > $o)).
% 2.83/1.66 thf(func_def_13, type, bNF_fstOp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > product_prod @ X0 @ X2 > product_prod @ X0 @ X1)).
% 2.83/1.66 thf(func_def_14, type, bNF_pick_middlep: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > X0 > X2 > X1)).
% 2.83/1.66 thf(func_def_15, type, bNF_sndOp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > product_prod @ X0 @ X2 > product_prod @ X1 @ X2)).
% 2.83/1.66 thf(func_def_16, type, bNF_Greatest_image2: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ X0 > (X0 > X1) > (X0 > X2) > set @ product_prod @ X1 @ X2)).
% 2.83/1.66 thf(func_def_17, type, finite_Fpow: !>[X0: $tType]:(set @ X0 > set @ set @ X0)).
% 2.83/1.66 thf(func_def_18, type, fun_in_rel: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > X0 > X1 > $o)).
% 2.83/1.66 thf(func_def_19, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.83/1.66 thf(func_def_20, type, order_Greatest: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.83/1.66 thf(func_def_21, type, order_strict_mono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 2.83/1.66 thf(func_def_22, type, top_top: !>[X0: $tType]:(X0)).
% 2.83/1.66 thf(func_def_23, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.83/1.66 thf(func_def_24, type, product_curry: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X0 @ X1 > X2) > X0 > X1 > X2)).
% 2.83/1.66 thf(func_def_25, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.83/1.66 thf(func_def_26, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 2.83/1.66 thf(func_def_27, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.83/1.66 thf(func_def_28, type, product_case_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.83/1.66 thf(func_def_29, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.83/1.66 thf(func_def_30, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.83/1.66 thf(func_def_31, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 2.83/1.66 thf(func_def_32, type, domain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X0)).
% 2.83/1.66 thf(func_def_33, type, range: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X1)).
% 2.83/1.66 thf(func_def_34, type, converse: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ product_prod @ X1 @ X0)).
% 2.83/1.66 thf(func_def_35, type, inv_image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X1 > X0) > set @ product_prod @ X1 @ X1)).
% 2.83/1.66 thf(func_def_36, type, inv_imagep: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > X1 > X1 > $o)).
% 2.83/1.66 thf(func_def_37, type, relcompp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > X0 > X2 > $o)).
% 2.83/1.66 thf(func_def_38, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.83/1.66 thf(func_def_39, type, pow: !>[X0: $tType]:(set @ X0 > set @ set @ X0)).
% 2.83/1.66 thf(func_def_40, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 2.83/1.66 thf(func_def_41, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.83/1.66 thf(func_def_42, type, c: set @ product_prod @ set @ a @ (a > e)).
% 2.83/1.66 thf(func_def_43, type, a1: set @ a).
% 2.83/1.66 thf(func_def_44, type, a2: set @ a).
% 2.83/1.66 thf(func_def_45, type, a3: set @ a).
% 2.83/1.66 thf(func_def_46, type, b1: a > e).
% 2.83/1.66 thf(func_def_47, type, b2: a > e).
% 2.83/1.66 thf(func_def_48, type, b3: a > e).
% 2.83/1.66 thf(func_def_66, type, sK0: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.83/1.66 thf(func_def_67, type, sK1: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 2.83/1.66 thf(func_def_68, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 2.83/1.66 thf(func_def_69, type, sK3: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 2.83/1.66 thf(func_def_70, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 2.83/1.66 thf(func_def_71, type, sK5: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 2.83/1.66 thf(func_def_72, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.83/1.66 thf(func_def_73, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.83/1.66 thf(func_def_74, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.83/1.66 thf(func_def_75, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.83/1.66 thf(func_def_76, type, sK10: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.83/1.66 thf(func_def_77, type, sK11: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.83/1.66 thf(func_def_78, type, sK12: !>[X0: $tType, X1: $tType]:((set @ X0 > $o) > set @ X1 > (X1 > X0) > set @ X0)).
% 2.83/1.66 thf(func_def_79, type, sK13: !>[X0: $tType, X1: $tType]:((set @ X0 > $o) > set @ X1 > (X1 > X0) > set @ X1)).
% 2.83/1.66 thf(func_def_80, type, sK14: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.83/1.66 thf(func_def_81, type, sK15: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.83/1.66 thf(func_def_82, type, sK16: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X0)).
% 2.83/1.66 thf(func_def_83, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.83/1.66 thf(func_def_84, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.83/1.66 thf(func_def_85, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.83/1.66 thf(func_def_86, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.83/1.66 thf(func_def_87, type, sK21: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.83/1.66 thf(func_def_88, type, sK22: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.83/1.66 thf(func_def_89, type, sK23: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.83/1.66 thf(func_def_90, type, sK24: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.83/1.66 thf(func_def_91, type, sK25: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.83/1.66 thf(func_def_92, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 2.83/1.66 thf(func_def_93, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 2.83/1.66 thf(func_def_94, type, sK28: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 2.83/1.66 thf(func_def_95, type, sK29: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 2.83/1.66 thf(func_def_96, type, sK30: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 2.83/1.66 thf(func_def_97, type, sK31: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 2.83/1.66 thf(func_def_98, type, sK32: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 2.83/1.66 thf(func_def_99, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 2.83/1.66 thf(func_def_100, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 2.83/1.66 thf(func_def_101, type, sK35: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 2.83/1.66 thf(func_def_102, type, sK36: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 2.83/1.66 thf(func_def_103, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 2.83/1.66 thf(func_def_104, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 2.83/1.66 thf(func_def_105, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 2.83/1.66 thf(func_def_106, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 2.83/1.66 thf(func_def_107, type, sK41: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 2.83/1.66 thf(func_def_108, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 2.83/1.66 thf(func_def_109, type, sK43: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.83/1.66 thf(func_def_110, type, sK44: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.83/1.66 thf(func_def_111, type, sK45: !>[X0: $tType, X1: $tType]:(set @ X1 > (X0 > X1) > set @ X0 > X0)).
% 2.83/1.66 thf(func_def_112, type, sK46: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0 > $o) > X1)).
% 2.83/1.66 thf(func_def_113, type, sK47: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0 > $o) > X0)).
% 2.83/1.66 thf(func_def_114, type, sK48: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > X0 > X1)).
% 2.83/1.66 thf(func_def_115, type, sK49: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X1)).
% 2.83/1.66 thf(func_def_116, type, sK50: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X0)).
% 2.83/1.66 thf(func_def_117, type, sK51: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > set @ X1)).
% 2.83/1.66 thf(func_def_118, type, sK52: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > (X1 > X0) > X1)).
% 2.83/1.66 thf(func_def_119, type, sK53: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.83/1.66 thf(func_def_120, type, sK54: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 2.83/1.66 thf(func_def_121, type, sK55: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.83/1.66 thf(func_def_122, type, sK56: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.83/1.66 thf(func_def_123, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.83/1.66 thf(func_def_124, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.83/1.66 thf(func_def_125, type, sK59: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.83/1.66 thf(func_def_126, type, sK60: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.83/1.66 thf(func_def_127, type, sK61: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.83/1.66 thf(func_def_128, type, sK62: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 2.83/1.66 thf(func_def_129, type, sK63: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.83/1.66 thf(func_def_130, type, sK64: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.83/1.66 thf(func_def_131, type, sK65: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.83/1.66 thf(func_def_132, type, sK66: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.83/1.66 thf(func_def_133, type, sK67: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.83/1.66 thf(func_def_134, type, sK68: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.83/1.66 thf(func_def_135, type, sK69: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 2.83/1.66 thf(func_def_136, type, sK70: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 2.83/1.66 thf(func_def_137, type, sK71: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 2.83/1.66 thf(func_def_138, type, sK72: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 2.83/1.66 thf(func_def_139, type, sK73: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X1)).
% 2.83/1.66 thf(func_def_140, type, sK74: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.83/1.66 thf(func_def_141, type, sK75: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 2.83/1.66 thf(func_def_142, type, sK76: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0 > X1)).
% 2.83/1.66 thf(func_def_143, type, sK77: set @ a).
% 2.83/1.66 thf(func_def_144, type, sK78: !>[X0: $tType, X1: $tType]:(set @ X0 > (X1 > X0) > set @ X1 > X1)).
% 2.83/1.66 thf(func_def_145, type, sK79: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.83/1.66 thf(func_def_146, type, sK80: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.83/1.66 thf(func_def_147, type, sK81: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > X0 > X1)).
% 2.83/1.66 thf(func_def_148, type, sK82: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > set @ X1)).
% 2.83/1.66 thf(func_def_150, type, ph84: !>[X0: $tType]:(X0)).
% 2.83/1.66 thf(f1098,plain,(
% 2.83/1.66 $false),
% 2.83/1.66 inference(avatar_sat_refutation,[status(thm)],[f1018,f1019,f1077])).
% 2.83/1.66 thf(f1077,plain,(
% 2.83/1.66 ~spl83_1),
% 2.83/1.66 inference(avatar_contradiction_clause,[status(thm)],[f1076])).
% 2.83/1.66 thf(f1076,plain,(
% 2.83/1.66 $false | ~spl83_1),
% 2.83/1.66 inference(subsumption_resolution,[status(thm)],[f1075,f964])).
% 2.83/1.66 thf(f964,plain,(
% 2.83/1.66 ((ord_less_eq @ set @ a @ a2 @ sK77) = $true)),
% 2.83/1.66 inference(cnf_transformation,[status(thm)],[f862])).
% 2.83/1.66 thf(f862,plain,(
% 2.83/1.66 ((ord_less_eq @ set @ a @ a1 @ sK77) = $true) & ((ord_less_eq @ set @ a @ a2 @ sK77) = $true) & ((ord_less_eq @ set @ a @ a3 @ sK77) = $true) & ((member @ set @ a @ sK77 @ (image @ product_prod @ set @ a @ (a > e) @ set @ a @ product_fst @ set @ a @ (a > e) @ c)) = $true)),
% 2.83/1.66 inference(skolemisation,[status(esa),new_symbols(skolem,[sK77])],[f460,f861])).
% 2.83/1.66 thf(f861,plain,(
% 2.83/1.66 ? [X0 : set @ a] : (((ord_less_eq @ set @ a @ a1 @ X0) = $true) & ((ord_less_eq @ set @ a @ a2 @ X0) = $true) & ((ord_less_eq @ set @ a @ a3 @ X0) = $true) & ((member @ set @ a @ X0 @ (image @ product_prod @ set @ a @ (a > e) @ set @ a @ product_fst @ set @ a @ (a > e) @ c)) = $true)) => (((ord_less_eq @ set @ a @ a1 @ sK77) = $true) & ((ord_less_eq @ set @ a @ a2 @ sK77) = $true) & ((ord_less_eq @ set @ a @ a3 @ sK77) = $true) & ((member @ set @ a @ sK77 @ (image @ product_prod @ set @ a @ (a > e) @ set @ a @ product_fst @ set @ a @ (a > e) @ c)) = $true))),
% 2.83/1.66 introduced(definition,[],[choice_axiom])).
% 2.83/1.66 thf(f460,plain,(
% 2.83/1.66 ? [X0 : set @ a] : (((ord_less_eq @ set @ a @ a1 @ X0) = $true) & ((ord_less_eq @ set @ a @ a2 @ X0) = $true) & ((ord_less_eq @ set @ a @ a3 @ X0) = $true) & ((member @ set @ a @ X0 @ (image @ product_prod @ set @ a @ (a > e) @ set @ a @ product_fst @ set @ a @ (a > e) @ c)) = $true))),
% 2.83/1.66 inference(fool_elimination,[status(thm)],[f459])).
% 2.83/1.66 thf(f459,plain,(
% 2.83/1.66 ? [X0 : set @ a] : ((ord_less_eq @ set @ a @ a3 @ X0) & (ord_less_eq @ set @ a @ a1 @ X0) & (ord_less_eq @ set @ a @ a2 @ X0) & (member @ set @ a @ X0 @ (image @ product_prod @ set @ a @ (a > e) @ set @ a @ product_fst @ set @ a @ (a > e) @ c)))),
% 2.83/1.66 inference(rectify,[status(thm)],[f2])).
% 2.83/1.66 thf(f2,axiom,(
% 2.83/1.66 ? [X7 : set @ a] : ((ord_less_eq @ set @ a @ a3 @ X7) & (ord_less_eq @ set @ a @ a1 @ X7) & (ord_less_eq @ set @ a @ a2 @ X7) & (member @ set @ a @ X7 @ (image @ product_prod @ set @ a @ (a > e) @ set @ a @ product_fst @ set @ a @ (a > e) @ c)))),
% 2.83/1.66 file('/export/starexec/sandbox2/tmp/tmp.HLLIQP6qlU/DTF2THF_11799.p',fact_1__092_060open_062_092_060exists_062a_092_060in_062fst_A_096_AC_O_Aa1_A_092_060subseteq_062_Aa_A_092_060and_062_Aa2_A_092_060subseteq_062_Aa_A_092_060and_062_Aa3_A_092_060subseteq_062_Aa_092_060close_062)).
% 2.83/1.66 thf(f1075,plain,(
% 2.83/1.66 ((ord_less_eq @ set @ a @ a2 @ sK77) != $true) | ~spl83_1),
% 2.83/1.66 inference(subsumption_resolution,[status(thm)],[f1074,f963])).
% 2.83/1.66 thf(f963,plain,(
% 2.83/1.66 ((ord_less_eq @ set @ a @ a3 @ sK77) = $true)),
% 2.83/1.66 inference(cnf_transformation,[status(thm)],[f862])).
% 2.83/1.66 thf(f1074,plain,(
% 2.83/1.66 ((ord_less_eq @ set @ a @ a3 @ sK77) != $true) | ((ord_less_eq @ set @ a @ a2 @ sK77) != $true) | ~spl83_1),
% 2.83/1.66 inference(subsumption_resolution,[status(thm)],[f1060,f965])).
% 2.83/1.66 thf(f965,plain,(
% 2.83/1.66 ((ord_less_eq @ set @ a @ a1 @ sK77) = $true)),
% 2.83/1.66 inference(cnf_transformation,[status(thm)],[f862])).
% 2.83/1.66 thf(f1060,plain,(
% 2.83/1.66 ((ord_less_eq @ set @ a @ a1 @ sK77) != $true) | ((ord_less_eq @ set @ a @ a3 @ sK77) != $true) | ((ord_less_eq @ set @ a @ a2 @ sK77) != $true) | ~spl83_1),
% 2.83/1.66 inference(trivial_inequality_removal,[status(thm)],[f1025])).
% 2.83/1.66 thf(f1025,plain,(
% 2.83/1.66 ((ord_less_eq @ set @ a @ a2 @ sK77) != $true) | ((ord_less_eq @ set @ a @ a1 @ sK77) != $true) | ((ord_less_eq @ set @ a @ a3 @ sK77) != $true) | ($true != $true) | ~spl83_1),
% 2.83/1.66 inference(superposition,[status(thm)],[f1013,f962])).
% 2.83/1.66 thf(f962,plain,(
% 2.83/1.66 ((member @ set @ a @ sK77 @ (image @ product_prod @ set @ a @ (a > e) @ set @ a @ product_fst @ set @ a @ (a > e) @ c)) = $true)),
% 2.83/1.66 inference(cnf_transformation,[status(thm)],[f862])).
% 2.83/1.66 thf(f1013,plain,(
% 2.83/1.66 ( ! [X0 : set @ a] : (((member @ set @ a @ X0 @ (image @ product_prod @ set @ a @ (a > e) @ set @ a @ product_fst @ set @ a @ (a > e) @ c)) != $true) | ((ord_less_eq @ set @ a @ a2 @ X0) != $true) | ((ord_less_eq @ set @ a @ a3 @ X0) != $true) | ((ord_less_eq @ set @ a @ a1 @ X0) != $true)) ) | ~spl83_1),
% 2.83/1.66 inference(avatar_component_clause,[status(thm)],[f1012])).
% 2.83/1.66 thf(f1012,definition,(
% 2.83/1.66 spl83_1 <=> ! [X0 : set @ a] : (((ord_less_eq @ set @ a @ a3 @ X0) != $true) | ((ord_less_eq @ set @ a @ a2 @ X0) != $true) | ((member @ set @ a @ X0 @ (image @ product_prod @ set @ a @ (a > e) @ set @ a @ product_fst @ set @ a @ (a > e) @ c)) != $true) | ((ord_less_eq @ set @ a @ a1 @ X0) != $true))),
% 2.83/1.66 introduced(definition,[new_symbols(naming,[spl83_1])],[avatar_definition])).
% 2.83/1.66 thf(f1019,plain,(
% 2.83/1.66 ~spl83_2),
% 2.83/1.66 inference(avatar_split_clause,[status(thm)],[f911,f1015])).
% 2.83/1.66 thf(f1015,definition,(
% 2.83/1.66 spl83_2 <=> (thesis = $true)),
% 2.83/1.66 introduced(definition,[new_symbols(naming,[spl83_2])],[avatar_definition])).
% 2.83/1.66 thf(f911,plain,(
% 2.83/1.66 (thesis != $true)),
% 2.83/1.66 inference(cnf_transformation,[status(thm)],[f693])).
% 2.83/1.66 thf(f693,plain,(
% 2.83/1.66 (thesis != $true)),
% 2.83/1.66 inference(flattening,[status(thm)],[f388])).
% 2.83/1.66 thf(f388,plain,(
% 2.83/1.66 ~(thesis = $true)),
% 2.83/1.66 inference(fool_elimination,[status(thm)],[f387])).
% 2.83/1.66 thf(f387,plain,(
% 2.83/1.66 ~thesis),
% 2.83/1.66 inference(rectify,[status(thm)],[f273])).
% 2.83/1.66 thf(f273,negated_conjecture,(
% 2.83/1.66 ~thesis),
% 2.83/1.66 inference(negated_conjecture,[status(cth)],[f272])).
% 2.83/1.66 thf(f272,conjecture,(
% 2.83/1.66 thesis),
% 2.83/1.66 file('/export/starexec/sandbox2/tmp/tmp.HLLIQP6qlU/DTF2THF_11799.p',conj_1)).
% 2.83/1.66 thf(f1018,plain,(
% 2.83/1.66 spl83_1 | spl83_2),
% 2.83/1.66 inference(avatar_split_clause,[status(thm)],[f926,f1015,f1012])).
% 2.83/1.66 thf(f926,plain,(
% 2.83/1.66 ( ! [X0 : set @ a] : (((ord_less_eq @ set @ a @ a3 @ X0) != $true) | (thesis = $true) | ((ord_less_eq @ set @ a @ a1 @ X0) != $true) | ((member @ set @ a @ X0 @ (image @ product_prod @ set @ a @ (a > e) @ set @ a @ product_fst @ set @ a @ (a > e) @ c)) != $true) | ((ord_less_eq @ set @ a @ a2 @ X0) != $true)) )),
% 2.83/1.66 inference(cnf_transformation,[status(thm)],[f778])).
% 2.83/1.66 thf(f778,plain,(
% 2.83/1.66 ! [X0 : set @ a] : (((ord_less_eq @ set @ a @ a3 @ X0) != $true) | (thesis = $true) | ((ord_less_eq @ set @ a @ a1 @ X0) != $true) | ((member @ set @ a @ X0 @ (image @ product_prod @ set @ a @ (a > e) @ set @ a @ product_fst @ set @ a @ (a > e) @ c)) != $true) | ((ord_less_eq @ set @ a @ a2 @ X0) != $true))),
% 2.83/1.66 inference(flattening,[status(thm)],[f777])).
% 2.83/1.66 thf(f777,plain,(
% 2.83/1.66 ! [X0 : set @ a] : (((((thesis = $true) | ((ord_less_eq @ set @ a @ a3 @ X0) != $true)) | ((ord_less_eq @ set @ a @ a2 @ X0) != $true)) | ((ord_less_eq @ set @ a @ a1 @ X0) != $true)) | ((member @ set @ a @ X0 @ (image @ product_prod @ set @ a @ (a > e) @ set @ a @ product_fst @ set @ a @ (a > e) @ c)) != $true))),
% 2.83/1.66 inference(ennf_transformation,[status(thm)],[f336])).
% 2.83/1.66 thf(f336,plain,(
% 2.83/1.66 ! [X0 : set @ a] : (((member @ set @ a @ X0 @ (image @ product_prod @ set @ a @ (a > e) @ set @ a @ product_fst @ set @ a @ (a > e) @ c)) = $true) => (((ord_less_eq @ set @ a @ a1 @ X0) = $true) => (((ord_less_eq @ set @ a @ a2 @ X0) = $true) => (((ord_less_eq @ set @ a @ a3 @ X0) = $true) => (thesis = $true)))))),
% 2.83/1.66 inference(fool_elimination,[status(thm)],[f335])).
% 2.83/1.66 thf(f335,plain,(
% 2.83/1.66 ! [X0 : set @ a] : ((member @ set @ a @ X0 @ (image @ product_prod @ set @ a @ (a > e) @ set @ a @ product_fst @ set @ a @ (a > e) @ c)) => ((ord_less_eq @ set @ a @ a1 @ X0) => ((ord_less_eq @ set @ a @ a2 @ X0) => ((ord_less_eq @ set @ a @ a3 @ X0) => thesis))))),
% 2.83/1.66 inference(rectify,[status(thm)],[f271])).
% 2.83/1.66 thf(f271,axiom,(
% 2.83/1.66 ! [X100 : set @ a] : ((member @ set @ a @ X100 @ (image @ product_prod @ set @ a @ (a > e) @ set @ a @ product_fst @ set @ a @ (a > e) @ c)) => ((ord_less_eq @ set @ a @ a1 @ X100) => ((ord_less_eq @ set @ a @ a2 @ X100) => ((ord_less_eq @ set @ a @ a3 @ X100) => thesis))))),
% 2.83/1.66 file('/export/starexec/sandbox2/tmp/tmp.HLLIQP6qlU/DTF2THF_11799.p',conj_0)).
% 2.83/1.66 % SZS output end Proof for DTF2THF_11799
% 2.83/1.66 % (12488)------------------------------
% 2.83/1.66 % (12488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.66 % (12488)Termination reason: Refutation
% 2.83/1.66
% 2.83/1.66 % (12488)Memory used [KB]: 7036
% 2.83/1.66 % (12488)Time elapsed: 0.065 s
% 2.83/1.66 % (12488)Instructions burned: 105 (million)
% 2.83/1.66 % (12488)------------------------------
% 2.83/1.66 % (12488)------------------------------
% 2.83/1.66 % (12472)Success in time 0.089 s
%------------------------------------------------------------------------------