%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM211^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n021.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:41:58 AM UTC 2026
% Result : Theorem 2.97s 1.65s
% Output : Refutation 2.97s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : COM211^1 : TPTP v9.2.1. Released v7.0.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n021.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 04:16:38 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.47 ---- Original DTF file ---
% 0.32/0.47 thf(spec,logic,$$dhol).
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 % File : COM211^1 : TPTP v9.2.1. Released v7.0.0.
% 0.32/0.47 % Domain : Computing Theory
% 0.32/0.47 % Problem : Parallel extension to grammars and languages 144
% 0.32/0.47 % Version : [Bla16] axioms : Especial.
% 0.32/0.47 % English :
% 0.32/0.47
% 0.32/0.47 % Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% 0.32/0.47 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.32/0.47 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.32/0.47 % Source : [Bla16]
% 0.32/0.47 % Names : parallel__144.p [Bla16]
% 0.32/0.47
% 0.32/0.47 % Status : Theorem
% 0.32/0.47 % Rating : 0.67 v9.0.0, 0.33 v8.1.0, 0.25 v7.5.0, 0.00 v7.2.0, 0.25 v7.1.0
% 0.32/0.47 % Syntax : Number of formulae : 339 ( 124 unt; 66 typ; 0 def)
% 0.32/0.47 % Number of atoms : 641 ( 246 equ; 0 cnn)
% 0.32/0.47 % Maximal formula atoms : 13 ( 2 avg)
% 0.32/0.47 % Number of connectives : 4584 ( 51 ~; 3 |; 23 &;4181 @)
% 0.32/0.47 % ( 0 <=>; 326 =>; 0 <=; 0 <~>)
% 0.32/0.47 % Maximal formula depth : 28 ( 9 avg)
% 0.32/0.47 % Number of types : 4 ( 3 usr)
% 0.32/0.47 % Number of type conns : 490 ( 490 >; 0 *; 0 +; 0 <<)
% 0.32/0.47 % Number of symbols : 66 ( 63 usr; 5 con; 0-9 aty)
% 0.32/0.47 % Number of variables : 1387 ( 59 ^;1218 !; 17 ?;1387 :)
% 0.32/0.47 % ( 93 !>; 0 ?*; 0 @-; 0 @+)
% 0.32/0.47 % SPC : TH1_THM_EQU_NAR
% 0.32/0.47
% 0.32/0.47 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.32/0.47 % 2016-07-13 14:39:25.594
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 %----Could-be-implicit typings (7)
% 0.32/0.47 thf(ty_t_Product__Type_Oprod,type,
% 0.32/0.47 product_prod: $tType > $tType > $tType ).
% 0.32/0.47
% 0.32/0.47 thf(ty_t_Sum__Type_Osum,type,
% 0.32/0.47 sum_sum: $tType > $tType > $tType ).
% 0.32/0.47
% 0.32/0.47 thf(ty_t_DTree_Odtree,type,
% 0.32/0.47 dtree: $tType ).
% 0.32/0.47
% 0.32/0.47 thf(ty_t_Set_Oset,type,
% 0.32/0.47 set: $tType > $tType ).
% 0.32/0.47
% 0.32/0.47 thf(ty_t_DTree_OT,type,
% 0.32/0.47 t: $tType ).
% 0.32/0.47
% 0.32/0.47 thf(ty_t_DTree_ON,type,
% 0.32/0.47 n: $tType ).
% 0.32/0.47
% 0.32/0.47 thf(ty_t_itself,type,
% 0.32/0.47 itself: $tType > $tType ).
% 0.32/0.47
% 0.32/0.47 %----Explicit typings (59)
% 0.32/0.47 thf(sy_cl_HOL_Otype,type,
% 0.32/0.47 type:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.32/0.47 ord:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.32/0.47 order:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.32/0.47 linorder:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.32/0.47 preorder:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Oorder__top,type,
% 0.32/0.47 order_top:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 0.32/0.47 boolean_algebra:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.32/0.47 ordered_ab_group_add:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.32/0.47 bNF_Ca1785829860lChain:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Ocsquare,type,
% 0.32/0.47 bNF_csquare:
% 0.32/0.47 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( set @ A ) > ( B > C ) > ( D > C ) > ( A > B ) > ( A > D ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_OfstOp,type,
% 0.32/0.47 bNF_fstOp:
% 0.32/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_Opick__middlep,type,
% 0.32/0.47 bNF_pick_middlep:
% 0.32/0.47 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Def_OsndOp,type,
% 0.32/0.47 bNF_sndOp:
% 0.32/0.47 !>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 0.32/0.47 bNF_Greatest_image2:
% 0.32/0.47 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_DTree_ONode,type,
% 0.32/0.47 node: n > ( set @ ( sum_sum @ t @ dtree ) ) > dtree ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_DTree_Ocont,type,
% 0.32/0.47 cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_DTree_Ocorec,type,
% 0.32/0.47 corec:
% 0.32/0.47 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ) ) > A > dtree ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_DTree_Odtree_Oroot,type,
% 0.32/0.47 root: dtree > n ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_DTree_Ounfold,type,
% 0.32/0.47 unfold:
% 0.32/0.47 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oinj__on,type,
% 0.32/0.47 inj_on:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Oswap,type,
% 0.32/0.47 swap:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun_Othe__inv__into,type,
% 0.32/0.47 the_inv_into:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Fun__Def_Oin__rel,type,
% 0.32/0.47 fun_in_rel:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > A > B > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.32/0.47 uminus_uminus:
% 0.32/0.47 !>[A: $tType] : ( A > A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.32/0.47 ord_less_eq:
% 0.32/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.32/0.47 top_top:
% 0.32/0.47 !>[A: $tType] : A ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_ONplus,type,
% 0.32/0.47 parall1518086719_Nplus: n > n > n ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar,type,
% 0.32/0.47 parall1899940088le_par: ( product_prod @ dtree @ dtree ) > dtree ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c,type,
% 0.32/0.47 parall1914194347_par_c: ( product_prod @ dtree @ dtree ) > ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r,type,
% 0.32/0.47 parall1914194362_par_r: ( product_prod @ dtree @ dtree ) > n ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r__rel,type,
% 0.32/0.47 parall556292031_r_rel: ( product_prod @ dtree @ dtree ) > ( product_prod @ dtree @ dtree ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Product__Type_OPair,type,
% 0.32/0.47 product_Pair:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Product__Type_Oapfst,type,
% 0.32/0.47 product_apfst:
% 0.32/0.47 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Product__Type_Oapsnd,type,
% 0.32/0.47 product_apsnd:
% 0.32/0.47 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Product__Type_Ocurry,type,
% 0.32/0.47 product_curry:
% 0.32/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.32/0.47 produc2004651681e_prod:
% 0.32/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Product__Type_Omap__prod,type,
% 0.32/0.47 product_map_prod:
% 0.32/0.47 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
% 0.32/0.47 product_rec_bool:
% 0.32/0.47 !>[T: $tType] : ( T > T > $o > T ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.32/0.47 product_rec_prod:
% 0.32/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.32/0.47 product_case_prod:
% 0.32/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.32/0.47 product_fst:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.32/0.47 product_snd:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.32/0.47 product_swap:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Product__Type_Oscomp,type,
% 0.32/0.47 product_scomp:
% 0.32/0.47 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Pure_Otype,type,
% 0.32/0.47 type2:
% 0.32/0.47 !>[A: $tType] : ( itself @ A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Relation_OPowp,type,
% 0.32/0.47 powp:
% 0.32/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Relation_Oinv__image,type,
% 0.32/0.47 inv_image:
% 0.32/0.47 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Relation_Orelcompp,type,
% 0.32/0.47 relcompp:
% 0.32/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Set_OCollect,type,
% 0.32/0.47 collect:
% 0.32/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Set_Oimage,type,
% 0.32/0.47 image:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Sum__Type_OInl,type,
% 0.32/0.47 sum_Inl:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Wellfounded_Oaccp,type,
% 0.32/0.47 accp:
% 0.32/0.47 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Wellfounded_Olex__prod,type,
% 0.32/0.47 lex_prod:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Wellfounded_Owf,type,
% 0.32/0.47 wf:
% 0.32/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Wfrec_Osame__fst,type,
% 0.32/0.47 same_fst:
% 0.32/0.47 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_member,type,
% 0.32/0.47 member:
% 0.32/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_v_tr1,type,
% 0.32/0.47 tr1: dtree ).
% 0.32/0.47
% 0.32/0.47 thf(sy_v_tr2,type,
% 0.32/0.47 tr2: dtree ).
% 0.32/0.47
% 0.32/0.47 thf(sy_v_tr3,type,
% 0.32/0.47 tr3: dtree ).
% 0.32/0.47
% 0.32/0.47 %----Relevant facts (256)
% 0.32/0.47 thf(fact_0__092_060open_062_092_060And_062trB_AtrA_O_A_092_060exists_062tr1_Atr2_Atr3_O_AtrA_A_061_A_Itr1_A_092_060parallel_062_Atr2_J_A_092_060parallel_062_Atr3_A_092_060and_062_AtrB_A_061_Atr1_A_092_060parallel_062_Atr2_A_092_060parallel_062_Atr3_A_092_060Longrightarrow_062_AtrA_A_061_AtrB_092_060close_062,axiom,
% 0.32/0.47 ! [TrA: dtree,TrB: dtree] :
% 0.32/0.47 ( ? [Tr1: dtree,Tr2: dtree,Tr3: dtree] :
% 0.32/0.47 ( ( TrA
% 0.32/0.47 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) @ Tr3 ) ) )
% 0.32/0.47 & ( TrB
% 0.32/0.47 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr1 @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr2 @ Tr3 ) ) ) ) ) )
% 0.32/0.47 => ( TrA = TrB ) ) ).
% 0.32/0.47
% 0.32/0.47 % \<open>\<And>trB trA. \<exists>tr1 tr2 tr3. trA = (tr1 \<parallel> tr2) \<parallel> tr3 \<and> trB = tr1 \<parallel> tr2 \<parallel> tr3 \<Longrightarrow> trA = trB\<close>
% 0.32/0.47 thf(fact_1_par__com,axiom,
% 0.32/0.47 ! [Tr12: dtree,Tr22: dtree] :
% 0.32/0.47 ( ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) )
% 0.32/0.47 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr22 @ Tr12 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % par_com
% 0.32/0.47 thf(fact_2_par__r_Ocases,axiom,
% 0.32/0.47 ! [X: product_prod @ dtree @ dtree] :
% 0.32/0.47 ~ ! [Tr13: dtree,Tr23: dtree] :
% 0.32/0.47 ( X
% 0.32/0.47 != ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) ) ).
% 0.32/0.47
% 0.32/0.47 % par_r.cases
% 0.32/0.47 thf(fact_3_par__r_Oinduct,axiom,
% 0.32/0.47 ! [P: ( product_prod @ dtree @ dtree ) > $o,A0: product_prod @ dtree @ dtree] :
% 0.32/0.47 ( ! [Tr13: dtree,Tr23: dtree] : ( P @ ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) )
% 0.32/0.47 => ( P @ A0 ) ) ).
% 0.32/0.47
% 0.32/0.47 % par_r.induct
% 0.32/0.47 thf(fact_4_prod_Oinject,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 0.32/0.47 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 0.32/0.47 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.32/0.47 = ( ( X1 = Y1 )
% 0.32/0.47 & ( X2 = Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod.inject
% 0.32/0.47 thf(fact_5_old_Oprod_Oinject,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.32/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.32/0.47 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.32/0.47 = ( ( A2 = A3 )
% 0.32/0.47 & ( B2 = B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % old.prod.inject
% 0.32/0.47 thf(fact_6_surj__pair,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.32/0.47 ? [X3: A,Y: B] :
% 0.32/0.47 ( P2
% 0.32/0.47 = ( product_Pair @ A @ B @ X3 @ Y ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_pair
% 0.32/0.47 thf(fact_7_prod__cases,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.32/0.47 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.32/0.47 => ( P @ P2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod_cases
% 0.32/0.47 thf(fact_8_Pair__inject,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 0.32/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.32/0.47 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.32/0.47 => ~ ( ( A2 = A3 )
% 0.32/0.47 => ( B2 != B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Pair_inject
% 0.32/0.47 thf(fact_9_prod__cases3,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.32/0.47 ~ ! [A4: A,B4: B,C2: C] :
% 0.32/0.47 ( Y3
% 0.32/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod_cases3
% 0.32/0.47 thf(fact_10_prod__cases4,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.32/0.47 ~ ! [A4: A,B4: B,C2: C,D2: D] :
% 0.32/0.47 ( Y3
% 0.32/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod_cases4
% 0.32/0.47 thf(fact_11_prod__cases5,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.32/0.47 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
% 0.32/0.47 ( Y3
% 0.32/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod_cases5
% 0.32/0.47 thf(fact_12_prod__cases6,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.32/0.47 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
% 0.32/0.47 ( Y3
% 0.32/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod_cases6
% 0.32/0.47 thf(fact_13_prod__cases7,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.32/0.47 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
% 0.32/0.47 ( Y3
% 0.32/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod_cases7
% 0.32/0.47 thf(fact_14_prod__induct3,axiom,
% 0.32/0.47 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.32/0.47 ( ! [A4: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 0.32/0.47 => ( P @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod_induct3
% 0.32/0.47 thf(fact_15_prod__induct4,axiom,
% 0.32/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.32/0.47 ( ! [A4: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 0.32/0.47 => ( P @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod_induct4
% 0.32/0.47 thf(fact_16_old_Oprod_Oinducts,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.32/0.47 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.32/0.47 => ( P @ Prod ) ) ).
% 0.32/0.47
% 0.32/0.47 % old.prod.inducts
% 0.32/0.47 thf(fact_17_old_Oprod_Oexhaust,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 0.32/0.47 ~ ! [A4: A,B4: B] :
% 0.32/0.47 ( Y3
% 0.32/0.47 != ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% 0.32/0.47
% 0.32/0.47 % old.prod.exhaust
% 0.32/0.47 thf(fact_18_prod__induct7,axiom,
% 0.32/0.47 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.32/0.47 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 0.32/0.47 => ( P @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod_induct7
% 0.32/0.47 thf(fact_19_prod__induct6,axiom,
% 0.32/0.47 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.32/0.47 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 0.32/0.47 => ( P @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod_induct6
% 0.32/0.47 thf(fact_20_prod__induct5,axiom,
% 0.32/0.47 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.32/0.47 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.32/0.47 => ( P @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod_induct5
% 0.32/0.47 thf(fact_21_old_Oprod_Orec,axiom,
% 0.32/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.32/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.32/0.47 = ( F1 @ A2 @ B2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % old.prod.rec
% 0.32/0.47 thf(fact_22_internal__case__prod__conv,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 0.32/0.47 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.32/0.47 = ( C3 @ A2 @ B2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % internal_case_prod_conv
% 0.32/0.47 thf(fact_23_ssubst__Pair__rhs,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,R: A,S: B,R2: set @ ( product_prod @ A @ B ),S2: B] :
% 0.32/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R2 )
% 0.32/0.47 => ( ( S2 = S )
% 0.32/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ssubst_Pair_rhs
% 0.32/0.47 thf(fact_24_curry__conv,axiom,
% 0.32/0.47 ! [A: $tType,C: $tType,B: $tType] :
% 0.32/0.47 ( ( product_curry @ B @ C @ A )
% 0.32/0.47 = ( ^ [F3: ( product_prod @ B @ C ) > A,A5: B,B5: C] : ( F3 @ ( product_Pair @ B @ C @ A5 @ B5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % curry_conv
% 0.32/0.47 thf(fact_25_curryI,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.32/0.47 ( ( F4 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.32/0.47 => ( product_curry @ A @ B @ $o @ F4 @ A2 @ B2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % curryI
% 0.32/0.47 thf(fact_26_swap__simp,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,X: B,Y3: A] :
% 0.32/0.47 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) )
% 0.32/0.47 = ( product_Pair @ A @ B @ Y3 @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % swap_simp
% 0.32/0.47 thf(fact_27_apsnd__conv,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,C: $tType,F4: C > B,X: A,Y3: C] :
% 0.32/0.47 ( ( product_apsnd @ C @ B @ A @ F4 @ ( product_Pair @ A @ C @ X @ Y3 ) )
% 0.32/0.47 = ( product_Pair @ A @ B @ X @ ( F4 @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % apsnd_conv
% 0.32/0.47 thf(fact_28_old_Obool_Osimps_I6_J,axiom,
% 0.32/0.47 ! [T: $tType,F1: T,F22: T] :
% 0.32/0.47 ( ( product_rec_bool @ T @ F1 @ F22 @ $false )
% 0.32/0.47 = F22 ) ).
% 0.32/0.47
% 0.32/0.47 % old.bool.simps(6)
% 0.32/0.47 thf(fact_29_old_Obool_Osimps_I5_J,axiom,
% 0.32/0.47 ! [T: $tType,F1: T,F22: T] :
% 0.32/0.47 ( ( product_rec_bool @ T @ F1 @ F22 @ $true )
% 0.32/0.47 = F1 ) ).
% 0.32/0.47
% 0.32/0.47 % old.bool.simps(5)
% 0.32/0.47 thf(fact_30_apfst__conv,axiom,
% 0.32/0.47 ! [C: $tType,A: $tType,B: $tType,F4: C > A,X: C,Y3: B] :
% 0.32/0.47 ( ( product_apfst @ C @ A @ B @ F4 @ ( product_Pair @ C @ B @ X @ Y3 ) )
% 0.32/0.47 = ( product_Pair @ A @ B @ ( F4 @ X ) @ Y3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % apfst_conv
% 0.32/0.47 thf(fact_31_swap__swap,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.32/0.47 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 0.32/0.47 = P2 ) ).
% 0.32/0.47
% 0.32/0.47 % swap_swap
% 0.32/0.47 thf(fact_32_apsnd__apfst__commute,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,F4: C > B,G3: D > A,P2: product_prod @ D @ C] :
% 0.32/0.47 ( ( product_apsnd @ C @ B @ A @ F4 @ ( product_apfst @ D @ A @ C @ G3 @ P2 ) )
% 0.32/0.47 = ( product_apfst @ D @ A @ B @ G3 @ ( product_apsnd @ C @ B @ D @ F4 @ P2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % apsnd_apfst_commute
% 0.32/0.47 thf(fact_33_curryD,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.32/0.47 ( ( product_curry @ A @ B @ $o @ F4 @ A2 @ B2 )
% 0.32/0.47 => ( F4 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % curryD
% 0.32/0.47 thf(fact_34_curryE,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.32/0.47 ( ( product_curry @ A @ B @ $o @ F4 @ A2 @ B2 )
% 0.32/0.47 => ( F4 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % curryE
% 0.32/0.47 thf(fact_35_pair__in__swap__image,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,Y3: A,X: B,A6: set @ ( product_prod @ B @ A )] :
% 0.32/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A6 ) )
% 0.32/0.47 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) @ A6 ) ) ).
% 0.32/0.47
% 0.32/0.47 % pair_in_swap_image
% 0.32/0.47 thf(fact_36_internal__case__prod__def,axiom,
% 0.32/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.32/0.47 ( ( produc2004651681e_prod @ A @ B @ C )
% 0.32/0.47 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.32/0.47
% 0.32/0.47 % internal_case_prod_def
% 0.32/0.47 thf(fact_37_case__prod__curry,axiom,
% 0.32/0.47 ! [C: $tType,B: $tType,A: $tType,F4: ( product_prod @ A @ B ) > C] :
% 0.32/0.47 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F4 ) )
% 0.32/0.47 = F4 ) ).
% 0.32/0.47
% 0.32/0.47 % case_prod_curry
% 0.32/0.47 thf(fact_38_curry__case__prod,axiom,
% 0.32/0.47 ! [C: $tType,B: $tType,A: $tType,F4: A > B > C] :
% 0.32/0.47 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F4 ) )
% 0.32/0.47 = F4 ) ).
% 0.32/0.47
% 0.32/0.47 % curry_case_prod
% 0.32/0.47 thf(fact_39_Pair__scomp,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,C: $tType,X: C,F4: C > A > B] :
% 0.32/0.47 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F4 )
% 0.32/0.47 = ( F4 @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % Pair_scomp
% 0.32/0.47 thf(fact_40_same__fstI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,P: A > $o,X: A,Y4: B,Y3: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.32/0.47 ( ( P @ X )
% 0.32/0.47 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y4 @ Y3 ) @ ( R2 @ X ) )
% 0.32/0.47 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( product_Pair @ A @ B @ X @ Y3 ) ) @ ( same_fst @ A @ B @ P @ R2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % same_fstI
% 0.32/0.47 thf(fact_41_root__par,axiom,
% 0.32/0.47 ! [Tr12: dtree,Tr22: dtree] :
% 0.32/0.47 ( ( root @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) )
% 0.32/0.47 = ( parall1518086719_Nplus @ ( root @ Tr12 ) @ ( root @ Tr22 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % root_par
% 0.32/0.47 thf(fact_42_in__lex__prod,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
% 0.32/0.47 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ A3 @ B3 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
% 0.32/0.47 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A3 ) @ R )
% 0.32/0.47 | ( ( A2 = A3 )
% 0.32/0.47 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B3 ) @ S ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % in_lex_prod
% 0.32/0.47 thf(fact_43_apfst__apsnd,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,D: $tType,C: $tType,F4: C > A,G3: D > B,X: product_prod @ C @ D] :
% 0.32/0.47 ( ( product_apfst @ C @ A @ B @ F4 @ ( product_apsnd @ D @ B @ C @ G3 @ X ) )
% 0.32/0.47 = ( product_Pair @ A @ B @ ( F4 @ ( product_fst @ C @ D @ X ) ) @ ( G3 @ ( product_snd @ C @ D @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % apfst_apsnd
% 0.32/0.47 thf(fact_44_scomp__apply,axiom,
% 0.32/0.47 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.32/0.47 ( ( product_scomp @ B @ C @ D @ A )
% 0.32/0.47 = ( ^ [F3: B > ( product_prod @ C @ D ),G4: C > D > A,X4: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F3 @ X4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % scomp_apply
% 0.32/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.32/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.32/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.32/0.47 = ( P @ A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % mem_Collect_eq
% 0.32/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A] :
% 0.32/0.47 ( ( collect @ A
% 0.32/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ A6 ) )
% 0.32/0.47 = A6 ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mem_eq
% 0.32/0.47 thf(fact_47_Collect__cong,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( P @ X3 )
% 0.32/0.47 = ( Q @ X3 ) )
% 0.32/0.47 => ( ( collect @ A @ P )
% 0.32/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_cong
% 0.32/0.47 thf(fact_48_ext,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F4: A > B,G3: A > B] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( F4 @ X3 )
% 0.32/0.47 = ( G3 @ X3 ) )
% 0.32/0.47 => ( F4 = G3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % ext
% 0.32/0.47 thf(fact_49_apfst__eq__conv,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,C: $tType,F4: C > A,X: product_prod @ C @ B,G3: C > A] :
% 0.32/0.47 ( ( ( product_apfst @ C @ A @ B @ F4 @ X )
% 0.32/0.47 = ( product_apfst @ C @ A @ B @ G3 @ X ) )
% 0.32/0.47 = ( ( F4 @ ( product_fst @ C @ B @ X ) )
% 0.32/0.47 = ( G3 @ ( product_fst @ C @ B @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % apfst_eq_conv
% 0.32/0.47 thf(fact_50_fst__apfst,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,C: $tType,F4: C > A,X: product_prod @ C @ B] :
% 0.32/0.47 ( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F4 @ X ) )
% 0.32/0.47 = ( F4 @ ( product_fst @ C @ B @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % fst_apfst
% 0.32/0.47 thf(fact_51_snd__apfst,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,C: $tType,F4: C > B,X: product_prod @ C @ A] :
% 0.32/0.47 ( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F4 @ X ) )
% 0.32/0.47 = ( product_snd @ C @ A @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % snd_apfst
% 0.32/0.47 thf(fact_52_fst__apsnd,axiom,
% 0.32/0.47 ! [B: $tType,C: $tType,A: $tType,F4: C > B,X: product_prod @ A @ C] :
% 0.32/0.47 ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F4 @ X ) )
% 0.32/0.47 = ( product_fst @ A @ C @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % fst_apsnd
% 0.32/0.47 thf(fact_53_apsnd__eq__conv,axiom,
% 0.32/0.47 ! [B: $tType,C: $tType,A: $tType,F4: C > B,X: product_prod @ A @ C,G3: C > B] :
% 0.32/0.47 ( ( ( product_apsnd @ C @ B @ A @ F4 @ X )
% 0.32/0.47 = ( product_apsnd @ C @ B @ A @ G3 @ X ) )
% 0.32/0.47 = ( ( F4 @ ( product_snd @ A @ C @ X ) )
% 0.32/0.47 = ( G3 @ ( product_snd @ A @ C @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % apsnd_eq_conv
% 0.32/0.47 thf(fact_54_snd__apsnd,axiom,
% 0.32/0.47 ! [A: $tType,C: $tType,B: $tType,F4: C > A,X: product_prod @ B @ C] :
% 0.32/0.47 ( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F4 @ X ) )
% 0.32/0.47 = ( F4 @ ( product_snd @ B @ C @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % snd_apsnd
% 0.32/0.47 thf(fact_55_prod_Ocollapse,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.32/0.47 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.32/0.47 = Prod ) ).
% 0.32/0.47
% 0.32/0.47 % prod.collapse
% 0.32/0.47 thf(fact_56_snd__swap,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 0.32/0.47 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 0.32/0.47 = ( product_fst @ A @ B @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % snd_swap
% 0.32/0.47 thf(fact_57_fst__swap,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 0.32/0.47 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 0.32/0.47 = ( product_snd @ B @ A @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % fst_swap
% 0.32/0.47 thf(fact_58_apsnd__apfst,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,F4: C > B,G3: D > A,X: product_prod @ D @ C] :
% 0.32/0.47 ( ( product_apsnd @ C @ B @ A @ F4 @ ( product_apfst @ D @ A @ C @ G3 @ X ) )
% 0.32/0.47 = ( product_Pair @ A @ B @ ( G3 @ ( product_fst @ D @ C @ X ) ) @ ( F4 @ ( product_snd @ D @ C @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % apsnd_apfst
% 0.32/0.47 thf(fact_59_Nplus__comm,axiom,
% 0.32/0.47 ( parall1518086719_Nplus
% 0.32/0.47 = ( ^ [A5: n,B5: n] : ( parall1518086719_Nplus @ B5 @ A5 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Nplus_comm
% 0.32/0.47 thf(fact_60_Nplus__assoc,axiom,
% 0.32/0.47 ! [A2: n,B2: n,C3: n] :
% 0.32/0.47 ( ( parall1518086719_Nplus @ ( parall1518086719_Nplus @ A2 @ B2 ) @ C3 )
% 0.32/0.47 = ( parall1518086719_Nplus @ A2 @ ( parall1518086719_Nplus @ B2 @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Nplus_assoc
% 0.32/0.47 thf(fact_61_surjective__pairing,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.32/0.47 ( T2
% 0.32/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjective_pairing
% 0.32/0.47 thf(fact_62_prod_Osplit__sel__asm,axiom,
% 0.32/0.47 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F4: A > B > C,Prod: product_prod @ A @ B] :
% 0.32/0.47 ( ( P @ ( product_case_prod @ A @ B @ C @ F4 @ Prod ) )
% 0.32/0.47 = ( ~ ( ( Prod
% 0.32/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.32/0.47 & ~ ( P @ ( F4 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod.split_sel_asm
% 0.32/0.47 thf(fact_63_Product__Type_OCollect__case__prodD,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: A > B > $o] :
% 0.32/0.47 ( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) )
% 0.32/0.47 => ( A6 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Product_Type.Collect_case_prodD
% 0.32/0.47 thf(fact_64_prod_Oexhaust__sel,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.32/0.47 ( Prod
% 0.32/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod.exhaust_sel
% 0.32/0.47 thf(fact_65_prod_Ocase__eq__if,axiom,
% 0.32/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.32/0.47 ( ( product_case_prod @ A @ B @ C )
% 0.32/0.47 = ( ^ [F3: A > B > C,Prod2: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ Prod2 ) @ ( product_snd @ A @ B @ Prod2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod.case_eq_if
% 0.32/0.47 thf(fact_66_prod_Osplit__sel,axiom,
% 0.32/0.47 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F4: A > B > C,Prod: product_prod @ A @ B] :
% 0.32/0.47 ( ( P @ ( product_case_prod @ A @ B @ C @ F4 @ Prod ) )
% 0.32/0.47 = ( ( Prod
% 0.32/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.32/0.47 => ( P @ ( F4 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod.split_sel
% 0.32/0.47 thf(fact_67_case__prod__beta,axiom,
% 0.32/0.47 ! [A: $tType,C: $tType,B: $tType] :
% 0.32/0.47 ( ( product_case_prod @ B @ C @ A )
% 0.32/0.47 = ( ^ [F3: B > C > A,P3: product_prod @ B @ C] : ( F3 @ ( product_fst @ B @ C @ P3 ) @ ( product_snd @ B @ C @ P3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % case_prod_beta
% 0.32/0.47 thf(fact_68_prod__eq__iff,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ^ [Y5: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y5 = Z ) )
% 0.32/0.47 = ( ^ [S3: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 0.32/0.47 ( ( ( product_fst @ A @ B @ S3 )
% 0.32/0.47 = ( product_fst @ A @ B @ T3 ) )
% 0.32/0.47 & ( ( product_snd @ A @ B @ S3 )
% 0.32/0.47 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod_eq_iff
% 0.32/0.47 thf(fact_69_prod_Oexpand,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod3: product_prod @ A @ B] :
% 0.32/0.47 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.32/0.47 = ( product_fst @ A @ B @ Prod3 ) )
% 0.32/0.47 & ( ( product_snd @ A @ B @ Prod )
% 0.32/0.47 = ( product_snd @ A @ B @ Prod3 ) ) )
% 0.32/0.47 => ( Prod = Prod3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod.expand
% 0.32/0.47 thf(fact_70_scomp__def,axiom,
% 0.32/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 0.32/0.47 ( ( product_scomp @ A @ B @ C @ D )
% 0.32/0.47 = ( ^ [F3: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F3 @ X4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % scomp_def
% 0.32/0.47 thf(fact_71_prod__eqI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 0.32/0.47 ( ( ( product_fst @ A @ B @ P2 )
% 0.32/0.47 = ( product_fst @ A @ B @ Q2 ) )
% 0.32/0.47 => ( ( ( product_snd @ A @ B @ P2 )
% 0.32/0.47 = ( product_snd @ A @ B @ Q2 ) )
% 0.32/0.47 => ( P2 = Q2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod_eqI
% 0.32/0.47 thf(fact_72_prod_Oswap__def,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( product_swap @ A @ B )
% 0.32/0.47 = ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod.swap_def
% 0.32/0.47 thf(fact_73_snd__conv,axiom,
% 0.32/0.47 ! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
% 0.32/0.47 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
% 0.32/0.47 = X2 ) ).
% 0.32/0.47
% 0.32/0.47 % snd_conv
% 0.32/0.47 thf(fact_74_snd__eqD,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,X: B,Y3: A,A2: A] :
% 0.32/0.47 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) )
% 0.32/0.47 = A2 )
% 0.32/0.47 => ( Y3 = A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % snd_eqD
% 0.32/0.47 thf(fact_75_fst__conv,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,X1: A,X2: B] :
% 0.32/0.47 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 0.32/0.47 = X1 ) ).
% 0.32/0.47
% 0.32/0.47 % fst_conv
% 0.32/0.47 thf(fact_76_fst__eqD,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,X: A,Y3: B,A2: A] :
% 0.32/0.47 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y3 ) )
% 0.32/0.47 = A2 )
% 0.32/0.47 => ( X = A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % fst_eqD
% 0.32/0.47 thf(fact_77_mem__case__prodE,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,C: $tType,Z2: A,C3: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 0.32/0.47 ( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P2 ) )
% 0.32/0.47 => ~ ! [X3: B,Y: C] :
% 0.32/0.47 ( ( P2
% 0.32/0.47 = ( product_Pair @ B @ C @ X3 @ Y ) )
% 0.32/0.47 => ~ ( member @ A @ Z2 @ ( C3 @ X3 @ Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % mem_case_prodE
% 0.32/0.47 thf(fact_78_old_Oprod_Ocase,axiom,
% 0.32/0.47 ! [A: $tType,C: $tType,B: $tType,F4: A > B > C,X1: A,X2: B] :
% 0.32/0.47 ( ( product_case_prod @ A @ B @ C @ F4 @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 0.32/0.47 = ( F4 @ X1 @ X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % old.prod.case
% 0.32/0.47 thf(fact_79_scomp__Pair,axiom,
% 0.32/0.47 ! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
% 0.32/0.47 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
% 0.32/0.47 = X ) ).
% 0.32/0.47
% 0.32/0.47 % scomp_Pair
% 0.32/0.47 thf(fact_80_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,P: A > B > $o,X: A,Y3: B,A2: product_prod @ A @ B] :
% 0.32/0.47 ( ( P @ X @ Y3 )
% 0.32/0.47 => ( ( A2
% 0.32/0.47 = ( product_Pair @ A @ B @ X @ Y3 ) )
% 0.32/0.47 => ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % BNF_Greatest_Fixpoint.subst_Pair
% 0.32/0.47 thf(fact_81_conjI__realizer,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
% 0.32/0.47 ( ( P @ P2 )
% 0.32/0.47 => ( ( Q @ Q2 )
% 0.32/0.47 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
% 0.32/0.47 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % conjI_realizer
% 0.32/0.47 thf(fact_82_exI__realizer,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,P: A > B > $o,Y3: A,X: B] :
% 0.32/0.47 ( ( P @ Y3 @ X )
% 0.32/0.47 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % exI_realizer
% 0.32/0.47 thf(fact_83_par__r_Osimps,axiom,
% 0.32/0.47 ! [Tr12: dtree,Tr22: dtree] :
% 0.32/0.47 ( ( parall1914194362_par_r @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) )
% 0.32/0.47 = ( parall1518086719_Nplus @ ( root @ Tr12 ) @ ( root @ Tr22 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % par_r.simps
% 0.32/0.47 thf(fact_84_par__r_Oelims,axiom,
% 0.32/0.47 ! [X: product_prod @ dtree @ dtree,Y3: n] :
% 0.32/0.47 ( ( ( parall1914194362_par_r @ X )
% 0.32/0.47 = Y3 )
% 0.32/0.47 => ~ ! [Tr13: dtree,Tr23: dtree] :
% 0.32/0.47 ( ( X
% 0.32/0.47 = ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) )
% 0.32/0.47 => ( Y3
% 0.32/0.47 != ( parall1518086719_Nplus @ ( root @ Tr13 ) @ ( root @ Tr23 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % par_r.elims
% 0.32/0.47 thf(fact_85_image__eqI,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,B2: A,F4: B > A,X: B,A6: set @ B] :
% 0.32/0.47 ( ( B2
% 0.32/0.47 = ( F4 @ X ) )
% 0.32/0.47 => ( ( member @ B @ X @ A6 )
% 0.32/0.47 => ( member @ A @ B2 @ ( image @ B @ A @ F4 @ A6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_eqI
% 0.32/0.47 thf(fact_86_exE__realizer_H,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
% 0.32/0.47 ( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
% 0.32/0.47 => ~ ! [X3: B,Y: A] :
% 0.32/0.47 ~ ( P @ Y @ X3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % exE_realizer'
% 0.32/0.47 thf(fact_87_sndI,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y3: A,Z2: B] :
% 0.32/0.47 ( ( X
% 0.32/0.47 = ( product_Pair @ A @ B @ Y3 @ Z2 ) )
% 0.32/0.47 => ( ( product_snd @ A @ B @ X )
% 0.32/0.47 = Z2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % sndI
% 0.32/0.47 thf(fact_88_rev__image__eqI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,X: A,A6: set @ A,B2: B,F4: A > B] :
% 0.32/0.47 ( ( member @ A @ X @ A6 )
% 0.32/0.47 => ( ( B2
% 0.32/0.47 = ( F4 @ X ) )
% 0.32/0.47 => ( member @ B @ B2 @ ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rev_image_eqI
% 0.32/0.47 thf(fact_89_ball__imageD,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P: A > $o] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.32/0.47 => ( P @ X3 ) )
% 0.32/0.47 => ! [X5: B] :
% 0.32/0.47 ( ( member @ B @ X5 @ A6 )
% 0.32/0.47 => ( P @ ( F4 @ X5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ball_imageD
% 0.32/0.47 thf(fact_90_image__cong,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F4: A > B,G3: A > B] :
% 0.32/0.47 ( ( M = N )
% 0.32/0.47 => ( ! [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ N )
% 0.32/0.47 => ( ( F4 @ X3 )
% 0.32/0.47 = ( G3 @ X3 ) ) )
% 0.32/0.47 => ( ( image @ A @ B @ F4 @ M )
% 0.32/0.47 = ( image @ A @ B @ G3 @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_cong
% 0.32/0.47 thf(fact_91_bex__imageD,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P: A > $o] :
% 0.32/0.47 ( ? [X5: A] :
% 0.32/0.47 ( ( member @ A @ X5 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.32/0.47 & ( P @ X5 ) )
% 0.32/0.47 => ? [X3: B] :
% 0.32/0.47 ( ( member @ B @ X3 @ A6 )
% 0.32/0.47 & ( P @ ( F4 @ X3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % bex_imageD
% 0.32/0.47 thf(fact_92_image__iff,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,Z2: A,F4: B > A,A6: set @ B] :
% 0.32/0.47 ( ( member @ A @ Z2 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.32/0.47 = ( ? [X4: B] :
% 0.32/0.47 ( ( member @ B @ X4 @ A6 )
% 0.32/0.47 & ( Z2
% 0.32/0.47 = ( F4 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_iff
% 0.32/0.47 thf(fact_93_imageI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,X: A,A6: set @ A,F4: A > B] :
% 0.32/0.47 ( ( member @ A @ X @ A6 )
% 0.32/0.47 => ( member @ B @ ( F4 @ X ) @ ( image @ A @ B @ F4 @ A6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % imageI
% 0.32/0.47 thf(fact_94_fstI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y3: A,Z2: B] :
% 0.32/0.47 ( ( X
% 0.32/0.47 = ( product_Pair @ A @ B @ Y3 @ Z2 ) )
% 0.32/0.47 => ( ( product_fst @ A @ B @ X )
% 0.32/0.47 = Y3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % fstI
% 0.32/0.47 thf(fact_95_eq__snd__iff,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
% 0.32/0.47 ( ( B2
% 0.32/0.47 = ( product_snd @ B @ A @ P2 ) )
% 0.32/0.47 = ( ? [A5: B] :
% 0.32/0.47 ( P2
% 0.32/0.47 = ( product_Pair @ B @ A @ A5 @ B2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_snd_iff
% 0.32/0.47 thf(fact_96_eq__fst__iff,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
% 0.32/0.47 ( ( A2
% 0.32/0.47 = ( product_fst @ A @ B @ P2 ) )
% 0.32/0.47 = ( ? [B5: B] :
% 0.32/0.47 ( P2
% 0.32/0.47 = ( product_Pair @ A @ B @ A2 @ B5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_fst_iff
% 0.32/0.47 thf(fact_97_split__cong,axiom,
% 0.32/0.47 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F4: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
% 0.32/0.47 ( ! [X3: A,Y: B] :
% 0.32/0.47 ( ( ( product_Pair @ A @ B @ X3 @ Y )
% 0.32/0.47 = Q2 )
% 0.32/0.47 => ( ( F4 @ X3 @ Y )
% 0.32/0.47 = ( G3 @ X3 @ Y ) ) )
% 0.32/0.47 => ( ( P2 = Q2 )
% 0.32/0.47 => ( ( product_case_prod @ A @ B @ C @ F4 @ P2 )
% 0.32/0.47 = ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % split_cong
% 0.32/0.47 thf(fact_98_par__r_Opelims,axiom,
% 0.32/0.47 ! [X: product_prod @ dtree @ dtree,Y3: n] :
% 0.32/0.47 ( ( ( parall1914194362_par_r @ X )
% 0.32/0.47 = Y3 )
% 0.32/0.47 => ( ( accp @ ( product_prod @ dtree @ dtree ) @ parall556292031_r_rel @ X )
% 0.32/0.47 => ~ ! [Tr13: dtree,Tr23: dtree] :
% 0.32/0.47 ( ( X
% 0.32/0.47 = ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) )
% 0.32/0.47 => ( ( Y3
% 0.32/0.47 = ( parall1518086719_Nplus @ ( root @ Tr13 ) @ ( root @ Tr23 ) ) )
% 0.32/0.47 => ~ ( accp @ ( product_prod @ dtree @ dtree ) @ parall556292031_r_rel @ ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % par_r.pelims
% 0.32/0.47 thf(fact_99_Collect__split__mono__strong,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,X6: set @ A,A6: set @ ( product_prod @ A @ B ),Y6: set @ B,P: A > B > $o,Q: A > B > $o] :
% 0.32/0.47 ( ( X6
% 0.32/0.47 = ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 ) )
% 0.32/0.47 => ( ( Y6
% 0.32/0.47 = ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) )
% 0.32/0.47 => ( ! [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ X6 )
% 0.32/0.47 => ! [Xa: B] :
% 0.32/0.47 ( ( member @ B @ Xa @ Y6 )
% 0.32/0.47 => ( ( P @ X3 @ Xa )
% 0.32/0.47 => ( Q @ X3 @ Xa ) ) ) )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
% 0.32/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_split_mono_strong
% 0.32/0.47 thf(fact_100_fstOp__def,axiom,
% 0.32/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.32/0.47 ( ( bNF_fstOp @ A @ B @ C )
% 0.32/0.47 = ( ^ [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 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % fstOp_def
% 0.32/0.47 thf(fact_101_subset__antisym,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
% 0.32/0.47 => ( A6 = B6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_antisym
% 0.32/0.47 thf(fact_102_subsetI,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ A6 )
% 0.32/0.47 => ( member @ A @ X3 @ B6 ) )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% 0.32/0.47
% 0.32/0.47 % subsetI
% 0.32/0.47 thf(fact_103_Collect__mono__iff,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.32/0.47 = ( ! [X4: A] :
% 0.32/0.47 ( ( P @ X4 )
% 0.32/0.47 => ( Q @ X4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mono_iff
% 0.32/0.47 thf(fact_104_contra__subsetD,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A,B6: set @ A,C3: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.32/0.47 => ( ~ ( member @ A @ C3 @ B6 )
% 0.32/0.47 => ~ ( member @ A @ C3 @ A6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % contra_subsetD
% 0.32/0.47 thf(fact_105_set__eq__subset,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ^ [Y5: set @ A,Z: set @ A] : ( Y5 = Z ) )
% 0.32/0.47 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.32/0.47 & ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % set_eq_subset
% 0.32/0.47 thf(fact_106_subset__trans,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_trans
% 0.32/0.47 thf(fact_107_Collect__mono,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( P @ X3 )
% 0.32/0.47 => ( Q @ X3 ) )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mono
% 0.32/0.47 thf(fact_108_subset__refl,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ A6 ) ).
% 0.32/0.47
% 0.32/0.47 % subset_refl
% 0.32/0.47 thf(fact_109_rev__subsetD,axiom,
% 0.32/0.47 ! [A: $tType,C3: A,A6: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( member @ A @ C3 @ A6 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.32/0.47 => ( member @ A @ C3 @ B6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rev_subsetD
% 0.32/0.47 thf(fact_110_subset__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.32/0.47 ! [T3: A] :
% 0.32/0.47 ( ( member @ A @ T3 @ A7 )
% 0.32/0.47 => ( member @ A @ T3 @ B7 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_iff
% 0.32/0.47 thf(fact_111_set__rev__mp,axiom,
% 0.32/0.47 ! [A: $tType,X: A,A6: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( member @ A @ X @ A6 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.32/0.47 => ( member @ A @ X @ B6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % set_rev_mp
% 0.32/0.47 thf(fact_112_equalityD2,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( A6 = B6 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ).
% 0.32/0.47
% 0.32/0.47 % equalityD2
% 0.32/0.47 thf(fact_113_equalityD1,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( A6 = B6 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% 0.32/0.47
% 0.32/0.47 % equalityD1
% 0.32/0.47 thf(fact_114_subset__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.32/0.47 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.32/0.47 ! [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ A7 )
% 0.32/0.47 => ( member @ A @ X4 @ B7 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_eq
% 0.32/0.47 thf(fact_115_equalityE,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( A6 = B6 )
% 0.32/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.32/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % equalityE
% 0.32/0.47 thf(fact_116_subsetCE,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A,B6: set @ A,C3: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.32/0.47 => ( ( member @ A @ C3 @ A6 )
% 0.32/0.47 => ( member @ A @ C3 @ B6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subsetCE
% 0.32/0.47 thf(fact_117_subsetD,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A,B6: set @ A,C3: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.32/0.47 => ( ( member @ A @ C3 @ A6 )
% 0.32/0.47 => ( member @ A @ C3 @ B6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subsetD
% 0.32/0.47 thf(fact_118_in__mono,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A,B6: set @ A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.32/0.47 => ( ( member @ A @ X @ A6 )
% 0.32/0.47 => ( member @ A @ X @ B6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % in_mono
% 0.32/0.47 thf(fact_119_set__mp,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A,B6: set @ A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.32/0.47 => ( ( member @ A @ X @ A6 )
% 0.32/0.47 => ( member @ A @ X @ B6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % set_mp
% 0.32/0.47 thf(fact_120_accp__induct__rule,axiom,
% 0.32/0.47 ! [A: $tType,R: A > A > $o,A2: A,P: A > $o] :
% 0.32/0.47 ( ( accp @ A @ R @ A2 )
% 0.32/0.47 => ( ! [X3: A] :
% 0.32/0.47 ( ( accp @ A @ R @ X3 )
% 0.32/0.47 => ( ! [Y7: A] :
% 0.32/0.47 ( ( R @ Y7 @ X3 )
% 0.32/0.47 => ( P @ Y7 ) )
% 0.32/0.47 => ( P @ X3 ) ) )
% 0.32/0.47 => ( P @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % accp_induct_rule
% 0.32/0.47 thf(fact_121_not__accp__down,axiom,
% 0.32/0.47 ! [A: $tType,R2: A > A > $o,X: A] :
% 0.32/0.47 ( ~ ( accp @ A @ R2 @ X )
% 0.32/0.47 => ~ ! [Z3: A] :
% 0.32/0.47 ( ( R2 @ Z3 @ X )
% 0.32/0.47 => ( accp @ A @ R2 @ Z3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_accp_down
% 0.32/0.47 thf(fact_122_accp__downward,axiom,
% 0.32/0.47 ! [A: $tType,R: A > A > $o,B2: A,A2: A] :
% 0.32/0.47 ( ( accp @ A @ R @ B2 )
% 0.32/0.47 => ( ( R @ A2 @ B2 )
% 0.32/0.47 => ( accp @ A @ R @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % accp_downward
% 0.32/0.47 thf(fact_123_accp_Oinducts,axiom,
% 0.32/0.47 ! [A: $tType,R: A > A > $o,X: A,P: A > $o] :
% 0.32/0.47 ( ( accp @ A @ R @ X )
% 0.32/0.47 => ( ! [X3: A] :
% 0.32/0.47 ( ! [Y7: A] :
% 0.32/0.47 ( ( R @ Y7 @ X3 )
% 0.32/0.47 => ( accp @ A @ R @ Y7 ) )
% 0.32/0.47 => ( ! [Y7: A] :
% 0.32/0.47 ( ( R @ Y7 @ X3 )
% 0.32/0.47 => ( P @ Y7 ) )
% 0.32/0.47 => ( P @ X3 ) ) )
% 0.32/0.47 => ( P @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % accp.inducts
% 0.32/0.47 thf(fact_124_accp__induct,axiom,
% 0.32/0.47 ! [A: $tType,R: A > A > $o,A2: A,P: A > $o] :
% 0.32/0.47 ( ( accp @ A @ R @ A2 )
% 0.32/0.47 => ( ! [X3: A] :
% 0.32/0.47 ( ( accp @ A @ R @ X3 )
% 0.32/0.47 => ( ! [Y7: A] :
% 0.32/0.47 ( ( R @ Y7 @ X3 )
% 0.32/0.47 => ( P @ Y7 ) )
% 0.32/0.47 => ( P @ X3 ) ) )
% 0.32/0.47 => ( P @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % accp_induct
% 0.32/0.47 thf(fact_125_accp_Ointros,axiom,
% 0.32/0.47 ! [A: $tType,R: A > A > $o,X: A] :
% 0.32/0.47 ( ! [Y: A] :
% 0.32/0.47 ( ( R @ Y @ X )
% 0.32/0.47 => ( accp @ A @ R @ Y ) )
% 0.32/0.47 => ( accp @ A @ R @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % accp.intros
% 0.32/0.47 thf(fact_126_accp_Osimps,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( accp @ A )
% 0.32/0.47 = ( ^ [R3: A > A > $o,A5: A] :
% 0.32/0.47 ? [X4: A] :
% 0.32/0.47 ( ( A5 = X4 )
% 0.32/0.47 & ! [Y8: A] :
% 0.32/0.47 ( ( R3 @ Y8 @ X4 )
% 0.32/0.47 => ( accp @ A @ R3 @ Y8 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % accp.simps
% 0.32/0.47 thf(fact_127_accp_Ocases,axiom,
% 0.32/0.47 ! [A: $tType,R: A > A > $o,A2: A] :
% 0.32/0.47 ( ( accp @ A @ R @ A2 )
% 0.32/0.47 => ! [Y7: A] :
% 0.32/0.47 ( ( R @ Y7 @ A2 )
% 0.32/0.47 => ( accp @ A @ R @ Y7 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % accp.cases
% 0.32/0.47 thf(fact_128_subset__image__iff,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,B6: set @ A,F4: B > A,A6: set @ B] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.32/0.47 = ( ? [AA: set @ B] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ B ) @ AA @ A6 )
% 0.32/0.47 & ( B6
% 0.32/0.47 = ( image @ B @ A @ F4 @ AA ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_image_iff
% 0.32/0.47 thf(fact_129_image__subset__iff,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,B6: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ B6 )
% 0.32/0.47 = ( ! [X4: B] :
% 0.32/0.47 ( ( member @ B @ X4 @ A6 )
% 0.32/0.47 => ( member @ A @ ( F4 @ X4 ) @ B6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_subset_iff
% 0.32/0.47 thf(fact_130_subset__imageE,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,B6: set @ A,F4: B > A,A6: set @ B] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.32/0.47 => ~ ! [C5: set @ B] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A6 )
% 0.32/0.47 => ( B6
% 0.32/0.47 != ( image @ B @ A @ F4 @ C5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_imageE
% 0.32/0.47 thf(fact_131_image__subsetI,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,A6: set @ A,F4: A > B,B6: set @ B] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ A6 )
% 0.32/0.47 => ( member @ B @ ( F4 @ X3 ) @ B6 ) )
% 0.32/0.47 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ B6 ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_subsetI
% 0.32/0.47 thf(fact_132_image__mono,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,F4: A > B] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_mono
% 0.32/0.47 thf(fact_133_sndOp__def,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,C: $tType] :
% 0.32/0.47 ( ( bNF_sndOp @ C @ A @ B )
% 0.32/0.47 = ( ^ [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 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sndOp_def
% 0.32/0.47 thf(fact_134_order__refl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_refl
% 0.32/0.47 thf(fact_135_subrelI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.32/0.47 ( ! [X3: A,Y: B] :
% 0.32/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R )
% 0.32/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ S ) )
% 0.32/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 0.32/0.47
% 0.32/0.47 % subrelI
% 0.32/0.47 thf(fact_136_relChain__def,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.32/0.47 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.32/0.47 = ( ^ [R3: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.32/0.47 ! [I: A,J: A] :
% 0.32/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R3 )
% 0.32/0.47 => ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % relChain_def
% 0.32/0.47 thf(fact_137_Collect__case__prod__in__rel__leE,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,X6: set @ ( product_prod @ A @ B ),Y6: set @ ( product_prod @ A @ B )] :
% 0.32/0.47 ( ( 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 @ Y6 ) ) ) )
% 0.32/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ Y6 ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_case_prod_in_rel_leE
% 0.32/0.47 thf(fact_138_accp__subset,axiom,
% 0.32/0.47 ! [A: $tType,R1: A > A > $o,R22: A > A > $o] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > A > $o ) @ R1 @ R22 )
% 0.32/0.47 => ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R22 ) @ ( accp @ A @ R1 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % accp_subset
% 0.32/0.47 thf(fact_139_accp__subset__induct,axiom,
% 0.32/0.47 ! [A: $tType,D3: A > $o,R2: A > A > $o,X: A,P: A > $o] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > $o ) @ D3 @ ( accp @ A @ R2 ) )
% 0.32/0.47 => ( ! [X3: A,Z3: A] :
% 0.32/0.47 ( ( D3 @ X3 )
% 0.32/0.47 => ( ( R2 @ Z3 @ X3 )
% 0.32/0.47 => ( D3 @ Z3 ) ) )
% 0.32/0.47 => ( ( D3 @ X )
% 0.32/0.47 => ( ! [X3: A] :
% 0.32/0.47 ( ( D3 @ X3 )
% 0.32/0.47 => ( ! [Z4: A] :
% 0.32/0.47 ( ( R2 @ Z4 @ X3 )
% 0.32/0.47 => ( P @ Z4 ) )
% 0.32/0.47 => ( P @ X3 ) ) )
% 0.32/0.47 => ( P @ X ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % accp_subset_induct
% 0.32/0.47 thf(fact_140_in__rel__def,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( fun_in_rel @ A @ B )
% 0.32/0.47 = ( ^ [R4: set @ ( product_prod @ A @ B ),X4: A,Y8: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y8 ) @ R4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % in_rel_def
% 0.32/0.47 thf(fact_141_in__rel__Collect__case__prod__eq,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,X6: A > B > $o] :
% 0.32/0.47 ( ( fun_in_rel @ A @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ X6 ) ) )
% 0.32/0.47 = X6 ) ).
% 0.32/0.47
% 0.32/0.47 % in_rel_Collect_case_prod_eq
% 0.32/0.47 thf(fact_142_Collect__case__prod__mono,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,A6: A > B > $o,B6: A > B > $o] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > B > $o ) @ A6 @ B6 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_case_prod_mono
% 0.32/0.47 thf(fact_143_dual__order_Oantisym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [B2: A,A2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.antisym
% 0.32/0.47 thf(fact_144_dual__order_Otrans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [B2: A,A2: A,C3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ C3 @ B2 )
% 0.32/0.47 => ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.trans
% 0.32/0.47 thf(fact_145_linorder__wlog,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.32/0.47 ( ! [A4: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.32/0.47 => ( P @ A4 @ B4 ) )
% 0.32/0.47 => ( ! [A4: A,B4: A] :
% 0.32/0.47 ( ( P @ B4 @ A4 )
% 0.32/0.47 => ( P @ A4 @ B4 ) )
% 0.32/0.47 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_wlog
% 0.32/0.47 thf(fact_146_dual__order_Orefl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.refl
% 0.32/0.47 thf(fact_147_order__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [X: A,Y3: A,Z2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_trans
% 0.32/0.47 thf(fact_148_order__class_Oorder_Oantisym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A,B2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.32/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.antisym
% 0.32/0.47 thf(fact_149_ord__le__eq__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A,B2: A,C3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/0.47 => ( ( B2 = C3 )
% 0.32/0.47 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_trans
% 0.32/0.47 thf(fact_150_ord__eq__le__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A,B2: A,C3: A] :
% 0.32/0.47 ( ( A2 = B2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 0.32/0.47 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_trans
% 0.32/0.47 thf(fact_151_antisym__conv,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Y3: A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ Y3 @ X )
% 0.32/0.47 => ( ( ord_less_eq @ A @ X @ Y3 )
% 0.32/0.47 = ( X = Y3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv
% 0.32/0.47 thf(fact_152_le__cases3,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [X: A,Y3: A,Z2: A] :
% 0.32/0.47 ( ( ( ord_less_eq @ A @ X @ Y3 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ Y3 @ X )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Y3 @ X ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 0.32/0.47 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases3
% 0.32/0.47 thf(fact_153_order_Otrans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A,B2: A,C3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 0.32/0.47 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.trans
% 0.32/0.47 thf(fact_154_le__cases,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [X: A,Y3: A] :
% 0.32/0.47 ( ~ ( ord_less_eq @ A @ X @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases
% 0.32/0.47 thf(fact_155_eq__refl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [X: A,Y3: A] :
% 0.32/0.47 ( ( X = Y3 )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_refl
% 0.32/0.47 thf(fact_156_linear,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [X: A,Y3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.32/0.47 | ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linear
% 0.32/0.47 thf(fact_157_antisym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [X: A,Y3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Y3 @ X )
% 0.32/0.47 => ( X = Y3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym
% 0.32/0.47 thf(fact_158_eq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.47 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 0.32/0.47 = ( ^ [X4: A,Y8: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X4 @ Y8 )
% 0.32/0.47 & ( ord_less_eq @ A @ Y8 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_iff
% 0.32/0.47 thf(fact_159_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.32/0.47 & ( ord @ A @ ( type2 @ A ) ) )
% 0.32/0.47 => ! [A2: A,B2: A,F4: A > B,C3: B] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/0.47 => ( ( ( F4 @ B2 )
% 0.32/0.47 = C3 )
% 0.32/0.47 => ( ! [X3: A,Y: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( F4 @ Y ) ) )
% 0.32/0.47 => ( ord_less_eq @ B @ ( F4 @ A2 ) @ C3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_160_ord__eq__le__subst,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.32/0.47 & ( ord @ A @ ( type2 @ A ) ) )
% 0.32/0.47 => ! [A2: A,F4: B > A,B2: B,C3: B] :
% 0.32/0.47 ( ( A2
% 0.32/0.47 = ( F4 @ B2 ) )
% 0.32/0.47 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 0.32/0.47 => ( ! [X3: B,Y: B] :
% 0.32/0.47 ( ( ord_less_eq @ B @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq @ A @ ( F4 @ X3 ) @ ( F4 @ Y ) ) )
% 0.32/0.47 => ( ord_less_eq @ A @ A2 @ ( F4 @ C3 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_subst
% 0.32/0.47 thf(fact_161_order__subst2,axiom,
% 0.32/0.47 ! [A: $tType,C: $tType] :
% 0.32/0.47 ( ( ( order @ C @ ( type2 @ C ) )
% 0.32/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.32/0.47 => ! [A2: A,B2: A,F4: A > C,C3: C] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/0.47 => ( ( ord_less_eq @ C @ ( F4 @ B2 ) @ C3 )
% 0.32/0.47 => ( ! [X3: A,Y: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq @ C @ ( F4 @ X3 ) @ ( F4 @ Y ) ) )
% 0.32/0.47 => ( ord_less_eq @ C @ ( F4 @ A2 ) @ C3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst2
% 0.32/0.47 thf(fact_162_order__subst1,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( order @ B @ ( type2 @ B ) )
% 0.32/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.32/0.47 => ! [A2: A,F4: B > A,B2: B,C3: B] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ ( F4 @ B2 ) )
% 0.32/0.47 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 0.32/0.47 => ( ! [X3: B,Y: B] :
% 0.32/0.47 ( ( ord_less_eq @ B @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq @ A @ ( F4 @ X3 ) @ ( F4 @ Y ) ) )
% 0.32/0.47 => ( ord_less_eq @ A @ A2 @ ( F4 @ C3 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst1
% 0.32/0.47 thf(fact_163_le__fun__def,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.32/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.32/0.47 = ( ^ [F3: A > B,G4: A > B] :
% 0.32/0.47 ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_fun_def
% 0.32/0.47 thf(fact_164_le__funI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.32/0.47 => ! [F4: A > B,G3: A > B] :
% 0.32/0.47 ( ! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G3 @ X3 ) )
% 0.32/0.47 => ( ord_less_eq @ ( A > B ) @ F4 @ G3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_funI
% 0.32/0.47 thf(fact_165_le__funE,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.32/0.47 => ! [F4: A > B,G3: A > B,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > B ) @ F4 @ G3 )
% 0.32/0.47 => ( ord_less_eq @ B @ ( F4 @ X ) @ ( G3 @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_funE
% 0.32/0.47 thf(fact_166_le__funD,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.32/0.47 => ! [F4: A > B,G3: A > B,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > B ) @ F4 @ G3 )
% 0.32/0.47 => ( ord_less_eq @ B @ ( F4 @ X ) @ ( G3 @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_funD
% 0.32/0.47 thf(fact_167_Collect__case__prod__in__rel__leI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,X6: set @ ( product_prod @ A @ B ),Y6: set @ ( product_prod @ A @ B )] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ Y6 )
% 0.32/0.47 => ( 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 @ Y6 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_case_prod_in_rel_leI
% 0.32/0.47 thf(fact_168_csquare__fstOp__sndOp,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,C: $tType,F4: ( A > B > $o ) > ( product_prod @ A @ B ) > $o,P: 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 ) @ ( F4 @ ( relcompp @ A @ C @ B @ P @ Q ) ) ) @ ( product_snd @ A @ C ) @ ( product_fst @ C @ B ) @ ( bNF_fstOp @ A @ C @ B @ P @ Q ) @ ( bNF_sndOp @ A @ C @ B @ P @ Q ) ) ).
% 0.32/0.47
% 0.32/0.47 % csquare_fstOp_sndOp
% 0.32/0.47 thf(fact_169_image2__eqI,axiom,
% 0.32/0.47 ! [A: $tType,C: $tType,B: $tType,B2: A,F4: B > A,X: B,C3: C,G3: B > C,A6: set @ B] :
% 0.32/0.47 ( ( B2
% 0.32/0.47 = ( F4 @ X ) )
% 0.32/0.47 => ( ( C3
% 0.32/0.47 = ( G3 @ X ) )
% 0.32/0.47 => ( ( member @ B @ X @ A6 )
% 0.32/0.47 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F4 @ G3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image2_eqI
% 0.32/0.47 thf(fact_170_surj__swap,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.32/0.47 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_swap
% 0.32/0.47 thf(fact_171_par__def,axiom,
% 0.32/0.47 ( parall1899940088le_par
% 0.32/0.47 = ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c ) ) ).
% 0.32/0.47
% 0.32/0.47 % par_def
% 0.32/0.47 thf(fact_172_UNIV__I,axiom,
% 0.32/0.47 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % UNIV_I
% 0.32/0.47 thf(fact_173_predicate2I,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 0.32/0.47 ( ! [X3: A,Y: B] :
% 0.32/0.47 ( ( P @ X3 @ Y )
% 0.32/0.47 => ( Q @ X3 @ Y ) )
% 0.32/0.47 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 0.32/0.47
% 0.32/0.47 % predicate2I
% 0.32/0.47 thf(fact_174_predicate1I,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( P @ X3 )
% 0.32/0.47 => ( Q @ X3 ) )
% 0.32/0.47 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 0.32/0.47
% 0.32/0.47 % predicate1I
% 0.32/0.47 thf(fact_175_range__fst,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % range_fst
% 0.32/0.47 thf(fact_176_range__snd,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % range_snd
% 0.32/0.47 thf(fact_177_refl__ge__eq,axiom,
% 0.32/0.47 ! [A: $tType,R2: A > A > $o] :
% 0.32/0.47 ( ! [X3: A] : ( R2 @ X3 @ X3 )
% 0.32/0.47 => ( ord_less_eq @ ( A > A > $o )
% 0.32/0.47 @ ^ [Y5: A,Z: A] : ( Y5 = Z )
% 0.32/0.47 @ R2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % refl_ge_eq
% 0.32/0.47 thf(fact_178_ge__eq__refl,axiom,
% 0.32/0.47 ! [A: $tType,R2: A > A > $o,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > A > $o )
% 0.32/0.47 @ ^ [Y5: A,Z: A] : ( Y5 = Z )
% 0.32/0.47 @ R2 )
% 0.32/0.47 => ( R2 @ X @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % ge_eq_refl
% 0.32/0.47 thf(fact_179_leq__OOI,axiom,
% 0.32/0.47 ! [A: $tType,R2: A > A > $o] :
% 0.32/0.47 ( ( R2
% 0.32/0.47 = ( ^ [Y5: A,Z: A] : ( Y5 = Z ) ) )
% 0.32/0.47 => ( ord_less_eq @ ( A > A > $o ) @ R2 @ ( relcompp @ A @ A @ A @ R2 @ R2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % leq_OOI
% 0.32/0.47 thf(fact_180_predicate1D,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,Q: A > $o,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.32/0.47 => ( ( P @ X )
% 0.32/0.47 => ( Q @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % predicate1D
% 0.32/0.47 thf(fact_181_predicate2D,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y3: B] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.32/0.47 => ( ( P @ X @ Y3 )
% 0.32/0.47 => ( Q @ X @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % predicate2D
% 0.32/0.47 thf(fact_182_relcompp__mono,axiom,
% 0.32/0.47 ! [A: $tType,C: $tType,B: $tType,R5: A > B > $o,R: A > B > $o,S2: B > C > $o,S: B > C > $o] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > B > $o ) @ R5 @ R )
% 0.32/0.47 => ( ( ord_less_eq @ ( B > C > $o ) @ S2 @ S )
% 0.32/0.47 => ( ord_less_eq @ ( A > C > $o ) @ ( relcompp @ A @ B @ C @ R5 @ S2 ) @ ( relcompp @ A @ B @ C @ R @ S ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % relcompp_mono
% 0.32/0.47 thf(fact_183_rev__predicate1D,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,X: A,Q: A > $o] :
% 0.32/0.47 ( ( P @ X )
% 0.32/0.47 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.32/0.47 => ( Q @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rev_predicate1D
% 0.32/0.47 thf(fact_184_rev__predicate2D,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,P: A > B > $o,X: A,Y3: B,Q: A > B > $o] :
% 0.32/0.47 ( ( P @ X @ Y3 )
% 0.32/0.47 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.32/0.47 => ( Q @ X @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rev_predicate2D
% 0.32/0.47 thf(fact_185_top__greatest,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order_top @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top_greatest
% 0.32/0.47 thf(fact_186_top_Oextremum__unique,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order_top @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 0.32/0.47 = ( A2
% 0.32/0.47 = ( top_top @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top.extremum_unique
% 0.32/0.47 thf(fact_187_top_Oextremum__uniqueI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order_top @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 0.32/0.47 => ( A2
% 0.32/0.47 = ( top_top @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top.extremum_uniqueI
% 0.32/0.47 thf(fact_188_rangeI,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F4: B > A,X: B] : ( member @ A @ ( F4 @ X ) @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % rangeI
% 0.32/0.47 thf(fact_189_range__eqI,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,B2: A,F4: B > A,X: B] :
% 0.32/0.47 ( ( B2
% 0.32/0.47 = ( F4 @ X ) )
% 0.32/0.47 => ( member @ A @ B2 @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % range_eqI
% 0.32/0.47 thf(fact_190_subset__UNIV,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_UNIV
% 0.32/0.47 thf(fact_191_pick__middlep,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,C: $tType,P: A > B > $o,Q: B > C > $o,A2: A,C3: C] :
% 0.32/0.47 ( ( relcompp @ A @ B @ C @ P @ Q @ A2 @ C3 )
% 0.32/0.47 => ( ( P @ A2 @ ( bNF_pick_middlep @ A @ B @ C @ P @ Q @ A2 @ C3 ) )
% 0.32/0.47 & ( Q @ ( bNF_pick_middlep @ A @ B @ C @ P @ Q @ A2 @ C3 ) @ C3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pick_middlep
% 0.32/0.47 thf(fact_192_UNIV__witness,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % UNIV_witness
% 0.32/0.47 thf(fact_193_UNIV__eq__I,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A] :
% 0.32/0.47 ( ! [X3: A] : ( member @ A @ X3 @ A6 )
% 0.32/0.47 => ( ( top_top @ ( set @ A ) )
% 0.32/0.47 = A6 ) ) ).
% 0.32/0.47
% 0.32/0.47 % UNIV_eq_I
% 0.32/0.47 thf(fact_194_nchotomy__relcomppE,axiom,
% 0.32/0.47 ! [C: $tType,B: $tType,A: $tType,D: $tType,F4: B > A,R: C > A > $o,S: A > D > $o,A2: C,C3: D] :
% 0.32/0.47 ( ! [Y: A] :
% 0.32/0.47 ? [X5: B] :
% 0.32/0.47 ( Y
% 0.32/0.47 = ( F4 @ X5 ) )
% 0.32/0.47 => ( ( relcompp @ C @ A @ D @ R @ S @ A2 @ C3 )
% 0.32/0.47 => ~ ! [B4: B] :
% 0.32/0.47 ( ( R @ A2 @ ( F4 @ B4 ) )
% 0.32/0.47 => ~ ( S @ ( F4 @ B4 ) @ C3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nchotomy_relcomppE
% 0.32/0.47 thf(fact_195_csquare__def,axiom,
% 0.32/0.47 ! [B: $tType,C: $tType,D: $tType,A: $tType] :
% 0.32/0.47 ( ( bNF_csquare @ A @ B @ C @ D )
% 0.32/0.47 = ( ^ [A7: set @ A,F12: B > C,F23: D > C,P1: A > B,P22: A > D] :
% 0.32/0.47 ! [X4: A] :
% 0.32/0.47 ( ( member @ A @ X4 @ A7 )
% 0.32/0.47 => ( ( F12 @ ( P1 @ X4 ) )
% 0.32/0.47 = ( F23 @ ( P22 @ X4 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % csquare_def
% 0.32/0.47 thf(fact_196_fstOp__in,axiom,
% 0.32/0.47 ! [B: $tType,C: $tType,A: $tType,Ac2: product_prod @ A @ B,P: A > C > $o,Q: C > B > $o] :
% 0.32/0.47 ( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P @ Q ) ) ) )
% 0.32/0.47 => ( member @ ( product_prod @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P @ Q @ Ac2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ P ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % fstOp_in
% 0.32/0.47 thf(fact_197_sndOp__in,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,C: $tType,Ac2: product_prod @ A @ B,P: A > C > $o,Q: C > B > $o] :
% 0.32/0.47 ( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P @ Q ) ) ) )
% 0.32/0.47 => ( member @ ( product_prod @ C @ B ) @ ( bNF_sndOp @ A @ C @ B @ P @ Q @ Ac2 ) @ ( collect @ ( product_prod @ C @ B ) @ ( product_case_prod @ C @ B @ $o @ Q ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % sndOp_in
% 0.32/0.47 thf(fact_198_unfold_I1_J,axiom,
% 0.32/0.47 ! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ A ) ),B2: A] :
% 0.32/0.47 ( ( root @ ( unfold @ A @ Rt @ Ct @ B2 ) )
% 0.32/0.47 = ( Rt @ B2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % unfold(1)
% 0.32/0.47 thf(fact_199_surj__def,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F4: B > A] :
% 0.32/0.47 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) )
% 0.32/0.47 = ( ! [Y8: A] :
% 0.32/0.47 ? [X4: B] :
% 0.32/0.47 ( Y8
% 0.32/0.47 = ( F4 @ X4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_def
% 0.32/0.47 thf(fact_200_surjI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,G3: B > A,F4: A > B] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( G3 @ ( F4 @ X3 ) )
% 0.32/0.47 = X3 )
% 0.32/0.47 => ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjI
% 0.32/0.47 thf(fact_201_top__set__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( top_top @ ( set @ A ) )
% 0.32/0.47 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % top_set_def
% 0.32/0.47 thf(fact_202_surjD,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F4: B > A,Y3: A] :
% 0.32/0.47 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ? [X3: B] :
% 0.32/0.47 ( Y3
% 0.32/0.47 = ( F4 @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjD
% 0.32/0.47 thf(fact_203_surjE,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F4: B > A,Y3: A] :
% 0.32/0.47 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ~ ! [X3: B] :
% 0.32/0.47 ( Y3
% 0.32/0.47 != ( F4 @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjE
% 0.32/0.47 thf(fact_204_map__prod__surj,axiom,
% 0.32/0.47 ! [A: $tType,C: $tType,D: $tType,B: $tType,F4: A > B,G3: C > D] :
% 0.32/0.47 ( ( ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 = ( top_top @ ( set @ B ) ) )
% 0.32/0.47 => ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
% 0.32/0.47 = ( top_top @ ( set @ D ) ) )
% 0.32/0.47 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F4 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 0.32/0.47 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % map_prod_surj
% 0.32/0.47 thf(fact_205_root__Node,axiom,
% 0.32/0.47 ! [N2: n,As2: set @ ( sum_sum @ t @ dtree )] :
% 0.32/0.47 ( ( root @ ( node @ N2 @ As2 ) )
% 0.32/0.47 = N2 ) ).
% 0.32/0.47
% 0.32/0.47 % root_Node
% 0.32/0.47 thf(fact_206_map__prod__simp,axiom,
% 0.32/0.47 ! [C: $tType,A: $tType,B: $tType,D: $tType,F4: C > A,G3: D > B,A2: C,B2: D] :
% 0.32/0.47 ( ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ ( product_Pair @ C @ D @ A2 @ B2 ) )
% 0.32/0.47 = ( product_Pair @ A @ B @ ( F4 @ A2 ) @ ( G3 @ B2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % map_prod_simp
% 0.32/0.47 thf(fact_207_fst__map__prod,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > A,G3: D > B,X: product_prod @ C @ D] :
% 0.32/0.47 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ X ) )
% 0.32/0.47 = ( F4 @ ( product_fst @ C @ D @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % fst_map_prod
% 0.32/0.47 thf(fact_208_snd__map__prod,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > B,G3: D > A,X: product_prod @ C @ D] :
% 0.32/0.47 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F4 @ G3 @ X ) )
% 0.32/0.47 = ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % snd_map_prod
% 0.32/0.47 thf(fact_209_map__prod__imageI,axiom,
% 0.32/0.47 ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),F4: A > C,G3: B > D] :
% 0.32/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
% 0.32/0.47 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F4 @ A2 ) @ ( G3 @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F4 @ G3 ) @ R2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % map_prod_imageI
% 0.32/0.47 thf(fact_210_prod__fun__imageE,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F4: C > A,G3: D > B,R2: set @ ( product_prod @ C @ D )] :
% 0.32/0.47 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 ) @ R2 ) )
% 0.32/0.47 => ~ ! [X3: C,Y: D] :
% 0.32/0.47 ( ( C3
% 0.32/0.47 = ( product_Pair @ A @ B @ ( F4 @ X3 ) @ ( G3 @ Y ) ) )
% 0.32/0.47 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y ) @ R2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % prod_fun_imageE
% 0.32/0.47 thf(fact_211_Node__root__cont,axiom,
% 0.32/0.47 ! [Tr: dtree] :
% 0.32/0.47 ( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
% 0.32/0.47 = Tr ) ).
% 0.32/0.47
% 0.32/0.47 % Node_root_cont
% 0.32/0.47 thf(fact_212_Powp__mono,axiom,
% 0.32/0.47 ! [A: $tType,A6: A > $o,B6: A > $o] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > $o ) @ A6 @ B6 )
% 0.32/0.47 => ( ord_less_eq @ ( ( set @ A ) > $o ) @ ( powp @ A @ A6 ) @ ( powp @ A @ B6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Powp_mono
% 0.32/0.47 thf(fact_213_dtree__cong,axiom,
% 0.32/0.47 ! [Tr: dtree,Tr4: dtree] :
% 0.32/0.47 ( ( ( root @ Tr )
% 0.32/0.47 = ( root @ Tr4 ) )
% 0.32/0.47 => ( ( ( cont @ Tr )
% 0.32/0.47 = ( cont @ Tr4 ) )
% 0.32/0.47 => ( Tr = Tr4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dtree_cong
% 0.32/0.47 thf(fact_214_predicate2D__conj,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R2: $o,X: A,Y3: B] :
% 0.32/0.47 ( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.32/0.47 & R2 )
% 0.32/0.47 => ( R2
% 0.32/0.47 & ( ( P @ X @ Y3 )
% 0.32/0.47 => ( Q @ X @ Y3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % predicate2D_conj
% 0.32/0.47 thf(fact_215_corec_I1_J,axiom,
% 0.32/0.47 ! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ),B2: A] :
% 0.32/0.47 ( ( root @ ( corec @ A @ Rt @ Ct @ B2 ) )
% 0.32/0.47 = ( Rt @ B2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % corec(1)
% 0.32/0.47 thf(fact_216_Inl__in__cont__par,axiom,
% 0.32/0.47 ! [T2: t,Tr12: dtree,Tr22: dtree] :
% 0.32/0.47 ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) )
% 0.32/0.47 = ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr12 ) )
% 0.32/0.47 | ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr22 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Inl_in_cont_par
% 0.32/0.47 thf(fact_217_surj__swap__iff,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,A2: B,B2: B,F4: B > A] :
% 0.32/0.47 ( ( ( image @ B @ A @ ( swap @ B @ A @ A2 @ B2 @ F4 ) @ ( top_top @ ( set @ B ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) )
% 0.32/0.47 = ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_swap_iff
% 0.32/0.47 thf(fact_218_swap__image__eq,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,A2: A,A6: set @ A,B2: A,F4: A > B] :
% 0.32/0.47 ( ( member @ A @ A2 @ A6 )
% 0.32/0.47 => ( ( member @ A @ B2 @ A6 )
% 0.32/0.47 => ( ( image @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F4 ) @ A6 )
% 0.32/0.47 = ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % swap_image_eq
% 0.32/0.47 thf(fact_219_surj__imp__surj__swap,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F4: B > A,A2: B,B2: B] :
% 0.32/0.47 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ( ( image @ B @ A @ ( swap @ B @ A @ A2 @ B2 @ F4 ) @ ( top_top @ ( set @ B ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_imp_surj_swap
% 0.32/0.47 thf(fact_220_Sup_OSUP__cong,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,A6: set @ B,B6: set @ B,C4: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 0.32/0.47 ( ( A6 = B6 )
% 0.32/0.47 => ( ! [X3: B] :
% 0.32/0.47 ( ( member @ B @ X3 @ B6 )
% 0.32/0.47 => ( ( C4 @ X3 )
% 0.32/0.47 = ( D3 @ X3 ) ) )
% 0.32/0.47 => ( ( Sup @ ( image @ B @ A @ C4 @ A6 ) )
% 0.32/0.47 = ( Sup @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Sup.SUP_cong
% 0.32/0.47 thf(fact_221_Inf_OINF__cong,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,A6: set @ B,B6: set @ B,C4: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 0.32/0.47 ( ( A6 = B6 )
% 0.32/0.47 => ( ! [X3: B] :
% 0.32/0.47 ( ( member @ B @ X3 @ B6 )
% 0.32/0.47 => ( ( C4 @ X3 )
% 0.32/0.47 = ( D3 @ X3 ) ) )
% 0.32/0.47 => ( ( Inf @ ( image @ B @ A @ C4 @ A6 ) )
% 0.32/0.47 = ( Inf @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Inf.INF_cong
% 0.32/0.47 thf(fact_222_surj__Compl__image__subset,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B] :
% 0.32/0.47 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.32/0.47 = ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) ) @ ( image @ B @ A @ F4 @ ( uminus_uminus @ ( set @ B ) @ A6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_Compl_image_subset
% 0.32/0.47 thf(fact_223_in__inv__image,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,X: A,Y3: A,R: set @ ( product_prod @ B @ B ),F4: A > B] :
% 0.32/0.47 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( inv_image @ B @ A @ R @ F4 ) )
% 0.32/0.47 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X ) @ ( F4 @ Y3 ) ) @ R ) ) ).
% 0.32/0.47
% 0.32/0.47 % in_inv_image
% 0.32/0.47 thf(fact_224_ComplI,axiom,
% 0.32/0.47 ! [A: $tType,C3: A,A6: set @ A] :
% 0.32/0.47 ( ~ ( member @ A @ C3 @ A6 )
% 0.32/0.47 => ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ComplI
% 0.32/0.47 thf(fact_225_Compl__iff,axiom,
% 0.32/0.47 ! [A: $tType,C3: A,A6: set @ A] :
% 0.32/0.47 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 0.32/0.47 = ( ~ ( member @ A @ C3 @ A6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Compl_iff
% 0.32/0.47 thf(fact_226_Compl__eq__Compl__iff,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( ( uminus_uminus @ ( set @ A ) @ A6 )
% 0.32/0.47 = ( uminus_uminus @ ( set @ A ) @ B6 ) )
% 0.32/0.47 = ( A6 = B6 ) ) ).
% 0.32/0.47
% 0.32/0.47 % Compl_eq_Compl_iff
% 0.32/0.47 thf(fact_227_Compl__subset__Compl__iff,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) )
% 0.32/0.47 = ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ).
% 0.32/0.47
% 0.32/0.47 % Compl_subset_Compl_iff
% 0.32/0.47 thf(fact_228_Compl__anti__mono,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.32/0.47 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Compl_anti_mono
% 0.32/0.47 thf(fact_229_ComplD,axiom,
% 0.32/0.47 ! [A: $tType,C3: A,A6: set @ A] :
% 0.32/0.47 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 0.32/0.47 => ~ ( member @ A @ C3 @ A6 ) ) ).
% 0.32/0.47
% 0.32/0.47 % ComplD
% 0.32/0.47 thf(fact_230_double__complement,axiom,
% 0.32/0.47 ! [A: $tType,A6: set @ A] :
% 0.32/0.47 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 0.32/0.47 = A6 ) ).
% 0.32/0.47
% 0.32/0.47 % double_complement
% 0.32/0.47 thf(fact_231_compl__le__compl__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [X: A,Y3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) )
% 0.32/0.47 = ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % compl_le_compl_iff
% 0.32/0.47 thf(fact_232_neg__le__iff__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [B2: A,A2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 0.32/0.47 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neg_le_iff_le
% 0.32/0.47 thf(fact_233_compl__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [X: A,Y3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % compl_mono
% 0.32/0.47 thf(fact_234_compl__le__swap1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Y3: A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X ) )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % compl_le_swap1
% 0.32/0.47 thf(fact_235_compl__le__swap2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Y3: A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X )
% 0.32/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % compl_le_swap2
% 0.32/0.47 thf(fact_236_le__minus__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A,B2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 0.32/0.47 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_minus_iff
% 0.32/0.47 thf(fact_237_minus__le__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A,B2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 0.32/0.47 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % minus_le_iff
% 0.32/0.47 thf(fact_238_le__imp__neg__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A,B2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_imp_neg_le
% 0.32/0.47 thf(fact_239_inj__image__Compl__subset,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F4: A > B,A6: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_image_Compl_subset
% 0.32/0.47 thf(fact_240_inj__apfst,axiom,
% 0.32/0.47 ! [B: $tType,C: $tType,A: $tType,F4: A > C] :
% 0.32/0.47 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.32/0.47 = ( inj_on @ A @ C @ F4 @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_apfst
% 0.32/0.47 thf(fact_241_inj__apsnd,axiom,
% 0.32/0.47 ! [A: $tType,C: $tType,B: $tType,F4: B > C] :
% 0.32/0.47 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.32/0.47 = ( inj_on @ B @ C @ F4 @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_apsnd
% 0.32/0.47 thf(fact_242_inj__image__subset__iff,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F4: A > B,A6: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B6 ) )
% 0.32/0.47 = ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_image_subset_iff
% 0.32/0.47 thf(fact_243_inj__image__mem__iff,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F4: A > B,A2: A,A6: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ( ( member @ B @ ( F4 @ A2 ) @ ( image @ A @ B @ F4 @ A6 ) )
% 0.32/0.47 = ( member @ A @ A2 @ A6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_image_mem_iff
% 0.32/0.47 thf(fact_244_inj__image__eq__iff,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F4: A > B,A6: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ( ( ( image @ A @ B @ F4 @ A6 )
% 0.32/0.47 = ( image @ A @ B @ F4 @ B6 ) )
% 0.32/0.47 = ( A6 = B6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_image_eq_iff
% 0.32/0.47 thf(fact_245_range__ex1__eq,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F4: A > B,B2: B] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ( ( member @ B @ B2 @ ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) ) )
% 0.32/0.47 = ( ? [X4: A] :
% 0.32/0.47 ( ( B2
% 0.32/0.47 = ( F4 @ X4 ) )
% 0.32/0.47 & ! [Y8: A] :
% 0.32/0.47 ( ( B2
% 0.32/0.47 = ( F4 @ Y8 ) )
% 0.32/0.47 => ( Y8 = X4 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % range_ex1_eq
% 0.32/0.47 thf(fact_246_inj__swap,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A6 ) ).
% 0.32/0.47
% 0.32/0.47 % inj_swap
% 0.32/0.47 thf(fact_247_inj__on__image__iff,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,A6: set @ A,G3: A > B,F4: A > A] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( member @ A @ X3 @ A6 )
% 0.32/0.47 => ! [Xa: A] :
% 0.32/0.47 ( ( member @ A @ Xa @ A6 )
% 0.32/0.47 => ( ( ( G3 @ ( F4 @ X3 ) )
% 0.32/0.47 = ( G3 @ ( F4 @ Xa ) ) )
% 0.32/0.47 = ( ( G3 @ X3 )
% 0.32/0.47 = ( G3 @ Xa ) ) ) ) )
% 0.32/0.47 => ( ( inj_on @ A @ A @ F4 @ A6 )
% 0.32/0.47 => ( ( inj_on @ A @ B @ G3 @ ( image @ A @ A @ F4 @ A6 ) )
% 0.32/0.47 = ( inj_on @ A @ B @ G3 @ A6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_image_iff
% 0.32/0.47 thf(fact_248_inj__on__subset,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F4: A > B,A6: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F4 @ A6 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
% 0.32/0.47 => ( inj_on @ A @ B @ F4 @ B6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_subset
% 0.32/0.47 thf(fact_249_subset__inj__on,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F4: A > B,B6: set @ A,A6: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F4 @ B6 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.32/0.47 => ( inj_on @ A @ B @ F4 @ A6 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % subset_inj_on
% 0.32/0.47 thf(fact_250_inj__on__image__mem__iff__alt,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F4: A > B,B6: set @ A,A6: set @ A,A2: A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F4 @ B6 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.32/0.47 => ( ( member @ B @ ( F4 @ A2 ) @ ( image @ A @ B @ F4 @ A6 ) )
% 0.32/0.47 => ( ( member @ A @ A2 @ B6 )
% 0.32/0.47 => ( member @ A @ A2 @ A6 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_image_mem_iff_alt
% 0.32/0.47 thf(fact_251_inj__on__image__mem__iff,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F4: A > B,B6: set @ A,A2: A,A6: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F4 @ B6 )
% 0.32/0.47 => ( ( member @ A @ A2 @ B6 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.32/0.47 => ( ( member @ B @ ( F4 @ A2 ) @ ( image @ A @ B @ F4 @ A6 ) )
% 0.32/0.47 = ( member @ A @ A2 @ A6 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_image_mem_iff
% 0.32/0.47 thf(fact_252_inj__on__image__eq__iff,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F4: A > B,C4: set @ A,A6: set @ A,B6: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F4 @ C4 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 0.32/0.47 => ( ( ( image @ A @ B @ F4 @ A6 )
% 0.32/0.47 = ( image @ A @ B @ F4 @ B6 ) )
% 0.32/0.47 = ( A6 = B6 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_image_eq_iff
% 0.32/0.47 thf(fact_253_the__inv__into__into,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,F4: A > B,A6: set @ A,X: B,B6: set @ A] :
% 0.32/0.47 ( ( inj_on @ A @ B @ F4 @ A6 )
% 0.32/0.47 => ( ( member @ B @ X @ ( image @ A @ B @ F4 @ A6 ) )
% 0.32/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.32/0.47 => ( member @ A @ ( the_inv_into @ A @ B @ A6 @ F4 @ X ) @ B6 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % the_inv_into_into
% 0.32/0.47 thf(fact_254_wf__map__prod__image,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F4: A > B] :
% 0.32/0.47 ( ( wf @ A @ R )
% 0.32/0.47 => ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 0.32/0.47 => ( wf @ B @ ( image @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( product_map_prod @ A @ B @ A @ B @ F4 @ F4 ) @ R ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % wf_map_prod_image
% 0.32/0.47 thf(fact_255_wf__inv__image,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ B ),F4: A > B] :
% 0.32/0.47 ( ( wf @ B @ R )
% 0.32/0.47 => ( wf @ A @ ( inv_image @ B @ A @ R @ F4 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % wf_inv_image
% 0.32/0.47
% 0.32/0.47 %----Type constructors (16)
% 0.32/0.47 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( boolean_algebra @ A9 @ ( type2 @ A9 ) )
% 0.32/0.47 => ( boolean_algebra @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( order_top @ A9 @ ( type2 @ A9 ) )
% 0.32/0.47 => ( order_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( preorder @ A9 @ ( type2 @ A9 ) )
% 0.32/0.47 => ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( order @ A9 @ ( type2 @ A9 ) )
% 0.32/0.47 => ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.32/0.47 ! [A8: $tType,A9: $tType] :
% 0.32/0.47 ( ( ord @ A9 @ ( type2 @ A9 ) )
% 0.32/0.47 => ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_1,axiom,
% 0.32/0.47 ! [A8: $tType] : ( boolean_algebra @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Orderings_Oorder__top_2,axiom,
% 0.32/0.47 ! [A8: $tType] : ( order_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
% 0.32/0.47 ! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Orderings_Oorder_4,axiom,
% 0.32/0.47 ! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
% 0.32/0.47 ! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_6,axiom,
% 0.32/0.47 boolean_algebra @ $o @ ( type2 @ $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Oorder__top_7,axiom,
% 0.32/0.47 order_top @ $o @ ( type2 @ $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
% 0.32/0.47 preorder @ $o @ ( type2 @ $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.32/0.47 linorder @ $o @ ( type2 @ $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 0.32/0.47 order @ $o @ ( type2 @ $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 0.32/0.47 ord @ $o @ ( type2 @ $o ) ).
% 0.32/0.47
% 0.32/0.47 %----Conjectures (1)
% 0.32/0.47 thf(conj_0,conjecture,
% 0.32/0.47 ( ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) @ tr3 ) )
% 0.32/0.47 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1 @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr2 @ tr3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 ------------------------
% 2.87/1.54 ---- Embedded in THF ---
% 2.87/1.54 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.87/1.54 %%% Generated on Tue Feb 24 04:16:39 EST 2026
% 2.87/1.54 %%% using '$$dhol' embedding, version 1.3.0.
% 2.87/1.54 %%% Logic specification used:
% 2.87/1.54 %%% thf(spec, logic, $$dhol).
% 2.87/1.54
% 2.87/1.54 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.XABAPK8w6b/DTF2DTF28763.p
% 2.87/1.54 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.87/1.54 thf(ty_t_Sum__Type_Osum, type, sum_sum: ($tType > ($tType > $tType))).
% 2.87/1.54 thf(ty_t_DTree_Odtree, type, dtree: $tType).
% 2.87/1.54 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.87/1.54 thf(ty_t_DTree_OT, type, t: $tType).
% 2.87/1.54 thf(ty_t_DTree_ON, type, n: $tType).
% 2.87/1.54 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.87/1.54 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.87/1.54 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.87/1.54 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.87/1.54 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.87/1.54 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.87/1.54 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.87/1.54 thf(sy_cl_Lattices_Oboolean__algebra, type, boolean_algebra: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.87/1.54 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.87/1.54 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca1785829860lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 2.87/1.54 thf(sy_c_BNF__Def_Ocsquare, type, bNF_csquare: (!> [A:$tType,B:$tType,C:$tType,D:$tType]: (((set @ A) > ((B > C) > ((D > C) > ((A > B) > ((A > D) > $o)))))))).
% 2.87/1.54 thf(sy_c_BNF__Def_OfstOp, type, bNF_fstOp: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((B > (C > $o)) > (((product_prod @ A) @ C) > ((product_prod @ A) @ B))))))).
% 2.87/1.54 thf(sy_c_BNF__Def_Opick__middlep, type, bNF_pick_middlep: (!> [B:$tType,A:$tType,C:$tType]: (((B > (A > $o)) > ((A > (C > $o)) > (B > (C > A))))))).
% 2.87/1.54 thf(sy_c_BNF__Def_OsndOp, type, bNF_sndOp: (!> [C:$tType,A:$tType,B:$tType]: (((C > (A > $o)) > ((A > (B > $o)) > (((product_prod @ C) @ B) > ((product_prod @ A) @ B))))))).
% 2.87/1.54 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2, type, bNF_Greatest_image2: (!> [C:$tType,A:$tType,B:$tType]: (((set @ C) > ((C > A) > ((C > B) > (set @ ((product_prod @ A) @ B)))))))).
% 2.87/1.54 thf(sy_c_DTree_ONode, type, node: (n > ((set @ ((sum_sum @ t) @ dtree)) > dtree))).
% 2.87/1.54 thf(sy_c_DTree_Ocont, type, cont: (dtree > (set @ ((sum_sum @ t) @ dtree)))).
% 2.87/1.54 thf(sy_c_DTree_Ocorec, type, corec: (!> [A:$tType]: (((A > n) > ((A > (set @ ((sum_sum @ t) @ ((sum_sum @ dtree) @ A)))) > (A > dtree)))))).
% 2.87/1.54 thf(sy_c_DTree_Odtree_Oroot, type, root: (dtree > n)).
% 2.87/1.54 thf(sy_c_DTree_Ounfold, type, unfold: (!> [A:$tType]: (((A > n) > ((A > (set @ ((sum_sum @ t) @ A))) > (A > dtree)))))).
% 2.87/1.54 thf(sy_c_Fun_Oinj__on, type, inj_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 2.87/1.54 thf(sy_c_Fun_Oswap, type, swap: (!> [A:$tType,B:$tType]: ((A > (A > ((A > B) > (A > B))))))).
% 2.87/1.54 thf(sy_c_Fun_Othe__inv__into, type, the_inv_into: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (B > A)))))).
% 2.87/1.54 thf(sy_c_Fun__Def_Oin__rel, type, fun_in_rel: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (A > (B > $o)))))).
% 2.87/1.54 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.87/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.87/1.54 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.87/1.54 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_ONplus, type, parall1518086719_Nplus: (n > (n > n))).
% 2.87/1.54 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar, type, parall1899940088le_par: (((product_prod @ dtree) @ dtree) > dtree)).
% 2.87/1.54 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c, type, parall1914194347_par_c: (((product_prod @ dtree) @ dtree) > (set @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree))))).
% 2.87/1.54 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r, type, parall1914194362_par_r: (((product_prod @ dtree) @ dtree) > n)).
% 2.87/1.54 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r__rel, type, parall556292031_r_rel: (((product_prod @ dtree) @ dtree) > (((product_prod @ dtree) @ dtree) > $o))).
% 2.87/1.54 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.87/1.54 thf(sy_c_Product__Type_Oapfst, type, product_apfst: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > (((product_prod @ A) @ B) > ((product_prod @ C) @ B)))))).
% 2.87/1.54 thf(sy_c_Product__Type_Oapsnd, type, product_apsnd: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > (((product_prod @ A) @ B) > ((product_prod @ A) @ C)))))).
% 2.87/1.54 thf(sy_c_Product__Type_Ocurry, type, product_curry: (!> [A:$tType,B:$tType,C:$tType]: (((((product_prod @ A) @ B) > C) > (A > (B > C)))))).
% 2.87/1.54 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.87/1.54 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 2.87/1.54 thf(sy_c_Product__Type_Oold_Obool_Orec__bool, type, product_rec_bool: (!> [T:$tType]: ((T > (T > ($o > T)))))).
% 2.87/1.54 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.87/1.54 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.87/1.54 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 2.87/1.54 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 2.87/1.54 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 2.87/1.54 thf(sy_c_Product__Type_Oscomp, type, product_scomp: (!> [A:$tType,B:$tType,C:$tType,D:$tType]: (((A > ((product_prod @ B) @ C)) > ((B > (C > D)) > (A > D)))))).
% 2.87/1.54 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.87/1.54 thf(sy_c_Relation_OPowp, type, powp: (!> [A:$tType]: (((A > $o) > ((set @ A) > $o))))).
% 2.87/1.54 thf(sy_c_Relation_Oinv__image, type, inv_image: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ B)) > ((A > B) > (set @ ((product_prod @ A) @ A))))))).
% 2.87/1.54 thf(sy_c_Relation_Orelcompp, type, relcompp: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((B > (C > $o)) > (A > (C > $o))))))).
% 2.87/1.54 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.87/1.54 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.87/1.54 thf(sy_c_Sum__Type_OInl, type, sum_Inl: (!> [A:$tType,B:$tType]: ((A > ((sum_sum @ A) @ B))))).
% 2.87/1.54 thf(sy_c_Wellfounded_Oaccp, type, accp: (!> [A:$tType]: (((A > (A > $o)) > (A > $o))))).
% 2.87/1.54 thf(sy_c_Wellfounded_Olex__prod, type, lex_prod: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 2.87/1.54 thf(sy_c_Wellfounded_Owf, type, wf: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 2.87/1.54 thf(sy_c_Wfrec_Osame__fst, type, same_fst: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > (set @ ((product_prod @ B) @ B))) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 2.87/1.54 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.87/1.54 thf(sy_v_tr1, type, tr1: dtree).
% 2.87/1.54 thf(sy_v_tr2, type, tr2: dtree).
% 2.87/1.54 thf(sy_v_tr3, type, tr3: dtree).
% 2.87/1.54 thf(fact_0__092_060open_062_092_060And_062trB_AtrA_O_A_092_060exists_062tr1_Atr2_Atr3_O_AtrA_A_061_A_Itr1_A_092_060parallel_062_Atr2_J_A_092_060parallel_062_Atr3_A_092_060and_062_AtrB_A_061_Atr1_A_092_060parallel_062_Atr2_A_092_060parallel_062_Atr3_A_092_060Longrightarrow_062_AtrA_A_061_AtrB_092_060close_062, axiom, (! [TrA:dtree,TrB:dtree]: (((? [Tr1:dtree,Tr2:dtree,Tr3:dtree]: (((TrA = (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2))) @ Tr3))) & (TrB = (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr1) @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr2) @ Tr3)))))))) => (TrA = TrB))))).
% 2.87/1.54 thf(fact_1_par__com, axiom, (! [Tr12:dtree,Tr22:dtree]: (((parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr12) @ Tr22)) = (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr22) @ Tr12)))))).
% 2.87/1.54 thf(fact_2_par__r_Ocases, axiom, (! [X:((product_prod @ dtree) @ dtree)]: ((~ ((! [Tr13:dtree,Tr23:dtree]: ((X != ((((product_Pair @ dtree) @ dtree) @ Tr13) @ Tr23))))))))).
% 2.87/1.54 thf(fact_3_par__r_Oinduct, axiom, (! [P:(((product_prod @ dtree) @ dtree) > $o),A0:((product_prod @ dtree) @ dtree)]: (((! [Tr13:dtree,Tr23:dtree]: ((P @ ((((product_Pair @ dtree) @ dtree) @ Tr13) @ Tr23)))) => (P @ A0))))).
% 2.87/1.54 thf(fact_4_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X2:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X2) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X2 = Y2)))))).
% 2.87/1.54 thf(fact_5_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((A2 = A3) & (B2 = B3)))))).
% 2.87/1.54 thf(fact_6_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X3:A,Y:B]: ((P2 = ((((product_Pair @ A) @ B) @ X3) @ Y))))))).
% 2.87/1.54 thf(fact_7_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A4:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A4) @ B4)))) => (P @ P2))))).
% 2.87/1.54 thf(fact_8_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A3) @ B3)) => (~ (((A2 = A3) => (B2 != B3)))))))).
% 2.87/1.54 thf(fact_9_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A4:A,B4:B,C2:C]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B4) @ C2)))))))))).
% 2.87/1.54 thf(fact_10_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2))))))))))).
% 2.87/1.54 thf(fact_11_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 2.87/1.54 thf(fact_12_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2))))))))))))).
% 2.87/1.54 thf(fact_13_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,G:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: ((~ ((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F,G2:G]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2)))))))))))))).
% 2.87/1.54 thf(fact_14_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A4:A,B4:B,C2:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Pair @ B) @ C) @ B4) @ C2))))) => (P @ X))))).
% 2.87/1.54 thf(fact_15_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A4:A,B4:B,C2:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C2) @ D2)))))) => (P @ X))))).
% 2.87/1.54 thf(fact_16_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A4:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A4) @ B4)))) => (P @ Prod))))).
% 2.87/1.54 thf(fact_17_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y3:((product_prod @ A) @ B)]: ((~ ((! [A4:A,B4:B]: ((Y3 != ((((product_Pair @ A) @ B) @ A4) @ B4))))))))).
% 2.87/1.54 thf(fact_18_prod__induct7, axiom, (! [G:$tType,F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F,G2:G]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2))))))))) => (P @ X))))).
% 2.87/1.54 thf(fact_19_prod__induct6, axiom, (! [F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E,F2:F]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2)))))))) => (P @ X))))).
% 2.87/1.54 thf(fact_20_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A4:A,B4:B,C2:C,D2:D,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A4) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P @ X))))).
% 2.87/1.54 thf(fact_21_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 2.87/1.54 thf(fact_22_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(B > (C > A)),A2:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C3) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((C3 @ A2) @ B2))))).
% 2.87/1.54 thf(fact_23_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R:A,S:B,R2:(set @ ((product_prod @ A) @ B)),S2:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S)) @ R2) => ((S2 = S) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S2)) @ R2)))))).
% 2.87/1.54 thf(fact_24_curry__conv, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_curry @ B) @ C) @ A) = (^ [F3:(((product_prod @ B) @ C) > A),A5:B,B5:C]: ((F3 @ ((((product_Pair @ B) @ C) @ A5) @ B5)))))))).
% 2.87/1.54 thf(fact_25_curryI, axiom, (! [A:$tType,B:$tType,F4:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: (((F4 @ ((((product_Pair @ A) @ B) @ A2) @ B2)) => ((((((product_curry @ A) @ B) @ $o) @ F4) @ A2) @ B2))))).
% 2.87/1.54 thf(fact_26_swap__simp, axiom, (! [A:$tType,B:$tType,X:B,Y3:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y3)) = ((((product_Pair @ A) @ B) @ Y3) @ X))))).
% 2.87/1.54 thf(fact_27_apsnd__conv, axiom, (! [A:$tType,B:$tType,C:$tType,F4:(C > B),X:A,Y3:C]: (((((((product_apsnd @ C) @ B) @ A) @ F4) @ ((((product_Pair @ A) @ C) @ X) @ Y3)) = ((((product_Pair @ A) @ B) @ X) @ (F4 @ Y3)))))).
% 2.87/1.54 thf(fact_28_old_Obool_Osimps_I6_J, axiom, (! [T:$tType,F1:T,F22:T]: ((((((product_rec_bool @ T) @ F1) @ F22) @ $false) = F22)))).
% 2.87/1.54 thf(fact_29_old_Obool_Osimps_I5_J, axiom, (! [T:$tType,F1:T,F22:T]: ((((((product_rec_bool @ T) @ F1) @ F22) @ $true) = F1)))).
% 2.87/1.54 thf(fact_30_apfst__conv, axiom, (! [C:$tType,A:$tType,B:$tType,F4:(C > A),X:C,Y3:B]: (((((((product_apfst @ C) @ A) @ B) @ F4) @ ((((product_Pair @ C) @ B) @ X) @ Y3)) = ((((product_Pair @ A) @ B) @ (F4 @ X)) @ Y3))))).
% 2.87/1.54 thf(fact_31_swap__swap, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P2)) = P2)))).
% 2.87/1.54 thf(fact_32_apsnd__apfst__commute, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,F4:(C > B),G3:(D > A),P2:((product_prod @ D) @ C)]: (((((((product_apsnd @ C) @ B) @ A) @ F4) @ (((((product_apfst @ D) @ A) @ C) @ G3) @ P2)) = (((((product_apfst @ D) @ A) @ B) @ G3) @ (((((product_apsnd @ C) @ B) @ D) @ F4) @ P2)))))).
% 2.87/1.54 thf(fact_33_curryD, axiom, (! [A:$tType,B:$tType,F4:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: ((((((((product_curry @ A) @ B) @ $o) @ F4) @ A2) @ B2) => (F4 @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 2.87/1.54 thf(fact_34_curryE, axiom, (! [A:$tType,B:$tType,F4:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: ((((((((product_curry @ A) @ B) @ $o) @ F4) @ A2) @ B2) => (F4 @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 2.87/1.54 thf(fact_35_pair__in__swap__image, axiom, (! [A:$tType,B:$tType,Y3:A,X:B,A6:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y3) @ X)) @ ((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ A6)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ Y3)) @ A6))))).
% 2.87/1.54 thf(fact_36_internal__case__prod__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((produc2004651681e_prod @ A) @ B) @ C) = (((product_case_prod @ A) @ B) @ C))))).
% 2.87/1.54 thf(fact_37_case__prod__curry, axiom, (! [C:$tType,B:$tType,A:$tType,F4:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ ((((product_curry @ A) @ B) @ C) @ F4)) = F4)))).
% 2.87/1.54 thf(fact_38_curry__case__prod, axiom, (! [C:$tType,B:$tType,A:$tType,F4:(A > (B > C))]: ((((((product_curry @ A) @ B) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F4)) = F4)))).
% 2.87/1.54 thf(fact_39_Pair__scomp, axiom, (! [A:$tType,B:$tType,C:$tType,X:C,F4:(C > (A > B))]: ((((((((product_scomp @ A) @ C) @ A) @ B) @ (((product_Pair @ C) @ A) @ X)) @ F4) = (F4 @ X))))).
% 2.87/1.54 thf(fact_40_same__fstI, axiom, (! [B:$tType,A:$tType,P:(A > $o),X:A,Y4:B,Y3:B,R2:(A > (set @ ((product_prod @ B) @ B)))]: (((P @ X) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y4) @ Y3)) @ (R2 @ X)) => (((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ ((((product_Pair @ A) @ B) @ X) @ Y3))) @ ((((same_fst @ A) @ B) @ P) @ R2))))))).
% 2.87/1.54 thf(fact_41_root__par, axiom, (! [Tr12:dtree,Tr22:dtree]: (((root @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr12) @ Tr22))) = ((parall1518086719_Nplus @ (root @ Tr12)) @ (root @ Tr22)))))).
% 2.87/1.54 thf(fact_42_in__lex__prod, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A3:A,B3:B,R:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Pair @ A) @ B) @ A3) @ B3))) @ ((((lex_prod @ A) @ B) @ R) @ S)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A3)) @ R) | ((A2 = A3) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B2) @ B3)) @ S))))))).
% 2.87/1.54 thf(fact_43_apfst__apsnd, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,F4:(C > A),G3:(D > B),X:((product_prod @ C) @ D)]: (((((((product_apfst @ C) @ A) @ B) @ F4) @ (((((product_apsnd @ D) @ B) @ C) @ G3) @ X)) = ((((product_Pair @ A) @ B) @ (F4 @ (((product_fst @ C) @ D) @ X))) @ (G3 @ (((product_snd @ C) @ D) @ X))))))).
% 2.87/1.54 thf(fact_44_scomp__apply, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType]: ((((((product_scomp @ B) @ C) @ D) @ A) = (^ [F3:(B > ((product_prod @ C) @ D)),G4:(C > (D > A)),X4:B]: ((((((product_case_prod @ C) @ D) @ A) @ G4) @ (F3 @ X4)))))))).
% 2.87/1.54 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.87/1.54 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A6:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A6)))) = A6)))).
% 2.87/1.54 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.87/1.54 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F4:(A > B),G3:(A > B)]: (((! [X3:A]: (((F4 @ X3) = (G3 @ X3)))) => (F4 = G3))))).
% 2.87/1.54 thf(fact_49_apfst__eq__conv, axiom, (! [A:$tType,B:$tType,C:$tType,F4:(C > A),X:((product_prod @ C) @ B),G3:(C > A)]: ((((((((product_apfst @ C) @ A) @ B) @ F4) @ X) = (((((product_apfst @ C) @ A) @ B) @ G3) @ X)) = ((F4 @ (((product_fst @ C) @ B) @ X)) = (G3 @ (((product_fst @ C) @ B) @ X))))))).
% 2.87/1.54 thf(fact_50_fst__apfst, axiom, (! [A:$tType,B:$tType,C:$tType,F4:(C > A),X:((product_prod @ C) @ B)]: (((((product_fst @ A) @ B) @ (((((product_apfst @ C) @ A) @ B) @ F4) @ X)) = (F4 @ (((product_fst @ C) @ B) @ X)))))).
% 2.87/1.54 thf(fact_51_snd__apfst, axiom, (! [B:$tType,A:$tType,C:$tType,F4:(C > B),X:((product_prod @ C) @ A)]: (((((product_snd @ B) @ A) @ (((((product_apfst @ C) @ B) @ A) @ F4) @ X)) = (((product_snd @ C) @ A) @ X))))).
% 2.87/1.54 thf(fact_52_fst__apsnd, axiom, (! [B:$tType,C:$tType,A:$tType,F4:(C > B),X:((product_prod @ A) @ C)]: (((((product_fst @ A) @ B) @ (((((product_apsnd @ C) @ B) @ A) @ F4) @ X)) = (((product_fst @ A) @ C) @ X))))).
% 2.87/1.54 thf(fact_53_apsnd__eq__conv, axiom, (! [B:$tType,C:$tType,A:$tType,F4:(C > B),X:((product_prod @ A) @ C),G3:(C > B)]: ((((((((product_apsnd @ C) @ B) @ A) @ F4) @ X) = (((((product_apsnd @ C) @ B) @ A) @ G3) @ X)) = ((F4 @ (((product_snd @ A) @ C) @ X)) = (G3 @ (((product_snd @ A) @ C) @ X))))))).
% 2.87/1.54 thf(fact_54_snd__apsnd, axiom, (! [A:$tType,C:$tType,B:$tType,F4:(C > A),X:((product_prod @ B) @ C)]: (((((product_snd @ B) @ A) @ (((((product_apsnd @ C) @ A) @ B) @ F4) @ X)) = (F4 @ (((product_snd @ B) @ C) @ X)))))).
% 2.87/1.54 thf(fact_55_prod_Ocollapse, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)) = Prod)))).
% 2.87/1.54 thf(fact_56_snd__swap, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B)]: (((((product_snd @ B) @ A) @ (((product_swap @ A) @ B) @ X)) = (((product_fst @ A) @ B) @ X))))).
% 2.87/1.54 thf(fact_57_fst__swap, axiom, (! [A:$tType,B:$tType,X:((product_prod @ B) @ A)]: (((((product_fst @ A) @ B) @ (((product_swap @ B) @ A) @ X)) = (((product_snd @ B) @ A) @ X))))).
% 2.87/1.54 thf(fact_58_apsnd__apfst, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,F4:(C > B),G3:(D > A),X:((product_prod @ D) @ C)]: (((((((product_apsnd @ C) @ B) @ A) @ F4) @ (((((product_apfst @ D) @ A) @ C) @ G3) @ X)) = ((((product_Pair @ A) @ B) @ (G3 @ (((product_fst @ D) @ C) @ X))) @ (F4 @ (((product_snd @ D) @ C) @ X))))))).
% 2.87/1.54 thf(fact_59_Nplus__comm, axiom, (parall1518086719_Nplus = (^ [A5:n,B5:n]: (((parall1518086719_Nplus @ B5) @ A5))))).
% 2.87/1.54 thf(fact_60_Nplus__assoc, axiom, (! [A2:n,B2:n,C3:n]: ((((parall1518086719_Nplus @ ((parall1518086719_Nplus @ A2) @ B2)) @ C3) = ((parall1518086719_Nplus @ A2) @ ((parall1518086719_Nplus @ B2) @ C3)))))).
% 2.87/1.54 thf(fact_61_surjective__pairing, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: ((T2 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T2)) @ (((product_snd @ A) @ B) @ T2)))))).
% 2.87/1.54 thf(fact_62_prod_Osplit__sel__asm, axiom, (! [C:$tType,B:$tType,A:$tType,P:(C > $o),F4:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P @ (((((product_case_prod @ A) @ B) @ C) @ F4) @ Prod)) = (~ (((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) & (~ ((P @ ((F4 @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))))))).
% 2.87/1.54 thf(fact_63_Product__Type_OCollect__case__prodD, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),A6:(A > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ X) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A6))) => ((A6 @ (((product_fst @ A) @ B) @ X)) @ (((product_snd @ A) @ B) @ X)))))).
% 2.87/1.54 thf(fact_64_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))).
% 2.87/1.54 thf(fact_65_prod_Ocase__eq__if, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [F3:(A > (B > C)),Prod2:((product_prod @ A) @ B)]: (((F3 @ (((product_fst @ A) @ B) @ Prod2)) @ (((product_snd @ A) @ B) @ Prod2)))))))).
% 2.87/1.54 thf(fact_66_prod_Osplit__sel, axiom, (! [C:$tType,B:$tType,A:$tType,P:(C > $o),F4:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P @ (((((product_case_prod @ A) @ B) @ C) @ F4) @ Prod)) = ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) => (P @ ((F4 @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))).
% 2.87/1.54 thf(fact_67_case__prod__beta, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ A) = (^ [F3:(B > (C > A)),P3:((product_prod @ B) @ C)]: (((F3 @ (((product_fst @ B) @ C) @ P3)) @ (((product_snd @ B) @ C) @ P3)))))))).
% 2.87/1.54 thf(fact_68_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y5:((product_prod @ A) @ B),Z:((product_prod @ A) @ B)]: ((Y5 = Z))) = (^ [S3:((product_prod @ A) @ B),T3:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S3) = (((product_fst @ A) @ B) @ T3)) & ((((product_snd @ A) @ B) @ S3) = (((product_snd @ A) @ B) @ T3))))))))).
% 2.87/1.54 thf(fact_69_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod3:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod3)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod3))) => (Prod = Prod3))))).
% 2.87/1.54 thf(fact_70_scomp__def, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType]: ((((((product_scomp @ A) @ B) @ C) @ D) = (^ [F3:(A > ((product_prod @ B) @ C)),G4:(B > (C > D)),X4:A]: ((((((product_case_prod @ B) @ C) @ D) @ G4) @ (F3 @ X4)))))))).
% 2.87/1.54 thf(fact_71_prod__eqI, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B),Q2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P2) = (((product_fst @ A) @ B) @ Q2)) => (((((product_snd @ A) @ B) @ P2) = (((product_snd @ A) @ B) @ Q2)) => (P2 = Q2)))))).
% 2.87/1.54 thf(fact_72_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)))))))).
% 2.87/1.54 thf(fact_73_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X2:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X2)) = X2)))).
% 2.87/1.54 thf(fact_74_snd__eqD, axiom, (! [B:$tType,A:$tType,X:B,Y3:A,A2:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y3)) = A2) => (Y3 = A2))))).
% 2.87/1.54 thf(fact_75_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X2:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X2)) = X1)))).
% 2.87/1.54 thf(fact_76_fst__eqD, axiom, (! [B:$tType,A:$tType,X:A,Y3:B,A2:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X) @ Y3)) = A2) => (X = A2))))).
% 2.87/1.54 thf(fact_77_mem__case__prodE, axiom, (! [B:$tType,A:$tType,C:$tType,Z2:A,C3:(B > (C > (set @ A))),P2:((product_prod @ B) @ C)]: (((((member @ A) @ Z2) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C3) @ P2)) => (~ ((! [X3:B,Y:C]: (((P2 = ((((product_Pair @ B) @ C) @ X3) @ Y)) => (~ ((((member @ A) @ Z2) @ ((C3 @ X3) @ Y))))))))))))).
% 2.87/1.54 thf(fact_78_old_Oprod_Ocase, axiom, (! [A:$tType,C:$tType,B:$tType,F4:(A > (B > C)),X1:A,X2:B]: (((((((product_case_prod @ A) @ B) @ C) @ F4) @ ((((product_Pair @ A) @ B) @ X1) @ X2)) = ((F4 @ X1) @ X2))))).
% 2.87/1.54 thf(fact_79_scomp__Pair, axiom, (! [C:$tType,B:$tType,A:$tType,X:(A > ((product_prod @ B) @ C))]: ((((((((product_scomp @ A) @ B) @ C) @ ((product_prod @ B) @ C)) @ X) @ ((product_Pair @ B) @ C)) = X)))).
% 2.87/1.54 thf(fact_80_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),X:A,Y3:B,A2:((product_prod @ A) @ B)]: ((((P @ X) @ Y3) => ((A2 = ((((product_Pair @ A) @ B) @ X) @ Y3)) => ((P @ (((product_fst @ A) @ B) @ A2)) @ (((product_snd @ A) @ B) @ A2))))))).
% 2.87/1.54 thf(fact_81_conjI__realizer, axiom, (! [A:$tType,B:$tType,P:(A > $o),P2:A,Q:(B > $o),Q2:B]: (((P @ P2) => ((Q @ Q2) => ((P @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q2))) & (Q @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q2))))))))).
% 2.87/1.54 thf(fact_82_exI__realizer, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Y3:A,X:B]: ((((P @ Y3) @ X) => ((P @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y3))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y3))))))).
% 2.87/1.54 thf(fact_83_par__r_Osimps, axiom, (! [Tr12:dtree,Tr22:dtree]: (((parall1914194362_par_r @ ((((product_Pair @ dtree) @ dtree) @ Tr12) @ Tr22)) = ((parall1518086719_Nplus @ (root @ Tr12)) @ (root @ Tr22)))))).
% 2.87/1.54 thf(fact_84_par__r_Oelims, axiom, (! [X:((product_prod @ dtree) @ dtree),Y3:n]: ((((parall1914194362_par_r @ X) = Y3) => (~ ((! [Tr13:dtree,Tr23:dtree]: (((X = ((((product_Pair @ dtree) @ dtree) @ Tr13) @ Tr23)) => (Y3 != ((parall1518086719_Nplus @ (root @ Tr13)) @ (root @ Tr23)))))))))))).
% 2.87/1.54 thf(fact_85_image__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F4:(B > A),X:B,A6:(set @ B)]: (((B2 = (F4 @ X)) => ((((member @ B) @ X) @ A6) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F4) @ A6))))))).
% 2.87/1.54 thf(fact_86_exE__realizer_H, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),P2:((product_prod @ B) @ A)]: ((((P @ (((product_snd @ B) @ A) @ P2)) @ (((product_fst @ B) @ A) @ P2)) => (~ ((! [X3:B,Y:A]: ((~ (((P @ Y) @ X3))))))))))).
% 2.87/1.54 thf(fact_87_sndI, axiom, (! [A:$tType,B:$tType,X:((product_prod @ A) @ B),Y3:A,Z2:B]: (((X = ((((product_Pair @ A) @ B) @ Y3) @ Z2)) => ((((product_snd @ A) @ B) @ X) = Z2))))).
% 2.87/1.54 thf(fact_88_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X:A,A6:(set @ A),B2:B,F4:(A > B)]: (((((member @ A) @ X) @ A6) => ((B2 = (F4 @ X)) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ F4) @ A6))))))).
% 2.87/1.54 thf(fact_89_ball__imageD, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B),P:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((((image @ B) @ A) @ F4) @ A6)) => (P @ X3)))) => (! [X5:B]: (((((member @ B) @ X5) @ A6) => (P @ (F4 @ X5))))))))).
% 2.87/1.54 thf(fact_90_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N:(set @ A),F4:(A > B),G3:(A > B)]: (((M = N) => ((! [X3:A]: (((((member @ A) @ X3) @ N) => ((F4 @ X3) = (G3 @ X3))))) => (((((image @ A) @ B) @ F4) @ M) = ((((image @ A) @ B) @ G3) @ N))))))).
% 2.87/1.54 thf(fact_91_bex__imageD, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B),P:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F4) @ A6)) & (P @ X5)))) => (? [X3:B]: (((((member @ B) @ X3) @ A6) & (P @ (F4 @ X3))))))))).
% 2.87/1.54 thf(fact_92_image__iff, axiom, (! [A:$tType,B:$tType,Z2:A,F4:(B > A),A6:(set @ B)]: (((((member @ A) @ Z2) @ ((((image @ B) @ A) @ F4) @ A6)) = (? [X4:B]: (((((member @ B) @ X4) @ A6) & (Z2 = (F4 @ X4))))))))).
% 2.87/1.54 thf(fact_93_imageI, axiom, (! [B:$tType,A:$tType,X:A,A6:(set @ A),F4:(A > B)]: (((((member @ A) @ X) @ A6) => (((member @ B) @ (F4 @ X)) @ ((((image @ A) @ B) @ F4) @ A6)))))).
% 2.87/1.54 thf(fact_94_fstI, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),Y3:A,Z2:B]: (((X = ((((product_Pair @ A) @ B) @ Y3) @ Z2)) => ((((product_fst @ A) @ B) @ X) = Y3))))).
% 2.87/1.54 thf(fact_95_eq__snd__iff, axiom, (! [A:$tType,B:$tType,B2:A,P2:((product_prod @ B) @ A)]: (((B2 = (((product_snd @ B) @ A) @ P2)) = (? [A5:B]: ((P2 = ((((product_Pair @ B) @ A) @ A5) @ B2)))))))).
% 2.87/1.54 thf(fact_96_eq__fst__iff, axiom, (! [A:$tType,B:$tType,A2:A,P2:((product_prod @ A) @ B)]: (((A2 = (((product_fst @ A) @ B) @ P2)) = (? [B5:B]: ((P2 = ((((product_Pair @ A) @ B) @ A2) @ B5)))))))).
% 2.87/1.54 thf(fact_97_split__cong, axiom, (! [C:$tType,B:$tType,A:$tType,Q2:((product_prod @ A) @ B),F4:(A > (B > C)),G3:(A > (B > C)),P2:((product_prod @ A) @ B)]: (((! [X3:A,Y:B]: (((((((product_Pair @ A) @ B) @ X3) @ Y) = Q2) => (((F4 @ X3) @ Y) = ((G3 @ X3) @ Y))))) => ((P2 = Q2) => ((((((product_case_prod @ A) @ B) @ C) @ F4) @ P2) = (((((product_case_prod @ A) @ B) @ C) @ G3) @ Q2))))))).
% 2.87/1.54 thf(fact_98_par__r_Opelims, axiom, (! [X:((product_prod @ dtree) @ dtree),Y3:n]: ((((parall1914194362_par_r @ X) = Y3) => ((((accp @ ((product_prod @ dtree) @ dtree)) @ parall556292031_r_rel) @ X) => (~ ((! [Tr13:dtree,Tr23:dtree]: (((X = ((((product_Pair @ dtree) @ dtree) @ Tr13) @ Tr23)) => ((Y3 = ((parall1518086719_Nplus @ (root @ Tr13)) @ (root @ Tr23))) => (~ ((((accp @ ((product_prod @ dtree) @ dtree)) @ parall556292031_r_rel) @ ((((product_Pair @ dtree) @ dtree) @ Tr13) @ Tr23))))))))))))))).
% 2.87/1.54 thf(fact_99_Collect__split__mono__strong, axiom, (! [B:$tType,A:$tType,X6:(set @ A),A6:(set @ ((product_prod @ A) @ B)),Y6:(set @ B),P:(A > (B > $o)),Q:(A > (B > $o))]: (((X6 = ((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ A6)) => ((Y6 = ((((image @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ A6)) => ((! [X3:A]: (((((member @ A) @ X3) @ X6) => (! [Xa:B]: (((((member @ B) @ Xa) @ Y6) => (((P @ X3) @ Xa) => ((Q @ X3) @ Xa)))))))) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A6) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ P))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A6) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ Q)))))))))).
% 2.87/1.54 thf(fact_100_fstOp__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((bNF_fstOp @ A) @ B) @ C) = (^ [P4:(A > (B > $o)),Q3:(B > (C > $o)),Ac:((product_prod @ A) @ C)]: (((((product_Pair @ A) @ B) @ (((product_fst @ A) @ C) @ Ac)) @ (((((((bNF_pick_middlep @ A) @ B) @ C) @ P4) @ Q3) @ (((product_fst @ A) @ C) @ Ac)) @ (((product_snd @ A) @ C) @ Ac))))))))).
% 2.87/1.54 thf(fact_101_subset__antisym, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ A6) => (A6 = B6)))))).
% 2.87/1.54 thf(fact_102_subsetI, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((member @ A) @ X3) @ B6)))) => (((ord_less_eq @ (set @ A)) @ A6) @ B6))))).
% 2.87/1.54 thf(fact_103_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X4:A]: (((P @ X4) => (Q @ X4)))))))).
% 2.87/1.54 thf(fact_104_contra__subsetD, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((~ ((((member @ A) @ C3) @ B6))) => (~ ((((member @ A) @ C3) @ A6)))))))).
% 2.87/1.54 thf(fact_105_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y5:(set @ A),Z:(set @ A)]: ((Y5 = Z))) = (^ [A7:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) & (((ord_less_eq @ (set @ A)) @ B7) @ A7)))))))).
% 2.87/1.54 thf(fact_106_subset__trans, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ C4) => (((ord_less_eq @ (set @ A)) @ A6) @ C4)))))).
% 2.87/1.54 thf(fact_107_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.87/1.54 thf(fact_108_subset__refl, axiom, (! [A:$tType,A6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A6) @ A6)))).
% 2.87/1.54 thf(fact_109_rev__subsetD, axiom, (! [A:$tType,C3:A,A6:(set @ A),B6:(set @ A)]: (((((member @ A) @ C3) @ A6) => ((((ord_less_eq @ (set @ A)) @ A6) @ B6) => (((member @ A) @ C3) @ B6)))))).
% 2.87/1.54 thf(fact_110_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A7) => (((member @ A) @ T3) @ B7)))))))))).
% 2.87/1.54 thf(fact_111_set__rev__mp, axiom, (! [A:$tType,X:A,A6:(set @ A),B6:(set @ A)]: (((((member @ A) @ X) @ A6) => ((((ord_less_eq @ (set @ A)) @ A6) @ B6) => (((member @ A) @ X) @ B6)))))).
% 2.87/1.54 thf(fact_112_equalityD2, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((A6 = B6) => (((ord_less_eq @ (set @ A)) @ B6) @ A6))))).
% 2.87/1.54 thf(fact_113_equalityD1, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((A6 = B6) => (((ord_less_eq @ (set @ A)) @ A6) @ B6))))).
% 2.87/1.54 thf(fact_114_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: ((! [X4:A]: (((((member @ A) @ X4) @ A7) => (((member @ A) @ X4) @ B7)))))))))).
% 2.87/1.54 thf(fact_115_equalityE, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((A6 = B6) => (~ (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => (~ ((((ord_less_eq @ (set @ A)) @ B6) @ A6)))))))))).
% 2.87/1.54 thf(fact_116_subsetCE, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((member @ A) @ C3) @ A6) => (((member @ A) @ C3) @ B6)))))).
% 2.87/1.54 thf(fact_117_subsetD, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((member @ A) @ C3) @ A6) => (((member @ A) @ C3) @ B6)))))).
% 2.87/1.54 thf(fact_118_in__mono, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((member @ A) @ X) @ A6) => (((member @ A) @ X) @ B6)))))).
% 2.87/1.54 thf(fact_119_set__mp, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((member @ A) @ X) @ A6) => (((member @ A) @ X) @ B6)))))).
% 2.87/1.54 thf(fact_120_accp__induct__rule, axiom, (! [A:$tType,R:(A > (A > $o)),A2:A,P:(A > $o)]: (((((accp @ A) @ R) @ A2) => ((! [X3:A]: (((((accp @ A) @ R) @ X3) => ((! [Y7:A]: ((((R @ Y7) @ X3) => (P @ Y7)))) => (P @ X3))))) => (P @ A2)))))).
% 2.87/1.54 thf(fact_121_not__accp__down, axiom, (! [A:$tType,R2:(A > (A > $o)),X:A]: (((~ ((((accp @ A) @ R2) @ X))) => (~ ((! [Z3:A]: ((((R2 @ Z3) @ X) => (((accp @ A) @ R2) @ Z3)))))))))).
% 2.87/1.54 thf(fact_122_accp__downward, axiom, (! [A:$tType,R:(A > (A > $o)),B2:A,A2:A]: (((((accp @ A) @ R) @ B2) => (((R @ A2) @ B2) => (((accp @ A) @ R) @ A2)))))).
% 2.87/1.54 thf(fact_123_accp_Oinducts, axiom, (! [A:$tType,R:(A > (A > $o)),X:A,P:(A > $o)]: (((((accp @ A) @ R) @ X) => ((! [X3:A]: (((! [Y7:A]: ((((R @ Y7) @ X3) => (((accp @ A) @ R) @ Y7)))) => ((! [Y7:A]: ((((R @ Y7) @ X3) => (P @ Y7)))) => (P @ X3))))) => (P @ X)))))).
% 2.87/1.54 thf(fact_124_accp__induct, axiom, (! [A:$tType,R:(A > (A > $o)),A2:A,P:(A > $o)]: (((((accp @ A) @ R) @ A2) => ((! [X3:A]: (((((accp @ A) @ R) @ X3) => ((! [Y7:A]: ((((R @ Y7) @ X3) => (P @ Y7)))) => (P @ X3))))) => (P @ A2)))))).
% 2.87/1.54 thf(fact_125_accp_Ointros, axiom, (! [A:$tType,R:(A > (A > $o)),X:A]: (((! [Y:A]: ((((R @ Y) @ X) => (((accp @ A) @ R) @ Y)))) => (((accp @ A) @ R) @ X))))).
% 2.87/1.54 thf(fact_126_accp_Osimps, axiom, (! [A:$tType]: (((accp @ A) = (^ [R3:(A > (A > $o)),A5:A]: ((? [X4:A]: (((A5 = X4) & (! [Y8:A]: ((((R3 @ Y8) @ X4) => (((accp @ A) @ R3) @ Y8))))))))))))).
% 2.87/1.54 thf(fact_127_accp_Ocases, axiom, (! [A:$tType,R:(A > (A > $o)),A2:A]: (((((accp @ A) @ R) @ A2) => (! [Y7:A]: ((((R @ Y7) @ A2) => (((accp @ A) @ R) @ Y7)))))))).
% 2.87/1.54 thf(fact_128_subset__image__iff, axiom, (! [A:$tType,B:$tType,B6:(set @ A),F4:(B > A),A6:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B6) @ ((((image @ B) @ A) @ F4) @ A6)) = (? [AA:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ AA) @ A6) & (B6 = ((((image @ B) @ A) @ F4) @ AA))))))))).
% 2.87/1.54 thf(fact_129_image__subset__iff, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F4) @ A6)) @ B6) = (! [X4:B]: (((((member @ B) @ X4) @ A6) => (((member @ A) @ (F4 @ X4)) @ B6)))))))).
% 2.87/1.54 thf(fact_130_subset__imageE, axiom, (! [A:$tType,B:$tType,B6:(set @ A),F4:(B > A),A6:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B6) @ ((((image @ B) @ A) @ F4) @ A6)) => (~ ((! [C5:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C5) @ A6) => (B6 != ((((image @ B) @ A) @ F4) @ C5))))))))))).
% 2.87/1.54 thf(fact_131_image__subsetI, axiom, (! [A:$tType,B:$tType,A6:(set @ A),F4:(A > B),B6:(set @ B)]: (((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((member @ B) @ (F4 @ X3)) @ B6)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F4) @ A6)) @ B6))))).
% 2.87/1.54 thf(fact_132_image__mono, axiom, (! [B:$tType,A:$tType,A6:(set @ A),B6:(set @ A),F4:(A > B)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F4) @ A6)) @ ((((image @ A) @ B) @ F4) @ B6)))))).
% 2.87/1.54 thf(fact_133_sndOp__def, axiom, (! [B:$tType,A:$tType,C:$tType]: (((((bNF_sndOp @ C) @ A) @ B) = (^ [P4:(C > (A > $o)),Q3:(A > (B > $o)),Ac:((product_prod @ C) @ B)]: (((((product_Pair @ A) @ B) @ (((((((bNF_pick_middlep @ C) @ A) @ B) @ P4) @ Q3) @ (((product_fst @ C) @ B) @ Ac)) @ (((product_snd @ C) @ B) @ Ac))) @ (((product_snd @ C) @ B) @ Ac)))))))).
% 2.87/1.54 thf(fact_134_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.87/1.54 thf(fact_135_subrelI, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((! [X3:A,Y:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y)) @ R) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y)) @ S)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S))))).
% 2.87/1.54 thf(fact_136_relChain__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (((bNF_Ca1785829860lChain @ A) @ B) = (^ [R3:(set @ ((product_prod @ A) @ A)),As:(A > B)]: ((! [I:A,J:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R3) => (((ord_less_eq @ B) @ (As @ I)) @ (As @ J)))))))))))).
% 2.87/1.54 thf(fact_137_Collect__case__prod__in__rel__leE, axiom, (! [B:$tType,A:$tType,X6:(set @ ((product_prod @ A) @ B)),Y6:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ X6) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (((fun_in_rel @ A) @ B) @ Y6)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ X6) @ Y6))))).
% 2.87/1.54 thf(fact_138_accp__subset, axiom, (! [A:$tType,R1:(A > (A > $o)),R22:(A > (A > $o))]: (((((ord_less_eq @ (A > (A > $o))) @ R1) @ R22) => (((ord_less_eq @ (A > $o)) @ ((accp @ A) @ R22)) @ ((accp @ A) @ R1)))))).
% 2.87/1.54 thf(fact_139_accp__subset__induct, axiom, (! [A:$tType,D3:(A > $o),R2:(A > (A > $o)),X:A,P:(A > $o)]: (((((ord_less_eq @ (A > $o)) @ D3) @ ((accp @ A) @ R2)) => ((! [X3:A,Z3:A]: (((D3 @ X3) => (((R2 @ Z3) @ X3) => (D3 @ Z3))))) => ((D3 @ X) => ((! [X3:A]: (((D3 @ X3) => ((! [Z4:A]: ((((R2 @ Z4) @ X3) => (P @ Z4)))) => (P @ X3))))) => (P @ X)))))))).
% 2.87/1.54 thf(fact_140_in__rel__def, axiom, (! [B:$tType,A:$tType]: ((((fun_in_rel @ A) @ B) = (^ [R4:(set @ ((product_prod @ A) @ B)),X4:A,Y8:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y8)) @ R4))))))).
% 2.87/1.54 thf(fact_141_in__rel__Collect__case__prod__eq, axiom, (! [B:$tType,A:$tType,X6:(A > (B > $o))]: (((((fun_in_rel @ A) @ B) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ X6))) = X6)))).
% 2.87/1.54 thf(fact_142_Collect__case__prod__mono, axiom, (! [B:$tType,A:$tType,A6:(A > (B > $o)),B6:(A > (B > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ A6) @ B6) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A6))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ B6))))))).
% 2.87/1.54 thf(fact_143_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.87/1.54 thf(fact_144_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C3:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C3) @ B2) => (((ord_less_eq @ A) @ C3) @ A2))))))))).
% 2.87/1.54 thf(fact_145_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((P @ A4) @ B4)))) => ((! [A4:A,B4:A]: ((((P @ B4) @ A4) => ((P @ A4) @ B4)))) => ((P @ A2) @ B2))))))))).
% 2.87/1.54 thf(fact_146_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.87/1.54 thf(fact_147_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y3:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z2) => (((ord_less_eq @ A) @ X) @ Z2))))))))).
% 2.87/1.54 thf(fact_148_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.87/1.54 thf(fact_149_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 2.87/1.54 thf(fact_150_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C3:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 2.87/1.54 thf(fact_151_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y3:A,X:A]: (((((ord_less_eq @ A) @ Y3) @ X) => ((((ord_less_eq @ A) @ X) @ Y3) = (X = Y3))))))))).
% 2.87/1.54 thf(fact_152_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y3:A,Z2:A]: ((((((ord_less_eq @ A) @ X) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ Z2)))) => (((((ord_less_eq @ A) @ Y3) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z2)))) => (((((ord_less_eq @ A) @ X) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y3)))) => (((((ord_less_eq @ A) @ Z2) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ X)))) => (((((ord_less_eq @ A) @ Y3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y3))))))))))))))))).
% 2.87/1.54 thf(fact_153_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C3) => (((ord_less_eq @ A) @ A2) @ C3))))))))).
% 2.87/1.54 thf(fact_154_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X)))))))).
% 2.87/1.54 thf(fact_155_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y3:A]: (((X = Y3) => (((ord_less_eq @ A) @ X) @ Y3)))))))).
% 2.87/1.54 thf(fact_156_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) | (((ord_less_eq @ A) @ Y3) @ X)))))))).
% 2.87/1.54 thf(fact_157_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ X) => (X = Y3))))))))).
% 2.87/1.54 thf(fact_158_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y5:A,Z:A]: ((Y5 = Z))) = (^ [X4:A,Y8:A]: (((((ord_less_eq @ A) @ X4) @ Y8) & (((ord_less_eq @ A) @ Y8) @ X4))))))))).
% 2.87/1.54 thf(fact_159_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F4:(A > B),C3:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F4 @ B2) = C3) => ((! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => (((ord_less_eq @ B) @ (F4 @ X3)) @ (F4 @ Y))))) => (((ord_less_eq @ B) @ (F4 @ A2)) @ C3)))))))))).
% 2.87/1.54 thf(fact_160_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F4:(B > A),B2:B,C3:B]: (((A2 = (F4 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C3) => ((! [X3:B,Y:B]: (((((ord_less_eq @ B) @ X3) @ Y) => (((ord_less_eq @ A) @ (F4 @ X3)) @ (F4 @ Y))))) => (((ord_less_eq @ A) @ A2) @ (F4 @ C3))))))))))).
% 2.87/1.54 thf(fact_161_order__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F4:(A > C),C3:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F4 @ B2)) @ C3) => ((! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => (((ord_less_eq @ C) @ (F4 @ X3)) @ (F4 @ Y))))) => (((ord_less_eq @ C) @ (F4 @ A2)) @ C3)))))))))).
% 2.87/1.54 thf(fact_162_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F4:(B > A),B2:B,C3:B]: (((((ord_less_eq @ A) @ A2) @ (F4 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C3) => ((! [X3:B,Y:B]: (((((ord_less_eq @ B) @ X3) @ Y) => (((ord_less_eq @ A) @ (F4 @ X3)) @ (F4 @ Y))))) => (((ord_less_eq @ A) @ A2) @ (F4 @ C3))))))))))).
% 2.87/1.54 thf(fact_163_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F3:(A > B),G4:(A > B)]: ((! [X4:A]: ((((ord_less_eq @ B) @ (F3 @ X4)) @ (G4 @ X4))))))))))).
% 2.87/1.54 thf(fact_164_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F4:(A > B),G3:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F4 @ X3)) @ (G3 @ X3)))) => (((ord_less_eq @ (A > B)) @ F4) @ G3)))))))).
% 2.87/1.54 thf(fact_165_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F4:(A > B),G3:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F4) @ G3) => (((ord_less_eq @ B) @ (F4 @ X)) @ (G3 @ X))))))))).
% 2.87/1.54 thf(fact_166_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F4:(A > B),G3:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F4) @ G3) => (((ord_less_eq @ B) @ (F4 @ X)) @ (G3 @ X))))))))).
% 2.87/1.54 thf(fact_167_Collect__case__prod__in__rel__leI, axiom, (! [B:$tType,A:$tType,X6:(set @ ((product_prod @ A) @ B)),Y6:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ X6) @ Y6) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ X6) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (((fun_in_rel @ A) @ B) @ Y6)))))))).
% 2.87/1.54 thf(fact_168_csquare__fstOp__sndOp, axiom, (! [A:$tType,B:$tType,C:$tType,F4:((A > (B > $o)) > (((product_prod @ A) @ B) > $o)),P:(A > (C > $o)),Q:(C > (B > $o))]: ((((((((((bNF_csquare @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C)) @ C) @ ((product_prod @ C) @ B)) @ ((collect @ ((product_prod @ A) @ B)) @ (F4 @ (((((relcompp @ A) @ C) @ B) @ P) @ Q)))) @ ((product_snd @ A) @ C)) @ ((product_fst @ C) @ B)) @ (((((bNF_fstOp @ A) @ C) @ B) @ P) @ Q)) @ (((((bNF_sndOp @ A) @ C) @ B) @ P) @ Q))))).
% 2.87/1.54 thf(fact_169_image2__eqI, axiom, (! [A:$tType,C:$tType,B:$tType,B2:A,F4:(B > A),X:B,C3:C,G3:(B > C),A6:(set @ B)]: (((B2 = (F4 @ X)) => ((C3 = (G3 @ X)) => ((((member @ B) @ X) @ A6) => (((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ B2) @ C3)) @ ((((((bNF_Greatest_image2 @ B) @ A) @ C) @ A6) @ F4) @ G3)))))))).
% 2.87/1.54 thf(fact_170_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))))))).
% 2.87/1.54 thf(fact_171_par__def, axiom, (parall1899940088le_par = (((unfold @ ((product_prod @ dtree) @ dtree)) @ parall1914194362_par_r) @ parall1914194347_par_c))).
% 2.87/1.54 thf(fact_172_UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 2.87/1.54 thf(fact_173_predicate2I, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Q:(A > (B > $o))]: (((! [X3:A,Y:B]: ((((P @ X3) @ Y) => ((Q @ X3) @ Y)))) => (((ord_less_eq @ (A > (B > $o))) @ P) @ Q))))).
% 2.87/1.54 thf(fact_174_predicate1I, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (A > $o)) @ P) @ Q))))).
% 2.87/1.54 thf(fact_175_range__fst, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = (top_top @ (set @ A)))))).
% 2.87/1.54 thf(fact_176_range__snd, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ A)))))).
% 2.87/1.54 thf(fact_177_refl__ge__eq, axiom, (! [A:$tType,R2:(A > (A > $o))]: (((! [X3:A]: (((R2 @ X3) @ X3))) => (((ord_less_eq @ (A > (A > $o))) @ (^ [Y5:A,Z:A]: ((Y5 = Z)))) @ R2))))).
% 2.87/1.54 thf(fact_178_ge__eq__refl, axiom, (! [A:$tType,R2:(A > (A > $o)),X:A]: (((((ord_less_eq @ (A > (A > $o))) @ (^ [Y5:A,Z:A]: ((Y5 = Z)))) @ R2) => ((R2 @ X) @ X))))).
% 2.87/1.54 thf(fact_179_leq__OOI, axiom, (! [A:$tType,R2:(A > (A > $o))]: (((R2 = (^ [Y5:A,Z:A]: ((Y5 = Z)))) => (((ord_less_eq @ (A > (A > $o))) @ R2) @ (((((relcompp @ A) @ A) @ A) @ R2) @ R2)))))).
% 2.87/1.54 thf(fact_180_predicate1D, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o),X:A]: (((((ord_less_eq @ (A > $o)) @ P) @ Q) => ((P @ X) => (Q @ X)))))).
% 2.87/1.54 thf(fact_181_predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Q:(A > (B > $o)),X:A,Y3:B]: (((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => (((P @ X) @ Y3) => ((Q @ X) @ Y3)))))).
% 2.87/1.54 thf(fact_182_relcompp__mono, axiom, (! [A:$tType,C:$tType,B:$tType,R5:(A > (B > $o)),R:(A > (B > $o)),S2:(B > (C > $o)),S:(B > (C > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ R5) @ R) => ((((ord_less_eq @ (B > (C > $o))) @ S2) @ S) => (((ord_less_eq @ (A > (C > $o))) @ (((((relcompp @ A) @ B) @ C) @ R5) @ S2)) @ (((((relcompp @ A) @ B) @ C) @ R) @ S))))))).
% 2.87/1.54 thf(fact_183_rev__predicate1D, axiom, (! [A:$tType,P:(A > $o),X:A,Q:(A > $o)]: (((P @ X) => ((((ord_less_eq @ (A > $o)) @ P) @ Q) => (Q @ X)))))).
% 2.87/1.54 thf(fact_184_rev__predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),X:A,Y3:B,Q:(A > (B > $o))]: ((((P @ X) @ Y3) => ((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => ((Q @ X) @ Y3)))))).
% 2.87/1.54 thf(fact_185_top__greatest, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ (top_top @ A)))))))).
% 2.87/1.54 thf(fact_186_top_Oextremum__unique, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) = (A2 = (top_top @ A))))))))).
% 2.87/1.54 thf(fact_187_top_Oextremum__uniqueI, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) => (A2 = (top_top @ A))))))))).
% 2.87/1.54 thf(fact_188_rangeI, axiom, (! [A:$tType,B:$tType,F4:(B > A),X:B]: ((((member @ A) @ (F4 @ X)) @ ((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))))))).
% 2.87/1.54 thf(fact_189_range__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F4:(B > A),X:B]: (((B2 = (F4 @ X)) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F4) @ (top_top @ (set @ B)))))))).
% 2.87/1.54 thf(fact_190_subset__UNIV, axiom, (! [A:$tType,A6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A6) @ (top_top @ (set @ A)))))).
% 2.87/1.54 thf(fact_191_pick__middlep, axiom, (! [B:$tType,A:$tType,C:$tType,P:(A > (B > $o)),Q:(B > (C > $o)),A2:A,C3:C]: (((((((((relcompp @ A) @ B) @ C) @ P) @ Q) @ A2) @ C3) => (((P @ A2) @ (((((((bNF_pick_middlep @ A) @ B) @ C) @ P) @ Q) @ A2) @ C3)) & ((Q @ (((((((bNF_pick_middlep @ A) @ B) @ C) @ P) @ Q) @ A2) @ C3)) @ C3)))))).
% 2.87/1.54 thf(fact_192_UNIV__witness, axiom, (! [A:$tType]: ((? [X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))))).
% 2.87/1.54 thf(fact_193_UNIV__eq__I, axiom, (! [A:$tType,A6:(set @ A)]: (((! [X3:A]: ((((member @ A) @ X3) @ A6))) => ((top_top @ (set @ A)) = A6))))).
% 2.87/1.54 thf(fact_194_nchotomy__relcomppE, axiom, (! [C:$tType,B:$tType,A:$tType,D:$tType,F4:(B > A),R:(C > (A > $o)),S:(A > (D > $o)),A2:C,C3:D]: (((! [Y:A]: ((? [X5:B]: ((Y = (F4 @ X5)))))) => ((((((((relcompp @ C) @ A) @ D) @ R) @ S) @ A2) @ C3) => (~ ((! [B4:B]: ((((R @ A2) @ (F4 @ B4)) => (~ (((S @ (F4 @ B4)) @ C3))))))))))))).
% 2.87/1.54 thf(fact_195_csquare__def, axiom, (! [B:$tType,C:$tType,D:$tType,A:$tType]: ((((((bNF_csquare @ A) @ B) @ C) @ D) = (^ [A7:(set @ A),F12:(B > C),F23:(D > C),P1:(A > B),P22:(A > D)]: ((! [X4:A]: (((((member @ A) @ X4) @ A7) => ((F12 @ (P1 @ X4)) = (F23 @ (P22 @ X4)))))))))))).
% 2.87/1.54 thf(fact_196_fstOp__in, axiom, (! [B:$tType,C:$tType,A:$tType,Ac2:((product_prod @ A) @ B),P:(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) @ P) @ Q)))) => (((member @ ((product_prod @ A) @ C)) @ ((((((bNF_fstOp @ A) @ C) @ B) @ P) @ Q) @ Ac2)) @ ((collect @ ((product_prod @ A) @ C)) @ ((((product_case_prod @ A) @ C) @ $o) @ P))))))).
% 2.87/1.54 thf(fact_197_sndOp__in, axiom, (! [A:$tType,B:$tType,C:$tType,Ac2:((product_prod @ A) @ B),P:(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) @ P) @ Q)))) => (((member @ ((product_prod @ C) @ B)) @ ((((((bNF_sndOp @ A) @ C) @ B) @ P) @ Q) @ Ac2)) @ ((collect @ ((product_prod @ C) @ B)) @ ((((product_case_prod @ C) @ B) @ $o) @ Q))))))).
% 2.87/1.54 thf(fact_198_unfold_I1_J, axiom, (! [A:$tType,Rt:(A > n),Ct:(A > (set @ ((sum_sum @ t) @ A))),B2:A]: (((root @ ((((unfold @ A) @ Rt) @ Ct) @ B2)) = (Rt @ B2))))).
% 2.87/1.54 thf(fact_199_surj__def, axiom, (! [B:$tType,A:$tType,F4:(B > A)]: (((((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y8:A]: ((? [X4:B]: ((Y8 = (F4 @ X4)))))))))).
% 2.87/1.54 thf(fact_200_surjI, axiom, (! [B:$tType,A:$tType,G3:(B > A),F4:(A > B)]: (((! [X3:A]: (((G3 @ (F4 @ X3)) = X3))) => (((((image @ B) @ A) @ G3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.87/1.54 thf(fact_201_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.87/1.54 thf(fact_202_surjD, axiom, (! [A:$tType,B:$tType,F4:(B > A),Y3:A]: (((((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X3:B]: ((Y3 = (F4 @ X3)))))))).
% 2.87/1.54 thf(fact_203_surjE, axiom, (! [A:$tType,B:$tType,F4:(B > A),Y3:A]: (((((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X3:B]: ((Y3 != (F4 @ X3)))))))))).
% 2.87/1.54 thf(fact_204_map__prod__surj, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F4:(A > B),G3:(C > D)]: (((((((image @ A) @ B) @ F4) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((((((image @ C) @ D) @ G3) @ (top_top @ (set @ C))) = (top_top @ (set @ D))) => (((((image @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)) @ ((((((product_map_prod @ A) @ B) @ C) @ D) @ F4) @ G3)) @ (top_top @ (set @ ((product_prod @ A) @ C)))) = (top_top @ (set @ ((product_prod @ B) @ D))))))))).
% 2.87/1.54 thf(fact_205_root__Node, axiom, (! [N2:n,As2:(set @ ((sum_sum @ t) @ dtree))]: (((root @ ((node @ N2) @ As2)) = N2)))).
% 2.87/1.54 thf(fact_206_map__prod__simp, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F4:(C > A),G3:(D > B),A2:C,B2:D]: (((((((((product_map_prod @ C) @ A) @ D) @ B) @ F4) @ G3) @ ((((product_Pair @ C) @ D) @ A2) @ B2)) = ((((product_Pair @ A) @ B) @ (F4 @ A2)) @ (G3 @ B2)))))).
% 2.87/1.54 thf(fact_207_fst__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F4:(C > A),G3:(D > B),X:((product_prod @ C) @ D)]: (((((product_fst @ A) @ B) @ (((((((product_map_prod @ C) @ A) @ D) @ B) @ F4) @ G3) @ X)) = (F4 @ (((product_fst @ C) @ D) @ X)))))).
% 2.87/1.54 thf(fact_208_snd__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F4:(C > B),G3:(D > A),X:((product_prod @ C) @ D)]: (((((product_snd @ B) @ A) @ (((((((product_map_prod @ C) @ B) @ D) @ A) @ F4) @ G3) @ X)) = (G3 @ (((product_snd @ C) @ D) @ X)))))).
% 2.87/1.54 thf(fact_209_map__prod__imageI, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,A2:A,B2:B,R2:(set @ ((product_prod @ A) @ B)),F4:(A > C),G3:(B > D)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R2) => (((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (F4 @ A2)) @ (G3 @ B2))) @ ((((image @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F4) @ G3)) @ R2)))))).
% 2.87/1.54 thf(fact_210_prod__fun__imageE, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,C3:((product_prod @ A) @ B),F4:(C > A),G3:(D > B),R2:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ A) @ B)) @ C3) @ ((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ A) @ D) @ B) @ F4) @ G3)) @ R2)) => (~ ((! [X3:C,Y:D]: (((C3 = ((((product_Pair @ A) @ B) @ (F4 @ X3)) @ (G3 @ Y))) => (~ ((((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ X3) @ Y)) @ R2)))))))))))).
% 2.87/1.54 thf(fact_211_Node__root__cont, axiom, (! [Tr:dtree]: ((((node @ (root @ Tr)) @ (cont @ Tr)) = Tr)))).
% 2.87/1.54 thf(fact_212_Powp__mono, axiom, (! [A:$tType,A6:(A > $o),B6:(A > $o)]: (((((ord_less_eq @ (A > $o)) @ A6) @ B6) => (((ord_less_eq @ ((set @ A) > $o)) @ ((powp @ A) @ A6)) @ ((powp @ A) @ B6)))))).
% 2.87/1.54 thf(fact_213_dtree__cong, axiom, (! [Tr:dtree,Tr4:dtree]: ((((root @ Tr) = (root @ Tr4)) => (((cont @ Tr) = (cont @ Tr4)) => (Tr = Tr4)))))).
% 2.87/1.54 thf(fact_214_predicate2D__conj, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Q:(A > (B > $o)),R2:$o,X:A,Y3:B]: ((((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) & R2) => (R2 & (((P @ X) @ Y3) => ((Q @ X) @ Y3))))))).
% 2.87/1.54 thf(fact_215_corec_I1_J, axiom, (! [A:$tType,Rt:(A > n),Ct:(A > (set @ ((sum_sum @ t) @ ((sum_sum @ dtree) @ A)))),B2:A]: (((root @ ((((corec @ A) @ Rt) @ Ct) @ B2)) = (Rt @ B2))))).
% 2.87/1.54 thf(fact_216_Inl__in__cont__par, axiom, (! [T2:t,Tr12:dtree,Tr22:dtree]: (((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ T2)) @ (cont @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr12) @ Tr22)))) = ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ T2)) @ (cont @ Tr12)) | (((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ T2)) @ (cont @ Tr22))))))).
% 2.87/1.54 thf(fact_217_surj__swap__iff, axiom, (! [B:$tType,A:$tType,A2:B,B2:B,F4:(B > A)]: (((((((image @ B) @ A) @ (((((swap @ B) @ A) @ A2) @ B2) @ F4)) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.87/1.54 thf(fact_218_swap__image__eq, axiom, (! [B:$tType,A:$tType,A2:A,A6:(set @ A),B2:A,F4:(A > B)]: (((((member @ A) @ A2) @ A6) => ((((member @ A) @ B2) @ A6) => (((((image @ A) @ B) @ (((((swap @ A) @ B) @ A2) @ B2) @ F4)) @ A6) = ((((image @ A) @ B) @ F4) @ A6))))))).
% 2.87/1.54 thf(fact_219_surj__imp__surj__swap, axiom, (! [B:$tType,A:$tType,F4:(B > A),A2:B,B2:B]: (((((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((((image @ B) @ A) @ (((((swap @ B) @ A) @ A2) @ B2) @ F4)) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.87/1.54 thf(fact_220_Sup_OSUP__cong, axiom, (! [A:$tType,B:$tType,A6:(set @ B),B6:(set @ B),C4:(B > A),D3:(B > A),Sup:((set @ A) > A)]: (((A6 = B6) => ((! [X3:B]: (((((member @ B) @ X3) @ B6) => ((C4 @ X3) = (D3 @ X3))))) => ((Sup @ ((((image @ B) @ A) @ C4) @ A6)) = (Sup @ ((((image @ B) @ A) @ D3) @ B6)))))))).
% 2.87/1.54 thf(fact_221_Inf_OINF__cong, axiom, (! [A:$tType,B:$tType,A6:(set @ B),B6:(set @ B),C4:(B > A),D3:(B > A),Inf:((set @ A) > A)]: (((A6 = B6) => ((! [X3:B]: (((((member @ B) @ X3) @ B6) => ((C4 @ X3) = (D3 @ X3))))) => ((Inf @ ((((image @ B) @ A) @ C4) @ A6)) = (Inf @ ((((image @ B) @ A) @ D3) @ B6)))))))).
% 2.87/1.54 thf(fact_222_surj__Compl__image__subset, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B)]: (((((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ ((((image @ B) @ A) @ F4) @ A6))) @ ((((image @ B) @ A) @ F4) @ ((uminus_uminus @ (set @ B)) @ A6))))))).
% 2.87/1.54 thf(fact_223_in__inv__image, axiom, (! [A:$tType,B:$tType,X:A,Y3:A,R:(set @ ((product_prod @ B) @ B)),F4:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y3)) @ ((((inv_image @ B) @ A) @ R) @ F4)) = (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F4 @ X)) @ (F4 @ Y3))) @ R))))).
% 2.87/1.54 thf(fact_224_ComplI, axiom, (! [A:$tType,C3:A,A6:(set @ A)]: (((~ ((((member @ A) @ C3) @ A6))) => (((member @ A) @ C3) @ ((uminus_uminus @ (set @ A)) @ A6)))))).
% 2.87/1.54 thf(fact_225_Compl__iff, axiom, (! [A:$tType,C3:A,A6:(set @ A)]: (((((member @ A) @ C3) @ ((uminus_uminus @ (set @ A)) @ A6)) = (~ ((((member @ A) @ C3) @ A6))))))).
% 2.87/1.54 thf(fact_226_Compl__eq__Compl__iff, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((((uminus_uminus @ (set @ A)) @ A6) = ((uminus_uminus @ (set @ A)) @ B6)) = (A6 = B6))))).
% 2.87/1.54 thf(fact_227_Compl__subset__Compl__iff, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A6)) @ ((uminus_uminus @ (set @ A)) @ B6)) = (((ord_less_eq @ (set @ A)) @ B6) @ A6))))).
% 2.87/1.54 thf(fact_228_Compl__anti__mono, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ B6)) @ ((uminus_uminus @ (set @ A)) @ A6)))))).
% 2.87/1.54 thf(fact_229_ComplD, axiom, (! [A:$tType,C3:A,A6:(set @ A)]: (((((member @ A) @ C3) @ ((uminus_uminus @ (set @ A)) @ A6)) => (~ ((((member @ A) @ C3) @ A6))))))).
% 2.87/1.54 thf(fact_230_double__complement, axiom, (! [A:$tType,A6:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A6)) = A6)))).
% 2.87/1.54 thf(fact_231_compl__le__compl__iff, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X)) @ ((uminus_uminus @ A) @ Y3)) = (((ord_less_eq @ A) @ Y3) @ X)))))))).
% 2.87/1.54 thf(fact_232_neg__le__iff__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.87/1.54 thf(fact_233_compl__mono, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y3)) @ ((uminus_uminus @ A) @ X))))))))).
% 2.87/1.54 thf(fact_234_compl__le__swap1, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [Y3:A,X:A]: (((((ord_less_eq @ A) @ Y3) @ ((uminus_uminus @ A) @ X)) => (((ord_less_eq @ A) @ X) @ ((uminus_uminus @ A) @ Y3))))))))).
% 2.87/1.54 thf(fact_235_compl__le__swap2, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [Y3:A,X:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y3)) @ X) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X)) @ Y3)))))))).
% 2.87/1.54 thf(fact_236_le__minus__iff, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ B2)) = (((ord_less_eq @ A) @ B2) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.87/1.54 thf(fact_237_minus__le__iff, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ A2)))))))).
% 2.87/1.54 thf(fact_238_le__imp__neg__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.87/1.54 thf(fact_239_inj__image__Compl__subset, axiom, (! [B:$tType,A:$tType,F4:(A > B),A6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ (top_top @ (set @ A))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F4) @ ((uminus_uminus @ (set @ A)) @ A6))) @ ((uminus_uminus @ (set @ B)) @ ((((image @ A) @ B) @ F4) @ A6))))))).
% 2.87/1.54 thf(fact_240_inj__apfst, axiom, (! [B:$tType,C:$tType,A:$tType,F4:(A > C)]: ((((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ B)) @ ((((product_apfst @ A) @ C) @ B) @ F4)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = ((((inj_on @ A) @ C) @ F4) @ (top_top @ (set @ A))))))).
% 2.87/1.54 thf(fact_241_inj__apsnd, axiom, (! [A:$tType,C:$tType,B:$tType,F4:(B > C)]: ((((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C)) @ ((((product_apsnd @ B) @ C) @ A) @ F4)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = ((((inj_on @ B) @ C) @ F4) @ (top_top @ (set @ B))))))).
% 2.87/1.54 thf(fact_242_inj__image__subset__iff, axiom, (! [B:$tType,A:$tType,F4:(A > B),A6:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ (top_top @ (set @ A))) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F4) @ A6)) @ ((((image @ A) @ B) @ F4) @ B6)) = (((ord_less_eq @ (set @ A)) @ A6) @ B6)))))).
% 2.87/1.54 thf(fact_243_inj__image__mem__iff, axiom, (! [B:$tType,A:$tType,F4:(A > B),A2:A,A6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ (top_top @ (set @ A))) => ((((member @ B) @ (F4 @ A2)) @ ((((image @ A) @ B) @ F4) @ A6)) = (((member @ A) @ A2) @ A6)))))).
% 2.87/1.54 thf(fact_244_inj__image__eq__iff, axiom, (! [B:$tType,A:$tType,F4:(A > B),A6:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ (top_top @ (set @ A))) => ((((((image @ A) @ B) @ F4) @ A6) = ((((image @ A) @ B) @ F4) @ B6)) = (A6 = B6)))))).
% 2.87/1.54 thf(fact_245_range__ex1__eq, axiom, (! [B:$tType,A:$tType,F4:(A > B),B2:B]: ((((((inj_on @ A) @ B) @ F4) @ (top_top @ (set @ A))) => ((((member @ B) @ B2) @ ((((image @ A) @ B) @ F4) @ (top_top @ (set @ A)))) = (? [X4:A]: (((B2 = (F4 @ X4)) & (! [Y8:A]: (((B2 = (F4 @ Y8)) => (Y8 = X4)))))))))))).
% 2.87/1.54 thf(fact_246_inj__swap, axiom, (! [B:$tType,A:$tType,A6:(set @ ((product_prod @ A) @ B))]: (((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ B) @ A)) @ ((product_swap @ A) @ B)) @ A6)))).
% 2.87/1.54 thf(fact_247_inj__on__image__iff, axiom, (! [B:$tType,A:$tType,A6:(set @ A),G3:(A > B),F4:(A > A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A6) => (! [Xa:A]: (((((member @ A) @ Xa) @ A6) => (((G3 @ (F4 @ X3)) = (G3 @ (F4 @ Xa))) = ((G3 @ X3) = (G3 @ Xa))))))))) => (((((inj_on @ A) @ A) @ F4) @ A6) => (((((inj_on @ A) @ B) @ G3) @ ((((image @ A) @ A) @ F4) @ A6)) = ((((inj_on @ A) @ B) @ G3) @ A6))))))).
% 2.87/1.54 thf(fact_248_inj__on__subset, axiom, (! [B:$tType,A:$tType,F4:(A > B),A6:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ A6) => ((((ord_less_eq @ (set @ A)) @ B6) @ A6) => ((((inj_on @ A) @ B) @ F4) @ B6)))))).
% 2.87/1.54 thf(fact_249_subset__inj__on, axiom, (! [B:$tType,A:$tType,F4:(A > B),B6:(set @ A),A6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ B6) => ((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((inj_on @ A) @ B) @ F4) @ A6)))))).
% 2.87/1.54 thf(fact_250_inj__on__image__mem__iff__alt, axiom, (! [B:$tType,A:$tType,F4:(A > B),B6:(set @ A),A6:(set @ A),A2:A]: ((((((inj_on @ A) @ B) @ F4) @ B6) => ((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((member @ B) @ (F4 @ A2)) @ ((((image @ A) @ B) @ F4) @ A6)) => ((((member @ A) @ A2) @ B6) => (((member @ A) @ A2) @ A6)))))))).
% 2.87/1.54 thf(fact_251_inj__on__image__mem__iff, axiom, (! [B:$tType,A:$tType,F4:(A > B),B6:(set @ A),A2:A,A6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ B6) => ((((member @ A) @ A2) @ B6) => ((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((member @ B) @ (F4 @ A2)) @ ((((image @ A) @ B) @ F4) @ A6)) = (((member @ A) @ A2) @ A6)))))))).
% 2.87/1.54 thf(fact_252_inj__on__image__eq__iff, axiom, (! [B:$tType,A:$tType,F4:(A > B),C4:(set @ A),A6:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ C4) => ((((ord_less_eq @ (set @ A)) @ A6) @ C4) => ((((ord_less_eq @ (set @ A)) @ B6) @ C4) => ((((((image @ A) @ B) @ F4) @ A6) = ((((image @ A) @ B) @ F4) @ B6)) = (A6 = B6)))))))).
% 2.87/1.54 thf(fact_253_the__inv__into__into, axiom, (! [B:$tType,A:$tType,F4:(A > B),A6:(set @ A),X:B,B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ A6) => ((((member @ B) @ X) @ ((((image @ A) @ B) @ F4) @ A6)) => ((((ord_less_eq @ (set @ A)) @ A6) @ B6) => (((member @ A) @ (((((the_inv_into @ A) @ B) @ A6) @ F4) @ X)) @ B6))))))).
% 2.87/1.54 thf(fact_254_wf__map__prod__image, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),F4:(A > B)]: ((((wf @ A) @ R) => (((((inj_on @ A) @ B) @ F4) @ (top_top @ (set @ A))) => ((wf @ B) @ ((((image @ ((product_prod @ A) @ A)) @ ((product_prod @ B) @ B)) @ ((((((product_map_prod @ A) @ B) @ A) @ B) @ F4) @ F4)) @ R))))))).
% 2.87/1.54 thf(fact_255_wf__inv__image, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ B) @ B)),F4:(A > B)]: ((((wf @ B) @ R) => ((wf @ A) @ ((((inv_image @ B) @ A) @ R) @ F4)))))).
% 2.87/1.54 thf(tcon_fun___Lattices_Oboolean__algebra, axiom, (! [A8:$tType,A9:$tType]: ((((boolean_algebra @ A9) @ (type2 @ A9)) => ((boolean_algebra @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.87/1.54 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A8:$tType,A9:$tType]: ((((order_top @ A9) @ (type2 @ A9)) => ((order_top @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.87/1.54 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: ((((preorder @ A9) @ (type2 @ A9)) => ((preorder @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.87/1.54 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: ((((order @ A9) @ (type2 @ A9)) => ((order @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.87/1.54 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: ((((ord @ A9) @ (type2 @ A9)) => ((ord @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 2.87/1.54 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_1, axiom, (! [A8:$tType]: (((boolean_algebra @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.87/1.54 thf(tcon_Set_Oset___Orderings_Oorder__top_2, axiom, (! [A8:$tType]: (((order_top @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.87/1.54 thf(tcon_Set_Oset___Orderings_Opreorder_3, axiom, (! [A8:$tType]: (((preorder @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.87/1.54 thf(tcon_Set_Oset___Orderings_Oorder_4, axiom, (! [A8:$tType]: (((order @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.87/1.54 thf(tcon_Set_Oset___Orderings_Oord_5, axiom, (! [A8:$tType]: (((ord @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 2.87/1.54 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_6, axiom, ((boolean_algebra @ $o) @ (type2 @ $o))).
% 2.87/1.54 thf(tcon_HOL_Obool___Orderings_Oorder__top_7, axiom, ((order_top @ $o) @ (type2 @ $o))).
% 2.87/1.54 thf(tcon_HOL_Obool___Orderings_Opreorder_8, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.87/1.54 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.87/1.54 thf(tcon_HOL_Obool___Orderings_Oorder_9, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.87/1.54 thf(tcon_HOL_Obool___Orderings_Oord_10, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.87/1.54 thf(conj_0, conjecture, ((parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ tr1) @ tr2))) @ tr3)) = (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ tr1) @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ tr2) @ tr3)))))).
% 2.87/1.54 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.XABAPK8w6b/DTF2DTF28763.p
% 2.87/1.54 ------------------------
% 2.87/1.57 ---- Cleaned THF ---
% 2.87/1.57 thf(ty_t_Product__Type_Oprod,type,
% 2.87/1.57 product_prod: $tType > $tType > $tType ).
% 2.87/1.57
% 2.87/1.57 thf(ty_t_Sum__Type_Osum,type,
% 2.87/1.57 sum_sum: $tType > $tType > $tType ).
% 2.87/1.57
% 2.87/1.57 thf(ty_t_DTree_Odtree,type,
% 2.87/1.57 dtree: $tType ).
% 2.87/1.57
% 2.87/1.57 thf(ty_t_Set_Oset,type,
% 2.87/1.57 set: $tType > $tType ).
% 2.87/1.57
% 2.87/1.57 thf(ty_t_DTree_OT,type,
% 2.87/1.57 t: $tType ).
% 2.87/1.57
% 2.87/1.57 thf(ty_t_DTree_ON,type,
% 2.87/1.57 n: $tType ).
% 2.87/1.57
% 2.87/1.57 thf(ty_t_itself,type,
% 2.87/1.57 itself: $tType > $tType ).
% 2.87/1.57
% 2.87/1.57 thf(sy_cl_HOL_Otype,type,
% 2.87/1.57 type:
% 2.87/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_cl_Orderings_Oord,type,
% 2.87/1.57 ord:
% 2.87/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_cl_Orderings_Oorder,type,
% 2.87/1.57 order:
% 2.87/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_cl_Orderings_Olinorder,type,
% 2.87/1.57 linorder:
% 2.87/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_cl_Orderings_Opreorder,type,
% 2.87/1.57 preorder:
% 2.87/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_cl_Orderings_Oorder__top,type,
% 2.87/1.57 order_top:
% 2.87/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 2.87/1.57 boolean_algebra:
% 2.87/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.87/1.57 ordered_ab_group_add:
% 2.87/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 2.87/1.57 bNF_Ca1785829860lChain:
% 2.87/1.57 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_BNF__Def_Ocsquare,type,
% 2.87/1.57 bNF_csquare:
% 2.87/1.57 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( set @ A ) > ( B > C ) > ( D > C ) > ( A > B ) > ( A > D ) > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_BNF__Def_OfstOp,type,
% 2.87/1.57 bNF_fstOp:
% 2.87/1.57 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_BNF__Def_Opick__middlep,type,
% 2.87/1.57 bNF_pick_middlep:
% 2.87/1.57 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_BNF__Def_OsndOp,type,
% 2.87/1.57 bNF_sndOp:
% 2.87/1.57 !>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 2.87/1.57 bNF_Greatest_image2:
% 2.87/1.57 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_DTree_ONode,type,
% 2.87/1.57 node: n > ( set @ ( sum_sum @ t @ dtree ) ) > dtree ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_DTree_Ocont,type,
% 2.87/1.57 cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_DTree_Ocorec,type,
% 2.87/1.57 corec:
% 2.87/1.57 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ) ) > A > dtree ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_DTree_Odtree_Oroot,type,
% 2.87/1.57 root: dtree > n ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_DTree_Ounfold,type,
% 2.87/1.57 unfold:
% 2.87/1.57 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Fun_Oinj__on,type,
% 2.87/1.57 inj_on:
% 2.87/1.57 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Fun_Oswap,type,
% 2.87/1.57 swap:
% 2.87/1.57 !>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Fun_Othe__inv__into,type,
% 2.87/1.57 the_inv_into:
% 2.87/1.57 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Fun__Def_Oin__rel,type,
% 2.87/1.57 fun_in_rel:
% 2.87/1.57 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > A > B > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 2.87/1.57 uminus_uminus:
% 2.87/1.57 !>[A: $tType] : ( A > A ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.87/1.57 ord_less_eq:
% 2.87/1.57 !>[A: $tType] : ( A > A > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.87/1.57 top_top:
% 2.87/1.57 !>[A: $tType] : A ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_ONplus,type,
% 2.87/1.57 parall1518086719_Nplus: n > n > n ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar,type,
% 2.87/1.57 parall1899940088le_par: ( product_prod @ dtree @ dtree ) > dtree ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c,type,
% 2.87/1.57 parall1914194347_par_c: ( product_prod @ dtree @ dtree ) > ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r,type,
% 2.87/1.57 parall1914194362_par_r: ( product_prod @ dtree @ dtree ) > n ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r__rel,type,
% 2.87/1.57 parall556292031_r_rel: ( product_prod @ dtree @ dtree ) > ( product_prod @ dtree @ dtree ) > $o ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Product__Type_OPair,type,
% 2.87/1.57 product_Pair:
% 2.87/1.57 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Product__Type_Oapfst,type,
% 2.87/1.57 product_apfst:
% 2.87/1.57 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Product__Type_Oapsnd,type,
% 2.87/1.57 product_apsnd:
% 2.87/1.57 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Product__Type_Ocurry,type,
% 2.87/1.57 product_curry:
% 2.87/1.57 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 2.87/1.57 produc2004651681e_prod:
% 2.87/1.57 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Product__Type_Omap__prod,type,
% 2.87/1.57 product_map_prod:
% 2.87/1.57 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
% 2.87/1.57 product_rec_bool:
% 2.87/1.57 !>[T: $tType] : ( T > T > $o > T ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.87/1.57 product_rec_prod:
% 2.87/1.57 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 2.87/1.57 product_case_prod:
% 2.87/1.57 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 2.87/1.57 product_fst:
% 2.87/1.57 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 2.87/1.57 product_snd:
% 2.87/1.57 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 2.87/1.57 product_swap:
% 2.87/1.57 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Product__Type_Oscomp,type,
% 2.87/1.57 product_scomp:
% 2.87/1.57 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Pure_Otype,type,
% 2.87/1.57 type2:
% 2.87/1.57 !>[A: $tType] : ( itself @ A ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Relation_OPowp,type,
% 2.87/1.57 powp:
% 2.87/1.57 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Relation_Oinv__image,type,
% 2.87/1.57 inv_image:
% 2.87/1.57 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Relation_Orelcompp,type,
% 2.87/1.57 relcompp:
% 2.87/1.57 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Set_OCollect,type,
% 2.87/1.57 collect:
% 2.87/1.57 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Set_Oimage,type,
% 2.87/1.57 image:
% 2.87/1.57 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Sum__Type_OInl,type,
% 2.87/1.57 sum_Inl:
% 2.87/1.57 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Wellfounded_Oaccp,type,
% 2.87/1.57 accp:
% 2.87/1.57 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Wellfounded_Olex__prod,type,
% 2.87/1.57 lex_prod:
% 2.87/1.57 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Wellfounded_Owf,type,
% 2.87/1.57 wf:
% 2.87/1.57 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_Wfrec_Osame__fst,type,
% 2.87/1.57 same_fst:
% 2.87/1.57 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_c_member,type,
% 2.87/1.57 member:
% 2.87/1.57 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.87/1.57
% 2.87/1.57 thf(sy_v_tr1,type,
% 2.87/1.57 tr1: dtree ).
% 2.87/1.57
% 2.87/1.57 thf(sy_v_tr2,type,
% 2.87/1.57 tr2: dtree ).
% 2.87/1.57
% 2.87/1.57 thf(sy_v_tr3,type,
% 2.87/1.57 tr3: dtree ).
% 2.87/1.57
% 2.87/1.57 thf(fact_0__092_060open_062_092_060And_062trB_AtrA_O_A_092_060exists_062tr1_Atr2_Atr3_O_AtrA_A_061_A_Itr1_A_092_060parallel_062_Atr2_J_A_092_060parallel_062_Atr3_A_092_060and_062_AtrB_A_061_Atr1_A_092_060parallel_062_Atr2_A_092_060parallel_062_Atr3_A_092_060Longrightarrow_062_AtrA_A_061_AtrB_092_060close_062,axiom,
% 2.87/1.57 ! [TrA: dtree,TrB: dtree] :
% 2.87/1.57 ( ? [Tr1: dtree,Tr2: dtree,Tr3: dtree] :
% 2.87/1.57 ( ( TrA
% 2.87/1.57 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) @ Tr3 ) ) )
% 2.87/1.57 & ( TrB
% 2.87/1.57 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr1 @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr2 @ Tr3 ) ) ) ) ) )
% 2.87/1.57 => ( TrA = TrB ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_1_par__com,axiom,
% 2.87/1.57 ! [Tr12: dtree,Tr22: dtree] :
% 2.87/1.57 ( ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) )
% 2.87/1.57 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr22 @ Tr12 ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_2_par__r_Ocases,axiom,
% 2.87/1.57 ! [X: product_prod @ dtree @ dtree] :
% 2.87/1.57 ~ ! [Tr13: dtree,Tr23: dtree] :
% 2.87/1.57 ( X
% 2.87/1.57 != ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_3_par__r_Oinduct,axiom,
% 2.87/1.57 ! [P: ( product_prod @ dtree @ dtree ) > $o,A0: product_prod @ dtree @ dtree] :
% 2.87/1.57 ( ! [Tr13: dtree,Tr23: dtree] : ( P @ ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) )
% 2.87/1.57 => ( P @ A0 ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_4_prod_Oinject,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 2.87/1.57 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 2.87/1.57 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 2.87/1.57 = ( ( X1 = Y1 )
% 2.87/1.57 & ( X2 = Y2 ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_5_old_Oprod_Oinject,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 2.87/1.57 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.87/1.57 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.87/1.57 = ( ( A2 = A3 )
% 2.87/1.57 & ( B2 = B3 ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_6_surj__pair,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 2.87/1.57 ? [X3: A,Y: B] :
% 2.87/1.57 ( P2
% 2.87/1.57 = ( product_Pair @ A @ B @ X3 @ Y ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_7_prod__cases,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 2.87/1.57 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 2.87/1.57 => ( P @ P2 ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_8_Pair__inject,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
% 2.87/1.57 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.87/1.57 = ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.87/1.57 => ~ ( ( A2 = A3 )
% 2.87/1.57 => ( B2 != B3 ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_9_prod__cases3,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.87/1.57 ~ ! [A4: A,B4: B,C2: C] :
% 2.87/1.57 ( Y3
% 2.87/1.57 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_10_prod__cases4,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.87/1.57 ~ ! [A4: A,B4: B,C2: C,D2: D] :
% 2.87/1.57 ( Y3
% 2.87/1.57 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_11_prod__cases5,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.87/1.57 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
% 2.87/1.57 ( Y3
% 2.87/1.57 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_12_prod__cases6,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 2.87/1.57 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
% 2.87/1.57 ( Y3
% 2.87/1.57 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_13_prod__cases7,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 2.87/1.57 ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
% 2.87/1.57 ( Y3
% 2.87/1.57 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_14_prod__induct3,axiom,
% 2.87/1.57 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.87/1.57 ( ! [A4: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
% 2.87/1.57 => ( P @ X ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_15_prod__induct4,axiom,
% 2.87/1.57 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.87/1.57 ( ! [A4: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 2.87/1.57 => ( P @ X ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_16_old_Oprod_Oinducts,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.87/1.57 ( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
% 2.87/1.57 => ( P @ Prod ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_17_old_Oprod_Oexhaust,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 2.87/1.57 ~ ! [A4: A,B4: B] :
% 2.87/1.57 ( Y3
% 2.87/1.57 != ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_18_prod__induct7,axiom,
% 2.87/1.57 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 2.87/1.57 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 2.87/1.57 => ( P @ X ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_19_prod__induct6,axiom,
% 2.87/1.57 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 2.87/1.57 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 2.87/1.57 => ( P @ X ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_20_prod__induct5,axiom,
% 2.87/1.57 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.87/1.57 ( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 2.87/1.57 => ( P @ X ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_21_old_Oprod_Orec,axiom,
% 2.87/1.57 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 2.87/1.57 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.87/1.57 = ( F1 @ A2 @ B2 ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_22_internal__case__prod__conv,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
% 2.87/1.57 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 2.87/1.57 = ( C3 @ A2 @ B2 ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_23_ssubst__Pair__rhs,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,R: A,S: B,R2: set @ ( product_prod @ A @ B ),S2: B] :
% 2.87/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R2 )
% 2.87/1.57 => ( ( S2 = S )
% 2.87/1.57 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R2 ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_24_curry__conv,axiom,
% 2.87/1.57 ! [A: $tType,C: $tType,B: $tType] :
% 2.87/1.57 ( ( product_curry @ B @ C @ A )
% 2.87/1.57 = ( ^ [F3: ( product_prod @ B @ C ) > A,A5: B,B5: C] : ( F3 @ ( product_Pair @ B @ C @ A5 @ B5 ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_25_curryI,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 2.87/1.57 ( ( F4 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.87/1.57 => ( product_curry @ A @ B @ $o @ F4 @ A2 @ B2 ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_26_swap__simp,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,X: B,Y3: A] :
% 2.87/1.57 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) )
% 2.87/1.57 = ( product_Pair @ A @ B @ Y3 @ X ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_27_apsnd__conv,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,C: $tType,F4: C > B,X: A,Y3: C] :
% 2.87/1.57 ( ( product_apsnd @ C @ B @ A @ F4 @ ( product_Pair @ A @ C @ X @ Y3 ) )
% 2.87/1.57 = ( product_Pair @ A @ B @ X @ ( F4 @ Y3 ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_28_old_Obool_Osimps_I6_J,axiom,
% 2.87/1.57 ! [T: $tType,F1: T,F22: T] :
% 2.87/1.57 ( ( product_rec_bool @ T @ F1 @ F22 @ $false )
% 2.87/1.57 = F22 ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_29_old_Obool_Osimps_I5_J,axiom,
% 2.87/1.57 ! [T: $tType,F1: T,F22: T] :
% 2.87/1.57 ( ( product_rec_bool @ T @ F1 @ F22 @ $true )
% 2.87/1.57 = F1 ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_30_apfst__conv,axiom,
% 2.87/1.57 ! [C: $tType,A: $tType,B: $tType,F4: C > A,X: C,Y3: B] :
% 2.87/1.57 ( ( product_apfst @ C @ A @ B @ F4 @ ( product_Pair @ C @ B @ X @ Y3 ) )
% 2.87/1.57 = ( product_Pair @ A @ B @ ( F4 @ X ) @ Y3 ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_31_swap__swap,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 2.87/1.57 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 2.87/1.57 = P2 ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_32_apsnd__apfst__commute,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,C: $tType,D: $tType,F4: C > B,G3: D > A,P2: product_prod @ D @ C] :
% 2.87/1.57 ( ( product_apsnd @ C @ B @ A @ F4 @ ( product_apfst @ D @ A @ C @ G3 @ P2 ) )
% 2.87/1.57 = ( product_apfst @ D @ A @ B @ G3 @ ( product_apsnd @ C @ B @ D @ F4 @ P2 ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_33_curryD,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 2.87/1.57 ( ( product_curry @ A @ B @ $o @ F4 @ A2 @ B2 )
% 2.87/1.57 => ( F4 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_34_curryE,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 2.87/1.57 ( ( product_curry @ A @ B @ $o @ F4 @ A2 @ B2 )
% 2.87/1.57 => ( F4 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_35_pair__in__swap__image,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,Y3: A,X: B,A6: set @ ( product_prod @ B @ A )] :
% 2.87/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A6 ) )
% 2.87/1.57 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) @ A6 ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_36_internal__case__prod__def,axiom,
% 2.87/1.57 ! [C: $tType,B: $tType,A: $tType] :
% 2.87/1.57 ( ( produc2004651681e_prod @ A @ B @ C )
% 2.87/1.57 = ( product_case_prod @ A @ B @ C ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_37_case__prod__curry,axiom,
% 2.87/1.57 ! [C: $tType,B: $tType,A: $tType,F4: ( product_prod @ A @ B ) > C] :
% 2.87/1.57 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F4 ) )
% 2.87/1.57 = F4 ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_38_curry__case__prod,axiom,
% 2.87/1.57 ! [C: $tType,B: $tType,A: $tType,F4: A > B > C] :
% 2.87/1.57 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F4 ) )
% 2.87/1.57 = F4 ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_39_Pair__scomp,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,C: $tType,X: C,F4: C > A > B] :
% 2.87/1.57 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F4 )
% 2.87/1.57 = ( F4 @ X ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_40_same__fstI,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,P: A > $o,X: A,Y4: B,Y3: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
% 2.87/1.57 ( ( P @ X )
% 2.87/1.57 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y4 @ Y3 ) @ ( R2 @ X ) )
% 2.87/1.57 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( product_Pair @ A @ B @ X @ Y3 ) ) @ ( same_fst @ A @ B @ P @ R2 ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_41_root__par,axiom,
% 2.87/1.57 ! [Tr12: dtree,Tr22: dtree] :
% 2.87/1.57 ( ( root @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) )
% 2.87/1.57 = ( parall1518086719_Nplus @ ( root @ Tr12 ) @ ( root @ Tr22 ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_42_in__lex__prod,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
% 2.87/1.57 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ A3 @ B3 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
% 2.87/1.57 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A3 ) @ R )
% 2.87/1.57 | ( ( A2 = A3 )
% 2.87/1.57 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B3 ) @ S ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_43_apfst__apsnd,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,D: $tType,C: $tType,F4: C > A,G3: D > B,X: product_prod @ C @ D] :
% 2.87/1.57 ( ( product_apfst @ C @ A @ B @ F4 @ ( product_apsnd @ D @ B @ C @ G3 @ X ) )
% 2.87/1.57 = ( product_Pair @ A @ B @ ( F4 @ ( product_fst @ C @ D @ X ) ) @ ( G3 @ ( product_snd @ C @ D @ X ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_44_scomp__apply,axiom,
% 2.87/1.57 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 2.87/1.57 ( ( product_scomp @ B @ C @ D @ A )
% 2.87/1.57 = ( ^ [F3: B > ( product_prod @ C @ D ),G4: C > D > A,X4: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F3 @ X4 ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_45_mem__Collect__eq,axiom,
% 2.87/1.57 ! [A: $tType,A2: A,P: A > $o] :
% 2.87/1.57 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.87/1.57 = ( P @ A2 ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_46_Collect__mem__eq,axiom,
% 2.87/1.57 ! [A: $tType,A6: set @ A] :
% 2.87/1.57 ( ( collect @ A
% 2.87/1.57 @ ^ [X4: A] : ( member @ A @ X4 @ A6 ) )
% 2.87/1.57 = A6 ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_47_Collect__cong,axiom,
% 2.87/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.87/1.57 ( ! [X3: A] :
% 2.87/1.57 ( ( P @ X3 )
% 2.87/1.57 = ( Q @ X3 ) )
% 2.87/1.57 => ( ( collect @ A @ P )
% 2.87/1.57 = ( collect @ A @ Q ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_48_ext,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,F4: A > B,G3: A > B] :
% 2.87/1.57 ( ! [X3: A] :
% 2.87/1.57 ( ( F4 @ X3 )
% 2.87/1.57 = ( G3 @ X3 ) )
% 2.87/1.57 => ( F4 = G3 ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_49_apfst__eq__conv,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,C: $tType,F4: C > A,X: product_prod @ C @ B,G3: C > A] :
% 2.87/1.57 ( ( ( product_apfst @ C @ A @ B @ F4 @ X )
% 2.87/1.57 = ( product_apfst @ C @ A @ B @ G3 @ X ) )
% 2.87/1.57 = ( ( F4 @ ( product_fst @ C @ B @ X ) )
% 2.87/1.57 = ( G3 @ ( product_fst @ C @ B @ X ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_50_fst__apfst,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,C: $tType,F4: C > A,X: product_prod @ C @ B] :
% 2.87/1.57 ( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F4 @ X ) )
% 2.87/1.57 = ( F4 @ ( product_fst @ C @ B @ X ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_51_snd__apfst,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,C: $tType,F4: C > B,X: product_prod @ C @ A] :
% 2.87/1.57 ( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F4 @ X ) )
% 2.87/1.57 = ( product_snd @ C @ A @ X ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_52_fst__apsnd,axiom,
% 2.87/1.57 ! [B: $tType,C: $tType,A: $tType,F4: C > B,X: product_prod @ A @ C] :
% 2.87/1.57 ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F4 @ X ) )
% 2.87/1.57 = ( product_fst @ A @ C @ X ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_53_apsnd__eq__conv,axiom,
% 2.87/1.57 ! [B: $tType,C: $tType,A: $tType,F4: C > B,X: product_prod @ A @ C,G3: C > B] :
% 2.87/1.57 ( ( ( product_apsnd @ C @ B @ A @ F4 @ X )
% 2.87/1.57 = ( product_apsnd @ C @ B @ A @ G3 @ X ) )
% 2.87/1.57 = ( ( F4 @ ( product_snd @ A @ C @ X ) )
% 2.87/1.57 = ( G3 @ ( product_snd @ A @ C @ X ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_54_snd__apsnd,axiom,
% 2.87/1.57 ! [A: $tType,C: $tType,B: $tType,F4: C > A,X: product_prod @ B @ C] :
% 2.87/1.57 ( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F4 @ X ) )
% 2.87/1.57 = ( F4 @ ( product_snd @ B @ C @ X ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_55_prod_Ocollapse,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.87/1.57 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 2.87/1.57 = Prod ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_56_snd__swap,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 2.87/1.57 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 2.87/1.57 = ( product_fst @ A @ B @ X ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_57_fst__swap,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 2.87/1.57 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 2.87/1.57 = ( product_snd @ B @ A @ X ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_58_apsnd__apfst,axiom,
% 2.87/1.57 ! [A: $tType,B: $tType,C: $tType,D: $tType,F4: C > B,G3: D > A,X: product_prod @ D @ C] :
% 2.87/1.57 ( ( product_apsnd @ C @ B @ A @ F4 @ ( product_apfst @ D @ A @ C @ G3 @ X ) )
% 2.87/1.57 = ( product_Pair @ A @ B @ ( G3 @ ( product_fst @ D @ C @ X ) ) @ ( F4 @ ( product_snd @ D @ C @ X ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_59_Nplus__comm,axiom,
% 2.87/1.57 ( parall1518086719_Nplus
% 2.87/1.57 = ( ^ [A5: n,B5: n] : ( parall1518086719_Nplus @ B5 @ A5 ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_60_Nplus__assoc,axiom,
% 2.87/1.57 ! [A2: n,B2: n,C3: n] :
% 2.87/1.57 ( ( parall1518086719_Nplus @ ( parall1518086719_Nplus @ A2 @ B2 ) @ C3 )
% 2.87/1.57 = ( parall1518086719_Nplus @ A2 @ ( parall1518086719_Nplus @ B2 @ C3 ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_61_surjective__pairing,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 2.87/1.57 ( T2
% 2.87/1.57 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_62_prod_Osplit__sel__asm,axiom,
% 2.87/1.57 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F4: A > B > C,Prod: product_prod @ A @ B] :
% 2.87/1.57 ( ( P @ ( product_case_prod @ A @ B @ C @ F4 @ Prod ) )
% 2.87/1.57 = ( ~ ( ( Prod
% 2.87/1.57 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 2.87/1.57 & ~ ( P @ ( F4 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_63_Product__Type_OCollect__case__prodD,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: A > B > $o] :
% 2.87/1.57 ( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) )
% 2.87/1.57 => ( A6 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_64_prod_Oexhaust__sel,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.87/1.57 ( Prod
% 2.87/1.57 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_65_prod_Ocase__eq__if,axiom,
% 2.87/1.57 ! [C: $tType,B: $tType,A: $tType] :
% 2.87/1.57 ( ( product_case_prod @ A @ B @ C )
% 2.87/1.57 = ( ^ [F3: A > B > C,Prod2: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ Prod2 ) @ ( product_snd @ A @ B @ Prod2 ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_66_prod_Osplit__sel,axiom,
% 2.87/1.57 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F4: A > B > C,Prod: product_prod @ A @ B] :
% 2.87/1.57 ( ( P @ ( product_case_prod @ A @ B @ C @ F4 @ Prod ) )
% 2.87/1.57 = ( ( Prod
% 2.87/1.57 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 2.87/1.57 => ( P @ ( F4 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_67_case__prod__beta,axiom,
% 2.87/1.57 ! [A: $tType,C: $tType,B: $tType] :
% 2.87/1.57 ( ( product_case_prod @ B @ C @ A )
% 2.87/1.57 = ( ^ [F3: B > C > A,P3: product_prod @ B @ C] : ( F3 @ ( product_fst @ B @ C @ P3 ) @ ( product_snd @ B @ C @ P3 ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_68_prod__eq__iff,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType] :
% 2.87/1.57 ( ( ^ [Y5: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y5 = Z ) )
% 2.87/1.57 = ( ^ [S3: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 2.87/1.57 ( ( ( product_fst @ A @ B @ S3 )
% 2.87/1.57 = ( product_fst @ A @ B @ T3 ) )
% 2.87/1.57 & ( ( product_snd @ A @ B @ S3 )
% 2.87/1.57 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_69_prod_Oexpand,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod3: product_prod @ A @ B] :
% 2.87/1.57 ( ( ( ( product_fst @ A @ B @ Prod )
% 2.87/1.57 = ( product_fst @ A @ B @ Prod3 ) )
% 2.87/1.57 & ( ( product_snd @ A @ B @ Prod )
% 2.87/1.57 = ( product_snd @ A @ B @ Prod3 ) ) )
% 2.87/1.57 => ( Prod = Prod3 ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_70_scomp__def,axiom,
% 2.87/1.57 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 2.87/1.57 ( ( product_scomp @ A @ B @ C @ D )
% 2.87/1.57 = ( ^ [F3: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F3 @ X4 ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_71_prod__eqI,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 2.87/1.57 ( ( ( product_fst @ A @ B @ P2 )
% 2.87/1.57 = ( product_fst @ A @ B @ Q2 ) )
% 2.87/1.57 => ( ( ( product_snd @ A @ B @ P2 )
% 2.87/1.57 = ( product_snd @ A @ B @ Q2 ) )
% 2.87/1.57 => ( P2 = Q2 ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_72_prod_Oswap__def,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType] :
% 2.87/1.57 ( ( product_swap @ A @ B )
% 2.87/1.57 = ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_73_snd__conv,axiom,
% 2.87/1.57 ! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
% 2.87/1.57 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
% 2.87/1.57 = X2 ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_74_snd__eqD,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,X: B,Y3: A,A2: A] :
% 2.87/1.57 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) )
% 2.87/1.57 = A2 )
% 2.87/1.57 => ( Y3 = A2 ) ) ).
% 2.87/1.57
% 2.87/1.57 thf(fact_75_fst__conv,axiom,
% 2.87/1.57 ! [B: $tType,A: $tType,X1: A,X2: B] :
% 2.87/1.57 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 2.87/1.58 = X1 ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_76_fst__eqD,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,X: A,Y3: B,A2: A] :
% 2.87/1.58 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y3 ) )
% 2.87/1.58 = A2 )
% 2.87/1.58 => ( X = A2 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_77_mem__case__prodE,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,C: $tType,Z2: A,C3: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 2.87/1.58 ( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P2 ) )
% 2.87/1.58 => ~ ! [X3: B,Y: C] :
% 2.87/1.58 ( ( P2
% 2.87/1.58 = ( product_Pair @ B @ C @ X3 @ Y ) )
% 2.87/1.58 => ~ ( member @ A @ Z2 @ ( C3 @ X3 @ Y ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_78_old_Oprod_Ocase,axiom,
% 2.87/1.58 ! [A: $tType,C: $tType,B: $tType,F4: A > B > C,X1: A,X2: B] :
% 2.87/1.58 ( ( product_case_prod @ A @ B @ C @ F4 @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 2.87/1.58 = ( F4 @ X1 @ X2 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_79_scomp__Pair,axiom,
% 2.87/1.58 ! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
% 2.87/1.58 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
% 2.87/1.58 = X ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_80_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,P: A > B > $o,X: A,Y3: B,A2: product_prod @ A @ B] :
% 2.87/1.58 ( ( P @ X @ Y3 )
% 2.87/1.58 => ( ( A2
% 2.87/1.58 = ( product_Pair @ A @ B @ X @ Y3 ) )
% 2.87/1.58 => ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_81_conjI__realizer,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
% 2.87/1.58 ( ( P @ P2 )
% 2.87/1.58 => ( ( Q @ Q2 )
% 2.87/1.58 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
% 2.87/1.58 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_82_exI__realizer,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,P: A > B > $o,Y3: A,X: B] :
% 2.87/1.58 ( ( P @ Y3 @ X )
% 2.87/1.58 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_83_par__r_Osimps,axiom,
% 2.87/1.58 ! [Tr12: dtree,Tr22: dtree] :
% 2.87/1.58 ( ( parall1914194362_par_r @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) )
% 2.87/1.58 = ( parall1518086719_Nplus @ ( root @ Tr12 ) @ ( root @ Tr22 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_84_par__r_Oelims,axiom,
% 2.87/1.58 ! [X: product_prod @ dtree @ dtree,Y3: n] :
% 2.87/1.58 ( ( ( parall1914194362_par_r @ X )
% 2.87/1.58 = Y3 )
% 2.87/1.58 => ~ ! [Tr13: dtree,Tr23: dtree] :
% 2.87/1.58 ( ( X
% 2.87/1.58 = ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) )
% 2.87/1.58 => ( Y3
% 2.87/1.58 != ( parall1518086719_Nplus @ ( root @ Tr13 ) @ ( root @ Tr23 ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_85_image__eqI,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,B2: A,F4: B > A,X: B,A6: set @ B] :
% 2.87/1.58 ( ( B2
% 2.87/1.58 = ( F4 @ X ) )
% 2.87/1.58 => ( ( member @ B @ X @ A6 )
% 2.87/1.58 => ( member @ A @ B2 @ ( image @ B @ A @ F4 @ A6 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_86_exE__realizer_H,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
% 2.87/1.58 ( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
% 2.87/1.58 => ~ ! [X3: B,Y: A] :
% 2.87/1.58 ~ ( P @ Y @ X3 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_87_sndI,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y3: A,Z2: B] :
% 2.87/1.58 ( ( X
% 2.87/1.58 = ( product_Pair @ A @ B @ Y3 @ Z2 ) )
% 2.87/1.58 => ( ( product_snd @ A @ B @ X )
% 2.87/1.58 = Z2 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_88_rev__image__eqI,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,X: A,A6: set @ A,B2: B,F4: A > B] :
% 2.87/1.58 ( ( member @ A @ X @ A6 )
% 2.87/1.58 => ( ( B2
% 2.87/1.58 = ( F4 @ X ) )
% 2.87/1.58 => ( member @ B @ B2 @ ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_89_ball__imageD,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P: A > $o] :
% 2.87/1.58 ( ! [X3: A] :
% 2.87/1.58 ( ( member @ A @ X3 @ ( image @ B @ A @ F4 @ A6 ) )
% 2.87/1.58 => ( P @ X3 ) )
% 2.87/1.58 => ! [X5: B] :
% 2.87/1.58 ( ( member @ B @ X5 @ A6 )
% 2.87/1.58 => ( P @ ( F4 @ X5 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_90_image__cong,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F4: A > B,G3: A > B] :
% 2.87/1.58 ( ( M = N )
% 2.87/1.58 => ( ! [X3: A] :
% 2.87/1.58 ( ( member @ A @ X3 @ N )
% 2.87/1.58 => ( ( F4 @ X3 )
% 2.87/1.58 = ( G3 @ X3 ) ) )
% 2.87/1.58 => ( ( image @ A @ B @ F4 @ M )
% 2.87/1.58 = ( image @ A @ B @ G3 @ N ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_91_bex__imageD,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P: A > $o] :
% 2.87/1.58 ( ? [X5: A] :
% 2.87/1.58 ( ( member @ A @ X5 @ ( image @ B @ A @ F4 @ A6 ) )
% 2.87/1.58 & ( P @ X5 ) )
% 2.87/1.58 => ? [X3: B] :
% 2.87/1.58 ( ( member @ B @ X3 @ A6 )
% 2.87/1.58 & ( P @ ( F4 @ X3 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_92_image__iff,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,Z2: A,F4: B > A,A6: set @ B] :
% 2.87/1.58 ( ( member @ A @ Z2 @ ( image @ B @ A @ F4 @ A6 ) )
% 2.87/1.58 = ( ? [X4: B] :
% 2.87/1.58 ( ( member @ B @ X4 @ A6 )
% 2.87/1.58 & ( Z2
% 2.87/1.58 = ( F4 @ X4 ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_93_imageI,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,X: A,A6: set @ A,F4: A > B] :
% 2.87/1.58 ( ( member @ A @ X @ A6 )
% 2.87/1.58 => ( member @ B @ ( F4 @ X ) @ ( image @ A @ B @ F4 @ A6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_94_fstI,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y3: A,Z2: B] :
% 2.87/1.58 ( ( X
% 2.87/1.58 = ( product_Pair @ A @ B @ Y3 @ Z2 ) )
% 2.87/1.58 => ( ( product_fst @ A @ B @ X )
% 2.87/1.58 = Y3 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_95_eq__snd__iff,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
% 2.87/1.58 ( ( B2
% 2.87/1.58 = ( product_snd @ B @ A @ P2 ) )
% 2.87/1.58 = ( ? [A5: B] :
% 2.87/1.58 ( P2
% 2.87/1.58 = ( product_Pair @ B @ A @ A5 @ B2 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_96_eq__fst__iff,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
% 2.87/1.58 ( ( A2
% 2.87/1.58 = ( product_fst @ A @ B @ P2 ) )
% 2.87/1.58 = ( ? [B5: B] :
% 2.87/1.58 ( P2
% 2.87/1.58 = ( product_Pair @ A @ B @ A2 @ B5 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_97_split__cong,axiom,
% 2.87/1.58 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F4: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
% 2.87/1.58 ( ! [X3: A,Y: B] :
% 2.87/1.58 ( ( ( product_Pair @ A @ B @ X3 @ Y )
% 2.87/1.58 = Q2 )
% 2.87/1.58 => ( ( F4 @ X3 @ Y )
% 2.87/1.58 = ( G3 @ X3 @ Y ) ) )
% 2.87/1.58 => ( ( P2 = Q2 )
% 2.87/1.58 => ( ( product_case_prod @ A @ B @ C @ F4 @ P2 )
% 2.87/1.58 = ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_98_par__r_Opelims,axiom,
% 2.87/1.58 ! [X: product_prod @ dtree @ dtree,Y3: n] :
% 2.87/1.58 ( ( ( parall1914194362_par_r @ X )
% 2.87/1.58 = Y3 )
% 2.87/1.58 => ( ( accp @ ( product_prod @ dtree @ dtree ) @ parall556292031_r_rel @ X )
% 2.87/1.58 => ~ ! [Tr13: dtree,Tr23: dtree] :
% 2.87/1.58 ( ( X
% 2.87/1.58 = ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) )
% 2.87/1.58 => ( ( Y3
% 2.87/1.58 = ( parall1518086719_Nplus @ ( root @ Tr13 ) @ ( root @ Tr23 ) ) )
% 2.87/1.58 => ~ ( accp @ ( product_prod @ dtree @ dtree ) @ parall556292031_r_rel @ ( product_Pair @ dtree @ dtree @ Tr13 @ Tr23 ) ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_99_Collect__split__mono__strong,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,X6: set @ A,A6: set @ ( product_prod @ A @ B ),Y6: set @ B,P: A > B > $o,Q: A > B > $o] :
% 2.87/1.58 ( ( X6
% 2.87/1.58 = ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 ) )
% 2.87/1.58 => ( ( Y6
% 2.87/1.58 = ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) )
% 2.87/1.58 => ( ! [X3: A] :
% 2.87/1.58 ( ( member @ A @ X3 @ X6 )
% 2.87/1.58 => ! [Xa: B] :
% 2.87/1.58 ( ( member @ B @ Xa @ Y6 )
% 2.87/1.58 => ( ( P @ X3 @ Xa )
% 2.87/1.58 => ( Q @ X3 @ Xa ) ) ) )
% 2.87/1.58 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
% 2.87/1.58 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_100_fstOp__def,axiom,
% 2.87/1.58 ! [C: $tType,B: $tType,A: $tType] :
% 2.87/1.58 ( ( bNF_fstOp @ A @ B @ C )
% 2.87/1.58 = ( ^ [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 ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_101_subset__antisym,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.87/1.58 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
% 2.87/1.58 => ( A6 = B6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_102_subsetI,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 2.87/1.58 ( ! [X3: A] :
% 2.87/1.58 ( ( member @ A @ X3 @ A6 )
% 2.87/1.58 => ( member @ A @ X3 @ B6 ) )
% 2.87/1.58 => ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_103_Collect__mono__iff,axiom,
% 2.87/1.58 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.87/1.58 = ( ! [X4: A] :
% 2.87/1.58 ( ( P @ X4 )
% 2.87/1.58 => ( Q @ X4 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_104_contra__subsetD,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A,B6: set @ A,C3: A] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.87/1.58 => ( ~ ( member @ A @ C3 @ B6 )
% 2.87/1.58 => ~ ( member @ A @ C3 @ A6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_105_set__eq__subset,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( ^ [Y5: set @ A,Z: set @ A] : ( Y5 = Z ) )
% 2.87/1.58 = ( ^ [A7: set @ A,B7: set @ A] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.87/1.58 & ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_106_subset__trans,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.87/1.58 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 2.87/1.58 => ( ord_less_eq @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_107_Collect__mono,axiom,
% 2.87/1.58 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.87/1.58 ( ! [X3: A] :
% 2.87/1.58 ( ( P @ X3 )
% 2.87/1.58 => ( Q @ X3 ) )
% 2.87/1.58 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_108_subset__refl,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ A6 ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_109_rev__subsetD,axiom,
% 2.87/1.58 ! [A: $tType,C3: A,A6: set @ A,B6: set @ A] :
% 2.87/1.58 ( ( member @ A @ C3 @ A6 )
% 2.87/1.58 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.87/1.58 => ( member @ A @ C3 @ B6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_110_subset__iff,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) )
% 2.87/1.58 = ( ^ [A7: set @ A,B7: set @ A] :
% 2.87/1.58 ! [T3: A] :
% 2.87/1.58 ( ( member @ A @ T3 @ A7 )
% 2.87/1.58 => ( member @ A @ T3 @ B7 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_111_set__rev__mp,axiom,
% 2.87/1.58 ! [A: $tType,X: A,A6: set @ A,B6: set @ A] :
% 2.87/1.58 ( ( member @ A @ X @ A6 )
% 2.87/1.58 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.87/1.58 => ( member @ A @ X @ B6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_112_equalityD2,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 2.87/1.58 ( ( A6 = B6 )
% 2.87/1.58 => ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_113_equalityD1,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 2.87/1.58 ( ( A6 = B6 )
% 2.87/1.58 => ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_114_subset__eq,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) )
% 2.87/1.58 = ( ^ [A7: set @ A,B7: set @ A] :
% 2.87/1.58 ! [X4: A] :
% 2.87/1.58 ( ( member @ A @ X4 @ A7 )
% 2.87/1.58 => ( member @ A @ X4 @ B7 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_115_equalityE,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 2.87/1.58 ( ( A6 = B6 )
% 2.87/1.58 => ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.87/1.58 => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_116_subsetCE,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A,B6: set @ A,C3: A] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.87/1.58 => ( ( member @ A @ C3 @ A6 )
% 2.87/1.58 => ( member @ A @ C3 @ B6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_117_subsetD,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A,B6: set @ A,C3: A] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.87/1.58 => ( ( member @ A @ C3 @ A6 )
% 2.87/1.58 => ( member @ A @ C3 @ B6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_118_in__mono,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A,B6: set @ A,X: A] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.87/1.58 => ( ( member @ A @ X @ A6 )
% 2.87/1.58 => ( member @ A @ X @ B6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_119_set__mp,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A,B6: set @ A,X: A] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.87/1.58 => ( ( member @ A @ X @ A6 )
% 2.87/1.58 => ( member @ A @ X @ B6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_120_accp__induct__rule,axiom,
% 2.87/1.58 ! [A: $tType,R: A > A > $o,A2: A,P: A > $o] :
% 2.87/1.58 ( ( accp @ A @ R @ A2 )
% 2.87/1.58 => ( ! [X3: A] :
% 2.87/1.58 ( ( accp @ A @ R @ X3 )
% 2.87/1.58 => ( ! [Y7: A] :
% 2.87/1.58 ( ( R @ Y7 @ X3 )
% 2.87/1.58 => ( P @ Y7 ) )
% 2.87/1.58 => ( P @ X3 ) ) )
% 2.87/1.58 => ( P @ A2 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_121_not__accp__down,axiom,
% 2.87/1.58 ! [A: $tType,R2: A > A > $o,X: A] :
% 2.87/1.58 ( ~ ( accp @ A @ R2 @ X )
% 2.87/1.58 => ~ ! [Z3: A] :
% 2.87/1.58 ( ( R2 @ Z3 @ X )
% 2.87/1.58 => ( accp @ A @ R2 @ Z3 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_122_accp__downward,axiom,
% 2.87/1.58 ! [A: $tType,R: A > A > $o,B2: A,A2: A] :
% 2.87/1.58 ( ( accp @ A @ R @ B2 )
% 2.87/1.58 => ( ( R @ A2 @ B2 )
% 2.87/1.58 => ( accp @ A @ R @ A2 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_123_accp_Oinducts,axiom,
% 2.87/1.58 ! [A: $tType,R: A > A > $o,X: A,P: A > $o] :
% 2.87/1.58 ( ( accp @ A @ R @ X )
% 2.87/1.58 => ( ! [X3: A] :
% 2.87/1.58 ( ! [Y7: A] :
% 2.87/1.58 ( ( R @ Y7 @ X3 )
% 2.87/1.58 => ( accp @ A @ R @ Y7 ) )
% 2.87/1.58 => ( ! [Y7: A] :
% 2.87/1.58 ( ( R @ Y7 @ X3 )
% 2.87/1.58 => ( P @ Y7 ) )
% 2.87/1.58 => ( P @ X3 ) ) )
% 2.87/1.58 => ( P @ X ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_124_accp__induct,axiom,
% 2.87/1.58 ! [A: $tType,R: A > A > $o,A2: A,P: A > $o] :
% 2.87/1.58 ( ( accp @ A @ R @ A2 )
% 2.87/1.58 => ( ! [X3: A] :
% 2.87/1.58 ( ( accp @ A @ R @ X3 )
% 2.87/1.58 => ( ! [Y7: A] :
% 2.87/1.58 ( ( R @ Y7 @ X3 )
% 2.87/1.58 => ( P @ Y7 ) )
% 2.87/1.58 => ( P @ X3 ) ) )
% 2.87/1.58 => ( P @ A2 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_125_accp_Ointros,axiom,
% 2.87/1.58 ! [A: $tType,R: A > A > $o,X: A] :
% 2.87/1.58 ( ! [Y: A] :
% 2.87/1.58 ( ( R @ Y @ X )
% 2.87/1.58 => ( accp @ A @ R @ Y ) )
% 2.87/1.58 => ( accp @ A @ R @ X ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_126_accp_Osimps,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( accp @ A )
% 2.87/1.58 = ( ^ [R3: A > A > $o,A5: A] :
% 2.87/1.58 ? [X4: A] :
% 2.87/1.58 ( ( A5 = X4 )
% 2.87/1.58 & ! [Y8: A] :
% 2.87/1.58 ( ( R3 @ Y8 @ X4 )
% 2.87/1.58 => ( accp @ A @ R3 @ Y8 ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_127_accp_Ocases,axiom,
% 2.87/1.58 ! [A: $tType,R: A > A > $o,A2: A] :
% 2.87/1.58 ( ( accp @ A @ R @ A2 )
% 2.87/1.58 => ! [Y7: A] :
% 2.87/1.58 ( ( R @ Y7 @ A2 )
% 2.87/1.58 => ( accp @ A @ R @ Y7 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_128_subset__image__iff,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,B6: set @ A,F4: B > A,A6: set @ B] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
% 2.87/1.58 = ( ? [AA: set @ B] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ B ) @ AA @ A6 )
% 2.87/1.58 & ( B6
% 2.87/1.58 = ( image @ B @ A @ F4 @ AA ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_129_image__subset__iff,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,B6: set @ A] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ B6 )
% 2.87/1.58 = ( ! [X4: B] :
% 2.87/1.58 ( ( member @ B @ X4 @ A6 )
% 2.87/1.58 => ( member @ A @ ( F4 @ X4 ) @ B6 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_130_subset__imageE,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,B6: set @ A,F4: B > A,A6: set @ B] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
% 2.87/1.58 => ~ ! [C5: set @ B] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A6 )
% 2.87/1.58 => ( B6
% 2.87/1.58 != ( image @ B @ A @ F4 @ C5 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_131_image__subsetI,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,A6: set @ A,F4: A > B,B6: set @ B] :
% 2.87/1.58 ( ! [X3: A] :
% 2.87/1.58 ( ( member @ A @ X3 @ A6 )
% 2.87/1.58 => ( member @ B @ ( F4 @ X3 ) @ B6 ) )
% 2.87/1.58 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ B6 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_132_image__mono,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,F4: A > B] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.87/1.58 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_133_sndOp__def,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,C: $tType] :
% 2.87/1.58 ( ( bNF_sndOp @ C @ A @ B )
% 2.87/1.58 = ( ^ [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 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_134_order__refl,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( preorder @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_135_subrelI,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.87/1.58 ( ! [X3: A,Y: B] :
% 2.87/1.58 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R )
% 2.87/1.58 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ S ) )
% 2.87/1.58 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_136_relChain__def,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType] :
% 2.87/1.58 ( ( ord @ B @ ( type2 @ B ) )
% 2.87/1.58 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 2.87/1.58 = ( ^ [R3: set @ ( product_prod @ A @ A ),As: A > B] :
% 2.87/1.58 ! [I: A,J: A] :
% 2.87/1.58 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R3 )
% 2.87/1.58 => ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_137_Collect__case__prod__in__rel__leE,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,X6: set @ ( product_prod @ A @ B ),Y6: set @ ( product_prod @ A @ B )] :
% 2.87/1.58 ( ( 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 @ Y6 ) ) ) )
% 2.87/1.58 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ Y6 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_138_accp__subset,axiom,
% 2.87/1.58 ! [A: $tType,R1: A > A > $o,R22: A > A > $o] :
% 2.87/1.58 ( ( ord_less_eq @ ( A > A > $o ) @ R1 @ R22 )
% 2.87/1.58 => ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R22 ) @ ( accp @ A @ R1 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_139_accp__subset__induct,axiom,
% 2.87/1.58 ! [A: $tType,D3: A > $o,R2: A > A > $o,X: A,P: A > $o] :
% 2.87/1.58 ( ( ord_less_eq @ ( A > $o ) @ D3 @ ( accp @ A @ R2 ) )
% 2.87/1.58 => ( ! [X3: A,Z3: A] :
% 2.87/1.58 ( ( D3 @ X3 )
% 2.87/1.58 => ( ( R2 @ Z3 @ X3 )
% 2.87/1.58 => ( D3 @ Z3 ) ) )
% 2.87/1.58 => ( ( D3 @ X )
% 2.87/1.58 => ( ! [X3: A] :
% 2.87/1.58 ( ( D3 @ X3 )
% 2.87/1.58 => ( ! [Z4: A] :
% 2.87/1.58 ( ( R2 @ Z4 @ X3 )
% 2.87/1.58 => ( P @ Z4 ) )
% 2.87/1.58 => ( P @ X3 ) ) )
% 2.87/1.58 => ( P @ X ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_140_in__rel__def,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType] :
% 2.87/1.58 ( ( fun_in_rel @ A @ B )
% 2.87/1.58 = ( ^ [R4: set @ ( product_prod @ A @ B ),X4: A,Y8: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y8 ) @ R4 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_141_in__rel__Collect__case__prod__eq,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,X6: A > B > $o] :
% 2.87/1.58 ( ( fun_in_rel @ A @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ X6 ) ) )
% 2.87/1.58 = X6 ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_142_Collect__case__prod__mono,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,A6: A > B > $o,B6: A > B > $o] :
% 2.87/1.58 ( ( ord_less_eq @ ( A > B > $o ) @ A6 @ B6 )
% 2.87/1.58 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B6 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_143_dual__order_Oantisym,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [B2: A,A2: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.87/1.58 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.87/1.58 => ( A2 = B2 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_144_dual__order_Otrans,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [B2: A,A2: A,C3: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.87/1.58 => ( ( ord_less_eq @ A @ C3 @ B2 )
% 2.87/1.58 => ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_145_linorder__wlog,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [P: A > A > $o,A2: A,B2: A] :
% 2.87/1.58 ( ! [A4: A,B4: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.87/1.58 => ( P @ A4 @ B4 ) )
% 2.87/1.58 => ( ! [A4: A,B4: A] :
% 2.87/1.58 ( ( P @ B4 @ A4 )
% 2.87/1.58 => ( P @ A4 @ B4 ) )
% 2.87/1.58 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_146_dual__order_Orefl,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_147_order__trans,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( preorder @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [X: A,Y3: A,Z2: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ X @ Y3 )
% 2.87/1.58 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 2.87/1.58 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_148_order__class_Oorder_Oantisym,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [A2: A,B2: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.87/1.58 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.87/1.58 => ( A2 = B2 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_149_ord__le__eq__trans,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( ord @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [A2: A,B2: A,C3: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.87/1.58 => ( ( B2 = C3 )
% 2.87/1.58 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_150_ord__eq__le__trans,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( ord @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [A2: A,B2: A,C3: A] :
% 2.87/1.58 ( ( A2 = B2 )
% 2.87/1.58 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 2.87/1.58 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_151_antisym__conv,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [Y3: A,X: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ Y3 @ X )
% 2.87/1.58 => ( ( ord_less_eq @ A @ X @ Y3 )
% 2.87/1.58 = ( X = Y3 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_152_le__cases3,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [X: A,Y3: A,Z2: A] :
% 2.87/1.58 ( ( ( ord_less_eq @ A @ X @ Y3 )
% 2.87/1.58 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 2.87/1.58 => ( ( ( ord_less_eq @ A @ Y3 @ X )
% 2.87/1.58 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 2.87/1.58 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 2.87/1.58 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 2.87/1.58 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 2.87/1.58 => ~ ( ord_less_eq @ A @ Y3 @ X ) )
% 2.87/1.58 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 2.87/1.58 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 2.87/1.58 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 2.87/1.58 => ~ ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_153_order_Otrans,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [A2: A,B2: A,C3: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.87/1.58 => ( ( ord_less_eq @ A @ B2 @ C3 )
% 2.87/1.58 => ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_154_le__cases,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [X: A,Y3: A] :
% 2.87/1.58 ( ~ ( ord_less_eq @ A @ X @ Y3 )
% 2.87/1.58 => ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_155_eq__refl,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( preorder @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [X: A,Y3: A] :
% 2.87/1.58 ( ( X = Y3 )
% 2.87/1.58 => ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_156_linear,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( linorder @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [X: A,Y3: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ X @ Y3 )
% 2.87/1.58 | ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_157_antisym,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [X: A,Y3: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ X @ Y3 )
% 2.87/1.58 => ( ( ord_less_eq @ A @ Y3 @ X )
% 2.87/1.58 => ( X = Y3 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_158_eq__iff,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( order @ A @ ( type2 @ A ) )
% 2.87/1.58 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 2.87/1.58 = ( ^ [X4: A,Y8: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ X4 @ Y8 )
% 2.87/1.58 & ( ord_less_eq @ A @ Y8 @ X4 ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_159_ord__le__eq__subst,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType] :
% 2.87/1.58 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.87/1.58 & ( ord @ A @ ( type2 @ A ) ) )
% 2.87/1.58 => ! [A2: A,B2: A,F4: A > B,C3: B] :
% 2.87/1.58 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.87/1.58 => ( ( ( F4 @ B2 )
% 2.87/1.58 = C3 )
% 2.87/1.58 => ( ! [X3: A,Y: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ X3 @ Y )
% 2.87/1.58 => ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( F4 @ Y ) ) )
% 2.87/1.58 => ( ord_less_eq @ B @ ( F4 @ A2 ) @ C3 ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_160_ord__eq__le__subst,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType] :
% 2.87/1.58 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.87/1.58 & ( ord @ A @ ( type2 @ A ) ) )
% 2.87/1.58 => ! [A2: A,F4: B > A,B2: B,C3: B] :
% 2.87/1.58 ( ( A2
% 2.87/1.58 = ( F4 @ B2 ) )
% 2.87/1.58 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 2.87/1.58 => ( ! [X3: B,Y: B] :
% 2.87/1.58 ( ( ord_less_eq @ B @ X3 @ Y )
% 2.87/1.58 => ( ord_less_eq @ A @ ( F4 @ X3 ) @ ( F4 @ Y ) ) )
% 2.87/1.58 => ( ord_less_eq @ A @ A2 @ ( F4 @ C3 ) ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_161_order__subst2,axiom,
% 2.87/1.58 ! [A: $tType,C: $tType] :
% 2.87/1.58 ( ( ( order @ C @ ( type2 @ C ) )
% 2.87/1.58 & ( order @ A @ ( type2 @ A ) ) )
% 2.87/1.58 => ! [A2: A,B2: A,F4: A > C,C3: C] :
% 2.87/1.58 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.87/1.58 => ( ( ord_less_eq @ C @ ( F4 @ B2 ) @ C3 )
% 2.87/1.58 => ( ! [X3: A,Y: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ X3 @ Y )
% 2.87/1.58 => ( ord_less_eq @ C @ ( F4 @ X3 ) @ ( F4 @ Y ) ) )
% 2.87/1.58 => ( ord_less_eq @ C @ ( F4 @ A2 ) @ C3 ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_162_order__subst1,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType] :
% 2.87/1.58 ( ( ( order @ B @ ( type2 @ B ) )
% 2.87/1.58 & ( order @ A @ ( type2 @ A ) ) )
% 2.87/1.58 => ! [A2: A,F4: B > A,B2: B,C3: B] :
% 2.87/1.58 ( ( ord_less_eq @ A @ A2 @ ( F4 @ B2 ) )
% 2.87/1.58 => ( ( ord_less_eq @ B @ B2 @ C3 )
% 2.87/1.58 => ( ! [X3: B,Y: B] :
% 2.87/1.58 ( ( ord_less_eq @ B @ X3 @ Y )
% 2.87/1.58 => ( ord_less_eq @ A @ ( F4 @ X3 ) @ ( F4 @ Y ) ) )
% 2.87/1.58 => ( ord_less_eq @ A @ A2 @ ( F4 @ C3 ) ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_163_le__fun__def,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType] :
% 2.87/1.58 ( ( ord @ B @ ( type2 @ B ) )
% 2.87/1.58 => ( ( ord_less_eq @ ( A > B ) )
% 2.87/1.58 = ( ^ [F3: A > B,G4: A > B] :
% 2.87/1.58 ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_164_le__funI,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType] :
% 2.87/1.58 ( ( ord @ B @ ( type2 @ B ) )
% 2.87/1.58 => ! [F4: A > B,G3: A > B] :
% 2.87/1.58 ( ! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G3 @ X3 ) )
% 2.87/1.58 => ( ord_less_eq @ ( A > B ) @ F4 @ G3 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_165_le__funE,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType] :
% 2.87/1.58 ( ( ord @ B @ ( type2 @ B ) )
% 2.87/1.58 => ! [F4: A > B,G3: A > B,X: A] :
% 2.87/1.58 ( ( ord_less_eq @ ( A > B ) @ F4 @ G3 )
% 2.87/1.58 => ( ord_less_eq @ B @ ( F4 @ X ) @ ( G3 @ X ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_166_le__funD,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType] :
% 2.87/1.58 ( ( ord @ B @ ( type2 @ B ) )
% 2.87/1.58 => ! [F4: A > B,G3: A > B,X: A] :
% 2.87/1.58 ( ( ord_less_eq @ ( A > B ) @ F4 @ G3 )
% 2.87/1.58 => ( ord_less_eq @ B @ ( F4 @ X ) @ ( G3 @ X ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_167_Collect__case__prod__in__rel__leI,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,X6: set @ ( product_prod @ A @ B ),Y6: set @ ( product_prod @ A @ B )] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ Y6 )
% 2.87/1.58 => ( 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 @ Y6 ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_168_csquare__fstOp__sndOp,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,C: $tType,F4: ( A > B > $o ) > ( product_prod @ A @ B ) > $o,P: 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 ) @ ( F4 @ ( relcompp @ A @ C @ B @ P @ Q ) ) ) @ ( product_snd @ A @ C ) @ ( product_fst @ C @ B ) @ ( bNF_fstOp @ A @ C @ B @ P @ Q ) @ ( bNF_sndOp @ A @ C @ B @ P @ Q ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_169_image2__eqI,axiom,
% 2.87/1.58 ! [A: $tType,C: $tType,B: $tType,B2: A,F4: B > A,X: B,C3: C,G3: B > C,A6: set @ B] :
% 2.87/1.58 ( ( B2
% 2.87/1.58 = ( F4 @ X ) )
% 2.87/1.58 => ( ( C3
% 2.87/1.58 = ( G3 @ X ) )
% 2.87/1.58 => ( ( member @ B @ X @ A6 )
% 2.87/1.58 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F4 @ G3 ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_170_surj__swap,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType] :
% 2.87/1.58 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.87/1.58 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_171_par__def,axiom,
% 2.87/1.58 ( parall1899940088le_par
% 2.87/1.58 = ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_172_UNIV__I,axiom,
% 2.87/1.58 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_173_predicate2I,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 2.87/1.58 ( ! [X3: A,Y: B] :
% 2.87/1.58 ( ( P @ X3 @ Y )
% 2.87/1.58 => ( Q @ X3 @ Y ) )
% 2.87/1.58 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_174_predicate1I,axiom,
% 2.87/1.58 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.87/1.58 ( ! [X3: A] :
% 2.87/1.58 ( ( P @ X3 )
% 2.87/1.58 => ( Q @ X3 ) )
% 2.87/1.58 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_175_range__fst,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType] :
% 2.87/1.58 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.87/1.58 = ( top_top @ ( set @ A ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_176_range__snd,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType] :
% 2.87/1.58 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.87/1.58 = ( top_top @ ( set @ A ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_177_refl__ge__eq,axiom,
% 2.87/1.58 ! [A: $tType,R2: A > A > $o] :
% 2.87/1.58 ( ! [X3: A] : ( R2 @ X3 @ X3 )
% 2.87/1.58 => ( ord_less_eq @ ( A > A > $o )
% 2.87/1.58 @ ^ [Y5: A,Z: A] : ( Y5 = Z )
% 2.87/1.58 @ R2 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_178_ge__eq__refl,axiom,
% 2.87/1.58 ! [A: $tType,R2: A > A > $o,X: A] :
% 2.87/1.58 ( ( ord_less_eq @ ( A > A > $o )
% 2.87/1.58 @ ^ [Y5: A,Z: A] : ( Y5 = Z )
% 2.87/1.58 @ R2 )
% 2.87/1.58 => ( R2 @ X @ X ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_179_leq__OOI,axiom,
% 2.87/1.58 ! [A: $tType,R2: A > A > $o] :
% 2.87/1.58 ( ( R2
% 2.87/1.58 = ( ^ [Y5: A,Z: A] : ( Y5 = Z ) ) )
% 2.87/1.58 => ( ord_less_eq @ ( A > A > $o ) @ R2 @ ( relcompp @ A @ A @ A @ R2 @ R2 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_180_predicate1D,axiom,
% 2.87/1.58 ! [A: $tType,P: A > $o,Q: A > $o,X: A] :
% 2.87/1.58 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 2.87/1.58 => ( ( P @ X )
% 2.87/1.58 => ( Q @ X ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_181_predicate2D,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y3: B] :
% 2.87/1.58 ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 2.87/1.58 => ( ( P @ X @ Y3 )
% 2.87/1.58 => ( Q @ X @ Y3 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_182_relcompp__mono,axiom,
% 2.87/1.58 ! [A: $tType,C: $tType,B: $tType,R5: A > B > $o,R: A > B > $o,S2: B > C > $o,S: B > C > $o] :
% 2.87/1.58 ( ( ord_less_eq @ ( A > B > $o ) @ R5 @ R )
% 2.87/1.58 => ( ( ord_less_eq @ ( B > C > $o ) @ S2 @ S )
% 2.87/1.58 => ( ord_less_eq @ ( A > C > $o ) @ ( relcompp @ A @ B @ C @ R5 @ S2 ) @ ( relcompp @ A @ B @ C @ R @ S ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_183_rev__predicate1D,axiom,
% 2.87/1.58 ! [A: $tType,P: A > $o,X: A,Q: A > $o] :
% 2.87/1.58 ( ( P @ X )
% 2.87/1.58 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 2.87/1.58 => ( Q @ X ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_184_rev__predicate2D,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,P: A > B > $o,X: A,Y3: B,Q: A > B > $o] :
% 2.87/1.58 ( ( P @ X @ Y3 )
% 2.87/1.58 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 2.87/1.58 => ( Q @ X @ Y3 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_185_top__greatest,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( order_top @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_186_top_Oextremum__unique,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( order_top @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [A2: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 2.87/1.58 = ( A2
% 2.87/1.58 = ( top_top @ A ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_187_top_Oextremum__uniqueI,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( order_top @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [A2: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 2.87/1.58 => ( A2
% 2.87/1.58 = ( top_top @ A ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_188_rangeI,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,F4: B > A,X: B] : ( member @ A @ ( F4 @ X ) @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_189_range__eqI,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,B2: A,F4: B > A,X: B] :
% 2.87/1.58 ( ( B2
% 2.87/1.58 = ( F4 @ X ) )
% 2.87/1.58 => ( member @ A @ B2 @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_190_subset__UNIV,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_191_pick__middlep,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,C: $tType,P: A > B > $o,Q: B > C > $o,A2: A,C3: C] :
% 2.87/1.58 ( ( relcompp @ A @ B @ C @ P @ Q @ A2 @ C3 )
% 2.87/1.58 => ( ( P @ A2 @ ( bNF_pick_middlep @ A @ B @ C @ P @ Q @ A2 @ C3 ) )
% 2.87/1.58 & ( Q @ ( bNF_pick_middlep @ A @ B @ C @ P @ Q @ A2 @ C3 ) @ C3 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_192_UNIV__witness,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_193_UNIV__eq__I,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A] :
% 2.87/1.58 ( ! [X3: A] : ( member @ A @ X3 @ A6 )
% 2.87/1.58 => ( ( top_top @ ( set @ A ) )
% 2.87/1.58 = A6 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_194_nchotomy__relcomppE,axiom,
% 2.87/1.58 ! [C: $tType,B: $tType,A: $tType,D: $tType,F4: B > A,R: C > A > $o,S: A > D > $o,A2: C,C3: D] :
% 2.87/1.58 ( ! [Y: A] :
% 2.87/1.58 ? [X5: B] :
% 2.87/1.58 ( Y
% 2.87/1.58 = ( F4 @ X5 ) )
% 2.87/1.58 => ( ( relcompp @ C @ A @ D @ R @ S @ A2 @ C3 )
% 2.87/1.58 => ~ ! [B4: B] :
% 2.87/1.58 ( ( R @ A2 @ ( F4 @ B4 ) )
% 2.87/1.58 => ~ ( S @ ( F4 @ B4 ) @ C3 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_195_csquare__def,axiom,
% 2.87/1.58 ! [B: $tType,C: $tType,D: $tType,A: $tType] :
% 2.87/1.58 ( ( bNF_csquare @ A @ B @ C @ D )
% 2.87/1.58 = ( ^ [A7: set @ A,F12: B > C,F23: D > C,P1: A > B,P22: A > D] :
% 2.87/1.58 ! [X4: A] :
% 2.87/1.58 ( ( member @ A @ X4 @ A7 )
% 2.87/1.58 => ( ( F12 @ ( P1 @ X4 ) )
% 2.87/1.58 = ( F23 @ ( P22 @ X4 ) ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_196_fstOp__in,axiom,
% 2.87/1.58 ! [B: $tType,C: $tType,A: $tType,Ac2: product_prod @ A @ B,P: A > C > $o,Q: C > B > $o] :
% 2.87/1.58 ( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P @ Q ) ) ) )
% 2.87/1.58 => ( member @ ( product_prod @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P @ Q @ Ac2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ P ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_197_sndOp__in,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,C: $tType,Ac2: product_prod @ A @ B,P: A > C > $o,Q: C > B > $o] :
% 2.87/1.58 ( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P @ Q ) ) ) )
% 2.87/1.58 => ( member @ ( product_prod @ C @ B ) @ ( bNF_sndOp @ A @ C @ B @ P @ Q @ Ac2 ) @ ( collect @ ( product_prod @ C @ B ) @ ( product_case_prod @ C @ B @ $o @ Q ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_198_unfold_I1_J,axiom,
% 2.87/1.58 ! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ A ) ),B2: A] :
% 2.87/1.58 ( ( root @ ( unfold @ A @ Rt @ Ct @ B2 ) )
% 2.87/1.58 = ( Rt @ B2 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_199_surj__def,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,F4: B > A] :
% 2.87/1.58 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 2.87/1.58 = ( top_top @ ( set @ A ) ) )
% 2.87/1.58 = ( ! [Y8: A] :
% 2.87/1.58 ? [X4: B] :
% 2.87/1.58 ( Y8
% 2.87/1.58 = ( F4 @ X4 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_200_surjI,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,G3: B > A,F4: A > B] :
% 2.87/1.58 ( ! [X3: A] :
% 2.87/1.58 ( ( G3 @ ( F4 @ X3 ) )
% 2.87/1.58 = X3 )
% 2.87/1.58 => ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
% 2.87/1.58 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_201_top__set__def,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( top_top @ ( set @ A ) )
% 2.87/1.58 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_202_surjD,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,F4: B > A,Y3: A] :
% 2.87/1.58 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 2.87/1.58 = ( top_top @ ( set @ A ) ) )
% 2.87/1.58 => ? [X3: B] :
% 2.87/1.58 ( Y3
% 2.87/1.58 = ( F4 @ X3 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_203_surjE,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,F4: B > A,Y3: A] :
% 2.87/1.58 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 2.87/1.58 = ( top_top @ ( set @ A ) ) )
% 2.87/1.58 => ~ ! [X3: B] :
% 2.87/1.58 ( Y3
% 2.87/1.58 != ( F4 @ X3 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_204_map__prod__surj,axiom,
% 2.87/1.58 ! [A: $tType,C: $tType,D: $tType,B: $tType,F4: A > B,G3: C > D] :
% 2.87/1.58 ( ( ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 2.87/1.58 = ( top_top @ ( set @ B ) ) )
% 2.87/1.58 => ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
% 2.87/1.58 = ( top_top @ ( set @ D ) ) )
% 2.87/1.58 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F4 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 2.87/1.58 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_205_root__Node,axiom,
% 2.87/1.58 ! [N2: n,As2: set @ ( sum_sum @ t @ dtree )] :
% 2.87/1.58 ( ( root @ ( node @ N2 @ As2 ) )
% 2.87/1.58 = N2 ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_206_map__prod__simp,axiom,
% 2.87/1.58 ! [C: $tType,A: $tType,B: $tType,D: $tType,F4: C > A,G3: D > B,A2: C,B2: D] :
% 2.87/1.58 ( ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ ( product_Pair @ C @ D @ A2 @ B2 ) )
% 2.87/1.58 = ( product_Pair @ A @ B @ ( F4 @ A2 ) @ ( G3 @ B2 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_207_fst__map__prod,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > A,G3: D > B,X: product_prod @ C @ D] :
% 2.87/1.58 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ X ) )
% 2.87/1.58 = ( F4 @ ( product_fst @ C @ D @ X ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_208_snd__map__prod,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > B,G3: D > A,X: product_prod @ C @ D] :
% 2.87/1.58 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F4 @ G3 @ X ) )
% 2.87/1.58 = ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_209_map__prod__imageI,axiom,
% 2.87/1.58 ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),F4: A > C,G3: B > D] :
% 2.87/1.58 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
% 2.87/1.58 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F4 @ A2 ) @ ( G3 @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F4 @ G3 ) @ R2 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_210_prod__fun__imageE,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F4: C > A,G3: D > B,R2: set @ ( product_prod @ C @ D )] :
% 2.87/1.58 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 ) @ R2 ) )
% 2.87/1.58 => ~ ! [X3: C,Y: D] :
% 2.87/1.58 ( ( C3
% 2.87/1.58 = ( product_Pair @ A @ B @ ( F4 @ X3 ) @ ( G3 @ Y ) ) )
% 2.87/1.58 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y ) @ R2 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_211_Node__root__cont,axiom,
% 2.87/1.58 ! [Tr: dtree] :
% 2.87/1.58 ( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
% 2.87/1.58 = Tr ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_212_Powp__mono,axiom,
% 2.87/1.58 ! [A: $tType,A6: A > $o,B6: A > $o] :
% 2.87/1.58 ( ( ord_less_eq @ ( A > $o ) @ A6 @ B6 )
% 2.87/1.58 => ( ord_less_eq @ ( ( set @ A ) > $o ) @ ( powp @ A @ A6 ) @ ( powp @ A @ B6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_213_dtree__cong,axiom,
% 2.87/1.58 ! [Tr: dtree,Tr4: dtree] :
% 2.87/1.58 ( ( ( root @ Tr )
% 2.87/1.58 = ( root @ Tr4 ) )
% 2.87/1.58 => ( ( ( cont @ Tr )
% 2.87/1.58 = ( cont @ Tr4 ) )
% 2.87/1.58 => ( Tr = Tr4 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_214_predicate2D__conj,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R2: $o,X: A,Y3: B] :
% 2.87/1.58 ( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 2.87/1.58 & R2 )
% 2.87/1.58 => ( R2
% 2.87/1.58 & ( ( P @ X @ Y3 )
% 2.87/1.58 => ( Q @ X @ Y3 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_215_corec_I1_J,axiom,
% 2.87/1.58 ! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ),B2: A] :
% 2.87/1.58 ( ( root @ ( corec @ A @ Rt @ Ct @ B2 ) )
% 2.87/1.58 = ( Rt @ B2 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_216_Inl__in__cont__par,axiom,
% 2.87/1.58 ! [T2: t,Tr12: dtree,Tr22: dtree] :
% 2.87/1.58 ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) )
% 2.87/1.58 = ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr12 ) )
% 2.87/1.58 | ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr22 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_217_surj__swap__iff,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,A2: B,B2: B,F4: B > A] :
% 2.87/1.58 ( ( ( image @ B @ A @ ( swap @ B @ A @ A2 @ B2 @ F4 ) @ ( top_top @ ( set @ B ) ) )
% 2.87/1.58 = ( top_top @ ( set @ A ) ) )
% 2.87/1.58 = ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 2.87/1.58 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_218_swap__image__eq,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,A2: A,A6: set @ A,B2: A,F4: A > B] :
% 2.87/1.58 ( ( member @ A @ A2 @ A6 )
% 2.87/1.58 => ( ( member @ A @ B2 @ A6 )
% 2.87/1.58 => ( ( image @ A @ B @ ( swap @ A @ B @ A2 @ B2 @ F4 ) @ A6 )
% 2.87/1.58 = ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_219_surj__imp__surj__swap,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,F4: B > A,A2: B,B2: B] :
% 2.87/1.58 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 2.87/1.58 = ( top_top @ ( set @ A ) ) )
% 2.87/1.58 => ( ( image @ B @ A @ ( swap @ B @ A @ A2 @ B2 @ F4 ) @ ( top_top @ ( set @ B ) ) )
% 2.87/1.58 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_220_Sup_OSUP__cong,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,A6: set @ B,B6: set @ B,C4: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 2.87/1.58 ( ( A6 = B6 )
% 2.87/1.58 => ( ! [X3: B] :
% 2.87/1.58 ( ( member @ B @ X3 @ B6 )
% 2.87/1.58 => ( ( C4 @ X3 )
% 2.87/1.58 = ( D3 @ X3 ) ) )
% 2.87/1.58 => ( ( Sup @ ( image @ B @ A @ C4 @ A6 ) )
% 2.87/1.58 = ( Sup @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_221_Inf_OINF__cong,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,A6: set @ B,B6: set @ B,C4: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 2.87/1.58 ( ( A6 = B6 )
% 2.87/1.58 => ( ! [X3: B] :
% 2.87/1.58 ( ( member @ B @ X3 @ B6 )
% 2.87/1.58 => ( ( C4 @ X3 )
% 2.87/1.58 = ( D3 @ X3 ) ) )
% 2.87/1.58 => ( ( Inf @ ( image @ B @ A @ C4 @ A6 ) )
% 2.87/1.58 = ( Inf @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_222_surj__Compl__image__subset,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B] :
% 2.87/1.58 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 2.87/1.58 = ( top_top @ ( set @ A ) ) )
% 2.87/1.58 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) ) @ ( image @ B @ A @ F4 @ ( uminus_uminus @ ( set @ B ) @ A6 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_223_in__inv__image,axiom,
% 2.87/1.58 ! [A: $tType,B: $tType,X: A,Y3: A,R: set @ ( product_prod @ B @ B ),F4: A > B] :
% 2.87/1.58 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( inv_image @ B @ A @ R @ F4 ) )
% 2.87/1.58 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X ) @ ( F4 @ Y3 ) ) @ R ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_224_ComplI,axiom,
% 2.87/1.58 ! [A: $tType,C3: A,A6: set @ A] :
% 2.87/1.58 ( ~ ( member @ A @ C3 @ A6 )
% 2.87/1.58 => ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_225_Compl__iff,axiom,
% 2.87/1.58 ! [A: $tType,C3: A,A6: set @ A] :
% 2.87/1.58 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 2.87/1.58 = ( ~ ( member @ A @ C3 @ A6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_226_Compl__eq__Compl__iff,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 2.87/1.58 ( ( ( uminus_uminus @ ( set @ A ) @ A6 )
% 2.87/1.58 = ( uminus_uminus @ ( set @ A ) @ B6 ) )
% 2.87/1.58 = ( A6 = B6 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_227_Compl__subset__Compl__iff,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) )
% 2.87/1.58 = ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_228_Compl__anti__mono,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 2.87/1.58 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.87/1.58 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_229_ComplD,axiom,
% 2.87/1.58 ! [A: $tType,C3: A,A6: set @ A] :
% 2.87/1.58 ( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 2.87/1.58 => ~ ( member @ A @ C3 @ A6 ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_230_double__complement,axiom,
% 2.87/1.58 ! [A: $tType,A6: set @ A] :
% 2.87/1.58 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
% 2.87/1.58 = A6 ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_231_compl__le__compl__iff,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [X: A,Y3: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) )
% 2.87/1.58 = ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_232_neg__le__iff__le,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [B2: A,A2: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 2.87/1.58 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_233_compl__mono,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [X: A,Y3: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ X @ Y3 )
% 2.87/1.58 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_234_compl__le__swap1,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [Y3: A,X: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X ) )
% 2.87/1.58 => ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_235_compl__le__swap2,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [Y3: A,X: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X )
% 2.87/1.58 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_236_le__minus__iff,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [A2: A,B2: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
% 2.87/1.58 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_237_minus__le__iff,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [A2: A,B2: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
% 2.87/1.58 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_238_le__imp__neg__le,axiom,
% 2.87/1.58 ! [A: $tType] :
% 2.87/1.58 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.87/1.58 => ! [A2: A,B2: A] :
% 2.87/1.58 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.87/1.58 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_239_inj__image__Compl__subset,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,F4: A > B,A6: set @ A] :
% 2.87/1.58 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 2.87/1.58 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_240_inj__apfst,axiom,
% 2.87/1.58 ! [B: $tType,C: $tType,A: $tType,F4: A > C] :
% 2.87/1.58 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.87/1.58 = ( inj_on @ A @ C @ F4 @ ( top_top @ ( set @ A ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_241_inj__apsnd,axiom,
% 2.87/1.58 ! [A: $tType,C: $tType,B: $tType,F4: B > C] :
% 2.87/1.58 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.87/1.58 = ( inj_on @ B @ C @ F4 @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_242_inj__image__subset__iff,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,F4: A > B,A6: set @ A,B6: set @ A] :
% 2.87/1.58 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 2.87/1.58 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B6 ) )
% 2.87/1.58 = ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_243_inj__image__mem__iff,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,F4: A > B,A2: A,A6: set @ A] :
% 2.87/1.58 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 2.87/1.58 => ( ( member @ B @ ( F4 @ A2 ) @ ( image @ A @ B @ F4 @ A6 ) )
% 2.87/1.58 = ( member @ A @ A2 @ A6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_244_inj__image__eq__iff,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,F4: A > B,A6: set @ A,B6: set @ A] :
% 2.87/1.58 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 2.87/1.58 => ( ( ( image @ A @ B @ F4 @ A6 )
% 2.87/1.58 = ( image @ A @ B @ F4 @ B6 ) )
% 2.87/1.58 = ( A6 = B6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_245_range__ex1__eq,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,F4: A > B,B2: B] :
% 2.87/1.58 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 2.87/1.58 => ( ( member @ B @ B2 @ ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) ) )
% 2.87/1.58 = ( ? [X4: A] :
% 2.87/1.58 ( ( B2
% 2.87/1.58 = ( F4 @ X4 ) )
% 2.87/1.58 & ! [Y8: A] :
% 2.87/1.58 ( ( B2
% 2.87/1.58 = ( F4 @ Y8 ) )
% 2.87/1.58 => ( Y8 = X4 ) ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_246_inj__swap,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A6 ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_247_inj__on__image__iff,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,A6: set @ A,G3: A > B,F4: A > A] :
% 2.87/1.58 ( ! [X3: A] :
% 2.87/1.58 ( ( member @ A @ X3 @ A6 )
% 2.87/1.58 => ! [Xa: A] :
% 2.87/1.58 ( ( member @ A @ Xa @ A6 )
% 2.87/1.58 => ( ( ( G3 @ ( F4 @ X3 ) )
% 2.87/1.58 = ( G3 @ ( F4 @ Xa ) ) )
% 2.87/1.58 = ( ( G3 @ X3 )
% 2.87/1.58 = ( G3 @ Xa ) ) ) ) )
% 2.87/1.58 => ( ( inj_on @ A @ A @ F4 @ A6 )
% 2.87/1.58 => ( ( inj_on @ A @ B @ G3 @ ( image @ A @ A @ F4 @ A6 ) )
% 2.87/1.58 = ( inj_on @ A @ B @ G3 @ A6 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_248_inj__on__subset,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,F4: A > B,A6: set @ A,B6: set @ A] :
% 2.87/1.58 ( ( inj_on @ A @ B @ F4 @ A6 )
% 2.87/1.58 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
% 2.87/1.58 => ( inj_on @ A @ B @ F4 @ B6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_249_subset__inj__on,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,F4: A > B,B6: set @ A,A6: set @ A] :
% 2.87/1.58 ( ( inj_on @ A @ B @ F4 @ B6 )
% 2.87/1.58 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.87/1.58 => ( inj_on @ A @ B @ F4 @ A6 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_250_inj__on__image__mem__iff__alt,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,F4: A > B,B6: set @ A,A6: set @ A,A2: A] :
% 2.87/1.58 ( ( inj_on @ A @ B @ F4 @ B6 )
% 2.87/1.58 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.87/1.58 => ( ( member @ B @ ( F4 @ A2 ) @ ( image @ A @ B @ F4 @ A6 ) )
% 2.87/1.58 => ( ( member @ A @ A2 @ B6 )
% 2.87/1.58 => ( member @ A @ A2 @ A6 ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_251_inj__on__image__mem__iff,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,F4: A > B,B6: set @ A,A2: A,A6: set @ A] :
% 2.87/1.58 ( ( inj_on @ A @ B @ F4 @ B6 )
% 2.87/1.58 => ( ( member @ A @ A2 @ B6 )
% 2.87/1.58 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.87/1.58 => ( ( member @ B @ ( F4 @ A2 ) @ ( image @ A @ B @ F4 @ A6 ) )
% 2.87/1.58 = ( member @ A @ A2 @ A6 ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_252_inj__on__image__eq__iff,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,F4: A > B,C4: set @ A,A6: set @ A,B6: set @ A] :
% 2.87/1.58 ( ( inj_on @ A @ B @ F4 @ C4 )
% 2.87/1.58 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
% 2.87/1.58 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 2.87/1.58 => ( ( ( image @ A @ B @ F4 @ A6 )
% 2.87/1.58 = ( image @ A @ B @ F4 @ B6 ) )
% 2.87/1.58 = ( A6 = B6 ) ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_253_the__inv__into__into,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,F4: A > B,A6: set @ A,X: B,B6: set @ A] :
% 2.87/1.58 ( ( inj_on @ A @ B @ F4 @ A6 )
% 2.87/1.58 => ( ( member @ B @ X @ ( image @ A @ B @ F4 @ A6 ) )
% 2.87/1.58 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 2.87/1.58 => ( member @ A @ ( the_inv_into @ A @ B @ A6 @ F4 @ X ) @ B6 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_254_wf__map__prod__image,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F4: A > B] :
% 2.87/1.58 ( ( wf @ A @ R )
% 2.87/1.58 => ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 2.87/1.58 => ( wf @ B @ ( image @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( product_map_prod @ A @ B @ A @ B @ F4 @ F4 ) @ R ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(fact_255_wf__inv__image,axiom,
% 2.87/1.58 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ B ),F4: A > B] :
% 2.87/1.58 ( ( wf @ B @ R )
% 2.87/1.58 => ( wf @ A @ ( inv_image @ B @ A @ R @ F4 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 2.87/1.58 ! [A8: $tType,A9: $tType] :
% 2.87/1.58 ( ( boolean_algebra @ A9 @ ( type2 @ A9 ) )
% 2.87/1.58 => ( boolean_algebra @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 2.87/1.58 ! [A8: $tType,A9: $tType] :
% 2.87/1.58 ( ( order_top @ A9 @ ( type2 @ A9 ) )
% 2.87/1.58 => ( order_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.87/1.58 ! [A8: $tType,A9: $tType] :
% 2.87/1.58 ( ( preorder @ A9 @ ( type2 @ A9 ) )
% 2.87/1.58 => ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.87/1.58 ! [A8: $tType,A9: $tType] :
% 2.87/1.58 ( ( order @ A9 @ ( type2 @ A9 ) )
% 2.87/1.58 => ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(tcon_fun___Orderings_Oord,axiom,
% 2.87/1.58 ! [A8: $tType,A9: $tType] :
% 2.87/1.58 ( ( ord @ A9 @ ( type2 @ A9 ) )
% 2.87/1.58 => ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_1,axiom,
% 2.87/1.58 ! [A8: $tType] : ( boolean_algebra @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(tcon_Set_Oset___Orderings_Oorder__top_2,axiom,
% 2.87/1.58 ! [A8: $tType] : ( order_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
% 2.87/1.58 ! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(tcon_Set_Oset___Orderings_Oorder_4,axiom,
% 2.87/1.58 ! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
% 2.87/1.58 ! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 2.87/1.58
% 2.87/1.58 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_6,axiom,
% 2.87/1.58 boolean_algebra @ $o @ ( type2 @ $o ) ).
% 2.87/1.58
% 2.87/1.58 thf(tcon_HOL_Obool___Orderings_Oorder__top_7,axiom,
% 2.87/1.58 order_top @ $o @ ( type2 @ $o ) ).
% 2.87/1.58
% 2.87/1.58 thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
% 2.87/1.58 preorder @ $o @ ( type2 @ $o ) ).
% 2.87/1.58
% 2.87/1.58 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.87/1.58 linorder @ $o @ ( type2 @ $o ) ).
% 2.87/1.58
% 2.87/1.58 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 2.87/1.58 order @ $o @ ( type2 @ $o ) ).
% 2.87/1.58
% 2.87/1.58 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 2.87/1.58 ord @ $o @ ( type2 @ $o ) ).
% 2.87/1.58
% 2.87/1.58 thf(conj_0,conjecture,
% 2.87/1.58 ( ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) @ tr3 ) )
% 2.87/1.58 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1 @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr2 @ tr3 ) ) ) ) ) ).
% 2.87/1.58 ------------------------
% 2.97/1.61 % (29093)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_28763 for (2999ds/183Mi)
% 2.97/1.61 % (29094)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_28763 for (2999ds/4Mi)
% 2.97/1.61 % (29094)Instruction limit reached!
% 2.97/1.61 % (29094)------------------------------
% 2.97/1.61 % (29094)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.61 % (29094)Termination reason: Unknown
% 2.97/1.61 % (29094)Termination phase: shuffling
% 2.97/1.61
% 2.97/1.61 % (29094)Memory used [KB]: 1535
% 2.97/1.61 % (29094)Time elapsed: 0.003 s
% 2.97/1.61 % (29094)Instructions burned: 5 (million)
% 2.97/1.61 % (29094)------------------------------
% 2.97/1.61 % (29094)------------------------------
% 2.97/1.62 % (29097)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_28763 for (2999ds/2Mi)
% 2.97/1.62 % (29095)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_28763 for (2999ds/27Mi)
% 2.97/1.62 % (29096)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_28763 for (2999ds/2Mi)
% 2.97/1.62 % (29098)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_28763 for (2999ds/275Mi)
% 2.97/1.62 % (29099)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_28763 for (2999ds/18Mi)
% 2.97/1.62 % (29097)Instruction limit reached!
% 2.97/1.62 % (29097)------------------------------
% 2.97/1.62 % (29097)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.62 % (29097)Termination reason: Unknown
% 2.97/1.62 % (29097)Termination phase: shuffling
% 2.97/1.62
% 2.97/1.62 % (29097)Memory used [KB]: 1407
% 2.97/1.62 % (29097)Time elapsed: 0.003 s
% 2.97/1.62 % (29097)Instructions burned: 3 (million)
% 2.97/1.62 % (29097)------------------------------
% 2.97/1.62 % (29097)------------------------------
% 2.97/1.62 % (29096)Instruction limit reached!
% 2.97/1.62 % (29096)------------------------------
% 2.97/1.62 % (29096)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.62 % (29096)Termination reason: Unknown
% 2.97/1.62 % (29096)Termination phase: shuffling
% 2.97/1.62
% 2.97/1.62 % (29096)Memory used [KB]: 1407
% 2.97/1.62 % (29096)Time elapsed: 0.004 s
% 2.97/1.62 % (29096)Instructions burned: 3 (million)
% 2.97/1.62 % (29096)------------------------------
% 2.97/1.62 % (29096)------------------------------
% 2.97/1.62 % (29100)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_28763 for (2999ds/3Mi)
% 2.97/1.62 % (29100)Instruction limit reached!
% 2.97/1.62 % (29100)------------------------------
% 2.97/1.62 % (29100)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.62 % (29100)Termination reason: Unknown
% 2.97/1.62 % (29100)Termination phase: shuffling
% 2.97/1.62
% 2.97/1.62 % (29100)Memory used [KB]: 1407
% 2.97/1.62 % (29100)Time elapsed: 0.003 s
% 2.97/1.62 % (29100)Instructions burned: 5 (million)
% 2.97/1.62 % (29100)------------------------------
% 2.97/1.62 % (29100)------------------------------
% 2.97/1.63 % (29099)Instruction limit reached!
% 2.97/1.63 % (29099)------------------------------
% 2.97/1.63 % (29099)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.63 % (29099)Termination reason: Unknown
% 2.97/1.63 % (29099)Termination phase: shuffling
% 2.97/1.63
% 2.97/1.63 % (29099)Memory used [KB]: 1791
% 2.97/1.63 % (29099)Time elapsed: 0.014 s
% 2.97/1.63 % (29099)Instructions burned: 18 (million)
% 2.97/1.63 % (29099)------------------------------
% 2.97/1.63 % (29099)------------------------------
% 2.97/1.63 % (29103)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_28763 for (2999ds/3Mi)
% 2.97/1.63 % (29101)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_28763 for (2999ds/37Mi)
% 2.97/1.63 % (29103)Instruction limit reached!
% 2.97/1.63 % (29103)------------------------------
% 2.97/1.63 % (29103)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.63 % (29103)Termination reason: Unknown
% 2.97/1.63 % (29103)Termination phase: shuffling
% 2.97/1.63
% 2.97/1.63 % (29103)Memory used [KB]: 1407
% 2.97/1.63 % (29103)Time elapsed: 0.003 s
% 2.97/1.63 % (29103)Instructions burned: 5 (million)
% 2.97/1.63 % (29103)------------------------------
% 2.97/1.63 % (29103)------------------------------
% 2.97/1.63 % (29095)Instruction limit reached!
% 2.97/1.63 % (29095)------------------------------
% 2.97/1.63 % (29095)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.63 % (29095)Termination reason: Unknown
% 2.97/1.63 % (29095)Termination phase: Property scanning
% 2.97/1.63
% 2.97/1.63 % (29095)Memory used [KB]: 1918
% 2.97/1.63 % (29095)Time elapsed: 0.019 s
% 2.97/1.63 % (29095)Instructions burned: 27 (million)
% 2.97/1.63 % (29095)------------------------------
% 2.97/1.63 % (29095)------------------------------
% 2.97/1.64 % (29102)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_28763 for (2999ds/15Mi)
% 2.97/1.64 % (29105)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_28763 for (2999ds/7Mi)
% 2.97/1.64 % (29101)Instruction limit reached!
% 2.97/1.64 % (29101)------------------------------
% 2.97/1.64 % (29101)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.64 % (29101)Termination reason: Unknown
% 2.97/1.64 % (29101)Termination phase: shuffling
% 2.97/1.64
% 2.97/1.64 % (29101)Memory used [KB]: 2174
% 2.97/1.64 % (29101)Time elapsed: 0.013 s
% 2.97/1.64 % (29101)Instructions burned: 38 (million)
% 2.97/1.64 % (29101)------------------------------
% 2.97/1.64 % (29101)------------------------------
% 2.97/1.64 % (29105)Instruction limit reached!
% 2.97/1.64 % (29105)------------------------------
% 2.97/1.64 % (29105)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.64 % (29105)Termination reason: Unknown
% 2.97/1.64 % (29105)Termination phase: shuffling
% 2.97/1.64
% 2.97/1.64 % (29105)Memory used [KB]: 1535
% 2.97/1.64 % (29105)Time elapsed: 0.004 s
% 2.97/1.64 % (29105)Instructions burned: 8 (million)
% 2.97/1.64 % (29105)------------------------------
% 2.97/1.64 % (29105)------------------------------
% 2.97/1.65 % (29102)Instruction limit reached!
% 2.97/1.65 % (29102)------------------------------
% 2.97/1.65 % (29102)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.65 % (29102)Termination reason: Unknown
% 2.97/1.65 % (29102)Termination phase: shuffling
% 2.97/1.65
% 2.97/1.65 % (29102)Memory used [KB]: 1791
% 2.97/1.65 % (29102)Time elapsed: 0.012 s
% 2.97/1.65 % (29102)Instructions burned: 15 (million)
% 2.97/1.65 % (29102)------------------------------
% 2.97/1.65 % (29102)------------------------------
% 2.97/1.65 % (29104)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_28763 for (2999ds/1041Mi)
% 2.97/1.65 % (29098)First to succeed.
% 2.97/1.65 % (29098)Refutation found. Thanks to Tanya!
% 2.97/1.65 % SZS status Theorem for DTF2THF_28763
% 2.97/1.65 % SZS output start Proof for DTF2THF_28763
% 2.97/1.65 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.97/1.65 thf(type_def_7, type, product_prod: ($tType * $tType) > $tType).
% 2.97/1.65 thf(type_def_8, type, set: ($tType) > $tType).
% 2.97/1.65 thf(type_def_9, type, n: $tType).
% 2.97/1.65 thf(type_def_10, type, t: $tType).
% 2.97/1.65 thf(type_def_11, type, dtree: $tType).
% 2.97/1.65 thf(type_def_12, type, sum_sum: ($tType * $tType) > $tType).
% 2.97/1.65 thf(func_def_0, type, product_prod: ($tType * $tType) > $tType).
% 2.97/1.65 thf(func_def_1, type, sum_sum: ($tType * $tType) > $tType).
% 2.97/1.65 thf(func_def_2, type, dtree: $tType).
% 2.97/1.65 thf(func_def_3, type, set: ($tType) > $tType).
% 2.97/1.65 thf(func_def_4, type, t: $tType).
% 2.97/1.65 thf(func_def_5, type, n: $tType).
% 2.97/1.65 thf(func_def_6, type, itself: ($tType) > $tType).
% 2.97/1.65 thf(func_def_7, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.65 thf(func_def_8, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.65 thf(func_def_9, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.65 thf(func_def_10, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.65 thf(func_def_11, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.65 thf(func_def_12, type, order_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.65 thf(func_def_13, type, boolean_algebra: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.65 thf(func_def_14, type, ordered_ab_group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.97/1.65 thf(func_def_15, type, bNF_Ca1785829860lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 2.97/1.65 thf(func_def_16, type, bNF_csquare: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(set @ X0 > (X1 > X2) > (X3 > X2) > (X0 > X1) > (X0 > X3) > $o)).
% 2.97/1.65 thf(func_def_17, type, bNF_fstOp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > product_prod @ X0 @ X2 > product_prod @ X0 @ X1)).
% 2.97/1.65 thf(func_def_18, type, bNF_pick_middlep: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > X0 > X2 > X1)).
% 2.97/1.65 thf(func_def_19, type, bNF_sndOp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > product_prod @ X0 @ X2 > product_prod @ X1 @ X2)).
% 2.97/1.65 thf(func_def_20, type, bNF_Greatest_image2: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ X0 > (X0 > X1) > (X0 > X2) > set @ product_prod @ X1 @ X2)).
% 2.97/1.65 thf(func_def_21, type, node: n > set @ sum_sum @ t @ dtree > dtree).
% 2.97/1.65 thf(func_def_22, type, cont: dtree > set @ sum_sum @ t @ dtree).
% 2.97/1.65 thf(func_def_23, type, corec: !>[X0: $tType]:((X0 > n) > (X0 > set @ sum_sum @ t @ sum_sum @ dtree @ X0) > X0 > dtree)).
% 2.97/1.65 thf(func_def_24, type, root: dtree > n).
% 2.97/1.65 thf(func_def_25, type, unfold: !>[X0: $tType]:((X0 > n) > (X0 > set @ sum_sum @ t @ X0) > X0 > dtree)).
% 2.97/1.65 thf(func_def_26, type, inj_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 2.97/1.65 thf(func_def_27, type, swap: !>[X0: $tType, X1: $tType]:(X0 > X0 > (X0 > X1) > X0 > X1)).
% 2.97/1.65 thf(func_def_28, type, the_inv_into: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X1 > X0)).
% 2.97/1.65 thf(func_def_29, type, fun_in_rel: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > X0 > X1 > $o)).
% 2.97/1.65 thf(func_def_30, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 2.97/1.65 thf(func_def_31, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.97/1.65 thf(func_def_32, type, top_top: !>[X0: $tType]:(X0)).
% 2.97/1.65 thf(func_def_33, type, parall1518086719_Nplus: n > n > n).
% 2.97/1.65 thf(func_def_34, type, parall1899940088le_par: product_prod @ dtree @ dtree > dtree).
% 2.97/1.65 thf(func_def_35, type, parall1914194347_par_c: product_prod @ dtree @ dtree > set @ sum_sum @ t @ product_prod @ dtree @ dtree).
% 2.97/1.65 thf(func_def_36, type, parall1914194362_par_r: product_prod @ dtree @ dtree > n).
% 2.97/1.65 thf(func_def_37, type, parall556292031_r_rel: product_prod @ dtree @ dtree > product_prod @ dtree @ dtree > $o).
% 2.97/1.65 thf(func_def_38, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.97/1.65 thf(func_def_39, type, product_apfst: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X0 @ X2 > product_prod @ X1 @ X2)).
% 2.97/1.65 thf(func_def_40, type, product_apsnd: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X2 @ X0 > product_prod @ X2 @ X1)).
% 2.97/1.65 thf(func_def_41, type, product_curry: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X0 @ X1 > X2) > X0 > X1 > X2)).
% 2.97/1.65 thf(func_def_42, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.97/1.65 thf(func_def_43, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 2.97/1.65 thf(func_def_44, type, product_rec_bool: !>[X0: $tType]:(X0 > X0 > $o > X0)).
% 2.97/1.65 thf(func_def_45, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.97/1.65 thf(func_def_46, type, product_case_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.97/1.65 thf(func_def_47, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.97/1.65 thf(func_def_48, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.97/1.65 thf(func_def_49, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 2.97/1.65 thf(func_def_50, type, product_scomp: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > product_prod @ X1 @ X2) > (X1 > X2 > X3) > X0 > X3)).
% 2.97/1.65 thf(func_def_51, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.97/1.65 thf(func_def_52, type, powp: !>[X0: $tType]:((X0 > $o) > set @ X0 > $o)).
% 2.97/1.65 thf(func_def_53, type, inv_image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X1 > X0) > set @ product_prod @ X1 @ X1)).
% 2.97/1.65 thf(func_def_54, type, relcompp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > X0 > X2 > $o)).
% 2.97/1.65 thf(func_def_55, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.97/1.65 thf(func_def_56, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 2.97/1.65 thf(func_def_57, type, sum_Inl: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X0 @ X1)).
% 2.97/1.65 thf(func_def_58, type, accp: !>[X0: $tType]:((X0 > X0 > $o) > X0 > $o)).
% 2.97/1.65 thf(func_def_59, type, lex_prod: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 2.97/1.65 thf(func_def_60, type, wf: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 2.97/1.65 thf(func_def_61, type, same_fst: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > set @ product_prod @ X1 @ X1) > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 2.97/1.65 thf(func_def_62, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.97/1.65 thf(func_def_63, type, tr1: dtree).
% 2.97/1.65 thf(func_def_64, type, tr2: dtree).
% 2.97/1.65 thf(func_def_65, type, tr3: dtree).
% 2.97/1.65 thf(func_def_81, type, sK0: !>[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)).
% 2.97/1.65 thf(func_def_82, type, sK1: !>[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)).
% 2.97/1.65 thf(func_def_83, type, sK2: !>[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)).
% 2.97/1.65 thf(func_def_84, type, sK3: !>[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)).
% 2.97/1.65 thf(func_def_85, type, sK4: !>[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)).
% 2.97/1.65 thf(func_def_86, type, sK5: !>[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)).
% 2.97/1.65 thf(func_def_87, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.97/1.65 thf(func_def_88, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.97/1.65 thf(func_def_89, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.97/1.65 thf(func_def_90, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.97/1.65 thf(func_def_91, type, sK10: product_prod @ dtree @ dtree > dtree).
% 2.97/1.65 thf(func_def_92, type, sK11: product_prod @ dtree @ dtree > dtree).
% 2.97/1.65 thf(func_def_93, type, sK12: !>[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)).
% 2.97/1.65 thf(func_def_94, type, sK13: !>[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)).
% 2.97/1.65 thf(func_def_95, type, sK14: !>[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)).
% 2.97/1.65 thf(func_def_96, type, sK15: !>[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)).
% 2.97/1.65 thf(func_def_97, type, sK16: !>[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)).
% 2.97/1.65 thf(func_def_98, type, sK17: !>[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)).
% 2.97/1.65 thf(func_def_99, type, sK18: !>[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)).
% 2.97/1.65 thf(func_def_100, type, sK19: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.97/1.65 thf(func_def_101, type, sK20: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.97/1.65 thf(func_def_102, type, sK21: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.97/1.65 thf(func_def_103, type, sK22: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.97/1.65 thf(func_def_104, type, sK23: !>[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)).
% 2.97/1.65 thf(func_def_105, type, sK24: !>[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)).
% 2.97/1.65 thf(func_def_106, type, sK25: !>[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)).
% 2.97/1.65 thf(func_def_107, type, sK26: !>[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)).
% 2.97/1.65 thf(func_def_108, type, sK27: !>[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)).
% 2.97/1.65 thf(func_def_109, type, sK28: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.97/1.65 thf(func_def_110, type, sK29: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.97/1.65 thf(func_def_111, type, sK30: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 2.97/1.65 thf(func_def_112, type, sK31: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 2.97/1.65 thf(func_def_113, type, sK32: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 2.97/1.65 thf(func_def_114, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 2.97/1.65 thf(func_def_115, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 2.97/1.65 thf(func_def_116, type, sK35: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.97/1.65 thf(func_def_117, type, sK36: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.97/1.65 thf(func_def_118, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 2.97/1.65 thf(func_def_119, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 2.97/1.65 thf(func_def_120, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 2.97/1.65 thf(func_def_121, type, sK40: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.97/1.65 thf(func_def_122, type, sK41: (product_prod @ dtree @ dtree > $o) > dtree).
% 2.97/1.65 thf(func_def_123, type, sK42: (product_prod @ dtree @ dtree > $o) > dtree).
% 2.97/1.65 thf(func_def_124, type, sK43: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 2.97/1.65 thf(func_def_125, type, sK44: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 2.97/1.65 thf(func_def_126, type, sK45: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 2.97/1.65 thf(func_def_127, type, sK46: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 2.97/1.65 thf(func_def_128, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.97/1.65 thf(func_def_129, type, sK48: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.97/1.65 thf(func_def_130, type, sK49: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.97/1.65 thf(func_def_131, type, sK50: !>[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)).
% 2.97/1.65 thf(func_def_132, type, sK51: !>[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)).
% 2.97/1.65 thf(func_def_133, type, sK52: !>[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)).
% 2.97/1.65 thf(func_def_134, type, sK53: !>[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)).
% 2.97/1.65 thf(func_def_135, type, sK54: !>[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)).
% 2.97/1.65 thf(func_def_136, type, sK55: !>[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)).
% 2.97/1.65 thf(func_def_137, type, sK56: !>[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)).
% 2.97/1.65 thf(func_def_138, type, sK57: !>[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)).
% 2.97/1.65 thf(func_def_139, type, sK58: !>[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)).
% 2.97/1.65 thf(func_def_140, type, sK59: !>[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)).
% 2.97/1.65 thf(func_def_141, type, sK60: !>[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)).
% 2.97/1.65 thf(func_def_142, type, sK61: !>[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)).
% 2.97/1.65 thf(func_def_143, type, sK62: !>[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)).
% 2.97/1.65 thf(func_def_145, type, ph64: !>[X0: $tType]:(X0)).
% 2.97/1.65 thf(f776,plain,(
% 2.97/1.65 $false),
% 2.97/1.65 inference(subsumption_resolution,[status(thm)],[f740,f771])).
% 2.97/1.65 thf(f771,plain,(
% 2.97/1.65 ( ! [X2 : dtree,X3 : dtree,X4 : dtree] : (((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X2 @ X3)) @ X4)) = (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X2 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X3 @ X4)))))) )),
% 2.97/1.65 inference(equality_resolution,[status(thm)],[f770])).
% 2.97/1.65 thf(f770,plain,(
% 2.97/1.65 ( ! [X2 : dtree,X3 : dtree,X0 : dtree,X4 : dtree] : (((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X2 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X3 @ X4)))) = X0) | ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X2 @ X3)) @ X4)) != X0)) )),
% 2.97/1.65 inference(equality_resolution,[status(thm)],[f749])).
% 2.97/1.65 thf(f749,plain,(
% 2.97/1.65 ( ! [X2 : dtree,X3 : dtree,X0 : dtree,X1 : dtree,X4 : dtree] : ((X0 = X1) | ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X2 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X3 @ X4)))) != X1) | ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X2 @ X3)) @ X4)) != X0)) )),
% 2.97/1.65 inference(cnf_transformation,[status(thm)],[f688])).
% 2.97/1.65 thf(f688,plain,(
% 2.97/1.65 ! [X0 : dtree,X1 : dtree] : ((X0 = X1) | ! [X2 : dtree,X3 : dtree,X4 : dtree] : (((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X2 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X3 @ X4)))) != X1) | ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X2 @ X3)) @ X4)) != X0)))),
% 2.97/1.65 inference(ennf_transformation,[status(thm)],[f672])).
% 2.97/1.65 thf(f672,plain,(
% 2.97/1.65 ! [X0 : dtree,X1 : dtree] : (? [X2 : dtree,X3 : dtree,X4 : dtree] : (((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X2 @ X3)) @ X4)) = X0) & ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X2 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X3 @ X4)))) = X1)) => (X0 = X1))),
% 2.97/1.65 inference(rectify,[status(thm)],[f1])).
% 2.97/1.65 thf(f1,axiom,(
% 2.97/1.65 ! [X5 : dtree,X6 : dtree] : (? [X7 : dtree,X8 : dtree,X9 : dtree] : (((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X7 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X8 @ X9)))) = X6) & ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X7 @ X8)) @ X9)) = X5)) => (X5 = X6))),
% 2.97/1.65 file('/export/starexec/sandbox/tmp/tmp.XABAPK8w6b/DTF2THF_28763.p',fact_0__092_060open_062_092_060And_062trB_AtrA_O_A_092_060exists_062tr1_Atr2_Atr3_O_AtrA_A_061_A_Itr1_A_092_060parallel_062_Atr2_J_A_092_060parallel_062_Atr3_A_092_060and_062_AtrB_A_061_Atr1_A_092_060parallel_062_Atr2_A_092_060parallel_062_Atr3_A_092_060Longrightarrow_062_AtrA_A_061_AtrB_092_060close_062)).
% 2.97/1.65 thf(f740,plain,(
% 2.97/1.65 ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)) @ tr3)) != (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr2 @ tr3)))))),
% 2.97/1.65 inference(cnf_transformation,[status(thm)],[f668])).
% 2.97/1.65 thf(f668,plain,(
% 2.97/1.65 ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)) @ tr3)) != (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr2 @ tr3)))))),
% 2.97/1.65 inference(flattening,[status(thm)],[f274])).
% 2.97/1.65 thf(f274,negated_conjecture,(
% 2.97/1.65 ~((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)) @ tr3)) = (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr2 @ tr3)))))),
% 2.97/1.65 inference(negated_conjecture,[status(cth)],[f273])).
% 2.97/1.65 thf(f273,conjecture,(
% 2.97/1.65 ((parall1899940088le_par @ (product_Pair @ dtree @ dtree @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)) @ tr3)) = (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr2 @ tr3)))))),
% 2.97/1.65 file('/export/starexec/sandbox/tmp/tmp.XABAPK8w6b/DTF2THF_28763.p',conj_0)).
% 2.97/1.65 % SZS output end Proof for DTF2THF_28763
% 2.97/1.65 % (29098)------------------------------
% 2.97/1.65 % (29098)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.65 % (29098)Termination reason: Refutation
% 2.97/1.65
% 2.97/1.65 % (29098)Memory used [KB]: 6652
% 2.97/1.65 % (29098)Time elapsed: 0.033 s
% 2.97/1.65 % (29098)Instructions burned: 52 (million)
% 2.97/1.65 % (29098)------------------------------
% 2.97/1.65 % (29098)------------------------------
% 2.97/1.65 % (29092)Success in time 0.062 s
%------------------------------------------------------------------------------