%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM204^1 : 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 : n007.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:41:57 AM UTC 2026
% Result : Theorem 5.32s 2.48s
% Output : Refutation 5.32s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : COM204^1 : TPTP v9.2.1. Released v7.0.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.34 % Computer : n007.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Tue Feb 24 04:16:08 EST 2026
% 0.18/0.34 % CPUTime :
% 0.18/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.35/0.49 ---- Original DTF file ---
% 0.35/0.49 thf(spec,logic,$$dhol).
% 0.35/0.49 %------------------------------------------------------------------------------
% 0.35/0.49 % File : COM204^1 : TPTP v9.2.1. Released v7.0.0.
% 0.35/0.49 % Domain : Computing Theory
% 0.35/0.49 % Problem : Parallel extension to grammars and languages 65
% 0.35/0.49 % Version : [Bla16] axioms : Especial.
% 0.35/0.49 % English :
% 0.35/0.49
% 0.35/0.49 % Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% 0.35/0.49 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.35/0.49 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.35/0.49 % Source : [Bla16]
% 0.35/0.49 % Names : parallel__65.p [Bla16]
% 0.35/0.49
% 0.35/0.49 % Status : Theorem
% 0.35/0.49 % Rating : 0.33 v8.1.0, 0.25 v7.5.0, 1.00 v7.1.0
% 0.35/0.49 % Syntax : Number of formulae : 327 ( 153 unt; 55 typ; 0 def)
% 0.35/0.49 % Number of atoms : 495 ( 281 equ; 0 cnn)
% 0.35/0.49 % Maximal formula atoms : 7 ( 1 avg)
% 0.35/0.49 % Number of connectives : 4633 ( 69 ~; 9 |; 29 &;4304 @)
% 0.35/0.49 % ( 0 <=>; 222 =>; 0 <=; 0 <~>)
% 0.35/0.49 % Maximal formula depth : 28 ( 8 avg)
% 0.35/0.49 % Number of types : 4 ( 3 usr)
% 0.35/0.49 % Number of type conns : 403 ( 403 >; 0 *; 0 +; 0 <<)
% 0.35/0.49 % Number of symbols : 55 ( 52 usr; 5 con; 0-7 aty)
% 0.35/0.49 % Number of variables : 1454 ( 177 ^;1185 !; 16 ?;1454 :)
% 0.35/0.49 % ( 76 !>; 0 ?*; 0 @-; 0 @+)
% 0.35/0.49 % SPC : TH1_THM_EQU_NAR
% 0.35/0.49
% 0.35/0.49 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.35/0.49 % 2016-07-13 14:38:53.195
% 0.35/0.49 %------------------------------------------------------------------------------
% 0.35/0.49 %----Could-be-implicit typings (7)
% 0.35/0.49 thf(ty_t_Product__Type_Oprod,type,
% 0.35/0.49 product_prod: $tType > $tType > $tType ).
% 0.35/0.49
% 0.35/0.49 thf(ty_t_Sum__Type_Osum,type,
% 0.35/0.49 sum_sum: $tType > $tType > $tType ).
% 0.35/0.49
% 0.35/0.49 thf(ty_t_DTree_Odtree,type,
% 0.35/0.49 dtree: $tType ).
% 0.35/0.49
% 0.35/0.49 thf(ty_t_Set_Oset,type,
% 0.35/0.49 set: $tType > $tType ).
% 0.35/0.49
% 0.35/0.49 thf(ty_t_DTree_OT,type,
% 0.35/0.49 t: $tType ).
% 0.35/0.49
% 0.35/0.49 thf(ty_t_DTree_ON,type,
% 0.35/0.49 n: $tType ).
% 0.35/0.49
% 0.35/0.49 thf(ty_t_itself,type,
% 0.35/0.49 itself: $tType > $tType ).
% 0.35/0.49
% 0.35/0.49 %----Explicit typings (48)
% 0.35/0.49 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.35/0.49 bounded_lattice:
% 0.35/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_cl_HOL_Otype,type,
% 0.35/0.49 type:
% 0.35/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_cl_Lattices_Olattice,type,
% 0.35/0.49 lattice:
% 0.35/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.35/0.49 semilattice_sup:
% 0.35/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_cl_Lattices_Obounded__lattice__top,type,
% 0.35/0.49 bounded_lattice_top:
% 0.35/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 0.35/0.49 bNF_Greatest_image2:
% 0.35/0.49 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_DTree_Ocont,type,
% 0.35/0.49 cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_DTree_Ounfold,type,
% 0.35/0.49 unfold:
% 0.35/0.49 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Fun__Def_Orp__inv__image,type,
% 0.35/0.49 fun_rp_inv_image:
% 0.35/0.49 !>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_HOL_OThe,type,
% 0.35/0.49 the:
% 0.35/0.49 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_If,type,
% 0.35/0.49 if:
% 0.35/0.49 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.35/0.49 sup_sup:
% 0.35/0.49 !>[A: $tType] : ( A > A > A ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.35/0.49 top_top:
% 0.35/0.49 !>[A: $tType] : A ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar,type,
% 0.35/0.49 parall1899940088le_par: ( product_prod @ dtree @ dtree ) > dtree ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c,type,
% 0.35/0.49 parall1914194347_par_c: ( product_prod @ dtree @ dtree ) > ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c__rel,type,
% 0.35/0.49 parall1418634510_c_rel: ( product_prod @ dtree @ dtree ) > ( product_prod @ dtree @ dtree ) > $o ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r,type,
% 0.35/0.49 parall1914194362_par_r: ( product_prod @ dtree @ dtree ) > n ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Product__Type_OPair,type,
% 0.35/0.49 product_Pair:
% 0.35/0.49 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Product__Type_OSigma,type,
% 0.35/0.49 product_Sigma:
% 0.35/0.49 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Product__Type_Ocurry,type,
% 0.35/0.49 product_curry:
% 0.35/0.49 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.35/0.49 produc2004651681e_prod:
% 0.35/0.49 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Product__Type_Omap__prod,type,
% 0.35/0.49 product_map_prod:
% 0.35/0.49 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.35/0.49 product_rec_prod:
% 0.35/0.49 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
% 0.35/0.49 product_rec_set_prod:
% 0.35/0.49 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.35/0.49 product_case_prod:
% 0.35/0.49 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.35/0.49 product_swap:
% 0.35/0.49 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Product__Type_Oproduct,type,
% 0.35/0.49 product_product:
% 0.35/0.49 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Pure_Otype,type,
% 0.35/0.49 type2:
% 0.35/0.49 !>[A: $tType] : ( itself @ A ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Relation_OId,type,
% 0.35/0.49 id:
% 0.35/0.49 !>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Relation_Oantisym,type,
% 0.35/0.49 antisym:
% 0.35/0.49 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Relation_Oinv__image,type,
% 0.35/0.49 inv_image:
% 0.35/0.49 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Set_OCollect,type,
% 0.35/0.49 collect:
% 0.35/0.49 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Set_Oimage,type,
% 0.35/0.49 image:
% 0.35/0.49 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Set_Ovimage,type,
% 0.35/0.49 vimage:
% 0.35/0.49 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Sum__Type_OInl,type,
% 0.35/0.49 sum_Inl:
% 0.35/0.49 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Sum__Type_OInr,type,
% 0.35/0.49 sum_Inr:
% 0.35/0.49 !>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Sum__Type_OPlus,type,
% 0.35/0.49 sum_Plus:
% 0.35/0.49 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Sum__Type_OSuml,type,
% 0.35/0.49 sum_Suml:
% 0.35/0.49 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( sum_sum @ A @ B ) > C ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Sum__Type_OSumr,type,
% 0.35/0.49 sum_Sumr:
% 0.35/0.49 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Sum__Type_Oold_Osum_Orec__set__sum,type,
% 0.35/0.49 sum_rec_set_sum:
% 0.35/0.49 !>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T > $o ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
% 0.35/0.49 sum_rec_sum:
% 0.35/0.49 !>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Wellfounded_Oaccp,type,
% 0.35/0.49 accp:
% 0.35/0.49 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Wellfounded_Olex__prod,type,
% 0.35/0.49 lex_prod:
% 0.35/0.49 !>[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 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_Wfrec_Osame__fst,type,
% 0.35/0.49 same_fst:
% 0.35/0.49 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_c_member,type,
% 0.35/0.49 member:
% 0.35/0.49 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.35/0.49
% 0.35/0.49 thf(sy_v_t,type,
% 0.35/0.49 t2: dtree ).
% 0.35/0.49
% 0.35/0.49 thf(sy_v_tr1,type,
% 0.35/0.49 tr1: dtree ).
% 0.35/0.49
% 0.35/0.49 thf(sy_v_tr2,type,
% 0.35/0.49 tr2: dtree ).
% 0.35/0.49
% 0.35/0.49 %----Relevant facts (256)
% 0.35/0.49 thf(fact_0_par__r_Ocases,axiom,
% 0.35/0.49 ! [X: product_prod @ dtree @ dtree] :
% 0.35/0.49 ~ ! [Tr1: dtree,Tr2: dtree] :
% 0.35/0.49 ( X
% 0.35/0.49 != ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ).
% 0.35/0.49
% 0.35/0.49 % par_r.cases
% 0.35/0.49 thf(fact_1_par__r_Oinduct,axiom,
% 0.35/0.49 ! [P: ( product_prod @ dtree @ dtree ) > $o,A0: product_prod @ dtree @ dtree] :
% 0.35/0.49 ( ! [Tr1: dtree,Tr2: dtree] : ( P @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 0.35/0.49 => ( P @ A0 ) ) ).
% 0.35/0.49
% 0.35/0.49 % par_r.induct
% 0.35/0.49 thf(fact_2__092_060open_062Inr_A_N_096_Acont_A_Itr1_A_092_060parallel_062_Atr2_J_A_061_Apar_A_096_A_IInr_A_N_096_Acont_Atr1_A_092_060times_062_AInr_A_N_096_Acont_Atr2_J_092_060close_062,axiom,
% 0.35/0.49 ( ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) ) )
% 0.35/0.49 = ( image @ ( product_prod @ dtree @ dtree ) @ dtree @ parall1899940088le_par
% 0.35/0.49 @ ( product_Sigma @ dtree @ dtree @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ tr1 ) )
% 0.35/0.49 @ ^ [Uu: dtree] : ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ tr2 ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % \<open>Inr -` cont (tr1 \<parallel> tr2) = par ` (Inr -` cont tr1 \<times> Inr -` cont tr2)\<close>
% 0.35/0.49 thf(fact_3_SigmaI,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,A2: A,A3: set @ A,B2: B,B3: A > ( set @ B )] :
% 0.35/0.49 ( ( member @ A @ A2 @ A3 )
% 0.35/0.49 => ( ( member @ B @ B2 @ ( B3 @ A2 ) )
% 0.35/0.49 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % SigmaI
% 0.35/0.49 thf(fact_4_mem__Sigma__iff,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] :
% 0.35/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 0.35/0.49 = ( ( member @ A @ A2 @ A3 )
% 0.35/0.49 & ( member @ B @ B2 @ ( B3 @ A2 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % mem_Sigma_iff
% 0.35/0.49 thf(fact_5_image__ident,axiom,
% 0.35/0.49 ! [A: $tType,Y: set @ A] :
% 0.35/0.49 ( ( image @ A @ A
% 0.35/0.49 @ ^ [X2: A] : X2
% 0.35/0.49 @ Y )
% 0.35/0.49 = Y ) ).
% 0.35/0.49
% 0.35/0.49 % image_ident
% 0.35/0.49 thf(fact_6_Inr__cont__par,axiom,
% 0.35/0.49 ! [Tr12: dtree,Tr22: dtree] :
% 0.35/0.49 ( ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) )
% 0.35/0.49 = ( image @ ( product_prod @ dtree @ dtree ) @ dtree @ parall1899940088le_par
% 0.35/0.49 @ ( product_Sigma @ dtree @ dtree @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr12 ) )
% 0.35/0.49 @ ^ [Uu: dtree] : ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr22 ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Inr_cont_par
% 0.35/0.49 thf(fact_7_sum_Oinject_I2_J,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,X22: B,Y2: B] :
% 0.35/0.49 ( ( ( sum_Inr @ B @ A @ X22 )
% 0.35/0.49 = ( sum_Inr @ B @ A @ Y2 ) )
% 0.35/0.49 = ( X22 = Y2 ) ) ).
% 0.35/0.49
% 0.35/0.49 % sum.inject(2)
% 0.35/0.49 thf(fact_8_old_Osum_Oinject_I2_J,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,B2: B,B4: B] :
% 0.35/0.49 ( ( ( sum_Inr @ B @ A @ B2 )
% 0.35/0.49 = ( sum_Inr @ B @ A @ B4 ) )
% 0.35/0.49 = ( B2 = B4 ) ) ).
% 0.35/0.49
% 0.35/0.49 % old.sum.inject(2)
% 0.35/0.49 thf(fact_9_image__eqI,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,B2: A,F: B > A,X: B,A3: set @ B] :
% 0.35/0.49 ( ( B2
% 0.35/0.49 = ( F @ X ) )
% 0.35/0.49 => ( ( member @ B @ X @ A3 )
% 0.35/0.49 => ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % image_eqI
% 0.35/0.49 thf(fact_10_prod_Oinject,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 0.35/0.49 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.35/0.49 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.35/0.49 = ( ( X1 = Y1 )
% 0.35/0.49 & ( X22 = Y2 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % prod.inject
% 0.35/0.49 thf(fact_11_old_Oprod_Oinject,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B4: B] :
% 0.35/0.49 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.35/0.49 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.35/0.49 = ( ( A2 = A4 )
% 0.35/0.49 & ( B2 = B4 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % old.prod.inject
% 0.35/0.49 thf(fact_12_SigmaE,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,C2: product_prod @ A @ B,A3: set @ A,B3: A > ( set @ B )] :
% 0.35/0.49 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 0.35/0.49 => ~ ! [X3: A] :
% 0.35/0.49 ( ( member @ A @ X3 @ A3 )
% 0.35/0.49 => ! [Y3: B] :
% 0.35/0.49 ( ( member @ B @ Y3 @ ( B3 @ X3 ) )
% 0.35/0.49 => ( C2
% 0.35/0.49 != ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % SigmaE
% 0.35/0.49 thf(fact_13_SigmaD1,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] :
% 0.35/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 0.35/0.49 => ( member @ A @ A2 @ A3 ) ) ).
% 0.35/0.49
% 0.35/0.49 % SigmaD1
% 0.35/0.49 thf(fact_14_vimageI,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,F: B > A,A2: B,B2: A,B3: set @ A] :
% 0.35/0.49 ( ( ( F @ A2 )
% 0.35/0.49 = B2 )
% 0.35/0.49 => ( ( member @ A @ B2 @ B3 )
% 0.35/0.49 => ( member @ B @ A2 @ ( vimage @ B @ A @ F @ B3 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % vimageI
% 0.35/0.49 thf(fact_15_vimage__eq,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,A2: A,F: A > B,B3: set @ B] :
% 0.35/0.49 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B3 ) )
% 0.35/0.49 = ( member @ B @ ( F @ A2 ) @ B3 ) ) ).
% 0.35/0.49
% 0.35/0.49 % vimage_eq
% 0.35/0.49 thf(fact_16_vimage__ident,axiom,
% 0.35/0.49 ! [A: $tType,Y: set @ A] :
% 0.35/0.49 ( ( vimage @ A @ A
% 0.35/0.49 @ ^ [X2: A] : X2
% 0.35/0.49 @ Y )
% 0.35/0.49 = Y ) ).
% 0.35/0.49
% 0.35/0.49 % vimage_ident
% 0.35/0.49 thf(fact_17_vimage__Collect__eq,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,F: A > B,P: B > $o] :
% 0.35/0.49 ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
% 0.35/0.49 = ( collect @ A
% 0.35/0.49 @ ^ [Y4: A] : ( P @ ( F @ Y4 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % vimage_Collect_eq
% 0.35/0.49 thf(fact_18_vimageD,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,A2: A,F: A > B,A3: set @ B] :
% 0.35/0.49 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ A3 ) )
% 0.35/0.49 => ( member @ B @ ( F @ A2 ) @ A3 ) ) ).
% 0.35/0.49
% 0.35/0.49 % vimageD
% 0.35/0.49 thf(fact_19_vimageE,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,A2: A,F: A > B,B3: set @ B] :
% 0.35/0.49 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B3 ) )
% 0.35/0.49 => ( member @ B @ ( F @ A2 ) @ B3 ) ) ).
% 0.35/0.49
% 0.35/0.49 % vimageE
% 0.35/0.49 thf(fact_20_vimageI2,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,F: B > A,A2: B,A3: set @ A] :
% 0.35/0.49 ( ( member @ A @ ( F @ A2 ) @ A3 )
% 0.35/0.49 => ( member @ B @ A2 @ ( vimage @ B @ A @ F @ A3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % vimageI2
% 0.35/0.49 thf(fact_21_vimage__def,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType] :
% 0.35/0.49 ( ( vimage @ A @ B )
% 0.35/0.49 = ( ^ [F2: A > B,B5: set @ B] :
% 0.35/0.49 ( collect @ A
% 0.35/0.49 @ ^ [X2: A] : ( member @ B @ ( F2 @ X2 ) @ B5 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % vimage_def
% 0.35/0.49 thf(fact_22_vimage__Collect,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,P: B > $o,F: A > B,Q: A > $o] :
% 0.35/0.49 ( ! [X3: A] :
% 0.35/0.49 ( ( P @ ( F @ X3 ) )
% 0.35/0.49 = ( Q @ X3 ) )
% 0.35/0.49 => ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
% 0.35/0.49 = ( collect @ A @ Q ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % vimage_Collect
% 0.35/0.49 thf(fact_23_old_Oprod_Oinducts,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.35/0.49 ( ! [A5: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B6 ) )
% 0.35/0.49 => ( P @ Prod ) ) ).
% 0.35/0.49
% 0.35/0.49 % old.prod.inducts
% 0.35/0.49 thf(fact_24_old_Oprod_Oexhaust,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,Y5: product_prod @ A @ B] :
% 0.35/0.49 ~ ! [A5: A,B6: B] :
% 0.35/0.49 ( Y5
% 0.35/0.49 != ( product_Pair @ A @ B @ A5 @ B6 ) ) ).
% 0.35/0.49
% 0.35/0.49 % old.prod.exhaust
% 0.35/0.49 thf(fact_25_prod__induct7,axiom,
% 0.35/0.49 ! [G: $tType,F3: $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 @ F3 @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) )] :
% 0.35/0.49 ( ! [A5: A,B6: B,C3: C,D2: D,E2: E,F4: F3,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G ) @ E2 @ ( product_Pair @ F3 @ G @ F4 @ G2 ) ) ) ) ) ) )
% 0.35/0.49 => ( P @ X ) ) ).
% 0.35/0.49
% 0.35/0.49 % prod_induct7
% 0.35/0.49 thf(fact_26_prod__induct6,axiom,
% 0.35/0.49 ! [F3: $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 @ F3 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 0.35/0.49 ( ! [A5: A,B6: B,C3: C,D2: D,E2: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
% 0.35/0.49 => ( P @ X ) ) ).
% 0.35/0.49
% 0.35/0.49 % prod_induct6
% 0.35/0.49 thf(fact_27_prod__induct5,axiom,
% 0.35/0.49 ! [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 ) ) )] :
% 0.35/0.49 ( ! [A5: A,B6: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.35/0.49 => ( P @ X ) ) ).
% 0.35/0.49
% 0.35/0.49 % prod_induct5
% 0.35/0.49 thf(fact_28_prod__induct4,axiom,
% 0.35/0.49 ! [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 ) )] :
% 0.35/0.49 ( ! [A5: A,B6: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B6 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 0.35/0.49 => ( P @ X ) ) ).
% 0.35/0.49
% 0.35/0.49 % prod_induct4
% 0.35/0.49 thf(fact_29_prod__induct3,axiom,
% 0.35/0.49 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.35/0.49 ( ! [A5: A,B6: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B6 @ C3 ) ) )
% 0.35/0.49 => ( P @ X ) ) ).
% 0.35/0.49
% 0.35/0.49 % prod_induct3
% 0.35/0.49 thf(fact_30_prod__cases7,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G: $tType,Y5: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) )] :
% 0.35/0.49 ~ ! [A5: A,B6: B,C3: C,D2: D,E2: E,F4: F3,G2: G] :
% 0.35/0.49 ( Y5
% 0.35/0.49 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G ) @ E2 @ ( product_Pair @ F3 @ G @ F4 @ G2 ) ) ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % prod_cases7
% 0.35/0.49 thf(fact_31_prod__cases6,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y5: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 0.35/0.49 ~ ! [A5: A,B6: B,C3: C,D2: D,E2: E,F4: F3] :
% 0.35/0.49 ( Y5
% 0.35/0.49 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % prod_cases6
% 0.35/0.49 thf(fact_32_prod__cases5,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y5: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.35/0.49 ~ ! [A5: A,B6: B,C3: C,D2: D,E2: E] :
% 0.35/0.49 ( Y5
% 0.35/0.49 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % prod_cases5
% 0.35/0.49 thf(fact_33_prod__cases4,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y5: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.35/0.49 ~ ! [A5: A,B6: B,C3: C,D2: D] :
% 0.35/0.49 ( Y5
% 0.35/0.49 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B6 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % prod_cases4
% 0.35/0.49 thf(fact_34_prod__cases3,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,C: $tType,Y5: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.35/0.49 ~ ! [A5: A,B6: B,C3: C] :
% 0.35/0.49 ( Y5
% 0.35/0.49 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B6 @ C3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % prod_cases3
% 0.35/0.49 thf(fact_35_Pair__inject,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B4: B] :
% 0.35/0.49 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.35/0.49 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.35/0.49 => ~ ( ( A2 = A4 )
% 0.35/0.49 => ( B2 != B4 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Pair_inject
% 0.35/0.49 thf(fact_36_prod__cases,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.35/0.49 ( ! [A5: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B6 ) )
% 0.35/0.49 => ( P @ P2 ) ) ).
% 0.35/0.49
% 0.35/0.49 % prod_cases
% 0.35/0.49 thf(fact_37_surj__pair,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.35/0.49 ? [X3: A,Y3: B] :
% 0.35/0.49 ( P2
% 0.35/0.49 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 0.35/0.49
% 0.35/0.49 % surj_pair
% 0.35/0.49 thf(fact_38_rev__image__eqI,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,X: A,A3: set @ A,B2: B,F: A > B] :
% 0.35/0.49 ( ( member @ A @ X @ A3 )
% 0.35/0.49 => ( ( B2
% 0.35/0.49 = ( F @ X ) )
% 0.35/0.49 => ( member @ B @ B2 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % rev_image_eqI
% 0.35/0.49 thf(fact_39_ball__imageD,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
% 0.35/0.49 ( ! [X3: A] :
% 0.35/0.49 ( ( member @ A @ X3 @ ( image @ B @ A @ F @ A3 ) )
% 0.35/0.49 => ( P @ X3 ) )
% 0.35/0.49 => ! [X4: B] :
% 0.35/0.49 ( ( member @ B @ X4 @ A3 )
% 0.35/0.49 => ( P @ ( F @ X4 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % ball_imageD
% 0.35/0.49 thf(fact_40_image__cong,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G3: A > B] :
% 0.35/0.49 ( ( M = N )
% 0.35/0.49 => ( ! [X3: A] :
% 0.35/0.49 ( ( member @ A @ X3 @ N )
% 0.35/0.49 => ( ( F @ X3 )
% 0.35/0.49 = ( G3 @ X3 ) ) )
% 0.35/0.49 => ( ( image @ A @ B @ F @ M )
% 0.35/0.49 = ( image @ A @ B @ G3 @ N ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % image_cong
% 0.35/0.49 thf(fact_41_bex__imageD,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
% 0.35/0.49 ( ? [X4: A] :
% 0.35/0.49 ( ( member @ A @ X4 @ ( image @ B @ A @ F @ A3 ) )
% 0.35/0.49 & ( P @ X4 ) )
% 0.35/0.49 => ? [X3: B] :
% 0.35/0.49 ( ( member @ B @ X3 @ A3 )
% 0.35/0.49 & ( P @ ( F @ X3 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % bex_imageD
% 0.35/0.49 thf(fact_42_image__iff,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,Z: A,F: B > A,A3: set @ B] :
% 0.35/0.49 ( ( member @ A @ Z @ ( image @ B @ A @ F @ A3 ) )
% 0.35/0.49 = ( ? [X2: B] :
% 0.35/0.49 ( ( member @ B @ X2 @ A3 )
% 0.35/0.49 & ( Z
% 0.35/0.49 = ( F @ X2 ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % image_iff
% 0.35/0.49 thf(fact_43_imageI,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,X: A,A3: set @ A,F: A > B] :
% 0.35/0.49 ( ( member @ A @ X @ A3 )
% 0.35/0.49 => ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % imageI
% 0.35/0.49 thf(fact_44_Inr__inject,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,X: B,Y5: B] :
% 0.35/0.49 ( ( ( sum_Inr @ B @ A @ X )
% 0.35/0.49 = ( sum_Inr @ B @ A @ Y5 ) )
% 0.35/0.49 => ( X = Y5 ) ) ).
% 0.35/0.49
% 0.35/0.49 % Inr_inject
% 0.35/0.49 thf(fact_45_mem__Collect__eq,axiom,
% 0.35/0.49 ! [A: $tType,A2: A,P: A > $o] :
% 0.35/0.49 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.35/0.49 = ( P @ A2 ) ) ).
% 0.35/0.49
% 0.35/0.49 % mem_Collect_eq
% 0.35/0.49 thf(fact_46_Collect__mem__eq,axiom,
% 0.35/0.49 ! [A: $tType,A3: set @ A] :
% 0.35/0.49 ( ( collect @ A
% 0.35/0.49 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 0.35/0.49 = A3 ) ).
% 0.35/0.49
% 0.35/0.49 % Collect_mem_eq
% 0.35/0.49 thf(fact_47_Collect__cong,axiom,
% 0.35/0.49 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.35/0.49 ( ! [X3: A] :
% 0.35/0.49 ( ( P @ X3 )
% 0.35/0.49 = ( Q @ X3 ) )
% 0.35/0.49 => ( ( collect @ A @ P )
% 0.35/0.49 = ( collect @ A @ Q ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Collect_cong
% 0.35/0.49 thf(fact_48_ext,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,F: A > B,G3: A > B] :
% 0.35/0.49 ( ! [X3: A] :
% 0.35/0.49 ( ( F @ X3 )
% 0.35/0.49 = ( G3 @ X3 ) )
% 0.35/0.49 => ( F = G3 ) ) ).
% 0.35/0.49
% 0.35/0.49 % ext
% 0.35/0.49 thf(fact_49_Compr__image__eq,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
% 0.35/0.49 ( ( collect @ A
% 0.35/0.49 @ ^ [X2: A] :
% 0.35/0.49 ( ( member @ A @ X2 @ ( image @ B @ A @ F @ A3 ) )
% 0.35/0.49 & ( P @ X2 ) ) )
% 0.35/0.49 = ( image @ B @ A @ F
% 0.35/0.49 @ ( collect @ B
% 0.35/0.49 @ ^ [X2: B] :
% 0.35/0.49 ( ( member @ B @ X2 @ A3 )
% 0.35/0.49 & ( P @ ( F @ X2 ) ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Compr_image_eq
% 0.35/0.49 thf(fact_50_image__image,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,C: $tType,F: B > A,G3: C > B,A3: set @ C] :
% 0.35/0.49 ( ( image @ B @ A @ F @ ( image @ C @ B @ G3 @ A3 ) )
% 0.35/0.49 = ( image @ C @ A
% 0.35/0.49 @ ^ [X2: C] : ( F @ ( G3 @ X2 ) )
% 0.35/0.49 @ A3 ) ) ).
% 0.35/0.49
% 0.35/0.49 % image_image
% 0.35/0.49 thf(fact_51_imageE,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,B2: A,F: B > A,A3: set @ B] :
% 0.35/0.49 ( ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) )
% 0.35/0.49 => ~ ! [X3: B] :
% 0.35/0.49 ( ( B2
% 0.35/0.49 = ( F @ X3 ) )
% 0.35/0.49 => ~ ( member @ B @ X3 @ A3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % imageE
% 0.35/0.49 thf(fact_52_Times__eq__cancel2,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,X: A,C4: set @ A,A3: set @ B,B3: set @ B] :
% 0.35/0.49 ( ( member @ A @ X @ C4 )
% 0.35/0.49 => ( ( ( product_Sigma @ B @ A @ A3
% 0.35/0.49 @ ^ [Uu: B] : C4 )
% 0.35/0.49 = ( product_Sigma @ B @ A @ B3
% 0.35/0.49 @ ^ [Uu: B] : C4 ) )
% 0.35/0.49 = ( A3 = B3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Times_eq_cancel2
% 0.35/0.49 thf(fact_53_Sigma__cong,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,C4: A > ( set @ B ),D3: A > ( set @ B )] :
% 0.35/0.49 ( ( A3 = B3 )
% 0.35/0.49 => ( ! [X3: A] :
% 0.35/0.49 ( ( member @ A @ X3 @ B3 )
% 0.35/0.49 => ( ( C4 @ X3 )
% 0.35/0.49 = ( D3 @ X3 ) ) )
% 0.35/0.49 => ( ( product_Sigma @ A @ B @ A3 @ C4 )
% 0.35/0.49 = ( product_Sigma @ A @ B @ B3 @ D3 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Sigma_cong
% 0.35/0.49 thf(fact_54_SigmaE2,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] :
% 0.35/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 0.35/0.49 => ~ ( ( member @ A @ A2 @ A3 )
% 0.35/0.49 => ~ ( member @ B @ B2 @ ( B3 @ A2 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % SigmaE2
% 0.35/0.49 thf(fact_55_SigmaD2,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] :
% 0.35/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 0.35/0.49 => ( member @ B @ B2 @ ( B3 @ A2 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % SigmaD2
% 0.35/0.49 thf(fact_56_old_Oprod_Orec,axiom,
% 0.35/0.49 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.35/0.49 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.35/0.49 = ( F1 @ A2 @ B2 ) ) ).
% 0.35/0.49
% 0.35/0.49 % old.prod.rec
% 0.35/0.49 thf(fact_57_old_Osum_Osimps_I8_J,axiom,
% 0.35/0.49 ! [A: $tType,T: $tType,B: $tType,F1: A > T,F22: B > T,B2: B] :
% 0.35/0.49 ( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inr @ B @ A @ B2 ) )
% 0.35/0.49 = ( F22 @ B2 ) ) ).
% 0.35/0.49
% 0.35/0.49 % old.sum.simps(8)
% 0.35/0.49 thf(fact_58_member__product,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A3: set @ A,B3: set @ B] :
% 0.35/0.49 ( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A3 @ B3 ) )
% 0.35/0.49 = ( member @ ( product_prod @ A @ B ) @ X
% 0.35/0.49 @ ( product_Sigma @ A @ B @ A3
% 0.35/0.49 @ ^ [Uu: A] : B3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % member_product
% 0.35/0.49 thf(fact_59_Product__Type_Oproduct__def,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType] :
% 0.35/0.49 ( ( product_product @ A @ B )
% 0.35/0.49 = ( ^ [A6: set @ A,B5: set @ B] :
% 0.35/0.49 ( product_Sigma @ A @ B @ A6
% 0.35/0.49 @ ^ [Uu: A] : B5 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Product_Type.product_def
% 0.35/0.49 thf(fact_60_Sup_OSUP__identity__eq,axiom,
% 0.35/0.49 ! [A: $tType,Sup: ( set @ A ) > A,A3: set @ A] :
% 0.35/0.49 ( ( Sup
% 0.35/0.49 @ ( image @ A @ A
% 0.35/0.49 @ ^ [X2: A] : X2
% 0.35/0.49 @ A3 ) )
% 0.35/0.49 = ( Sup @ A3 ) ) ).
% 0.35/0.49
% 0.35/0.49 % Sup.SUP_identity_eq
% 0.35/0.49 thf(fact_61_Inf_OINF__identity__eq,axiom,
% 0.35/0.49 ! [A: $tType,Inf: ( set @ A ) > A,A3: set @ A] :
% 0.35/0.49 ( ( Inf
% 0.35/0.49 @ ( image @ A @ A
% 0.35/0.49 @ ^ [X2: A] : X2
% 0.35/0.49 @ A3 ) )
% 0.35/0.49 = ( Inf @ A3 ) ) ).
% 0.35/0.49
% 0.35/0.49 % Inf.INF_identity_eq
% 0.35/0.49 thf(fact_62_pred__equals__eq2,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.35/0.49 ( ( ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R ) )
% 0.35/0.49 = ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ S ) ) )
% 0.35/0.49 = ( R = S ) ) ).
% 0.35/0.49
% 0.35/0.49 % pred_equals_eq2
% 0.35/0.49 thf(fact_63_internal__case__prod__conv,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
% 0.35/0.49 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.35/0.49 = ( C2 @ A2 @ B2 ) ) ).
% 0.35/0.49
% 0.35/0.49 % internal_case_prod_conv
% 0.35/0.49 thf(fact_64_Sumr_Osimps,axiom,
% 0.35/0.49 ! [A: $tType,C: $tType,B: $tType,F: B > C,X: B] :
% 0.35/0.49 ( ( sum_Sumr @ B @ C @ A @ F @ ( sum_Inr @ B @ A @ X ) )
% 0.35/0.49 = ( F @ X ) ) ).
% 0.35/0.49
% 0.35/0.49 % Sumr.simps
% 0.35/0.49 thf(fact_65_not__arg__cong__Inr,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,X: A,Y5: A] :
% 0.35/0.49 ( ( X != Y5 )
% 0.35/0.49 => ( ( sum_Inr @ A @ B @ X )
% 0.35/0.49 != ( sum_Inr @ A @ B @ Y5 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % not_arg_cong_Inr
% 0.35/0.49 thf(fact_66_Inl__in__cont__par,axiom,
% 0.35/0.49 ! [T2: t,Tr12: dtree,Tr22: dtree] :
% 0.35/0.49 ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) )
% 0.35/0.49 = ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr12 ) )
% 0.35/0.49 | ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr22 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Inl_in_cont_par
% 0.35/0.49 thf(fact_67_sum_Oinject_I1_J,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,X1: A,Y1: A] :
% 0.35/0.49 ( ( ( sum_Inl @ A @ B @ X1 )
% 0.35/0.49 = ( sum_Inl @ A @ B @ Y1 ) )
% 0.35/0.49 = ( X1 = Y1 ) ) ).
% 0.35/0.49
% 0.35/0.49 % sum.inject(1)
% 0.35/0.49 thf(fact_68_old_Osum_Oinject_I1_J,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,A2: A,A4: A] :
% 0.35/0.49 ( ( ( sum_Inl @ A @ B @ A2 )
% 0.35/0.49 = ( sum_Inl @ A @ B @ A4 ) )
% 0.35/0.49 = ( A2 = A4 ) ) ).
% 0.35/0.49
% 0.35/0.49 % old.sum.inject(1)
% 0.35/0.49 thf(fact_69_old_Osum_Osimps_I7_J,axiom,
% 0.35/0.49 ! [B: $tType,T: $tType,A: $tType,F1: A > T,F22: B > T,A2: A] :
% 0.35/0.49 ( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inl @ A @ B @ A2 ) )
% 0.35/0.49 = ( F1 @ A2 ) ) ).
% 0.35/0.49
% 0.35/0.49 % old.sum.simps(7)
% 0.35/0.49 thf(fact_70_Inl__inject,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,X: A,Y5: A] :
% 0.35/0.49 ( ( ( sum_Inl @ A @ B @ X )
% 0.35/0.49 = ( sum_Inl @ A @ B @ Y5 ) )
% 0.35/0.49 => ( X = Y5 ) ) ).
% 0.35/0.49
% 0.35/0.49 % Inl_inject
% 0.35/0.49 thf(fact_71_obj__sumE,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
% 0.35/0.49 ( ! [X3: A] :
% 0.35/0.49 ( S2
% 0.35/0.49 != ( sum_Inl @ A @ B @ X3 ) )
% 0.35/0.49 => ~ ! [X3: B] :
% 0.35/0.49 ( S2
% 0.35/0.49 != ( sum_Inr @ B @ A @ X3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % obj_sumE
% 0.35/0.49 thf(fact_72_Sumr__inject,axiom,
% 0.35/0.49 ! [A: $tType,C: $tType,B: $tType,F: B > C,G3: B > C] :
% 0.35/0.49 ( ( ( sum_Sumr @ B @ C @ A @ F )
% 0.35/0.49 = ( sum_Sumr @ B @ C @ A @ G3 ) )
% 0.35/0.49 => ( F = G3 ) ) ).
% 0.35/0.49
% 0.35/0.49 % Sumr_inject
% 0.35/0.49 thf(fact_73_sum_Odistinct_I1_J,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,X1: A,X22: B] :
% 0.35/0.49 ( ( sum_Inl @ A @ B @ X1 )
% 0.35/0.49 != ( sum_Inr @ B @ A @ X22 ) ) ).
% 0.35/0.49
% 0.35/0.49 % sum.distinct(1)
% 0.35/0.49 thf(fact_74_old_Osum_Odistinct_I2_J,axiom,
% 0.35/0.49 ! [B7: $tType,A7: $tType,B8: B7,A8: A7] :
% 0.35/0.49 ( ( sum_Inr @ B7 @ A7 @ B8 )
% 0.35/0.49 != ( sum_Inl @ A7 @ B7 @ A8 ) ) ).
% 0.35/0.49
% 0.35/0.49 % old.sum.distinct(2)
% 0.35/0.49 thf(fact_75_old_Osum_Odistinct_I1_J,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,A2: A,B4: B] :
% 0.35/0.49 ( ( sum_Inl @ A @ B @ A2 )
% 0.35/0.49 != ( sum_Inr @ B @ A @ B4 ) ) ).
% 0.35/0.49
% 0.35/0.49 % old.sum.distinct(1)
% 0.35/0.49 thf(fact_76_sumE,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
% 0.35/0.49 ( ! [X3: A] :
% 0.35/0.49 ( S2
% 0.35/0.49 != ( sum_Inl @ A @ B @ X3 ) )
% 0.35/0.49 => ~ ! [Y3: B] :
% 0.35/0.49 ( S2
% 0.35/0.49 != ( sum_Inr @ B @ A @ Y3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % sumE
% 0.35/0.49 thf(fact_77_Inr__not__Inl,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,B2: B,A2: A] :
% 0.35/0.49 ( ( sum_Inr @ B @ A @ B2 )
% 0.35/0.49 != ( sum_Inl @ A @ B @ A2 ) ) ).
% 0.35/0.49
% 0.35/0.49 % Inr_not_Inl
% 0.35/0.49 thf(fact_78_split__sum__ex,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType] :
% 0.35/0.49 ( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 0.35/0.49 ? [X5: sum_sum @ A @ B] : ( P3 @ X5 ) )
% 0.35/0.49 = ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
% 0.35/0.49 ( ? [X2: A] : ( P4 @ ( sum_Inl @ A @ B @ X2 ) )
% 0.35/0.49 | ? [X2: B] : ( P4 @ ( sum_Inr @ B @ A @ X2 ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % split_sum_ex
% 0.35/0.49 thf(fact_79_split__sum__all,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType] :
% 0.35/0.49 ( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 0.35/0.49 ! [X5: sum_sum @ A @ B] : ( P3 @ X5 ) )
% 0.35/0.49 = ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
% 0.35/0.49 ( ! [X2: A] : ( P4 @ ( sum_Inl @ A @ B @ X2 ) )
% 0.35/0.49 & ! [X2: B] : ( P4 @ ( sum_Inr @ B @ A @ X2 ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % split_sum_all
% 0.35/0.49 thf(fact_80_old_Osum_Oexhaust,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,Y5: sum_sum @ A @ B] :
% 0.35/0.49 ( ! [A5: A] :
% 0.35/0.49 ( Y5
% 0.35/0.49 != ( sum_Inl @ A @ B @ A5 ) )
% 0.35/0.49 => ~ ! [B6: B] :
% 0.35/0.49 ( Y5
% 0.35/0.49 != ( sum_Inr @ B @ A @ B6 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % old.sum.exhaust
% 0.35/0.49 thf(fact_81_old_Osum_Oinducts,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
% 0.35/0.49 ( ! [A5: A] : ( P @ ( sum_Inl @ A @ B @ A5 ) )
% 0.35/0.49 => ( ! [B6: B] : ( P @ ( sum_Inr @ B @ A @ B6 ) )
% 0.35/0.49 => ( P @ Sum ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % old.sum.inducts
% 0.35/0.49 thf(fact_82_Inf_OINF__cong,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,A3: set @ B,B3: set @ B,C4: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 0.35/0.49 ( ( A3 = B3 )
% 0.35/0.49 => ( ! [X3: B] :
% 0.35/0.49 ( ( member @ B @ X3 @ B3 )
% 0.35/0.49 => ( ( C4 @ X3 )
% 0.35/0.49 = ( D3 @ X3 ) ) )
% 0.35/0.49 => ( ( Inf @ ( image @ B @ A @ C4 @ A3 ) )
% 0.35/0.49 = ( Inf @ ( image @ B @ A @ D3 @ B3 ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Inf.INF_cong
% 0.35/0.49 thf(fact_83_Sup_OSUP__cong,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,A3: set @ B,B3: set @ B,C4: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 0.35/0.49 ( ( A3 = B3 )
% 0.35/0.49 => ( ! [X3: B] :
% 0.35/0.49 ( ( member @ B @ X3 @ B3 )
% 0.35/0.49 => ( ( C4 @ X3 )
% 0.35/0.49 = ( D3 @ X3 ) ) )
% 0.35/0.49 => ( ( Sup @ ( image @ B @ A @ C4 @ A3 ) )
% 0.35/0.49 = ( Sup @ ( image @ B @ A @ D3 @ B3 ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Sup.SUP_cong
% 0.35/0.49 thf(fact_84_Inl__Inr__False,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,X: A,Y5: B] :
% 0.35/0.49 ( ( sum_Inl @ A @ B @ X )
% 0.35/0.49 != ( sum_Inr @ B @ A @ Y5 ) ) ).
% 0.35/0.49
% 0.35/0.49 % Inl_Inr_False
% 0.35/0.49 thf(fact_85_Inr__Inl__False,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,X: B,Y5: A] :
% 0.35/0.49 ( ( sum_Inr @ B @ A @ X )
% 0.35/0.49 != ( sum_Inl @ A @ B @ Y5 ) ) ).
% 0.35/0.49
% 0.35/0.49 % Inr_Inl_False
% 0.35/0.49 thf(fact_86_Inl__Inr__image__cong,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,A3: set @ ( sum_sum @ A @ B ),B3: set @ ( sum_sum @ A @ B )] :
% 0.35/0.49 ( ( ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A3 )
% 0.35/0.49 = ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ B3 ) )
% 0.35/0.49 => ( ( ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ A3 )
% 0.35/0.49 = ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B3 ) )
% 0.35/0.49 => ( A3 = B3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Inl_Inr_image_cong
% 0.35/0.49 thf(fact_87_obj__sumE__f,axiom,
% 0.35/0.49 ! [A: $tType,C: $tType,B: $tType,S2: B,F: ( sum_sum @ A @ C ) > B,P: $o] :
% 0.35/0.49 ( ! [X3: A] :
% 0.35/0.49 ( ( S2
% 0.35/0.49 = ( F @ ( sum_Inl @ A @ C @ X3 ) ) )
% 0.35/0.49 => P )
% 0.35/0.49 => ( ! [X3: C] :
% 0.35/0.49 ( ( S2
% 0.35/0.49 = ( F @ ( sum_Inr @ C @ A @ X3 ) ) )
% 0.35/0.49 => P )
% 0.35/0.49 => ! [X4: sum_sum @ A @ C] :
% 0.35/0.49 ( ( S2
% 0.35/0.49 = ( F @ X4 ) )
% 0.35/0.49 => P ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % obj_sumE_f
% 0.35/0.49 thf(fact_88_Inl__cont__par,axiom,
% 0.35/0.49 ! [Tr12: dtree,Tr22: dtree] :
% 0.35/0.49 ( ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) )
% 0.35/0.49 = ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( sup_sup @ ( set @ ( sum_sum @ t @ dtree ) ) @ ( cont @ Tr12 ) @ ( cont @ Tr22 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Inl_cont_par
% 0.35/0.49 thf(fact_89_Suml_Osimps,axiom,
% 0.35/0.49 ! [B: $tType,C: $tType,A: $tType,F: A > C,X: A] :
% 0.35/0.49 ( ( sum_Suml @ A @ C @ B @ F @ ( sum_Inl @ A @ B @ X ) )
% 0.35/0.49 = ( F @ X ) ) ).
% 0.35/0.49
% 0.35/0.49 % Suml.simps
% 0.35/0.49 thf(fact_90_image2__eqI,axiom,
% 0.35/0.49 ! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X: B,C2: C,G3: B > C,A3: set @ B] :
% 0.35/0.49 ( ( B2
% 0.35/0.49 = ( F @ X ) )
% 0.35/0.49 => ( ( C2
% 0.35/0.49 = ( G3 @ X ) )
% 0.35/0.49 => ( ( member @ B @ X @ A3 )
% 0.35/0.49 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A3 @ F @ G3 ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % image2_eqI
% 0.35/0.49 thf(fact_91_Un__iff,axiom,
% 0.35/0.49 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.35/0.49 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 0.35/0.49 = ( ( member @ A @ C2 @ A3 )
% 0.35/0.49 | ( member @ A @ C2 @ B3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Un_iff
% 0.35/0.49 thf(fact_92_UnCI,axiom,
% 0.35/0.49 ! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
% 0.35/0.49 ( ( ~ ( member @ A @ C2 @ B3 )
% 0.35/0.49 => ( member @ A @ C2 @ A3 ) )
% 0.35/0.49 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % UnCI
% 0.35/0.49 thf(fact_93_vimage__Un,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,F: A > B,A3: set @ B,B3: set @ B] :
% 0.35/0.49 ( ( vimage @ A @ B @ F @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
% 0.35/0.49 = ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F @ A3 ) @ ( vimage @ A @ B @ F @ B3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % vimage_Un
% 0.35/0.49 thf(fact_94_Sigma__Un__distrib1,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,I: set @ A,J: set @ A,C4: A > ( set @ B )] :
% 0.35/0.49 ( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ I @ J ) @ C4 )
% 0.35/0.49 = ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ C4 ) @ ( product_Sigma @ A @ B @ J @ C4 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Sigma_Un_distrib1
% 0.35/0.49 thf(fact_95_Collect__disj__eq,axiom,
% 0.35/0.49 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.35/0.49 ( ( collect @ A
% 0.35/0.49 @ ^ [X2: A] :
% 0.35/0.49 ( ( P @ X2 )
% 0.35/0.49 | ( Q @ X2 ) ) )
% 0.35/0.49 = ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Collect_disj_eq
% 0.35/0.49 thf(fact_96_Un__def,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( sup_sup @ ( set @ A ) )
% 0.35/0.49 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.35/0.49 ( collect @ A
% 0.35/0.49 @ ^ [X2: A] :
% 0.35/0.49 ( ( member @ A @ X2 @ A6 )
% 0.35/0.49 | ( member @ A @ X2 @ B5 ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Un_def
% 0.35/0.49 thf(fact_97_Sigma__Un__distrib2,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,I: set @ A,A3: A > ( set @ B ),B3: A > ( set @ B )] :
% 0.35/0.49 ( ( product_Sigma @ A @ B @ I
% 0.35/0.49 @ ^ [I2: A] : ( sup_sup @ ( set @ B ) @ ( A3 @ I2 ) @ ( B3 @ I2 ) ) )
% 0.35/0.49 = ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ A3 ) @ ( product_Sigma @ A @ B @ I @ B3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Sigma_Un_distrib2
% 0.35/0.49 thf(fact_98_Times__Un__distrib1,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,C4: set @ B] :
% 0.35/0.49 ( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A3 @ B3 )
% 0.35/0.49 @ ^ [Uu: A] : C4 )
% 0.35/0.49 = ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
% 0.35/0.49 @ ( product_Sigma @ A @ B @ A3
% 0.35/0.49 @ ^ [Uu: A] : C4 )
% 0.35/0.49 @ ( product_Sigma @ A @ B @ B3
% 0.35/0.49 @ ^ [Uu: A] : C4 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Times_Un_distrib1
% 0.35/0.49 thf(fact_99_Un__left__commute,axiom,
% 0.35/0.49 ! [A: $tType,A3: set @ A,B3: set @ A,C4: set @ A] :
% 0.35/0.49 ( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C4 ) )
% 0.35/0.49 = ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Un_left_commute
% 0.35/0.49 thf(fact_100_Un__left__absorb,axiom,
% 0.35/0.49 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.35/0.49 ( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 0.35/0.49 = ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.35/0.49
% 0.35/0.49 % Un_left_absorb
% 0.35/0.49 thf(fact_101_Un__commute,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( sup_sup @ ( set @ A ) )
% 0.35/0.49 = ( ^ [A6: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Un_commute
% 0.35/0.49 thf(fact_102_Un__absorb,axiom,
% 0.35/0.49 ! [A: $tType,A3: set @ A] :
% 0.35/0.49 ( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
% 0.35/0.49 = A3 ) ).
% 0.35/0.49
% 0.35/0.49 % Un_absorb
% 0.35/0.49 thf(fact_103_Un__assoc,axiom,
% 0.35/0.49 ! [A: $tType,A3: set @ A,B3: set @ A,C4: set @ A] :
% 0.35/0.49 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C4 )
% 0.35/0.49 = ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C4 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % Un_assoc
% 0.35/0.49 thf(fact_104_ball__Un,axiom,
% 0.35/0.49 ! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
% 0.35/0.49 ( ( ! [X2: A] :
% 0.35/0.49 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 0.35/0.49 => ( P @ X2 ) ) )
% 0.35/0.49 = ( ! [X2: A] :
% 0.35/0.49 ( ( member @ A @ X2 @ A3 )
% 0.35/0.49 => ( P @ X2 ) )
% 0.35/0.49 & ! [X2: A] :
% 0.35/0.49 ( ( member @ A @ X2 @ B3 )
% 0.35/0.49 => ( P @ X2 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % ball_Un
% 0.35/0.49 thf(fact_105_bex__Un,axiom,
% 0.35/0.49 ! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
% 0.35/0.49 ( ( ? [X2: A] :
% 0.35/0.49 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 0.35/0.49 & ( P @ X2 ) ) )
% 0.35/0.49 = ( ? [X2: A] :
% 0.35/0.49 ( ( member @ A @ X2 @ A3 )
% 0.35/0.49 & ( P @ X2 ) )
% 0.35/0.49 | ? [X2: A] :
% 0.35/0.49 ( ( member @ A @ X2 @ B3 )
% 0.35/0.49 & ( P @ X2 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % bex_Un
% 0.35/0.49 thf(fact_106_UnI2,axiom,
% 0.35/0.49 ! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
% 0.35/0.49 ( ( member @ A @ C2 @ B3 )
% 0.35/0.49 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % UnI2
% 0.35/0.49 thf(fact_107_UnI1,axiom,
% 0.35/0.49 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.35/0.49 ( ( member @ A @ C2 @ A3 )
% 0.35/0.49 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % UnI1
% 0.35/0.49 thf(fact_108_UnE,axiom,
% 0.35/0.49 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.35/0.49 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 0.35/0.49 => ( ~ ( member @ A @ C2 @ A3 )
% 0.35/0.49 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % UnE
% 0.35/0.49 thf(fact_109_Suml__inject,axiom,
% 0.35/0.49 ! [B: $tType,C: $tType,A: $tType,F: A > C,G3: A > C] :
% 0.35/0.49 ( ( ( sum_Suml @ A @ C @ B @ F )
% 0.35/0.49 = ( sum_Suml @ A @ C @ B @ G3 ) )
% 0.35/0.49 => ( F = G3 ) ) ).
% 0.35/0.49
% 0.35/0.49 % Suml_inject
% 0.35/0.49 thf(fact_110_image__Un,axiom,
% 0.35/0.49 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,B3: set @ B] :
% 0.35/0.49 ( ( image @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
% 0.35/0.49 = ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F @ A3 ) @ ( image @ B @ A @ F @ B3 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % image_Un
% 0.35/0.49 thf(fact_111_sup__apply,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType] :
% 0.35/0.49 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 0.35/0.49 => ( ( sup_sup @ ( A > B ) )
% 0.35/0.49 = ( ^ [F2: A > B,G4: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G4 @ X2 ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % sup_apply
% 0.35/0.49 thf(fact_112_sup_Oidem,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.35/0.49 => ! [A2: A] :
% 0.35/0.49 ( ( sup_sup @ A @ A2 @ A2 )
% 0.35/0.49 = A2 ) ) ).
% 0.35/0.49
% 0.35/0.49 % sup.idem
% 0.35/0.49 thf(fact_113_sup__idem,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.35/0.49 => ! [X: A] :
% 0.35/0.49 ( ( sup_sup @ A @ X @ X )
% 0.35/0.49 = X ) ) ).
% 0.35/0.49
% 0.35/0.49 % sup_idem
% 0.35/0.49 thf(fact_114_sup_Oleft__idem,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.35/0.49 => ! [A2: A,B2: A] :
% 0.35/0.49 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
% 0.35/0.49 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % sup.left_idem
% 0.35/0.49 thf(fact_115_sup__left__idem,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.35/0.49 => ! [X: A,Y5: A] :
% 0.35/0.49 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y5 ) )
% 0.35/0.49 = ( sup_sup @ A @ X @ Y5 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % sup_left_idem
% 0.35/0.49 thf(fact_116_sup_Oright__idem,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.35/0.49 => ! [A2: A,B2: A] :
% 0.35/0.49 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
% 0.35/0.49 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % sup.right_idem
% 0.35/0.49 thf(fact_117_sup__Un__eq2,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.35/0.49 ( ( sup_sup @ ( A > B > $o )
% 0.35/0.49 @ ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R )
% 0.35/0.49 @ ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ S ) )
% 0.35/0.49 = ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % sup_Un_eq2
% 0.35/0.49 thf(fact_118_sup__set__def,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( sup_sup @ ( set @ A ) )
% 0.35/0.49 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.35/0.49 ( collect @ A
% 0.35/0.49 @ ( sup_sup @ ( A > $o )
% 0.35/0.49 @ ^ [X2: A] : ( member @ A @ X2 @ A6 )
% 0.35/0.49 @ ^ [X2: A] : ( member @ A @ X2 @ B5 ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % sup_set_def
% 0.35/0.49 thf(fact_119_sup__Un__eq,axiom,
% 0.35/0.49 ! [A: $tType,R: set @ A,S: set @ A] :
% 0.35/0.49 ( ( sup_sup @ ( A > $o )
% 0.35/0.49 @ ^ [X2: A] : ( member @ A @ X2 @ R )
% 0.35/0.49 @ ^ [X2: A] : ( member @ A @ X2 @ S ) )
% 0.35/0.49 = ( ^ [X2: A] : ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ R @ S ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % sup_Un_eq
% 0.35/0.49 thf(fact_120_sup__left__commute,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.35/0.49 => ! [X: A,Y5: A,Z: A] :
% 0.35/0.49 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y5 @ Z ) )
% 0.35/0.49 = ( sup_sup @ A @ Y5 @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % sup_left_commute
% 0.35/0.49 thf(fact_121_sup_Oleft__commute,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.35/0.49 => ! [B2: A,A2: A,C2: A] :
% 0.35/0.49 ( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C2 ) )
% 0.35/0.49 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % sup.left_commute
% 0.35/0.49 thf(fact_122_sup__commute,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.35/0.49 => ( ( sup_sup @ A )
% 0.35/0.49 = ( ^ [X2: A,Y4: A] : ( sup_sup @ A @ Y4 @ X2 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % sup_commute
% 0.35/0.49 thf(fact_123_sup_Ocommute,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.35/0.49 => ( ( sup_sup @ A )
% 0.35/0.49 = ( ^ [A9: A,B9: A] : ( sup_sup @ A @ B9 @ A9 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % sup.commute
% 0.35/0.49 thf(fact_124_sup__assoc,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.35/0.49 => ! [X: A,Y5: A,Z: A] :
% 0.35/0.49 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y5 ) @ Z )
% 0.35/0.49 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y5 @ Z ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % sup_assoc
% 0.35/0.49 thf(fact_125_sup_Oassoc,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.35/0.49 => ! [A2: A,B2: A,C2: A] :
% 0.35/0.49 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C2 )
% 0.35/0.49 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % sup.assoc
% 0.35/0.49 thf(fact_126_sup__fun__def,axiom,
% 0.35/0.49 ! [B: $tType,A: $tType] :
% 0.35/0.49 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 0.35/0.49 => ( ( sup_sup @ ( A > B ) )
% 0.35/0.49 = ( ^ [F2: A > B,G4: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G4 @ X2 ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % sup_fun_def
% 0.35/0.49 thf(fact_127_inf__sup__aci_I5_J,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( lattice @ A @ ( type2 @ A ) )
% 0.35/0.49 => ( ( sup_sup @ A )
% 0.35/0.49 = ( ^ [X2: A,Y4: A] : ( sup_sup @ A @ Y4 @ X2 ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % inf_sup_aci(5)
% 0.35/0.49 thf(fact_128_inf__sup__aci_I6_J,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( lattice @ A @ ( type2 @ A ) )
% 0.35/0.49 => ! [X: A,Y5: A,Z: A] :
% 0.35/0.49 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y5 ) @ Z )
% 0.35/0.49 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y5 @ Z ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % inf_sup_aci(6)
% 0.35/0.49 thf(fact_129_inf__sup__aci_I7_J,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( lattice @ A @ ( type2 @ A ) )
% 0.35/0.49 => ! [X: A,Y5: A,Z: A] :
% 0.35/0.49 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y5 @ Z ) )
% 0.35/0.49 = ( sup_sup @ A @ Y5 @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % inf_sup_aci(7)
% 0.35/0.49 thf(fact_130_inf__sup__aci_I8_J,axiom,
% 0.35/0.49 ! [A: $tType] :
% 0.35/0.49 ( ( lattice @ A @ ( type2 @ A ) )
% 0.35/0.49 => ! [X: A,Y5: A] :
% 0.35/0.49 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y5 ) )
% 0.35/0.49 = ( sup_sup @ A @ X @ Y5 ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % inf_sup_aci(8)
% 0.35/0.49 thf(fact_131_par__c_Oelims,axiom,
% 0.35/0.49 ! [X: product_prod @ dtree @ dtree,Y5: set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) )] :
% 0.35/0.49 ( ( ( parall1914194347_par_c @ X )
% 0.35/0.49 = Y5 )
% 0.35/0.49 => ~ ! [Tr1: dtree,Tr2: dtree] :
% 0.35/0.49 ( ( X
% 0.35/0.49 = ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 0.35/0.49 => ( Y5
% 0.35/0.49 != ( sup_sup @ ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) @ ( image @ t @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inl @ t @ ( product_prod @ dtree @ dtree ) ) @ ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( sup_sup @ ( set @ ( sum_sum @ t @ dtree ) ) @ ( cont @ Tr1 ) @ ( cont @ Tr2 ) ) ) )
% 0.35/0.49 @ ( image @ ( product_prod @ dtree @ dtree ) @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inr @ ( product_prod @ dtree @ dtree ) @ t )
% 0.35/0.49 @ ( product_Sigma @ dtree @ dtree @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr1 ) )
% 0.35/0.49 @ ^ [Uu: dtree] : ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr2 ) ) ) ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % par_c.elims
% 0.35/0.49 thf(fact_132_par__c_Osimps,axiom,
% 0.35/0.49 ! [Tr12: dtree,Tr22: dtree] :
% 0.35/0.49 ( ( parall1914194347_par_c @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) )
% 0.35/0.49 = ( sup_sup @ ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) @ ( image @ t @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inl @ t @ ( product_prod @ dtree @ dtree ) ) @ ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( sup_sup @ ( set @ ( sum_sum @ t @ dtree ) ) @ ( cont @ Tr12 ) @ ( cont @ Tr22 ) ) ) )
% 0.35/0.49 @ ( image @ ( product_prod @ dtree @ dtree ) @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inr @ ( product_prod @ dtree @ dtree ) @ t )
% 0.35/0.49 @ ( product_Sigma @ dtree @ dtree @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr12 ) )
% 0.35/0.49 @ ^ [Uu: dtree] : ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr22 ) ) ) ) ) ) ).
% 0.35/0.49
% 0.35/0.49 % par_c.simps
% 0.35/0.49 thf(fact_133_par__c_Opelims,axiom,
% 0.35/0.49 ! [X: product_prod @ dtree @ dtree,Y5: set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) )] :
% 0.35/0.49 ( ( ( parall1914194347_par_c @ X )
% 0.35/0.49 = Y5 )
% 0.35/0.49 => ( ( accp @ ( product_prod @ dtree @ dtree ) @ parall1418634510_c_rel @ X )
% 0.35/0.49 => ~ ! [Tr1: dtree,Tr2: dtree] :
% 0.35/0.49 ( ( X
% 0.35/0.49 = ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 0.35/0.49 => ( ( Y5
% 0.35/0.49 = ( sup_sup @ ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) @ ( image @ t @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inl @ t @ ( product_prod @ dtree @ dtree ) ) @ ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( sup_sup @ ( set @ ( sum_sum @ t @ dtree ) ) @ ( cont @ Tr1 ) @ ( cont @ Tr2 ) ) ) )
% 0.35/0.50 @ ( image @ ( product_prod @ dtree @ dtree ) @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inr @ ( product_prod @ dtree @ dtree ) @ t )
% 0.35/0.50 @ ( product_Sigma @ dtree @ dtree @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr1 ) )
% 0.35/0.50 @ ^ [Uu: dtree] : ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr2 ) ) ) ) ) )
% 0.35/0.50 => ~ ( accp @ ( product_prod @ dtree @ dtree ) @ parall1418634510_c_rel @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % par_c.pelims
% 0.35/0.50 thf(fact_134_Plus__def,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType] :
% 0.35/0.50 ( ( sum_Plus @ A @ B )
% 0.35/0.50 = ( ^ [A6: set @ A,B5: set @ B] : ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A6 ) @ ( image @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B5 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % Plus_def
% 0.35/0.50 thf(fact_135_sup2CI,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,B3: A > B > $o,X: A,Y5: B,A3: A > B > $o] :
% 0.35/0.50 ( ( ~ ( B3 @ X @ Y5 )
% 0.35/0.50 => ( A3 @ X @ Y5 ) )
% 0.35/0.50 => ( sup_sup @ ( A > B > $o ) @ A3 @ B3 @ X @ Y5 ) ) ).
% 0.35/0.50
% 0.35/0.50 % sup2CI
% 0.35/0.50 thf(fact_136_sup1CI,axiom,
% 0.35/0.50 ! [A: $tType,B3: A > $o,X: A,A3: A > $o] :
% 0.35/0.50 ( ( ~ ( B3 @ X )
% 0.35/0.50 => ( A3 @ X ) )
% 0.35/0.50 => ( sup_sup @ ( A > $o ) @ A3 @ B3 @ X ) ) ).
% 0.35/0.50
% 0.35/0.50 % sup1CI
% 0.35/0.50 thf(fact_137_InrI,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,B2: A,B3: set @ A,A3: set @ B] :
% 0.35/0.50 ( ( member @ A @ B2 @ B3 )
% 0.35/0.50 => ( member @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B @ B2 ) @ ( sum_Plus @ B @ A @ A3 @ B3 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % InrI
% 0.35/0.50 thf(fact_138_InlI,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,A2: A,A3: set @ A,B3: set @ B] :
% 0.35/0.50 ( ( member @ A @ A2 @ A3 )
% 0.35/0.50 => ( member @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ A2 ) @ ( sum_Plus @ A @ B @ A3 @ B3 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % InlI
% 0.35/0.50 thf(fact_139_sup1E,axiom,
% 0.35/0.50 ! [A: $tType,A3: A > $o,B3: A > $o,X: A] :
% 0.35/0.50 ( ( sup_sup @ ( A > $o ) @ A3 @ B3 @ X )
% 0.35/0.50 => ( ~ ( A3 @ X )
% 0.35/0.50 => ( B3 @ X ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % sup1E
% 0.35/0.50 thf(fact_140_sup2E,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,A3: A > B > $o,B3: A > B > $o,X: A,Y5: B] :
% 0.35/0.50 ( ( sup_sup @ ( A > B > $o ) @ A3 @ B3 @ X @ Y5 )
% 0.35/0.50 => ( ~ ( A3 @ X @ Y5 )
% 0.35/0.50 => ( B3 @ X @ Y5 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % sup2E
% 0.35/0.50 thf(fact_141_sup1I1,axiom,
% 0.35/0.50 ! [A: $tType,A3: A > $o,X: A,B3: A > $o] :
% 0.35/0.50 ( ( A3 @ X )
% 0.35/0.50 => ( sup_sup @ ( A > $o ) @ A3 @ B3 @ X ) ) ).
% 0.35/0.50
% 0.35/0.50 % sup1I1
% 0.35/0.50 thf(fact_142_sup1I2,axiom,
% 0.35/0.50 ! [A: $tType,B3: A > $o,X: A,A3: A > $o] :
% 0.35/0.50 ( ( B3 @ X )
% 0.35/0.50 => ( sup_sup @ ( A > $o ) @ A3 @ B3 @ X ) ) ).
% 0.35/0.50
% 0.35/0.50 % sup1I2
% 0.35/0.50 thf(fact_143_sup2I1,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,A3: A > B > $o,X: A,Y5: B,B3: A > B > $o] :
% 0.35/0.50 ( ( A3 @ X @ Y5 )
% 0.35/0.50 => ( sup_sup @ ( A > B > $o ) @ A3 @ B3 @ X @ Y5 ) ) ).
% 0.35/0.50
% 0.35/0.50 % sup2I1
% 0.35/0.50 thf(fact_144_sup2I2,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,B3: A > B > $o,X: A,Y5: B,A3: A > B > $o] :
% 0.35/0.50 ( ( B3 @ X @ Y5 )
% 0.35/0.50 => ( sup_sup @ ( A > B > $o ) @ A3 @ B3 @ X @ Y5 ) ) ).
% 0.35/0.50
% 0.35/0.50 % sup2I2
% 0.35/0.50 thf(fact_145_PlusE,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,U: sum_sum @ A @ B,A3: set @ A,B3: set @ B] :
% 0.35/0.50 ( ( member @ ( sum_sum @ A @ B ) @ U @ ( sum_Plus @ A @ B @ A3 @ B3 ) )
% 0.35/0.50 => ( ! [X3: A] :
% 0.35/0.50 ( ( member @ A @ X3 @ A3 )
% 0.35/0.50 => ( U
% 0.35/0.50 != ( sum_Inl @ A @ B @ X3 ) ) )
% 0.35/0.50 => ~ ! [Y3: B] :
% 0.35/0.50 ( ( member @ B @ Y3 @ B3 )
% 0.35/0.50 => ( U
% 0.35/0.50 != ( sum_Inr @ B @ A @ Y3 ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % PlusE
% 0.35/0.50 thf(fact_146_reflclp__idemp,axiom,
% 0.35/0.50 ! [A: $tType,P: A > A > $o] :
% 0.35/0.50 ( ( sup_sup @ ( A > A > $o )
% 0.35/0.50 @ ( sup_sup @ ( A > A > $o ) @ P
% 0.35/0.50 @ ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) )
% 0.35/0.50 @ ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) )
% 0.35/0.50 = ( sup_sup @ ( A > A > $o ) @ P
% 0.35/0.50 @ ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % reflclp_idemp
% 0.35/0.50 thf(fact_147_accp_Ocases,axiom,
% 0.35/0.50 ! [A: $tType,R2: A > A > $o,A2: A] :
% 0.35/0.50 ( ( accp @ A @ R2 @ A2 )
% 0.35/0.50 => ! [Y7: A] :
% 0.35/0.50 ( ( R2 @ Y7 @ A2 )
% 0.35/0.50 => ( accp @ A @ R2 @ Y7 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % accp.cases
% 0.35/0.50 thf(fact_148_accp_Osimps,axiom,
% 0.35/0.50 ! [A: $tType] :
% 0.35/0.50 ( ( accp @ A )
% 0.35/0.50 = ( ^ [R3: A > A > $o,A9: A] :
% 0.35/0.50 ? [X2: A] :
% 0.35/0.50 ( ( A9 = X2 )
% 0.35/0.50 & ! [Y4: A] :
% 0.35/0.50 ( ( R3 @ Y4 @ X2 )
% 0.35/0.50 => ( accp @ A @ R3 @ Y4 ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % accp.simps
% 0.35/0.50 thf(fact_149_accp_Ointros,axiom,
% 0.35/0.50 ! [A: $tType,R2: A > A > $o,X: A] :
% 0.35/0.50 ( ! [Y3: A] :
% 0.35/0.50 ( ( R2 @ Y3 @ X )
% 0.35/0.50 => ( accp @ A @ R2 @ Y3 ) )
% 0.35/0.50 => ( accp @ A @ R2 @ X ) ) ).
% 0.35/0.50
% 0.35/0.50 % accp.intros
% 0.35/0.50 thf(fact_150_accp__induct__rule,axiom,
% 0.35/0.50 ! [A: $tType,R2: A > A > $o,A2: A,P: A > $o] :
% 0.35/0.50 ( ( accp @ A @ R2 @ A2 )
% 0.35/0.50 => ( ! [X3: A] :
% 0.35/0.50 ( ( accp @ A @ R2 @ X3 )
% 0.35/0.50 => ( ! [Y7: A] :
% 0.35/0.50 ( ( R2 @ Y7 @ X3 )
% 0.35/0.50 => ( P @ Y7 ) )
% 0.35/0.50 => ( P @ X3 ) ) )
% 0.35/0.50 => ( P @ A2 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % accp_induct_rule
% 0.35/0.50 thf(fact_151_not__accp__down,axiom,
% 0.35/0.50 ! [A: $tType,R: A > A > $o,X: A] :
% 0.35/0.50 ( ~ ( accp @ A @ R @ X )
% 0.35/0.50 => ~ ! [Z3: A] :
% 0.35/0.50 ( ( R @ Z3 @ X )
% 0.35/0.50 => ( accp @ A @ R @ Z3 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % not_accp_down
% 0.35/0.50 thf(fact_152_accp__downward,axiom,
% 0.35/0.50 ! [A: $tType,R2: A > A > $o,B2: A,A2: A] :
% 0.35/0.50 ( ( accp @ A @ R2 @ B2 )
% 0.35/0.50 => ( ( R2 @ A2 @ B2 )
% 0.35/0.50 => ( accp @ A @ R2 @ A2 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % accp_downward
% 0.35/0.50 thf(fact_153_accp_Oinducts,axiom,
% 0.35/0.50 ! [A: $tType,R2: A > A > $o,X: A,P: A > $o] :
% 0.35/0.50 ( ( accp @ A @ R2 @ X )
% 0.35/0.50 => ( ! [X3: A] :
% 0.35/0.50 ( ! [Y7: A] :
% 0.35/0.50 ( ( R2 @ Y7 @ X3 )
% 0.35/0.50 => ( accp @ A @ R2 @ Y7 ) )
% 0.35/0.50 => ( ! [Y7: A] :
% 0.35/0.50 ( ( R2 @ Y7 @ X3 )
% 0.35/0.50 => ( P @ Y7 ) )
% 0.35/0.50 => ( P @ X3 ) ) )
% 0.35/0.50 => ( P @ X ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % accp.inducts
% 0.35/0.50 thf(fact_154_accp__induct,axiom,
% 0.35/0.50 ! [A: $tType,R2: A > A > $o,A2: A,P: A > $o] :
% 0.35/0.50 ( ( accp @ A @ R2 @ A2 )
% 0.35/0.50 => ( ! [X3: A] :
% 0.35/0.50 ( ( accp @ A @ R2 @ X3 )
% 0.35/0.50 => ( ! [Y7: A] :
% 0.35/0.50 ( ( R2 @ Y7 @ X3 )
% 0.35/0.50 => ( P @ Y7 ) )
% 0.35/0.50 => ( P @ X3 ) ) )
% 0.35/0.50 => ( P @ A2 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % accp_induct
% 0.35/0.50 thf(fact_155_in__lex__prod,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B4: B,R2: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B )] :
% 0.35/0.50 ( ( 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 @ A4 @ B4 ) ) @ ( lex_prod @ A @ B @ R2 @ S2 ) )
% 0.35/0.50 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A4 ) @ R2 )
% 0.35/0.50 | ( ( A2 = A4 )
% 0.35/0.50 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B4 ) @ S2 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % in_lex_prod
% 0.35/0.50 thf(fact_156_reflcl__set__eq,axiom,
% 0.35/0.50 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.35/0.50 ( ( sup_sup @ ( A > A > $o )
% 0.35/0.50 @ ^ [X2: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R2 )
% 0.35/0.50 @ ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) )
% 0.35/0.50 = ( ^ [X2: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id @ A ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % reflcl_set_eq
% 0.35/0.50 thf(fact_157_par__def,axiom,
% 0.35/0.50 ( parall1899940088le_par
% 0.35/0.50 = ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c ) ) ).
% 0.35/0.50
% 0.35/0.50 % par_def
% 0.35/0.50 thf(fact_158_IdI,axiom,
% 0.35/0.50 ! [A: $tType,A2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id @ A ) ) ).
% 0.35/0.50
% 0.35/0.50 % IdI
% 0.35/0.50 thf(fact_159_pair__in__Id__conv,axiom,
% 0.35/0.50 ! [A: $tType,A2: A,B2: A] :
% 0.35/0.50 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id @ A ) )
% 0.35/0.50 = ( A2 = B2 ) ) ).
% 0.35/0.50
% 0.35/0.50 % pair_in_Id_conv
% 0.35/0.50 thf(fact_160_IdE,axiom,
% 0.35/0.50 ! [A: $tType,P2: product_prod @ A @ A] :
% 0.35/0.50 ( ( member @ ( product_prod @ A @ A ) @ P2 @ ( id @ A ) )
% 0.35/0.50 => ~ ! [X3: A] :
% 0.35/0.50 ( P2
% 0.35/0.50 != ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % IdE
% 0.35/0.50 thf(fact_161_IdD,axiom,
% 0.35/0.50 ! [A: $tType,A2: A,B2: A] :
% 0.35/0.50 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id @ A ) )
% 0.35/0.50 => ( A2 = B2 ) ) ).
% 0.35/0.50
% 0.35/0.50 % IdD
% 0.35/0.50 thf(fact_162_same__fstI,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,P: A > $o,X: A,Y8: B,Y5: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.35/0.50 ( ( P @ X )
% 0.35/0.50 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y5 ) @ ( R @ X ) )
% 0.35/0.50 => ( 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 @ Y8 ) @ ( product_Pair @ A @ B @ X @ Y5 ) ) @ ( same_fst @ A @ B @ P @ R ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % same_fstI
% 0.35/0.50 thf(fact_163_antisym__reflcl,axiom,
% 0.35/0.50 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.35/0.50 ( ( antisym @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id @ A ) ) )
% 0.35/0.50 = ( antisym @ A @ R2 ) ) ).
% 0.35/0.50
% 0.35/0.50 % antisym_reflcl
% 0.35/0.50 thf(fact_164_pair__imageI,axiom,
% 0.35/0.50 ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set @ ( product_prod @ A @ B ),F: A > B > C] :
% 0.35/0.50 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ A3 )
% 0.35/0.50 => ( member @ C @ ( F @ A2 @ B2 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ A3 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % pair_imageI
% 0.35/0.50 thf(fact_165_case__prodI,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
% 0.35/0.50 ( ( F @ A2 @ B2 )
% 0.35/0.50 => ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % case_prodI
% 0.35/0.50 thf(fact_166_case__prodI2,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,C2: A > B > $o] :
% 0.35/0.50 ( ! [A5: A,B6: B] :
% 0.35/0.50 ( ( P2
% 0.35/0.50 = ( product_Pair @ A @ B @ A5 @ B6 ) )
% 0.35/0.50 => ( C2 @ A5 @ B6 ) )
% 0.35/0.50 => ( product_case_prod @ A @ B @ $o @ C2 @ P2 ) ) ).
% 0.35/0.50
% 0.35/0.50 % case_prodI2
% 0.35/0.50 thf(fact_167_mem__case__prodI,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),A2: B,B2: C] :
% 0.35/0.50 ( ( member @ A @ Z @ ( C2 @ A2 @ B2 ) )
% 0.35/0.50 => ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % mem_case_prodI
% 0.35/0.50 thf(fact_168_mem__case__prodI2,axiom,
% 0.35/0.50 ! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z: C,C2: A > B > ( set @ C )] :
% 0.35/0.50 ( ! [A5: A,B6: B] :
% 0.35/0.50 ( ( P2
% 0.35/0.50 = ( product_Pair @ A @ B @ A5 @ B6 ) )
% 0.35/0.50 => ( member @ C @ Z @ ( C2 @ A5 @ B6 ) ) )
% 0.35/0.50 => ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P2 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % mem_case_prodI2
% 0.35/0.50 thf(fact_169_case__prodI2_H,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
% 0.35/0.50 ( ! [A5: A,B6: B] :
% 0.35/0.50 ( ( ( product_Pair @ A @ B @ A5 @ B6 )
% 0.35/0.50 = P2 )
% 0.35/0.50 => ( C2 @ A5 @ B6 @ X ) )
% 0.35/0.50 => ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P2 @ X ) ) ).
% 0.35/0.50
% 0.35/0.50 % case_prodI2'
% 0.35/0.50 thf(fact_170_Collect__case__prod,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,P: A > $o,Q: B > $o] :
% 0.35/0.50 ( ( collect @ ( product_prod @ A @ B )
% 0.35/0.50 @ ( product_case_prod @ A @ B @ $o
% 0.35/0.50 @ ^ [A9: A,B9: B] :
% 0.35/0.50 ( ( P @ A9 )
% 0.35/0.50 & ( Q @ B9 ) ) ) )
% 0.35/0.50 = ( product_Sigma @ A @ B @ ( collect @ A @ P )
% 0.35/0.50 @ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % Collect_case_prod
% 0.35/0.50 thf(fact_171_case__prod__conv,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B2: C] :
% 0.35/0.50 ( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.35/0.50 = ( F @ A2 @ B2 ) ) ).
% 0.35/0.50
% 0.35/0.50 % case_prod_conv
% 0.35/0.50 thf(fact_172_lex__prod__def,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType] :
% 0.35/0.50 ( ( lex_prod @ A @ B )
% 0.35/0.50 = ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
% 0.35/0.50 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 0.35/0.50 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 0.35/0.50 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 0.35/0.50 @ ^ [A9: A,B9: B] :
% 0.35/0.50 ( product_case_prod @ A @ B @ $o
% 0.35/0.50 @ ^ [A10: A,B10: B] :
% 0.35/0.50 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A9 @ A10 ) @ Ra )
% 0.35/0.50 | ( ( A9 = A10 )
% 0.35/0.50 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B9 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % lex_prod_def
% 0.35/0.50 thf(fact_173_case__prodD_H,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A2: A,B2: B,C2: C] :
% 0.35/0.50 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 )
% 0.35/0.50 => ( R @ A2 @ B2 @ C2 ) ) ).
% 0.35/0.50
% 0.35/0.50 % case_prodD'
% 0.35/0.50 thf(fact_174_case__prodE_H,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P2: product_prod @ A @ B,Z: C] :
% 0.35/0.50 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P2 @ Z )
% 0.35/0.50 => ~ ! [X3: A,Y3: B] :
% 0.35/0.50 ( ( P2
% 0.35/0.50 = ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 0.35/0.50 => ~ ( C2 @ X3 @ Y3 @ Z ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % case_prodE'
% 0.35/0.50 thf(fact_175_case__prodE2,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z: product_prod @ B @ C] :
% 0.35/0.50 ( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z ) )
% 0.35/0.50 => ~ ! [X3: B,Y3: C] :
% 0.35/0.50 ( ( Z
% 0.35/0.50 = ( product_Pair @ B @ C @ X3 @ Y3 ) )
% 0.35/0.50 => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % case_prodE2
% 0.35/0.50 thf(fact_176_case__prod__eta,axiom,
% 0.35/0.50 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.35/0.50 ( ( product_case_prod @ A @ B @ C
% 0.35/0.50 @ ^ [X2: A,Y4: B] : ( F @ ( product_Pair @ A @ B @ X2 @ Y4 ) ) )
% 0.35/0.50 = F ) ).
% 0.35/0.50
% 0.35/0.50 % case_prod_eta
% 0.35/0.50 thf(fact_177_cond__case__prod__eta,axiom,
% 0.35/0.50 ! [C: $tType,B: $tType,A: $tType,F: A > B > C,G3: ( product_prod @ A @ B ) > C] :
% 0.35/0.50 ( ! [X3: A,Y3: B] :
% 0.35/0.50 ( ( F @ X3 @ Y3 )
% 0.35/0.50 = ( G3 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
% 0.35/0.50 => ( ( product_case_prod @ A @ B @ C @ F )
% 0.35/0.50 = G3 ) ) ).
% 0.35/0.50
% 0.35/0.50 % cond_case_prod_eta
% 0.35/0.50 thf(fact_178_mem__case__prodE,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 0.35/0.50 ( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P2 ) )
% 0.35/0.50 => ~ ! [X3: B,Y3: C] :
% 0.35/0.50 ( ( P2
% 0.35/0.50 = ( product_Pair @ B @ C @ X3 @ Y3 ) )
% 0.35/0.50 => ~ ( member @ A @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % mem_case_prodE
% 0.35/0.50 thf(fact_179_old_Oprod_Ocase,axiom,
% 0.35/0.50 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
% 0.35/0.50 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.35/0.50 = ( F @ X1 @ X22 ) ) ).
% 0.35/0.50
% 0.35/0.50 % old.prod.case
% 0.35/0.50 thf(fact_180_same__fst__def,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType] :
% 0.35/0.50 ( ( same_fst @ A @ B )
% 0.35/0.50 = ( ^ [P4: A > $o,R4: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.35/0.50 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 0.35/0.50 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 0.35/0.50 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 0.35/0.50 @ ^ [X6: A,Y9: B] :
% 0.35/0.50 ( product_case_prod @ A @ B @ $o
% 0.35/0.50 @ ^ [X2: A,Y4: B] :
% 0.35/0.50 ( ( X6 = X2 )
% 0.35/0.50 & ( P4 @ X2 )
% 0.35/0.50 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y9 @ Y4 ) @ ( R4 @ X2 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % same_fst_def
% 0.35/0.50 thf(fact_181_antisym__Id,axiom,
% 0.35/0.50 ! [A: $tType] : ( antisym @ A @ ( id @ A ) ) ).
% 0.35/0.50
% 0.35/0.50 % antisym_Id
% 0.35/0.50 thf(fact_182_antisymD,axiom,
% 0.35/0.50 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
% 0.35/0.50 ( ( antisym @ A @ R2 )
% 0.35/0.50 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
% 0.35/0.50 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A2 ) @ R2 )
% 0.35/0.50 => ( A2 = B2 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % antisymD
% 0.35/0.50 thf(fact_183_antisymI,axiom,
% 0.35/0.50 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.35/0.50 ( ! [X3: A,Y3: A] :
% 0.35/0.50 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
% 0.35/0.50 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R2 )
% 0.35/0.50 => ( X3 = Y3 ) ) )
% 0.35/0.50 => ( antisym @ A @ R2 ) ) ).
% 0.35/0.50
% 0.35/0.50 % antisymI
% 0.35/0.50 thf(fact_184_antisym__def,axiom,
% 0.35/0.50 ! [A: $tType] :
% 0.35/0.50 ( ( antisym @ A )
% 0.35/0.50 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.35/0.50 ! [X2: A,Y4: A] :
% 0.35/0.50 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R3 )
% 0.35/0.50 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ R3 )
% 0.35/0.50 => ( X2 = Y4 ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % antisym_def
% 0.35/0.50 thf(fact_185_case__prod__Pair__iden,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.35/0.50 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
% 0.35/0.50 = P2 ) ).
% 0.35/0.50
% 0.35/0.50 % case_prod_Pair_iden
% 0.35/0.50 thf(fact_186_case__prodD,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
% 0.35/0.50 ( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.35/0.50 => ( F @ A2 @ B2 ) ) ).
% 0.35/0.50
% 0.35/0.50 % case_prodD
% 0.35/0.50 thf(fact_187_case__prodE,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,C2: A > B > $o,P2: product_prod @ A @ B] :
% 0.35/0.50 ( ( product_case_prod @ A @ B @ $o @ C2 @ P2 )
% 0.35/0.50 => ~ ! [X3: A,Y3: B] :
% 0.35/0.50 ( ( P2
% 0.35/0.50 = ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 0.35/0.50 => ~ ( C2 @ X3 @ Y3 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % case_prodE
% 0.35/0.50 thf(fact_188_Collect__case__prod__Sigma,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] :
% 0.35/0.50 ( ( collect @ ( product_prod @ A @ B )
% 0.35/0.50 @ ( product_case_prod @ A @ B @ $o
% 0.35/0.50 @ ^ [X2: A,Y4: B] :
% 0.35/0.50 ( ( P @ X2 )
% 0.35/0.50 & ( Q @ X2 @ Y4 ) ) ) )
% 0.35/0.50 = ( product_Sigma @ A @ B @ ( collect @ A @ P )
% 0.35/0.50 @ ^ [X2: A] : ( collect @ B @ ( Q @ X2 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % Collect_case_prod_Sigma
% 0.35/0.50 thf(fact_189_case__prod__app,axiom,
% 0.35/0.50 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.35/0.50 ( ( product_case_prod @ B @ C @ ( D > A ) )
% 0.35/0.50 = ( ^ [F2: B > C > D > A,X2: product_prod @ B @ C,Y4: D] :
% 0.35/0.50 ( product_case_prod @ B @ C @ A
% 0.35/0.50 @ ^ [L: B,R3: C] : ( F2 @ L @ R3 @ Y4 )
% 0.35/0.50 @ X2 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % case_prod_app
% 0.35/0.50 thf(fact_190_prod_Ocase__distrib,axiom,
% 0.35/0.50 ! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.35/0.50 ( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.35/0.50 = ( product_case_prod @ A @ B @ D
% 0.35/0.50 @ ^ [X12: A,X23: B] : ( H @ ( F @ X12 @ X23 ) )
% 0.35/0.50 @ Prod ) ) ).
% 0.35/0.50
% 0.35/0.50 % prod.case_distrib
% 0.35/0.50 thf(fact_191_antisymP__equality,axiom,
% 0.35/0.50 ! [A: $tType] :
% 0.35/0.50 ( antisym @ A
% 0.35/0.50 @ ( collect @ ( product_prod @ A @ A )
% 0.35/0.50 @ ( product_case_prod @ A @ A @ $o
% 0.35/0.50 @ ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % antisymP_equality
% 0.35/0.50 thf(fact_192_internal__case__prod__def,axiom,
% 0.35/0.50 ! [C: $tType,B: $tType,A: $tType] :
% 0.35/0.50 ( ( produc2004651681e_prod @ A @ B @ C )
% 0.35/0.50 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.35/0.50
% 0.35/0.50 % internal_case_prod_def
% 0.35/0.50 thf(fact_193_swap__product,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,A3: set @ B,B3: set @ A] :
% 0.35/0.50 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
% 0.35/0.50 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
% 0.35/0.50 @ ^ [I2: B,J2: A] : ( product_Pair @ A @ B @ J2 @ I2 ) )
% 0.35/0.50 @ ( product_Sigma @ B @ A @ A3
% 0.35/0.50 @ ^ [Uu: B] : B3 ) )
% 0.35/0.50 = ( product_Sigma @ A @ B @ B3
% 0.35/0.50 @ ^ [Uu: A] : A3 ) ) ).
% 0.35/0.50
% 0.35/0.50 % swap_product
% 0.35/0.50 thf(fact_194_split__cong,axiom,
% 0.35/0.50 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
% 0.35/0.50 ( ! [X3: A,Y3: B] :
% 0.35/0.50 ( ( ( product_Pair @ A @ B @ X3 @ Y3 )
% 0.35/0.50 = Q2 )
% 0.35/0.50 => ( ( F @ X3 @ Y3 )
% 0.35/0.50 = ( G3 @ X3 @ Y3 ) ) )
% 0.35/0.50 => ( ( P2 = Q2 )
% 0.35/0.50 => ( ( product_case_prod @ A @ B @ C @ F @ P2 )
% 0.35/0.50 = ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % split_cong
% 0.35/0.50 thf(fact_195_case__swap,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,C: $tType,F: C > B > A,P2: product_prod @ C @ B] :
% 0.35/0.50 ( ( product_case_prod @ B @ C @ A
% 0.35/0.50 @ ^ [Y4: B,X2: C] : ( F @ X2 @ Y4 )
% 0.35/0.50 @ ( product_swap @ C @ B @ P2 ) )
% 0.35/0.50 = ( product_case_prod @ C @ B @ A @ F @ P2 ) ) ).
% 0.35/0.50
% 0.35/0.50 % case_swap
% 0.35/0.50 thf(fact_196_case__prod__curry,axiom,
% 0.35/0.50 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.35/0.50 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 0.35/0.50 = F ) ).
% 0.35/0.50
% 0.35/0.50 % case_prod_curry
% 0.35/0.50 thf(fact_197_curryI,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.35/0.50 ( ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.35/0.50 => ( product_curry @ A @ B @ $o @ F @ A2 @ B2 ) ) ).
% 0.35/0.50
% 0.35/0.50 % curryI
% 0.35/0.50 thf(fact_198_swap__swap,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.35/0.50 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 0.35/0.50 = P2 ) ).
% 0.35/0.50
% 0.35/0.50 % swap_swap
% 0.35/0.50 thf(fact_199_split__part,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
% 0.35/0.50 ( ( product_case_prod @ A @ B @ $o
% 0.35/0.50 @ ^ [A9: A,B9: B] :
% 0.35/0.50 ( P
% 0.35/0.50 & ( Q @ A9 @ B9 ) ) )
% 0.35/0.50 = ( ^ [Ab: product_prod @ A @ B] :
% 0.35/0.50 ( P
% 0.35/0.50 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % split_part
% 0.35/0.50 thf(fact_200_swap__simp,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,X: B,Y5: A] :
% 0.35/0.50 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y5 ) )
% 0.35/0.50 = ( product_Pair @ A @ B @ Y5 @ X ) ) ).
% 0.35/0.50
% 0.35/0.50 % swap_simp
% 0.35/0.50 thf(fact_201_curry__conv,axiom,
% 0.35/0.50 ! [A: $tType,C: $tType,B: $tType] :
% 0.35/0.50 ( ( product_curry @ B @ C @ A )
% 0.35/0.50 = ( ^ [F2: ( product_prod @ B @ C ) > A,A9: B,B9: C] : ( F2 @ ( product_Pair @ B @ C @ A9 @ B9 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % curry_conv
% 0.35/0.50 thf(fact_202_curry__case__prod,axiom,
% 0.35/0.50 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 0.35/0.50 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
% 0.35/0.50 = F ) ).
% 0.35/0.50
% 0.35/0.50 % curry_case_prod
% 0.35/0.50 thf(fact_203_pair__in__swap__image,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,Y5: A,X: B,A3: set @ ( product_prod @ B @ A )] :
% 0.35/0.50 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y5 @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A3 ) )
% 0.35/0.50 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y5 ) @ A3 ) ) ).
% 0.35/0.50
% 0.35/0.50 % pair_in_swap_image
% 0.35/0.50 thf(fact_204_prod_Odisc__eq__case,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.35/0.50 ( product_case_prod @ A @ B @ $o
% 0.35/0.50 @ ^ [Uu: A,Uv: B] : $true
% 0.35/0.50 @ Prod ) ).
% 0.35/0.50
% 0.35/0.50 % prod.disc_eq_case
% 0.35/0.50 thf(fact_205_curry__K,axiom,
% 0.35/0.50 ! [B: $tType,C: $tType,A: $tType,C2: C] :
% 0.35/0.50 ( ( product_curry @ A @ B @ C
% 0.35/0.50 @ ^ [X2: product_prod @ A @ B] : C2 )
% 0.35/0.50 = ( ^ [X2: A,Y4: B] : C2 ) ) ).
% 0.35/0.50
% 0.35/0.50 % curry_K
% 0.35/0.50 thf(fact_206_curryE,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.35/0.50 ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
% 0.35/0.50 => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % curryE
% 0.35/0.50 thf(fact_207_curryD,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 0.35/0.50 ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
% 0.35/0.50 => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % curryD
% 0.35/0.50 thf(fact_208_product__swap,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,A3: set @ B,B3: set @ A] :
% 0.35/0.50 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
% 0.35/0.50 @ ( product_Sigma @ B @ A @ A3
% 0.35/0.50 @ ^ [Uu: B] : B3 ) )
% 0.35/0.50 = ( product_Sigma @ A @ B @ B3
% 0.35/0.50 @ ^ [Uu: A] : A3 ) ) ).
% 0.35/0.50
% 0.35/0.50 % product_swap
% 0.35/0.50 thf(fact_209_curry__def,axiom,
% 0.35/0.50 ! [C: $tType,B: $tType,A: $tType] :
% 0.35/0.50 ( ( product_curry @ A @ B @ C )
% 0.35/0.50 = ( ^ [C5: ( product_prod @ A @ B ) > C,X2: A,Y4: B] : ( C5 @ ( product_Pair @ A @ B @ X2 @ Y4 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % curry_def
% 0.35/0.50 thf(fact_210_inv__image__def,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType] :
% 0.35/0.50 ( ( inv_image @ B @ A )
% 0.35/0.50 = ( ^ [R3: set @ ( product_prod @ B @ B ),F2: A > B] :
% 0.35/0.50 ( collect @ ( product_prod @ A @ A )
% 0.35/0.50 @ ( product_case_prod @ A @ A @ $o
% 0.35/0.50 @ ^ [X2: A,Y4: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) @ R3 ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % inv_image_def
% 0.35/0.50 thf(fact_211_map__prod__surj__on,axiom,
% 0.35/0.50 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: B > A,A3: set @ B,A11: set @ A,G3: D > C,B3: set @ D,B11: set @ C] :
% 0.35/0.50 ( ( ( image @ B @ A @ F @ A3 )
% 0.35/0.50 = A11 )
% 0.35/0.50 => ( ( ( image @ D @ C @ G3 @ B3 )
% 0.35/0.50 = B11 )
% 0.35/0.50 => ( ( image @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F @ G3 )
% 0.35/0.50 @ ( product_Sigma @ B @ D @ A3
% 0.35/0.50 @ ^ [Uu: B] : B3 ) )
% 0.35/0.50 = ( product_Sigma @ A @ C @ A11
% 0.35/0.50 @ ^ [Uu: A] : B11 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % map_prod_surj_on
% 0.35/0.50 thf(fact_212_map__prod__ident,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType] :
% 0.35/0.50 ( ( product_map_prod @ A @ A @ B @ B
% 0.35/0.50 @ ^ [X2: A] : X2
% 0.35/0.50 @ ^ [Y4: B] : Y4 )
% 0.35/0.50 = ( ^ [Z4: product_prod @ A @ B] : Z4 ) ) ).
% 0.35/0.50
% 0.35/0.50 % map_prod_ident
% 0.35/0.50 thf(fact_213_map__prod__simp,axiom,
% 0.35/0.50 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G3: D > B,A2: C,B2: D] :
% 0.35/0.50 ( ( product_map_prod @ C @ A @ D @ B @ F @ G3 @ ( product_Pair @ C @ D @ A2 @ B2 ) )
% 0.35/0.50 = ( product_Pair @ A @ B @ ( F @ A2 ) @ ( G3 @ B2 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % map_prod_simp
% 0.35/0.50 thf(fact_214_in__inv__image,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,X: A,Y5: A,R2: set @ ( product_prod @ B @ B ),F: A > B] :
% 0.35/0.50 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ ( inv_image @ B @ A @ R2 @ F ) )
% 0.35/0.50 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y5 ) ) @ R2 ) ) ).
% 0.35/0.50
% 0.35/0.50 % in_inv_image
% 0.35/0.50 thf(fact_215_map__prod__imageI,axiom,
% 0.35/0.50 ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R: set @ ( product_prod @ A @ B ),F: A > C,G3: B > D] :
% 0.35/0.50 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R )
% 0.35/0.50 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A2 ) @ ( G3 @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G3 ) @ R ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % map_prod_imageI
% 0.35/0.50 thf(fact_216_case__prod__map__prod,axiom,
% 0.35/0.50 ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F: D > B,G3: E > C,X: product_prod @ D @ E] :
% 0.35/0.50 ( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F @ G3 @ X ) )
% 0.35/0.50 = ( product_case_prod @ D @ E @ A
% 0.35/0.50 @ ^ [L: D,R3: E] : ( H @ ( F @ L ) @ ( G3 @ R3 ) )
% 0.35/0.50 @ X ) ) ).
% 0.35/0.50
% 0.35/0.50 % case_prod_map_prod
% 0.35/0.50 thf(fact_217_prod__fun__imageE,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F: C > A,G3: D > B,R: set @ ( product_prod @ C @ D )] :
% 0.35/0.50 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G3 ) @ R ) )
% 0.35/0.50 => ~ ! [X3: C,Y3: D] :
% 0.35/0.50 ( ( C2
% 0.35/0.50 = ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G3 @ Y3 ) ) )
% 0.35/0.50 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % prod_fun_imageE
% 0.35/0.50 thf(fact_218_map__prod__def,axiom,
% 0.35/0.50 ! [B: $tType,D: $tType,C: $tType,A: $tType] :
% 0.35/0.50 ( ( product_map_prod @ A @ C @ B @ D )
% 0.35/0.50 = ( ^ [F2: A > C,G4: B > D] :
% 0.35/0.50 ( product_case_prod @ A @ B @ ( product_prod @ C @ D )
% 0.35/0.50 @ ^ [X2: A,Y4: B] : ( product_Pair @ C @ D @ ( F2 @ X2 ) @ ( G4 @ Y4 ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % map_prod_def
% 0.35/0.50 thf(fact_219_rp__inv__image__def,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType] :
% 0.35/0.50 ( ( fun_rp_inv_image @ A @ B )
% 0.35/0.50 = ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
% 0.35/0.50 @ ^ [R4: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A ),F2: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R4 @ F2 ) @ ( inv_image @ A @ B @ S3 @ F2 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % rp_inv_image_def
% 0.35/0.50 thf(fact_220_prod_Omap__ident,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.35/0.50 ( ( product_map_prod @ A @ A @ B @ B
% 0.35/0.50 @ ^ [X2: A] : X2
% 0.35/0.50 @ ^ [X2: B] : X2
% 0.35/0.50 @ T2 )
% 0.35/0.50 = T2 ) ).
% 0.35/0.50
% 0.35/0.50 % prod.map_ident
% 0.35/0.50 thf(fact_221_surj__swap,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType] :
% 0.35/0.50 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.35/0.50 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % surj_swap
% 0.35/0.50 thf(fact_222_The__split__eq,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,X: A,Y5: B] :
% 0.35/0.50 ( ( the @ ( product_prod @ A @ B )
% 0.35/0.50 @ ( product_case_prod @ A @ B @ $o
% 0.35/0.50 @ ^ [X6: A,Y9: B] :
% 0.35/0.50 ( ( X = X6 )
% 0.35/0.50 & ( Y5 = Y9 ) ) ) )
% 0.35/0.50 = ( product_Pair @ A @ B @ X @ Y5 ) ) ).
% 0.35/0.50
% 0.35/0.50 % The_split_eq
% 0.35/0.50 thf(fact_223_UNIV__I,axiom,
% 0.35/0.50 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % UNIV_I
% 0.35/0.50 thf(fact_224_sup__top__right,axiom,
% 0.35/0.50 ! [A: $tType] :
% 0.35/0.50 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 0.35/0.50 => ! [X: A] :
% 0.35/0.50 ( ( sup_sup @ A @ X @ ( top_top @ A ) )
% 0.35/0.50 = ( top_top @ A ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % sup_top_right
% 0.35/0.50 thf(fact_225_sup__top__left,axiom,
% 0.35/0.50 ! [A: $tType] :
% 0.35/0.50 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 0.35/0.50 => ! [X: A] :
% 0.35/0.50 ( ( sup_sup @ A @ ( top_top @ A ) @ X )
% 0.35/0.50 = ( top_top @ A ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % sup_top_left
% 0.35/0.50 thf(fact_226_vimage__UNIV,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType,F: A > B] :
% 0.35/0.50 ( ( vimage @ A @ B @ F @ ( top_top @ ( set @ B ) ) )
% 0.35/0.50 = ( top_top @ ( set @ A ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % vimage_UNIV
% 0.35/0.50 thf(fact_227_UNIV__Plus__UNIV,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType] :
% 0.35/0.50 ( ( sum_Plus @ A @ B @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.35/0.50 = ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % UNIV_Plus_UNIV
% 0.35/0.50 thf(fact_228_UNIV__Times__UNIV,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType] :
% 0.35/0.50 ( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 0.35/0.50 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
% 0.35/0.50 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % UNIV_Times_UNIV
% 0.35/0.50 thf(fact_229_UNIV__def,axiom,
% 0.35/0.50 ! [A: $tType] :
% 0.35/0.50 ( ( top_top @ ( set @ A ) )
% 0.35/0.50 = ( collect @ A
% 0.35/0.50 @ ^ [X2: A] : $true ) ) ).
% 0.35/0.50
% 0.35/0.50 % UNIV_def
% 0.35/0.50 thf(fact_230_rangeE,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,B2: A,F: B > A] :
% 0.35/0.50 ( ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
% 0.35/0.50 => ~ ! [X3: B] :
% 0.35/0.50 ( B2
% 0.35/0.50 != ( F @ X3 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % rangeE
% 0.35/0.50 thf(fact_231_range__composition,axiom,
% 0.35/0.50 ! [A: $tType,C: $tType,B: $tType,F: C > A,G3: B > C] :
% 0.35/0.50 ( ( image @ B @ A
% 0.35/0.50 @ ^ [X2: B] : ( F @ ( G3 @ X2 ) )
% 0.35/0.50 @ ( top_top @ ( set @ B ) ) )
% 0.35/0.50 = ( image @ C @ A @ F @ ( image @ B @ C @ G3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % range_composition
% 0.35/0.50 thf(fact_232_rangeI,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % rangeI
% 0.35/0.50 thf(fact_233_range__eqI,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType,B2: A,F: B > A,X: B] :
% 0.35/0.50 ( ( B2
% 0.35/0.50 = ( F @ X ) )
% 0.35/0.50 => ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % range_eqI
% 0.35/0.50 thf(fact_234_Un__UNIV__left,axiom,
% 0.35/0.50 ! [A: $tType,B3: set @ A] :
% 0.35/0.50 ( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 )
% 0.35/0.50 = ( top_top @ ( set @ A ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % Un_UNIV_left
% 0.35/0.50 thf(fact_235_Un__UNIV__right,axiom,
% 0.35/0.50 ! [A: $tType,A3: set @ A] :
% 0.35/0.50 ( ( sup_sup @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
% 0.35/0.50 = ( top_top @ ( set @ A ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % Un_UNIV_right
% 0.35/0.50 thf(fact_236_UNIV__eq__I,axiom,
% 0.35/0.50 ! [A: $tType,A3: set @ A] :
% 0.35/0.50 ( ! [X3: A] : ( member @ A @ X3 @ A3 )
% 0.35/0.50 => ( ( top_top @ ( set @ A ) )
% 0.35/0.50 = A3 ) ) ).
% 0.35/0.50
% 0.35/0.50 % UNIV_eq_I
% 0.35/0.50 thf(fact_237_UNIV__witness,axiom,
% 0.35/0.50 ! [A: $tType] :
% 0.35/0.50 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % UNIV_witness
% 0.35/0.50 thf(fact_238_UNIV__sum,axiom,
% 0.35/0.50 ! [A: $tType,B: $tType] :
% 0.35/0.50 ( ( top_top @ ( set @ ( sum_sum @ A @ B ) ) )
% 0.35/0.50 = ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ ( top_top @ ( set @ A ) ) ) @ ( image @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % UNIV_sum
% 0.35/0.50 thf(fact_239_map__prod__surj,axiom,
% 0.35/0.50 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G3: C > D] :
% 0.35/0.50 ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.35/0.50 = ( top_top @ ( set @ B ) ) )
% 0.35/0.50 => ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
% 0.35/0.50 = ( top_top @ ( set @ D ) ) )
% 0.35/0.50 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 0.35/0.50 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % map_prod_surj
% 0.35/0.50 thf(fact_240_old_Orec__sum__def,axiom,
% 0.35/0.50 ! [B: $tType,T: $tType,A: $tType] :
% 0.35/0.50 ( ( sum_rec_sum @ A @ T @ B )
% 0.35/0.50 = ( ^ [F12: A > T,F23: B > T,X2: sum_sum @ A @ B] : ( the @ T @ ( sum_rec_set_sum @ A @ T @ B @ F12 @ F23 @ X2 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % old.rec_sum_def
% 0.35/0.50 thf(fact_241_old_Orec__prod__def,axiom,
% 0.35/0.50 ! [T: $tType,B: $tType,A: $tType] :
% 0.35/0.50 ( ( product_rec_prod @ A @ B @ T )
% 0.35/0.50 = ( ^ [F12: A > B > T,X2: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X2 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % old.rec_prod_def
% 0.35/0.50 thf(fact_242_top__empty__eq2,axiom,
% 0.35/0.50 ! [B: $tType,A: $tType] :
% 0.35/0.50 ( ( top_top @ ( A > B > $o ) )
% 0.35/0.50 = ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % top_empty_eq2
% 0.35/0.50 thf(fact_243_top__set__def,axiom,
% 0.35/0.50 ! [A: $tType] :
% 0.35/0.50 ( ( top_top @ ( set @ A ) )
% 0.35/0.50 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % top_set_def
% 0.35/0.50 thf(fact_244_top__empty__eq,axiom,
% 0.35/0.50 ! [A: $tType] :
% 0.35/0.50 ( ( top_top @ ( A > $o ) )
% 0.35/0.50 = ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % top_empty_eq
% 0.35/0.50 thf(fact_245_the__sym__eq__trivial,axiom,
% 0.35/0.50 ! [A: $tType,X: A] :
% 0.35/0.50 ( ( the @ A
% 0.35/0.50 @ ( ^ [Y6: A,Z2: A] : ( Y6 = Z2 )
% 0.35/0.50 @ X ) )
% 0.35/0.50 = X ) ).
% 0.35/0.50
% 0.35/0.50 % the_sym_eq_trivial
% 0.35/0.50 thf(fact_246_the__eq__trivial,axiom,
% 0.35/0.50 ! [A: $tType,A2: A] :
% 0.35/0.50 ( ( the @ A
% 0.35/0.50 @ ^ [X2: A] : ( X2 = A2 ) )
% 0.35/0.50 = A2 ) ).
% 0.35/0.50
% 0.35/0.50 % the_eq_trivial
% 0.35/0.50 thf(fact_247_the__equality,axiom,
% 0.35/0.50 ! [A: $tType,P: A > $o,A2: A] :
% 0.35/0.50 ( ( P @ A2 )
% 0.35/0.50 => ( ! [X3: A] :
% 0.35/0.50 ( ( P @ X3 )
% 0.35/0.50 => ( X3 = A2 ) )
% 0.35/0.50 => ( ( the @ A @ P )
% 0.35/0.50 = A2 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % the_equality
% 0.35/0.50 thf(fact_248_top__conj_I2_J,axiom,
% 0.35/0.50 ! [A: $tType,P: $o,X: A] :
% 0.35/0.50 ( ( P
% 0.35/0.50 & ( top_top @ ( A > $o ) @ X ) )
% 0.35/0.50 = P ) ).
% 0.35/0.50
% 0.35/0.50 % top_conj(2)
% 0.35/0.50 thf(fact_249_top__conj_I1_J,axiom,
% 0.35/0.50 ! [A: $tType,X: A,P: $o] :
% 0.35/0.50 ( ( ( top_top @ ( A > $o ) @ X )
% 0.35/0.50 & P )
% 0.35/0.50 = P ) ).
% 0.35/0.50
% 0.35/0.50 % top_conj(1)
% 0.35/0.50 thf(fact_250_theI,axiom,
% 0.35/0.50 ! [A: $tType,P: A > $o,A2: A] :
% 0.35/0.50 ( ( P @ A2 )
% 0.35/0.50 => ( ! [X3: A] :
% 0.35/0.50 ( ( P @ X3 )
% 0.35/0.50 => ( X3 = A2 ) )
% 0.35/0.50 => ( P @ ( the @ A @ P ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % theI
% 0.35/0.50 thf(fact_251_theI_H,axiom,
% 0.35/0.50 ! [A: $tType,P: A > $o] :
% 0.35/0.50 ( ? [X4: A] :
% 0.35/0.50 ( ( P @ X4 )
% 0.35/0.50 & ! [Y3: A] :
% 0.35/0.50 ( ( P @ Y3 )
% 0.35/0.50 => ( Y3 = X4 ) ) )
% 0.35/0.50 => ( P @ ( the @ A @ P ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % theI'
% 0.35/0.50 thf(fact_252_theI2,axiom,
% 0.35/0.50 ! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
% 0.35/0.50 ( ( P @ A2 )
% 0.35/0.50 => ( ! [X3: A] :
% 0.35/0.50 ( ( P @ X3 )
% 0.35/0.50 => ( X3 = A2 ) )
% 0.35/0.50 => ( ! [X3: A] :
% 0.35/0.50 ( ( P @ X3 )
% 0.35/0.50 => ( Q @ X3 ) )
% 0.35/0.50 => ( Q @ ( the @ A @ P ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % theI2
% 0.35/0.50 thf(fact_253_If__def,axiom,
% 0.35/0.50 ! [A: $tType] :
% 0.35/0.50 ( ( if @ A )
% 0.35/0.50 = ( ^ [P4: $o,X2: A,Y4: A] :
% 0.35/0.50 ( the @ A
% 0.35/0.50 @ ^ [Z4: A] :
% 0.35/0.50 ( ( P4
% 0.35/0.50 => ( Z4 = X2 ) )
% 0.35/0.50 & ( ~ P4
% 0.35/0.50 => ( Z4 = Y4 ) ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % If_def
% 0.35/0.50 thf(fact_254_the1I2,axiom,
% 0.35/0.50 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.35/0.50 ( ? [X4: A] :
% 0.35/0.50 ( ( P @ X4 )
% 0.35/0.50 & ! [Y3: A] :
% 0.35/0.50 ( ( P @ Y3 )
% 0.35/0.50 => ( Y3 = X4 ) ) )
% 0.35/0.50 => ( ! [X3: A] :
% 0.35/0.50 ( ( P @ X3 )
% 0.35/0.50 => ( Q @ X3 ) )
% 0.35/0.50 => ( Q @ ( the @ A @ P ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % the1I2
% 0.35/0.50 thf(fact_255_the1__equality,axiom,
% 0.35/0.50 ! [A: $tType,P: A > $o,A2: A] :
% 0.35/0.50 ( ? [X4: A] :
% 0.35/0.50 ( ( P @ X4 )
% 0.35/0.50 & ! [Y3: A] :
% 0.35/0.50 ( ( P @ Y3 )
% 0.35/0.50 => ( Y3 = X4 ) ) )
% 0.35/0.50 => ( ( P @ A2 )
% 0.35/0.50 => ( ( the @ A @ P )
% 0.35/0.50 = A2 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 % the1_equality
% 0.35/0.50
% 0.35/0.50 %----Type constructors (12)
% 0.35/0.50 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 0.35/0.50 bounded_lattice @ $o @ ( type2 @ $o ) ).
% 0.35/0.50
% 0.35/0.50 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 0.35/0.50 ! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 0.35/0.50 ! [A7: $tType,A12: $tType] :
% 0.35/0.50 ( ( bounded_lattice @ A12 @ ( type2 @ A12 ) )
% 0.35/0.50 => ( bounded_lattice @ ( A7 > A12 ) @ ( type2 @ ( A7 > A12 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
% 0.35/0.50 ! [A7: $tType,A12: $tType] :
% 0.35/0.50 ( ( bounded_lattice @ A12 @ ( type2 @ A12 ) )
% 0.35/0.50 => ( bounded_lattice_top @ ( A7 > A12 ) @ ( type2 @ ( A7 > A12 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.35/0.50 ! [A7: $tType,A12: $tType] :
% 0.35/0.50 ( ( semilattice_sup @ A12 @ ( type2 @ A12 ) )
% 0.35/0.50 => ( semilattice_sup @ ( A7 > A12 ) @ ( type2 @ ( A7 > A12 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.35/0.50 ! [A7: $tType,A12: $tType] :
% 0.35/0.50 ( ( lattice @ A12 @ ( type2 @ A12 ) )
% 0.35/0.50 => ( lattice @ ( A7 > A12 ) @ ( type2 @ ( A7 > A12 ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_3,axiom,
% 0.35/0.50 ! [A7: $tType] : ( bounded_lattice_top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_4,axiom,
% 0.35/0.50 ! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 thf(tcon_Set_Oset___Lattices_Olattice_5,axiom,
% 0.35/0.50 ! [A7: $tType] : ( lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.35/0.50
% 0.35/0.50 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_6,axiom,
% 0.35/0.50 bounded_lattice_top @ $o @ ( type2 @ $o ) ).
% 0.35/0.50
% 0.35/0.50 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_7,axiom,
% 0.35/0.50 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 0.35/0.50
% 0.35/0.50 thf(tcon_HOL_Obool___Lattices_Olattice_8,axiom,
% 0.35/0.50 lattice @ $o @ ( type2 @ $o ) ).
% 0.35/0.50
% 0.35/0.50 %----Helper facts (3)
% 0.35/0.50 thf(help_If_3_1_T,axiom,
% 0.35/0.50 ! [P: $o] :
% 0.35/0.50 ( ( P = $true )
% 0.35/0.50 | ( P = $false ) ) ).
% 0.35/0.50
% 0.35/0.50 thf(help_If_2_1_T,axiom,
% 0.35/0.50 ! [A: $tType,X: A,Y5: A] :
% 0.35/0.50 ( ( if @ A @ $false @ X @ Y5 )
% 0.35/0.50 = Y5 ) ).
% 0.35/0.50
% 0.35/0.50 thf(help_If_1_1_T,axiom,
% 0.35/0.50 ! [A: $tType,X: A,Y5: A] :
% 0.35/0.50 ( ( if @ A @ $true @ X @ Y5 )
% 0.35/0.50 = X ) ).
% 0.35/0.50
% 0.35/0.50 %----Conjectures (1)
% 0.35/0.50 thf(conj_0,conjecture,
% 0.35/0.50 ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ t2 ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) ) )
% 0.35/0.50 = ( member @ dtree @ t2
% 0.35/0.50 @ ( image @ ( product_prod @ dtree @ dtree ) @ dtree @ parall1899940088le_par
% 0.35/0.50 @ ( product_Sigma @ dtree @ dtree
% 0.35/0.50 @ ( collect @ dtree
% 0.35/0.50 @ ^ [X2: dtree] : ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ X2 ) @ ( cont @ tr1 ) ) )
% 0.35/0.50 @ ^ [Uu: dtree] :
% 0.35/0.50 ( collect @ dtree
% 0.35/0.50 @ ^ [X2: dtree] : ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ X2 ) @ ( cont @ tr2 ) ) ) ) ) ) ) ).
% 0.35/0.50
% 0.35/0.50 %------------------------------------------------------------------------------
% 0.35/0.50 ------------------------
% 3.41/1.87 ---- Embedded in THF ---
% 3.41/1.87 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.41/1.87 %%% Generated on Tue Feb 24 04:16:09 EST 2026
% 3.41/1.87 %%% using '$$dhol' embedding, version 1.3.0.
% 3.41/1.87 %%% Logic specification used:
% 3.41/1.87 %%% thf(spec, logic, $$dhol).
% 3.41/1.87
% 3.41/1.87 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.ssUgw0micP/DTF2DTF15689.p
% 3.41/1.87 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 3.41/1.87 thf(ty_t_Sum__Type_Osum, type, sum_sum: ($tType > ($tType > $tType))).
% 3.41/1.87 thf(ty_t_DTree_Odtree, type, dtree: $tType).
% 3.41/1.87 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.41/1.87 thf(ty_t_DTree_OT, type, t: $tType).
% 3.41/1.87 thf(ty_t_DTree_ON, type, n: $tType).
% 3.41/1.87 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 3.41/1.87 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.41/1.87 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.41/1.87 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.41/1.87 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.41/1.87 thf(sy_cl_Lattices_Obounded__lattice__top, type, bounded_lattice_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.41/1.87 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2, type, bNF_Greatest_image2: (!> [C:$tType,A:$tType,B:$tType]: (((set @ C) > ((C > A) > ((C > B) > (set @ ((product_prod @ A) @ B)))))))).
% 3.41/1.87 thf(sy_c_DTree_Ocont, type, cont: (dtree > (set @ ((sum_sum @ t) @ dtree)))).
% 3.41/1.87 thf(sy_c_DTree_Ounfold, type, unfold: (!> [A:$tType]: (((A > n) > ((A > (set @ ((sum_sum @ t) @ A))) > (A > dtree)))))).
% 3.41/1.87 thf(sy_c_Fun__Def_Orp__inv__image, type, fun_rp_inv_image: (!> [A:$tType,B:$tType]: ((((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) > ((B > A) > ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B)))))))).
% 3.41/1.87 thf(sy_c_HOL_OThe, type, the: (!> [A:$tType]: (((A > $o) > A)))).
% 3.41/1.87 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 3.41/1.87 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 3.41/1.87 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 3.41/1.87 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar, type, parall1899940088le_par: (((product_prod @ dtree) @ dtree) > dtree)).
% 3.41/1.87 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c, type, parall1914194347_par_c: (((product_prod @ dtree) @ dtree) > (set @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree))))).
% 3.41/1.87 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c__rel, type, parall1418634510_c_rel: (((product_prod @ dtree) @ dtree) > (((product_prod @ dtree) @ dtree) > $o))).
% 3.41/1.87 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r, type, parall1914194362_par_r: (((product_prod @ dtree) @ dtree) > n)).
% 3.41/1.87 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 3.41/1.87 thf(sy_c_Product__Type_OSigma, type, product_Sigma: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ ((product_prod @ A) @ B))))))).
% 3.41/1.87 thf(sy_c_Product__Type_Ocurry, type, product_curry: (!> [A:$tType,B:$tType,C:$tType]: (((((product_prod @ A) @ B) > C) > (A > (B > C)))))).
% 3.41/1.87 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 3.41/1.87 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 3.41/1.87 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 3.41/1.87 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod, type, product_rec_set_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > (T > $o)))))).
% 3.41/1.87 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 3.41/1.87 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 3.41/1.87 thf(sy_c_Product__Type_Oproduct, type, product_product: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ B) > (set @ ((product_prod @ A) @ B))))))).
% 3.41/1.87 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 3.41/1.87 thf(sy_c_Relation_OId, type, id: (!> [A:$tType]: ((set @ ((product_prod @ A) @ A))))).
% 3.41/1.87 thf(sy_c_Relation_Oantisym, type, antisym: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 3.41/1.87 thf(sy_c_Relation_Oinv__image, type, inv_image: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ B)) > ((A > B) > (set @ ((product_prod @ A) @ A))))))).
% 3.41/1.87 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.41/1.87 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 3.41/1.87 thf(sy_c_Set_Ovimage, type, vimage: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ B) > (set @ A)))))).
% 3.41/1.87 thf(sy_c_Sum__Type_OInl, type, sum_Inl: (!> [A:$tType,B:$tType]: ((A > ((sum_sum @ A) @ B))))).
% 3.41/1.87 thf(sy_c_Sum__Type_OInr, type, sum_Inr: (!> [B:$tType,A:$tType]: ((B > ((sum_sum @ A) @ B))))).
% 3.41/1.87 thf(sy_c_Sum__Type_OPlus, type, sum_Plus: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ B) > (set @ ((sum_sum @ A) @ B))))))).
% 3.41/1.87 thf(sy_c_Sum__Type_OSuml, type, sum_Suml: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > (((sum_sum @ A) @ B) > C))))).
% 3.41/1.87 thf(sy_c_Sum__Type_OSumr, type, sum_Sumr: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > (((sum_sum @ A) @ B) > C))))).
% 3.41/1.87 thf(sy_c_Sum__Type_Oold_Osum_Orec__set__sum, type, sum_rec_set_sum: (!> [A:$tType,T:$tType,B:$tType]: (((A > T) > ((B > T) > (((sum_sum @ A) @ B) > (T > $o))))))).
% 3.41/1.87 thf(sy_c_Sum__Type_Oold_Osum_Orec__sum, type, sum_rec_sum: (!> [A:$tType,T:$tType,B:$tType]: (((A > T) > ((B > T) > (((sum_sum @ A) @ B) > T)))))).
% 3.41/1.87 thf(sy_c_Wellfounded_Oaccp, type, accp: (!> [A:$tType]: (((A > (A > $o)) > (A > $o))))).
% 3.41/1.87 thf(sy_c_Wellfounded_Olex__prod, type, lex_prod: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 3.41/1.87 thf(sy_c_Wfrec_Osame__fst, type, same_fst: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > (set @ ((product_prod @ B) @ B))) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 3.41/1.87 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.41/1.87 thf(sy_v_t, type, t2: dtree).
% 3.41/1.87 thf(sy_v_tr1, type, tr1: dtree).
% 3.41/1.87 thf(sy_v_tr2, type, tr2: dtree).
% 3.41/1.87 thf(fact_0_par__r_Ocases, axiom, (! [X:((product_prod @ dtree) @ dtree)]: ((~ ((! [Tr1:dtree,Tr2:dtree]: ((X != ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2))))))))).
% 3.41/1.87 thf(fact_1_par__r_Oinduct, axiom, (! [P:(((product_prod @ dtree) @ dtree) > $o),A0:((product_prod @ dtree) @ dtree)]: (((! [Tr1:dtree,Tr2:dtree]: ((P @ ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2)))) => (P @ A0))))).
% 3.41/1.87 thf(fact_2__092_060open_062Inr_A_N_096_Acont_A_Itr1_A_092_060parallel_062_Atr2_J_A_061_Apar_A_096_A_IInr_A_N_096_Acont_Atr1_A_092_060times_062_AInr_A_N_096_Acont_Atr2_J_092_060close_062, axiom, (((((vimage @ dtree) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inr @ dtree) @ t)) @ (cont @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ tr1) @ tr2)))) = ((((image @ ((product_prod @ dtree) @ dtree)) @ dtree) @ parall1899940088le_par) @ ((((product_Sigma @ dtree) @ dtree) @ ((((vimage @ dtree) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inr @ dtree) @ t)) @ (cont @ tr1))) @ (^ [Uu:dtree]: (((((vimage @ dtree) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inr @ dtree) @ t)) @ (cont @ tr2)))))))).
% 3.41/1.87 thf(fact_3_SigmaI, axiom, (! [B:$tType,A:$tType,A2:A,A3:(set @ A),B2:B,B3:(A > (set @ B))]: (((((member @ A) @ A2) @ A3) => ((((member @ B) @ B2) @ (B3 @ A2)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3))))))).
% 3.41/1.87 thf(fact_4_mem__Sigma__iff, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,A3:(set @ A),B3:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) = ((((member @ A) @ A2) @ A3) & (((member @ B) @ B2) @ (B3 @ A2))))))).
% 3.41/1.87 thf(fact_5_image__ident, axiom, (! [A:$tType,Y:(set @ A)]: ((((((image @ A) @ A) @ (^ [X2:A]: (X2))) @ Y) = Y)))).
% 3.41/1.87 thf(fact_6_Inr__cont__par, axiom, (! [Tr12:dtree,Tr22:dtree]: ((((((vimage @ dtree) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inr @ dtree) @ t)) @ (cont @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr12) @ Tr22)))) = ((((image @ ((product_prod @ dtree) @ dtree)) @ dtree) @ parall1899940088le_par) @ ((((product_Sigma @ dtree) @ dtree) @ ((((vimage @ dtree) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inr @ dtree) @ t)) @ (cont @ Tr12))) @ (^ [Uu:dtree]: (((((vimage @ dtree) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inr @ dtree) @ t)) @ (cont @ Tr22)))))))))).
% 3.41/1.87 thf(fact_7_sum_Oinject_I2_J, axiom, (! [A:$tType,B:$tType,X22:B,Y2:B]: ((((((sum_Inr @ B) @ A) @ X22) = (((sum_Inr @ B) @ A) @ Y2)) = (X22 = Y2))))).
% 3.41/1.87 thf(fact_8_old_Osum_Oinject_I2_J, axiom, (! [A:$tType,B:$tType,B2:B,B4:B]: ((((((sum_Inr @ B) @ A) @ B2) = (((sum_Inr @ B) @ A) @ B4)) = (B2 = B4))))).
% 3.41/1.87 thf(fact_9_image__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),X:B,A3:(set @ B)]: (((B2 = (F @ X)) => ((((member @ B) @ X) @ A3) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ A3))))))).
% 3.41/1.87 thf(fact_10_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X22 = Y2)))))).
% 3.41/1.87 thf(fact_11_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A4:A,B4:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A4) @ B4)) = ((A2 = A4) & (B2 = B4)))))).
% 3.41/1.87 thf(fact_12_SigmaE, axiom, (! [A:$tType,B:$tType,C2:((product_prod @ A) @ B),A3:(set @ A),B3:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ C2) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) => (~ ((! [X3:A]: (((((member @ A) @ X3) @ A3) => (! [Y3:B]: (((((member @ B) @ Y3) @ (B3 @ X3)) => (C2 != ((((product_Pair @ A) @ B) @ X3) @ Y3)))))))))))))).
% 3.41/1.87 thf(fact_13_SigmaD1, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,A3:(set @ A),B3:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) => (((member @ A) @ A2) @ A3))))).
% 3.41/1.87 thf(fact_14_vimageI, axiom, (! [B:$tType,A:$tType,F:(B > A),A2:B,B2:A,B3:(set @ A)]: ((((F @ A2) = B2) => ((((member @ A) @ B2) @ B3) => (((member @ B) @ A2) @ ((((vimage @ B) @ A) @ F) @ B3))))))).
% 3.41/1.87 thf(fact_15_vimage__eq, axiom, (! [A:$tType,B:$tType,A2:A,F:(A > B),B3:(set @ B)]: (((((member @ A) @ A2) @ ((((vimage @ A) @ B) @ F) @ B3)) = (((member @ B) @ (F @ A2)) @ B3))))).
% 3.41/1.87 thf(fact_16_vimage__ident, axiom, (! [A:$tType,Y:(set @ A)]: ((((((vimage @ A) @ A) @ (^ [X2:A]: (X2))) @ Y) = Y)))).
% 3.41/1.87 thf(fact_17_vimage__Collect__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),P:(B > $o)]: ((((((vimage @ A) @ B) @ F) @ ((collect @ B) @ P)) = ((collect @ A) @ (^ [Y4:A]: ((P @ (F @ Y4))))))))).
% 3.41/1.87 thf(fact_18_vimageD, axiom, (! [A:$tType,B:$tType,A2:A,F:(A > B),A3:(set @ B)]: (((((member @ A) @ A2) @ ((((vimage @ A) @ B) @ F) @ A3)) => (((member @ B) @ (F @ A2)) @ A3))))).
% 3.41/1.87 thf(fact_19_vimageE, axiom, (! [A:$tType,B:$tType,A2:A,F:(A > B),B3:(set @ B)]: (((((member @ A) @ A2) @ ((((vimage @ A) @ B) @ F) @ B3)) => (((member @ B) @ (F @ A2)) @ B3))))).
% 3.41/1.87 thf(fact_20_vimageI2, axiom, (! [B:$tType,A:$tType,F:(B > A),A2:B,A3:(set @ A)]: (((((member @ A) @ (F @ A2)) @ A3) => (((member @ B) @ A2) @ ((((vimage @ B) @ A) @ F) @ A3)))))).
% 3.41/1.87 thf(fact_21_vimage__def, axiom, (! [B:$tType,A:$tType]: ((((vimage @ A) @ B) = (^ [F2:(A > B),B5:(set @ B)]: (((collect @ A) @ (^ [X2:A]: ((((member @ B) @ (F2 @ X2)) @ B5)))))))))).
% 3.41/1.87 thf(fact_22_vimage__Collect, axiom, (! [B:$tType,A:$tType,P:(B > $o),F:(A > B),Q:(A > $o)]: (((! [X3:A]: (((P @ (F @ X3)) = (Q @ X3)))) => (((((vimage @ A) @ B) @ F) @ ((collect @ B) @ P)) = ((collect @ A) @ Q)))))).
% 3.41/1.87 thf(fact_23_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B6:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B6)))) => (P @ Prod))))).
% 3.41/1.87 thf(fact_24_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y5:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B6:B]: ((Y5 != ((((product_Pair @ A) @ B) @ A5) @ B6))))))))).
% 3.41/1.87 thf(fact_25_prod__induct7, axiom, (! [G:$tType,F3:$tType,E:$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 @ F3) @ G)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G))))))]: (((! [A5:A,B6:B,C3:C,D2:D,E2:E,F4:F3,G2:G]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G))))) @ B6) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F3) @ G)) @ E2) @ ((((product_Pair @ F3) @ G) @ F4) @ G2))))))))) => (P @ X))))).
% 3.41/1.87 thf(fact_26_prod__induct6, axiom, (! [F3:$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) @ F3))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))))]: (((! [A5:A,B6:B,C3:C,D2:D,E2:E,F4:F3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))) @ B6) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F3)) @ D2) @ ((((product_Pair @ E) @ F3) @ E2) @ F4)))))))) => (P @ X))))).
% 3.41/1.87 thf(fact_27_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A5:A,B6:B,C3:C,D2:D,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B6) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C3) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P @ X))))).
% 3.41/1.87 thf(fact_28_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A5:A,B6:B,C3:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B6) @ ((((product_Pair @ C) @ D) @ C3) @ D2)))))) => (P @ X))))).
% 3.41/1.87 thf(fact_29_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B6:B,C3:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B6) @ C3))))) => (P @ X))))).
% 3.41/1.87 thf(fact_30_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F3:$tType,G:$tType,Y5:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G))))))]: ((~ ((! [A5:A,B6:B,C3:C,D2:D,E2:E,F4:F3,G2:G]: ((Y5 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G))))) @ B6) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F3) @ G)) @ E2) @ ((((product_Pair @ F3) @ G) @ F4) @ G2)))))))))))))).
% 3.41/1.87 thf(fact_31_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F3:$tType,Y5:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))))]: ((~ ((! [A5:A,B6:B,C3:C,D2:D,E2:E,F4:F3]: ((Y5 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))) @ B6) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F3)) @ D2) @ ((((product_Pair @ E) @ F3) @ E2) @ F4))))))))))))).
% 3.41/1.87 thf(fact_32_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y5:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A5:A,B6:B,C3:C,D2:D,E2:E]: ((Y5 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B6) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C3) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 3.41/1.87 thf(fact_33_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y5:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A5:A,B6:B,C3:C,D2:D]: ((Y5 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B6) @ ((((product_Pair @ C) @ D) @ C3) @ D2))))))))))).
% 3.41/1.87 thf(fact_34_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y5:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B6:B,C3:C]: ((Y5 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B6) @ C3)))))))))).
% 3.41/1.87 thf(fact_35_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A4:A,B4:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A4) @ B4)) => (~ (((A2 = A4) => (B2 != B4)))))))).
% 3.41/1.87 thf(fact_36_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A5:A,B6:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B6)))) => (P @ P2))))).
% 3.41/1.87 thf(fact_37_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X3:A,Y3:B]: ((P2 = ((((product_Pair @ A) @ B) @ X3) @ Y3))))))).
% 3.41/1.87 thf(fact_38_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X:A,A3:(set @ A),B2:B,F:(A > B)]: (((((member @ A) @ X) @ A3) => ((B2 = (F @ X)) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ F) @ A3))))))).
% 3.41/1.87 thf(fact_39_ball__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:(set @ B),P:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((((image @ B) @ A) @ F) @ A3)) => (P @ X3)))) => (! [X4:B]: (((((member @ B) @ X4) @ A3) => (P @ (F @ X4))))))))).
% 3.41/1.87 thf(fact_40_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N:(set @ A),F:(A > B),G3:(A > B)]: (((M = N) => ((! [X3:A]: (((((member @ A) @ X3) @ N) => ((F @ X3) = (G3 @ X3))))) => (((((image @ A) @ B) @ F) @ M) = ((((image @ A) @ B) @ G3) @ N))))))).
% 3.41/1.87 thf(fact_41_bex__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:(set @ B),P:(A > $o)]: (((? [X4:A]: (((((member @ A) @ X4) @ ((((image @ B) @ A) @ F) @ A3)) & (P @ X4)))) => (? [X3:B]: (((((member @ B) @ X3) @ A3) & (P @ (F @ X3))))))))).
% 3.41/1.87 thf(fact_42_image__iff, axiom, (! [A:$tType,B:$tType,Z:A,F:(B > A),A3:(set @ B)]: (((((member @ A) @ Z) @ ((((image @ B) @ A) @ F) @ A3)) = (? [X2:B]: (((((member @ B) @ X2) @ A3) & (Z = (F @ X2))))))))).
% 3.41/1.87 thf(fact_43_imageI, axiom, (! [B:$tType,A:$tType,X:A,A3:(set @ A),F:(A > B)]: (((((member @ A) @ X) @ A3) => (((member @ B) @ (F @ X)) @ ((((image @ A) @ B) @ F) @ A3)))))).
% 3.41/1.87 thf(fact_44_Inr__inject, axiom, (! [A:$tType,B:$tType,X:B,Y5:B]: ((((((sum_Inr @ B) @ A) @ X) = (((sum_Inr @ B) @ A) @ Y5)) => (X = Y5))))).
% 3.41/1.87 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 3.41/1.87 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A3)))) = A3)))).
% 3.41/1.87 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 3.41/1.87 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G3:(A > B)]: (((! [X3:A]: (((F @ X3) = (G3 @ X3)))) => (F = G3))))).
% 3.41/1.87 thf(fact_49_Compr__image__eq, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:(set @ B),P:(A > $o)]: ((((collect @ A) @ (^ [X2:A]: (((((member @ A) @ X2) @ ((((image @ B) @ A) @ F) @ A3)) & (P @ X2))))) = ((((image @ B) @ A) @ F) @ ((collect @ B) @ (^ [X2:B]: (((((member @ B) @ X2) @ A3) & (P @ (F @ X2))))))))))).
% 3.41/1.87 thf(fact_50_image__image, axiom, (! [A:$tType,B:$tType,C:$tType,F:(B > A),G3:(C > B),A3:(set @ C)]: ((((((image @ B) @ A) @ F) @ ((((image @ C) @ B) @ G3) @ A3)) = ((((image @ C) @ A) @ (^ [X2:C]: ((F @ (G3 @ X2))))) @ A3))))).
% 3.41/1.87 thf(fact_51_imageE, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),A3:(set @ B)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ A3)) => (~ ((! [X3:B]: (((B2 = (F @ X3)) => (~ ((((member @ B) @ X3) @ A3)))))))))))).
% 3.41/1.87 thf(fact_52_Times__eq__cancel2, axiom, (! [A:$tType,B:$tType,X:A,C4:(set @ A),A3:(set @ B),B3:(set @ B)]: (((((member @ A) @ X) @ C4) => ((((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu:B]: (C4))) = ((((product_Sigma @ B) @ A) @ B3) @ (^ [Uu:B]: (C4)))) = (A3 = B3)))))).
% 3.41/1.87 thf(fact_53_Sigma__cong, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ A),C4:(A > (set @ B)),D3:(A > (set @ B))]: (((A3 = B3) => ((! [X3:A]: (((((member @ A) @ X3) @ B3) => ((C4 @ X3) = (D3 @ X3))))) => (((((product_Sigma @ A) @ B) @ A3) @ C4) = ((((product_Sigma @ A) @ B) @ B3) @ D3))))))).
% 3.41/1.87 thf(fact_54_SigmaE2, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,A3:(set @ A),B3:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) => (~ (((((member @ A) @ A2) @ A3) => (~ ((((member @ B) @ B2) @ (B3 @ A2))))))))))).
% 3.41/1.87 thf(fact_55_SigmaD2, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,A3:(set @ A),B3:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) => (((member @ B) @ B2) @ (B3 @ A2)))))).
% 3.41/1.87 thf(fact_56_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 3.41/1.87 thf(fact_57_old_Osum_Osimps_I8_J, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > T),F22:(B > T),B2:B]: ((((((((sum_rec_sum @ A) @ T) @ B) @ F1) @ F22) @ (((sum_Inr @ B) @ A) @ B2)) = (F22 @ B2))))).
% 3.41/1.87 thf(fact_58_member__product, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),A3:(set @ A),B3:(set @ B)]: (((((member @ ((product_prod @ A) @ B)) @ X) @ ((((product_product @ A) @ B) @ A3) @ B3)) = (((member @ ((product_prod @ A) @ B)) @ X) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B3)))))))).
% 3.41/1.87 thf(fact_59_Product__Type_Oproduct__def, axiom, (! [B:$tType,A:$tType]: ((((product_product @ A) @ B) = (^ [A6:(set @ A),B5:(set @ B)]: (((((product_Sigma @ A) @ B) @ A6) @ (^ [Uu:A]: (B5))))))))).
% 3.41/1.87 thf(fact_60_Sup_OSUP__identity__eq, axiom, (! [A:$tType,Sup:((set @ A) > A),A3:(set @ A)]: (((Sup @ ((((image @ A) @ A) @ (^ [X2:A]: (X2))) @ A3)) = (Sup @ A3))))).
% 3.41/1.87 thf(fact_61_Inf_OINF__identity__eq, axiom, (! [A:$tType,Inf:((set @ A) > A),A3:(set @ A)]: (((Inf @ ((((image @ A) @ A) @ (^ [X2:A]: (X2))) @ A3)) = (Inf @ A3))))).
% 3.41/1.87 thf(fact_62_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: ((((^ [X2:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y4)) @ R))) = (^ [X2:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y4)) @ S)))) = (R = S))))).
% 3.41/1.87 thf(fact_63_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(B > (C > A)),A2:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C2) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((C2 @ A2) @ B2))))).
% 3.41/1.87 thf(fact_64_Sumr_Osimps, axiom, (! [A:$tType,C:$tType,B:$tType,F:(B > C),X:B]: (((((((sum_Sumr @ B) @ C) @ A) @ F) @ (((sum_Inr @ B) @ A) @ X)) = (F @ X))))).
% 3.41/1.87 thf(fact_65_not__arg__cong__Inr, axiom, (! [B:$tType,A:$tType,X:A,Y5:A]: (((X != Y5) => ((((sum_Inr @ A) @ B) @ X) != (((sum_Inr @ A) @ B) @ Y5)))))).
% 3.41/1.87 thf(fact_66_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))))))).
% 3.41/1.87 thf(fact_67_sum_Oinject_I1_J, axiom, (! [B:$tType,A:$tType,X1:A,Y1:A]: ((((((sum_Inl @ A) @ B) @ X1) = (((sum_Inl @ A) @ B) @ Y1)) = (X1 = Y1))))).
% 3.41/1.87 thf(fact_68_old_Osum_Oinject_I1_J, axiom, (! [B:$tType,A:$tType,A2:A,A4:A]: ((((((sum_Inl @ A) @ B) @ A2) = (((sum_Inl @ A) @ B) @ A4)) = (A2 = A4))))).
% 3.41/1.87 thf(fact_69_old_Osum_Osimps_I7_J, axiom, (! [B:$tType,T:$tType,A:$tType,F1:(A > T),F22:(B > T),A2:A]: ((((((((sum_rec_sum @ A) @ T) @ B) @ F1) @ F22) @ (((sum_Inl @ A) @ B) @ A2)) = (F1 @ A2))))).
% 3.41/1.87 thf(fact_70_Inl__inject, axiom, (! [B:$tType,A:$tType,X:A,Y5:A]: ((((((sum_Inl @ A) @ B) @ X) = (((sum_Inl @ A) @ B) @ Y5)) => (X = Y5))))).
% 3.41/1.87 thf(fact_71_obj__sumE, axiom, (! [A:$tType,B:$tType,S2:((sum_sum @ A) @ B)]: (((! [X3:A]: ((S2 != (((sum_Inl @ A) @ B) @ X3)))) => (~ ((! [X3:B]: ((S2 != (((sum_Inr @ B) @ A) @ X3)))))))))).
% 3.41/1.87 thf(fact_72_Sumr__inject, axiom, (! [A:$tType,C:$tType,B:$tType,F:(B > C),G3:(B > C)]: (((((((sum_Sumr @ B) @ C) @ A) @ F) = ((((sum_Sumr @ B) @ C) @ A) @ G3)) => (F = G3))))).
% 3.41/1.87 thf(fact_73_sum_Odistinct_I1_J, axiom, (! [A:$tType,B:$tType,X1:A,X22:B]: (((((sum_Inl @ A) @ B) @ X1) != (((sum_Inr @ B) @ A) @ X22))))).
% 3.41/1.87 thf(fact_74_old_Osum_Odistinct_I2_J, axiom, (! [B7:$tType,A7:$tType,B8:B7,A8:A7]: (((((sum_Inr @ B7) @ A7) @ B8) != (((sum_Inl @ A7) @ B7) @ A8))))).
% 3.41/1.87 thf(fact_75_old_Osum_Odistinct_I1_J, axiom, (! [A:$tType,B:$tType,A2:A,B4:B]: (((((sum_Inl @ A) @ B) @ A2) != (((sum_Inr @ B) @ A) @ B4))))).
% 3.41/1.87 thf(fact_76_sumE, axiom, (! [A:$tType,B:$tType,S2:((sum_sum @ A) @ B)]: (((! [X3:A]: ((S2 != (((sum_Inl @ A) @ B) @ X3)))) => (~ ((! [Y3:B]: ((S2 != (((sum_Inr @ B) @ A) @ Y3)))))))))).
% 3.41/1.87 thf(fact_77_Inr__not__Inl, axiom, (! [B:$tType,A:$tType,B2:B,A2:A]: (((((sum_Inr @ B) @ A) @ B2) != (((sum_Inl @ A) @ B) @ A2))))).
% 3.41/1.87 thf(fact_78_split__sum__ex, axiom, (! [B:$tType,A:$tType]: (((^ [P3:(((sum_sum @ A) @ B) > $o)]: ((? [X5:((sum_sum @ A) @ B)]: ((P3 @ X5))))) = (^ [P4:(((sum_sum @ A) @ B) > $o)]: (((? [X2:A]: ((P4 @ (((sum_Inl @ A) @ B) @ X2)))) | (? [X2:B]: ((P4 @ (((sum_Inr @ B) @ A) @ X2))))))))))).
% 3.41/1.87 thf(fact_79_split__sum__all, axiom, (! [B:$tType,A:$tType]: (((^ [P3:(((sum_sum @ A) @ B) > $o)]: ((! [X5:((sum_sum @ A) @ B)]: ((P3 @ X5))))) = (^ [P4:(((sum_sum @ A) @ B) > $o)]: (((! [X2:A]: ((P4 @ (((sum_Inl @ A) @ B) @ X2)))) & (! [X2:B]: ((P4 @ (((sum_Inr @ B) @ A) @ X2))))))))))).
% 3.41/1.87 thf(fact_80_old_Osum_Oexhaust, axiom, (! [A:$tType,B:$tType,Y5:((sum_sum @ A) @ B)]: (((! [A5:A]: ((Y5 != (((sum_Inl @ A) @ B) @ A5)))) => (~ ((! [B6:B]: ((Y5 != (((sum_Inr @ B) @ A) @ B6)))))))))).
% 3.41/1.87 thf(fact_81_old_Osum_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((sum_sum @ A) @ B) > $o),Sum:((sum_sum @ A) @ B)]: (((! [A5:A]: ((P @ (((sum_Inl @ A) @ B) @ A5)))) => ((! [B6:B]: ((P @ (((sum_Inr @ B) @ A) @ B6)))) => (P @ Sum)))))).
% 3.41/1.87 thf(fact_82_Inf_OINF__cong, axiom, (! [A:$tType,B:$tType,A3:(set @ B),B3:(set @ B),C4:(B > A),D3:(B > A),Inf:((set @ A) > A)]: (((A3 = B3) => ((! [X3:B]: (((((member @ B) @ X3) @ B3) => ((C4 @ X3) = (D3 @ X3))))) => ((Inf @ ((((image @ B) @ A) @ C4) @ A3)) = (Inf @ ((((image @ B) @ A) @ D3) @ B3)))))))).
% 3.41/1.87 thf(fact_83_Sup_OSUP__cong, axiom, (! [A:$tType,B:$tType,A3:(set @ B),B3:(set @ B),C4:(B > A),D3:(B > A),Sup:((set @ A) > A)]: (((A3 = B3) => ((! [X3:B]: (((((member @ B) @ X3) @ B3) => ((C4 @ X3) = (D3 @ X3))))) => ((Sup @ ((((image @ B) @ A) @ C4) @ A3)) = (Sup @ ((((image @ B) @ A) @ D3) @ B3)))))))).
% 3.41/1.87 thf(fact_84_Inl__Inr__False, axiom, (! [A:$tType,B:$tType,X:A,Y5:B]: (((((sum_Inl @ A) @ B) @ X) != (((sum_Inr @ B) @ A) @ Y5))))).
% 3.41/1.87 thf(fact_85_Inr__Inl__False, axiom, (! [B:$tType,A:$tType,X:B,Y5:A]: (((((sum_Inr @ B) @ A) @ X) != (((sum_Inl @ A) @ B) @ Y5))))).
% 3.41/1.87 thf(fact_86_Inl__Inr__image__cong, axiom, (! [B:$tType,A:$tType,A3:(set @ ((sum_sum @ A) @ B)),B3:(set @ ((sum_sum @ A) @ B))]: (((((((vimage @ A) @ ((sum_sum @ A) @ B)) @ ((sum_Inl @ A) @ B)) @ A3) = ((((vimage @ A) @ ((sum_sum @ A) @ B)) @ ((sum_Inl @ A) @ B)) @ B3)) => ((((((vimage @ B) @ ((sum_sum @ A) @ B)) @ ((sum_Inr @ B) @ A)) @ A3) = ((((vimage @ B) @ ((sum_sum @ A) @ B)) @ ((sum_Inr @ B) @ A)) @ B3)) => (A3 = B3)))))).
% 3.41/1.87 thf(fact_87_obj__sumE__f, axiom, (! [A:$tType,C:$tType,B:$tType,S2:B,F:(((sum_sum @ A) @ C) > B),P:$o]: (((! [X3:A]: (((S2 = (F @ (((sum_Inl @ A) @ C) @ X3))) => P))) => ((! [X3:C]: (((S2 = (F @ (((sum_Inr @ C) @ A) @ X3))) => P))) => (! [X4:((sum_sum @ A) @ C)]: (((S2 = (F @ X4)) => P)))))))).
% 3.41/1.87 thf(fact_88_Inl__cont__par, axiom, (! [Tr12:dtree,Tr22:dtree]: ((((((vimage @ t) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inl @ t) @ dtree)) @ (cont @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr12) @ Tr22)))) = ((((vimage @ t) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inl @ t) @ dtree)) @ (((sup_sup @ (set @ ((sum_sum @ t) @ dtree))) @ (cont @ Tr12)) @ (cont @ Tr22))))))).
% 3.41/1.87 thf(fact_89_Suml_Osimps, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > C),X:A]: (((((((sum_Suml @ A) @ C) @ B) @ F) @ (((sum_Inl @ A) @ B) @ X)) = (F @ X))))).
% 3.41/1.87 thf(fact_90_image2__eqI, axiom, (! [A:$tType,C:$tType,B:$tType,B2:A,F:(B > A),X:B,C2:C,G3:(B > C),A3:(set @ B)]: (((B2 = (F @ X)) => ((C2 = (G3 @ X)) => ((((member @ B) @ X) @ A3) => (((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ B2) @ C2)) @ ((((((bNF_Greatest_image2 @ B) @ A) @ C) @ A3) @ F) @ G3)))))))).
% 3.41/1.87 thf(fact_91_Un__iff, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = ((((member @ A) @ C2) @ A3) | (((member @ A) @ C2) @ B3)))))).
% 3.41/1.87 thf(fact_92_UnCI, axiom, (! [A:$tType,C2:A,B3:(set @ A),A3:(set @ A)]: ((((~ ((((member @ A) @ C2) @ B3))) => (((member @ A) @ C2) @ A3)) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)))))).
% 3.41/1.87 thf(fact_93_vimage__Un, axiom, (! [A:$tType,B:$tType,F:(A > B),A3:(set @ B),B3:(set @ B)]: ((((((vimage @ A) @ B) @ F) @ (((sup_sup @ (set @ B)) @ A3) @ B3)) = (((sup_sup @ (set @ A)) @ ((((vimage @ A) @ B) @ F) @ A3)) @ ((((vimage @ A) @ B) @ F) @ B3)))))).
% 3.41/1.87 thf(fact_94_Sigma__Un__distrib1, axiom, (! [B:$tType,A:$tType,I:(set @ A),J:(set @ A),C4:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ (((sup_sup @ (set @ A)) @ I) @ J)) @ C4) = (((sup_sup @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ I) @ C4)) @ ((((product_Sigma @ A) @ B) @ J) @ C4)))))).
% 3.41/1.87 thf(fact_95_Collect__disj__eq, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((collect @ A) @ (^ [X2:A]: (((P @ X2) | (Q @ X2))))) = (((sup_sup @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 3.41/1.87 thf(fact_96_Un__def, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: (((collect @ A) @ (^ [X2:A]: (((((member @ A) @ X2) @ A6) | (((member @ A) @ X2) @ B5))))))))))).
% 3.41/1.87 thf(fact_97_Sigma__Un__distrib2, axiom, (! [B:$tType,A:$tType,I:(set @ A),A3:(A > (set @ B)),B3:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ I) @ (^ [I2:A]: ((((sup_sup @ (set @ B)) @ (A3 @ I2)) @ (B3 @ I2))))) = (((sup_sup @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ I) @ A3)) @ ((((product_Sigma @ A) @ B) @ I) @ B3)))))).
% 3.41/1.87 thf(fact_98_Times__Un__distrib1, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ A),C4:(set @ B)]: ((((((product_Sigma @ A) @ B) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) @ (^ [Uu:A]: (C4))) = (((sup_sup @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (C4)))) @ ((((product_Sigma @ A) @ B) @ B3) @ (^ [Uu:A]: (C4)))))))).
% 3.41/1.87 thf(fact_99_Un__left__commute, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ B3) @ C4)) = (((sup_sup @ (set @ A)) @ B3) @ (((sup_sup @ (set @ A)) @ A3) @ C4)))))).
% 3.41/1.87 thf(fact_100_Un__left__absorb, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = (((sup_sup @ (set @ A)) @ A3) @ B3))))).
% 3.41/1.87 thf(fact_101_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: ((((sup_sup @ (set @ A)) @ B5) @ A6))))))).
% 3.41/1.87 thf(fact_102_Un__absorb, axiom, (! [A:$tType,A3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ A3) = A3)))).
% 3.41/1.87 thf(fact_103_Un__assoc, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C4:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) @ C4) = (((sup_sup @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ B3) @ C4)))))).
% 3.41/1.87 thf(fact_104_ball__Un, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),P:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) => (P @ X2)))) = ((! [X2:A]: (((((member @ A) @ X2) @ A3) => (P @ X2)))) & (! [X2:A]: (((((member @ A) @ X2) @ B3) => (P @ X2))))))))).
% 3.41/1.87 thf(fact_105_bex__Un, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),P:(A > $o)]: (((? [X2:A]: (((((member @ A) @ X2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) & (P @ X2)))) = ((? [X2:A]: (((((member @ A) @ X2) @ A3) & (P @ X2)))) | (? [X2:A]: (((((member @ A) @ X2) @ B3) & (P @ X2))))))))).
% 3.41/1.87 thf(fact_106_UnI2, axiom, (! [A:$tType,C2:A,B3:(set @ A),A3:(set @ A)]: (((((member @ A) @ C2) @ B3) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)))))).
% 3.41/1.87 thf(fact_107_UnI1, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)))))).
% 3.41/1.87 thf(fact_108_UnE, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) => ((~ ((((member @ A) @ C2) @ A3))) => (((member @ A) @ C2) @ B3)))))).
% 3.41/1.87 thf(fact_109_Suml__inject, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > C),G3:(A > C)]: (((((((sum_Suml @ A) @ C) @ B) @ F) = ((((sum_Suml @ A) @ C) @ B) @ G3)) => (F = G3))))).
% 3.41/1.87 thf(fact_110_image__Un, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:(set @ B),B3:(set @ B)]: ((((((image @ B) @ A) @ F) @ (((sup_sup @ (set @ B)) @ A3) @ B3)) = (((sup_sup @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A3)) @ ((((image @ B) @ A) @ F) @ B3)))))).
% 3.41/1.87 thf(fact_111_sup__apply, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ B) @ (type2 @ B)) => ((sup_sup @ (A > B)) = (^ [F2:(A > B),G4:(A > B),X2:A]: ((((sup_sup @ B) @ (F2 @ X2)) @ (G4 @ X2))))))))).
% 3.41/1.87 thf(fact_112_sup_Oidem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A]: (((((sup_sup @ A) @ A2) @ A2) = A2))))))).
% 3.41/1.87 thf(fact_113_sup__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ X) @ X) = X))))))).
% 3.41/1.87 thf(fact_114_sup_Oleft__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ A2) @ B2)) = (((sup_sup @ A) @ A2) @ B2)))))))).
% 3.41/1.87 thf(fact_115_sup__left__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y5:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y5)) = (((sup_sup @ A) @ X) @ Y5)))))))).
% 3.41/1.87 thf(fact_116_sup_Oright__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ B2) = (((sup_sup @ A) @ A2) @ B2)))))))).
% 3.41/1.87 thf(fact_117_sup__Un__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((((sup_sup @ (A > (B > $o))) @ (^ [X2:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y4)) @ R)))) @ (^ [X2:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y4)) @ S)))) = (^ [X2:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y4)) @ (((sup_sup @ (set @ ((product_prod @ A) @ B))) @ R) @ S)))))))).
% 3.41/1.87 thf(fact_118_sup__set__def, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: (((collect @ A) @ (((sup_sup @ (A > $o)) @ (^ [X2:A]: ((((member @ A) @ X2) @ A6)))) @ (^ [X2:A]: ((((member @ A) @ X2) @ B5))))))))))).
% 3.41/1.87 thf(fact_119_sup__Un__eq, axiom, (! [A:$tType,R:(set @ A),S:(set @ A)]: (((((sup_sup @ (A > $o)) @ (^ [X2:A]: ((((member @ A) @ X2) @ R)))) @ (^ [X2:A]: ((((member @ A) @ X2) @ S)))) = (^ [X2:A]: ((((member @ A) @ X2) @ (((sup_sup @ (set @ A)) @ R) @ S)))))))).
% 3.41/1.87 thf(fact_120_sup__left__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y5:A,Z:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y5) @ Z)) = (((sup_sup @ A) @ Y5) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 3.41/1.87 thf(fact_121_sup_Oleft__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C2:A]: (((((sup_sup @ A) @ B2) @ (((sup_sup @ A) @ A2) @ C2)) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B2) @ C2))))))))).
% 3.41/1.87 thf(fact_122_sup__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [X2:A,Y4:A]: ((((sup_sup @ A) @ Y4) @ X2)))))))).
% 3.41/1.87 thf(fact_123_sup_Ocommute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [A9:A,B9:A]: ((((sup_sup @ A) @ B9) @ A9)))))))).
% 3.41/1.87 thf(fact_124_sup__assoc, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y5:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y5)) @ Z) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y5) @ Z))))))))).
% 3.41/1.87 thf(fact_125_sup_Oassoc, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ C2) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B2) @ C2))))))))).
% 3.41/1.87 thf(fact_126_sup__fun__def, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ B) @ (type2 @ B)) => ((sup_sup @ (A > B)) = (^ [F2:(A > B),G4:(A > B),X2:A]: ((((sup_sup @ B) @ (F2 @ X2)) @ (G4 @ X2))))))))).
% 3.41/1.87 thf(fact_127_inf__sup__aci_I5_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [X2:A,Y4:A]: ((((sup_sup @ A) @ Y4) @ X2)))))))).
% 3.41/1.87 thf(fact_128_inf__sup__aci_I6_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y5:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y5)) @ Z) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y5) @ Z))))))))).
% 3.41/1.87 thf(fact_129_inf__sup__aci_I7_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y5:A,Z:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y5) @ Z)) = (((sup_sup @ A) @ Y5) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 3.41/1.87 thf(fact_130_inf__sup__aci_I8_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y5:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y5)) = (((sup_sup @ A) @ X) @ Y5)))))))).
% 3.41/1.87 thf(fact_131_par__c_Oelims, axiom, (! [X:((product_prod @ dtree) @ dtree),Y5:(set @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree)))]: ((((parall1914194347_par_c @ X) = Y5) => (~ ((! [Tr1:dtree,Tr2:dtree]: (((X = ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2)) => (Y5 != (((sup_sup @ (set @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree)))) @ ((((image @ t) @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree))) @ ((sum_Inl @ t) @ ((product_prod @ dtree) @ dtree))) @ ((((vimage @ t) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inl @ t) @ dtree)) @ (((sup_sup @ (set @ ((sum_sum @ t) @ dtree))) @ (cont @ Tr1)) @ (cont @ Tr2))))) @ ((((image @ ((product_prod @ dtree) @ dtree)) @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree))) @ ((sum_Inr @ ((product_prod @ dtree) @ dtree)) @ t)) @ ((((product_Sigma @ dtree) @ dtree) @ ((((vimage @ dtree) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inr @ dtree) @ t)) @ (cont @ Tr1))) @ (^ [Uu:dtree]: (((((vimage @ dtree) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inr @ dtree) @ t)) @ (cont @ Tr2))))))))))))))))).
% 3.41/1.87 thf(fact_132_par__c_Osimps, axiom, (! [Tr12:dtree,Tr22:dtree]: (((parall1914194347_par_c @ ((((product_Pair @ dtree) @ dtree) @ Tr12) @ Tr22)) = (((sup_sup @ (set @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree)))) @ ((((image @ t) @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree))) @ ((sum_Inl @ t) @ ((product_prod @ dtree) @ dtree))) @ ((((vimage @ t) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inl @ t) @ dtree)) @ (((sup_sup @ (set @ ((sum_sum @ t) @ dtree))) @ (cont @ Tr12)) @ (cont @ Tr22))))) @ ((((image @ ((product_prod @ dtree) @ dtree)) @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree))) @ ((sum_Inr @ ((product_prod @ dtree) @ dtree)) @ t)) @ ((((product_Sigma @ dtree) @ dtree) @ ((((vimage @ dtree) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inr @ dtree) @ t)) @ (cont @ Tr12))) @ (^ [Uu:dtree]: (((((vimage @ dtree) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inr @ dtree) @ t)) @ (cont @ Tr22))))))))))).
% 3.41/1.87 thf(fact_133_par__c_Opelims, axiom, (! [X:((product_prod @ dtree) @ dtree),Y5:(set @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree)))]: ((((parall1914194347_par_c @ X) = Y5) => ((((accp @ ((product_prod @ dtree) @ dtree)) @ parall1418634510_c_rel) @ X) => (~ ((! [Tr1:dtree,Tr2:dtree]: (((X = ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2)) => ((Y5 = (((sup_sup @ (set @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree)))) @ ((((image @ t) @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree))) @ ((sum_Inl @ t) @ ((product_prod @ dtree) @ dtree))) @ ((((vimage @ t) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inl @ t) @ dtree)) @ (((sup_sup @ (set @ ((sum_sum @ t) @ dtree))) @ (cont @ Tr1)) @ (cont @ Tr2))))) @ ((((image @ ((product_prod @ dtree) @ dtree)) @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree))) @ ((sum_Inr @ ((product_prod @ dtree) @ dtree)) @ t)) @ ((((product_Sigma @ dtree) @ dtree) @ ((((vimage @ dtree) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inr @ dtree) @ t)) @ (cont @ Tr1))) @ (^ [Uu:dtree]: (((((vimage @ dtree) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inr @ dtree) @ t)) @ (cont @ Tr2)))))))) => (~ ((((accp @ ((product_prod @ dtree) @ dtree)) @ parall1418634510_c_rel) @ ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2))))))))))))))).
% 3.41/1.87 thf(fact_134_Plus__def, axiom, (! [B:$tType,A:$tType]: ((((sum_Plus @ A) @ B) = (^ [A6:(set @ A),B5:(set @ B)]: ((((sup_sup @ (set @ ((sum_sum @ A) @ B))) @ ((((image @ A) @ ((sum_sum @ A) @ B)) @ ((sum_Inl @ A) @ B)) @ A6)) @ ((((image @ B) @ ((sum_sum @ A) @ B)) @ ((sum_Inr @ B) @ A)) @ B5)))))))).
% 3.41/1.87 thf(fact_135_sup2CI, axiom, (! [A:$tType,B:$tType,B3:(A > (B > $o)),X:A,Y5:B,A3:(A > (B > $o))]: ((((~ (((B3 @ X) @ Y5))) => ((A3 @ X) @ Y5)) => (((((sup_sup @ (A > (B > $o))) @ A3) @ B3) @ X) @ Y5))))).
% 3.41/1.87 thf(fact_136_sup1CI, axiom, (! [A:$tType,B3:(A > $o),X:A,A3:(A > $o)]: ((((~ ((B3 @ X))) => (A3 @ X)) => ((((sup_sup @ (A > $o)) @ A3) @ B3) @ X))))).
% 3.41/1.87 thf(fact_137_InrI, axiom, (! [B:$tType,A:$tType,B2:A,B3:(set @ A),A3:(set @ B)]: (((((member @ A) @ B2) @ B3) => (((member @ ((sum_sum @ B) @ A)) @ (((sum_Inr @ A) @ B) @ B2)) @ ((((sum_Plus @ B) @ A) @ A3) @ B3)))))).
% 3.41/1.87 thf(fact_138_InlI, axiom, (! [A:$tType,B:$tType,A2:A,A3:(set @ A),B3:(set @ B)]: (((((member @ A) @ A2) @ A3) => (((member @ ((sum_sum @ A) @ B)) @ (((sum_Inl @ A) @ B) @ A2)) @ ((((sum_Plus @ A) @ B) @ A3) @ B3)))))).
% 3.41/1.87 thf(fact_139_sup1E, axiom, (! [A:$tType,A3:(A > $o),B3:(A > $o),X:A]: ((((((sup_sup @ (A > $o)) @ A3) @ B3) @ X) => ((~ ((A3 @ X))) => (B3 @ X)))))).
% 3.41/1.87 thf(fact_140_sup2E, axiom, (! [A:$tType,B:$tType,A3:(A > (B > $o)),B3:(A > (B > $o)),X:A,Y5:B]: (((((((sup_sup @ (A > (B > $o))) @ A3) @ B3) @ X) @ Y5) => ((~ (((A3 @ X) @ Y5))) => ((B3 @ X) @ Y5)))))).
% 3.41/1.87 thf(fact_141_sup1I1, axiom, (! [A:$tType,A3:(A > $o),X:A,B3:(A > $o)]: (((A3 @ X) => ((((sup_sup @ (A > $o)) @ A3) @ B3) @ X))))).
% 3.41/1.87 thf(fact_142_sup1I2, axiom, (! [A:$tType,B3:(A > $o),X:A,A3:(A > $o)]: (((B3 @ X) => ((((sup_sup @ (A > $o)) @ A3) @ B3) @ X))))).
% 3.41/1.87 thf(fact_143_sup2I1, axiom, (! [A:$tType,B:$tType,A3:(A > (B > $o)),X:A,Y5:B,B3:(A > (B > $o))]: ((((A3 @ X) @ Y5) => (((((sup_sup @ (A > (B > $o))) @ A3) @ B3) @ X) @ Y5))))).
% 3.41/1.87 thf(fact_144_sup2I2, axiom, (! [A:$tType,B:$tType,B3:(A > (B > $o)),X:A,Y5:B,A3:(A > (B > $o))]: ((((B3 @ X) @ Y5) => (((((sup_sup @ (A > (B > $o))) @ A3) @ B3) @ X) @ Y5))))).
% 3.41/1.87 thf(fact_145_PlusE, axiom, (! [A:$tType,B:$tType,U:((sum_sum @ A) @ B),A3:(set @ A),B3:(set @ B)]: (((((member @ ((sum_sum @ A) @ B)) @ U) @ ((((sum_Plus @ A) @ B) @ A3) @ B3)) => ((! [X3:A]: (((((member @ A) @ X3) @ A3) => (U != (((sum_Inl @ A) @ B) @ X3))))) => (~ ((! [Y3:B]: (((((member @ B) @ Y3) @ B3) => (U != (((sum_Inr @ B) @ A) @ Y3)))))))))))).
% 3.41/1.87 thf(fact_146_reflclp__idemp, axiom, (! [A:$tType,P:(A > (A > $o))]: (((((sup_sup @ (A > (A > $o))) @ (((sup_sup @ (A > (A > $o))) @ P) @ (^ [Y6:A,Z2:A]: ((Y6 = Z2))))) @ (^ [Y6:A,Z2:A]: ((Y6 = Z2)))) = (((sup_sup @ (A > (A > $o))) @ P) @ (^ [Y6:A,Z2:A]: ((Y6 = Z2)))))))).
% 3.41/1.87 thf(fact_147_accp_Ocases, axiom, (! [A:$tType,R2:(A > (A > $o)),A2:A]: (((((accp @ A) @ R2) @ A2) => (! [Y7:A]: ((((R2 @ Y7) @ A2) => (((accp @ A) @ R2) @ Y7)))))))).
% 3.41/1.87 thf(fact_148_accp_Osimps, axiom, (! [A:$tType]: (((accp @ A) = (^ [R3:(A > (A > $o)),A9:A]: ((? [X2:A]: (((A9 = X2) & (! [Y4:A]: ((((R3 @ Y4) @ X2) => (((accp @ A) @ R3) @ Y4))))))))))))).
% 3.41/1.87 thf(fact_149_accp_Ointros, axiom, (! [A:$tType,R2:(A > (A > $o)),X:A]: (((! [Y3:A]: ((((R2 @ Y3) @ X) => (((accp @ A) @ R2) @ Y3)))) => (((accp @ A) @ R2) @ X))))).
% 3.41/1.87 thf(fact_150_accp__induct__rule, axiom, (! [A:$tType,R2:(A > (A > $o)),A2:A,P:(A > $o)]: (((((accp @ A) @ R2) @ A2) => ((! [X3:A]: (((((accp @ A) @ R2) @ X3) => ((! [Y7:A]: ((((R2 @ Y7) @ X3) => (P @ Y7)))) => (P @ X3))))) => (P @ A2)))))).
% 3.41/1.87 thf(fact_151_not__accp__down, axiom, (! [A:$tType,R:(A > (A > $o)),X:A]: (((~ ((((accp @ A) @ R) @ X))) => (~ ((! [Z3:A]: ((((R @ Z3) @ X) => (((accp @ A) @ R) @ Z3)))))))))).
% 3.41/1.87 thf(fact_152_accp__downward, axiom, (! [A:$tType,R2:(A > (A > $o)),B2:A,A2:A]: (((((accp @ A) @ R2) @ B2) => (((R2 @ A2) @ B2) => (((accp @ A) @ R2) @ A2)))))).
% 3.41/1.87 thf(fact_153_accp_Oinducts, axiom, (! [A:$tType,R2:(A > (A > $o)),X:A,P:(A > $o)]: (((((accp @ A) @ R2) @ X) => ((! [X3:A]: (((! [Y7:A]: ((((R2 @ Y7) @ X3) => (((accp @ A) @ R2) @ Y7)))) => ((! [Y7:A]: ((((R2 @ Y7) @ X3) => (P @ Y7)))) => (P @ X3))))) => (P @ X)))))).
% 3.41/1.87 thf(fact_154_accp__induct, axiom, (! [A:$tType,R2:(A > (A > $o)),A2:A,P:(A > $o)]: (((((accp @ A) @ R2) @ A2) => ((! [X3:A]: (((((accp @ A) @ R2) @ X3) => ((! [Y7:A]: ((((R2 @ Y7) @ X3) => (P @ Y7)))) => (P @ X3))))) => (P @ A2)))))).
% 3.41/1.87 thf(fact_155_in__lex__prod, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A4:A,B4:B,R2:(set @ ((product_prod @ A) @ A)),S2:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Pair @ A) @ B) @ A4) @ B4))) @ ((((lex_prod @ A) @ B) @ R2) @ S2)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A4)) @ R2) | ((A2 = A4) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B2) @ B4)) @ S2))))))).
% 3.41/1.87 thf(fact_156_reflcl__set__eq, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((((sup_sup @ (A > (A > $o))) @ (^ [X2:A,Y4:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ R2)))) @ (^ [Y6:A,Z2:A]: ((Y6 = Z2)))) = (^ [X2:A,Y4:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ R2) @ (id @ A))))))))).
% 3.41/1.87 thf(fact_157_par__def, axiom, (parall1899940088le_par = (((unfold @ ((product_prod @ dtree) @ dtree)) @ parall1914194362_par_r) @ parall1914194347_par_c))).
% 3.41/1.87 thf(fact_158_IdI, axiom, (! [A:$tType,A2:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ (id @ A))))).
% 3.41/1.87 thf(fact_159_pair__in__Id__conv, axiom, (! [A:$tType,A2:A,B2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ (id @ A)) = (A2 = B2))))).
% 3.41/1.87 thf(fact_160_IdE, axiom, (! [A:$tType,P2:((product_prod @ A) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ P2) @ (id @ A)) => (~ ((! [X3:A]: ((P2 != ((((product_Pair @ A) @ A) @ X3) @ X3)))))))))).
% 3.41/1.87 thf(fact_161_IdD, axiom, (! [A:$tType,A2:A,B2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ (id @ A)) => (A2 = B2))))).
% 3.41/1.87 thf(fact_162_same__fstI, axiom, (! [B:$tType,A:$tType,P:(A > $o),X:A,Y8:B,Y5:B,R:(A > (set @ ((product_prod @ B) @ B)))]: (((P @ X) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y8) @ Y5)) @ (R @ X)) => (((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y8)) @ ((((product_Pair @ A) @ B) @ X) @ Y5))) @ ((((same_fst @ A) @ B) @ P) @ R))))))).
% 3.41/1.87 thf(fact_163_antisym__reflcl, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((antisym @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ R2) @ (id @ A))) = ((antisym @ A) @ R2))))).
% 3.41/1.87 thf(fact_164_pair__imageI, axiom, (! [C:$tType,B:$tType,A:$tType,A2:A,B2:B,A3:(set @ ((product_prod @ A) @ B)),F:(A > (B > C))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ A3) => (((member @ C) @ ((F @ A2) @ B2)) @ ((((image @ ((product_prod @ A) @ B)) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F)) @ A3)))))).
% 3.41/1.87 thf(fact_165_case__prodI, axiom, (! [A:$tType,B:$tType,F:(A > (B > $o)),A2:A,B2:B]: ((((F @ A2) @ B2) => (((((product_case_prod @ A) @ B) @ $o) @ F) @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 3.41/1.87 thf(fact_166_case__prodI2, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B),C2:(A > (B > $o))]: (((! [A5:A,B6:B]: (((P2 = ((((product_Pair @ A) @ B) @ A5) @ B6)) => ((C2 @ A5) @ B6)))) => (((((product_case_prod @ A) @ B) @ $o) @ C2) @ P2))))).
% 3.41/1.87 thf(fact_167_mem__case__prodI, axiom, (! [A:$tType,B:$tType,C:$tType,Z:A,C2:(B > (C > (set @ A))),A2:B,B2:C]: (((((member @ A) @ Z) @ ((C2 @ A2) @ B2)) => (((member @ A) @ Z) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C2) @ ((((product_Pair @ B) @ C) @ A2) @ B2))))))).
% 3.41/1.87 thf(fact_168_mem__case__prodI2, axiom, (! [C:$tType,B:$tType,A:$tType,P2:((product_prod @ A) @ B),Z:C,C2:(A > (B > (set @ C)))]: (((! [A5:A,B6:B]: (((P2 = ((((product_Pair @ A) @ B) @ A5) @ B6)) => (((member @ C) @ Z) @ ((C2 @ A5) @ B6))))) => (((member @ C) @ Z) @ (((((product_case_prod @ A) @ B) @ (set @ C)) @ C2) @ P2)))))).
% 3.41/1.87 thf(fact_169_case__prodI2_H, axiom, (! [A:$tType,B:$tType,C:$tType,P2:((product_prod @ A) @ B),C2:(A > (B > (C > $o))),X:C]: (((! [A5:A,B6:B]: (((((((product_Pair @ A) @ B) @ A5) @ B6) = P2) => (((C2 @ A5) @ B6) @ X)))) => ((((((product_case_prod @ A) @ B) @ (C > $o)) @ C2) @ P2) @ X))))).
% 3.41/1.87 thf(fact_170_Collect__case__prod, axiom, (! [B:$tType,A:$tType,P:(A > $o),Q:(B > $o)]: ((((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [A9:A,B9:B]: (((P @ A9) & (Q @ B9)))))) = ((((product_Sigma @ A) @ B) @ ((collect @ A) @ P)) @ (^ [Uu:A]: (((collect @ B) @ Q)))))))).
% 3.41/1.87 thf(fact_171_case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > (C > A)),A2:B,B2:C]: (((((((product_case_prod @ B) @ C) @ A) @ F) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((F @ A2) @ B2))))).
% 3.41/1.87 thf(fact_172_lex__prod__def, axiom, (! [B:$tType,A:$tType]: ((((lex_prod @ A) @ B) = (^ [Ra:(set @ ((product_prod @ A) @ A)),Rb:(set @ ((product_prod @ B) @ B))]: (((collect @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_case_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ $o) @ ((((product_case_prod @ A) @ B) @ (((product_prod @ A) @ B) > $o)) @ (^ [A9:A,B9:B]: (((((product_case_prod @ A) @ B) @ $o) @ (^ [A10:A,B10:B]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A9) @ A10)) @ Ra) | ((A9 = A10) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B9) @ B10)) @ Rb))))))))))))))))).
% 3.41/1.87 thf(fact_173_case__prodD_H, axiom, (! [B:$tType,A:$tType,C:$tType,R:(A > (B > (C > $o))),A2:A,B2:B,C2:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ R) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ C2) => (((R @ A2) @ B2) @ C2))))).
% 3.41/1.87 thf(fact_174_case__prodE_H, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(A > (B > (C > $o))),P2:((product_prod @ A) @ B),Z:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ C2) @ P2) @ Z) => (~ ((! [X3:A,Y3:B]: (((P2 = ((((product_Pair @ A) @ B) @ X3) @ Y3)) => (~ ((((C2 @ X3) @ Y3) @ Z)))))))))))).
% 3.41/1.87 thf(fact_175_case__prodE2, axiom, (! [B:$tType,A:$tType,C:$tType,Q:(A > $o),P:(B > (C > A)),Z:((product_prod @ B) @ C)]: (((Q @ (((((product_case_prod @ B) @ C) @ A) @ P) @ Z)) => (~ ((! [X3:B,Y3:C]: (((Z = ((((product_Pair @ B) @ C) @ X3) @ Y3)) => (~ ((Q @ ((P @ X3) @ Y3))))))))))))).
% 3.41/1.87 thf(fact_176_case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ (^ [X2:A,Y4:B]: ((F @ ((((product_Pair @ A) @ B) @ X2) @ Y4))))) = F)))).
% 3.41/1.87 thf(fact_177_cond__case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > C)),G3:(((product_prod @ A) @ B) > C)]: (((! [X3:A,Y3:B]: ((((F @ X3) @ Y3) = (G3 @ ((((product_Pair @ A) @ B) @ X3) @ Y3))))) => (((((product_case_prod @ A) @ B) @ C) @ F) = G3))))).
% 3.41/1.87 thf(fact_178_mem__case__prodE, axiom, (! [B:$tType,A:$tType,C:$tType,Z:A,C2:(B > (C > (set @ A))),P2:((product_prod @ B) @ C)]: (((((member @ A) @ Z) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C2) @ P2)) => (~ ((! [X3:B,Y3:C]: (((P2 = ((((product_Pair @ B) @ C) @ X3) @ Y3)) => (~ ((((member @ A) @ Z) @ ((C2 @ X3) @ Y3))))))))))))).
% 3.41/1.87 thf(fact_179_old_Oprod_Ocase, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > (B > C)),X1:A,X22:B]: (((((((product_case_prod @ A) @ B) @ C) @ F) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = ((F @ X1) @ X22))))).
% 3.41/1.87 thf(fact_180_same__fst__def, axiom, (! [B:$tType,A:$tType]: ((((same_fst @ A) @ B) = (^ [P4:(A > $o),R4:(A > (set @ ((product_prod @ B) @ B)))]: (((collect @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_case_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ $o) @ ((((product_case_prod @ A) @ B) @ (((product_prod @ A) @ B) > $o)) @ (^ [X6:A,Y9:B]: (((((product_case_prod @ A) @ B) @ $o) @ (^ [X2:A,Y4:B]: (((X6 = X2) & ((P4 @ X2) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y9) @ Y4)) @ (R4 @ X2)))))))))))))))))).
% 3.41/1.87 thf(fact_181_antisym__Id, axiom, (! [A:$tType]: (((antisym @ A) @ (id @ A))))).
% 3.41/1.87 thf(fact_182_antisymD, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A2:A,B2:A]: ((((antisym @ A) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ A2)) @ R2) => (A2 = B2))))))).
% 3.41/1.87 thf(fact_183_antisymI, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((! [X3:A,Y3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X3)) @ R2) => (X3 = Y3))))) => ((antisym @ A) @ R2))))).
% 3.41/1.87 thf(fact_184_antisym__def, axiom, (! [A:$tType]: (((antisym @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [X2:A,Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ R3) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X2)) @ R3) => (X2 = Y4))))))))))).
% 3.41/1.87 thf(fact_185_case__prod__Pair__iden, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ ((product_prod @ A) @ B)) @ ((product_Pair @ A) @ B)) @ P2) = P2)))).
% 3.41/1.87 thf(fact_186_case__prodD, axiom, (! [A:$tType,B:$tType,F:(A > (B > $o)),A2:A,B2:B]: (((((((product_case_prod @ A) @ B) @ $o) @ F) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) => ((F @ A2) @ B2))))).
% 3.41/1.87 thf(fact_187_case__prodE, axiom, (! [A:$tType,B:$tType,C2:(A > (B > $o)),P2:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ $o) @ C2) @ P2) => (~ ((! [X3:A,Y3:B]: (((P2 = ((((product_Pair @ A) @ B) @ X3) @ Y3)) => (~ (((C2 @ X3) @ Y3)))))))))))).
% 3.41/1.87 thf(fact_188_Collect__case__prod__Sigma, axiom, (! [B:$tType,A:$tType,P:(A > $o),Q:(A > (B > $o))]: ((((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X2:A,Y4:B]: (((P @ X2) & ((Q @ X2) @ Y4)))))) = ((((product_Sigma @ A) @ B) @ ((collect @ A) @ P)) @ (^ [X2:A]: (((collect @ B) @ (Q @ X2))))))))).
% 3.41/1.87 thf(fact_189_case__prod__app, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ (D > A)) = (^ [F2:(B > (C > (D > A))),X2:((product_prod @ B) @ C),Y4:D]: ((((((product_case_prod @ B) @ C) @ A) @ (^ [L:B,R3:C]: ((((F2 @ L) @ R3) @ Y4)))) @ X2))))))).
% 3.41/1.87 thf(fact_190_prod_Ocase__distrib, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,H:(C > D),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((H @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = (((((product_case_prod @ A) @ B) @ D) @ (^ [X12:A,X23:B]: ((H @ ((F @ X12) @ X23))))) @ Prod))))).
% 3.41/1.87 thf(fact_191_antisymP__equality, axiom, (! [A:$tType]: (((antisym @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [Y6:A,Z2:A]: ((Y6 = Z2))))))))).
% 3.41/1.87 thf(fact_192_internal__case__prod__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((produc2004651681e_prod @ A) @ B) @ C) = (((product_case_prod @ A) @ B) @ C))))).
% 3.41/1.87 thf(fact_193_swap__product, axiom, (! [B:$tType,A:$tType,A3:(set @ B),B3:(set @ A)]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ B) @ A) @ ((product_prod @ A) @ B)) @ (^ [I2:B,J2:A]: (((((product_Pair @ A) @ B) @ J2) @ I2))))) @ ((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu:B]: (B3)))) = ((((product_Sigma @ A) @ B) @ B3) @ (^ [Uu:A]: (A3))))))).
% 3.41/1.87 thf(fact_194_split__cong, axiom, (! [C:$tType,B:$tType,A:$tType,Q2:((product_prod @ A) @ B),F:(A > (B > C)),G3:(A > (B > C)),P2:((product_prod @ A) @ B)]: (((! [X3:A,Y3:B]: (((((((product_Pair @ A) @ B) @ X3) @ Y3) = Q2) => (((F @ X3) @ Y3) = ((G3 @ X3) @ Y3))))) => ((P2 = Q2) => ((((((product_case_prod @ A) @ B) @ C) @ F) @ P2) = (((((product_case_prod @ A) @ B) @ C) @ G3) @ Q2))))))).
% 3.41/1.87 thf(fact_195_case__swap, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > (B > A)),P2:((product_prod @ C) @ B)]: (((((((product_case_prod @ B) @ C) @ A) @ (^ [Y4:B,X2:C]: (((F @ X2) @ Y4)))) @ (((product_swap @ C) @ B) @ P2)) = (((((product_case_prod @ C) @ B) @ A) @ F) @ P2))))).
% 3.41/1.87 thf(fact_196_case__prod__curry, axiom, (! [C:$tType,B:$tType,A:$tType,F:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ ((((product_curry @ A) @ B) @ C) @ F)) = F)))).
% 3.41/1.87 thf(fact_197_curryI, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: (((F @ ((((product_Pair @ A) @ B) @ A2) @ B2)) => ((((((product_curry @ A) @ B) @ $o) @ F) @ A2) @ B2))))).
% 3.41/1.87 thf(fact_198_swap__swap, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P2)) = P2)))).
% 3.41/1.87 thf(fact_199_split__part, axiom, (! [B:$tType,A:$tType,P:$o,Q:(A > (B > $o))]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [A9:A,B9:B]: ((P & ((Q @ A9) @ B9))))) = (^ [Ab:((product_prod @ A) @ B)]: ((P & (((((product_case_prod @ A) @ B) @ $o) @ Q) @ Ab)))))))).
% 3.41/1.87 thf(fact_200_swap__simp, axiom, (! [A:$tType,B:$tType,X:B,Y5:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y5)) = ((((product_Pair @ A) @ B) @ Y5) @ X))))).
% 3.41/1.87 thf(fact_201_curry__conv, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_curry @ B) @ C) @ A) = (^ [F2:(((product_prod @ B) @ C) > A),A9:B,B9:C]: ((F2 @ ((((product_Pair @ B) @ C) @ A9) @ B9)))))))).
% 3.41/1.87 thf(fact_202_curry__case__prod, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > C))]: ((((((product_curry @ A) @ B) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F)) = F)))).
% 3.41/1.87 thf(fact_203_pair__in__swap__image, axiom, (! [A:$tType,B:$tType,Y5:A,X:B,A3:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y5) @ X)) @ ((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ A3)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ Y5)) @ A3))))).
% 3.41/1.87 thf(fact_204_prod_Odisc__eq__case, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [Uu:A,Uv:B]: ($true))) @ Prod)))).
% 3.41/1.87 thf(fact_205_curry__K, axiom, (! [B:$tType,C:$tType,A:$tType,C2:C]: ((((((product_curry @ A) @ B) @ C) @ (^ [X2:((product_prod @ A) @ B)]: (C2))) = (^ [X2:A,Y4:B]: (C2)))))).
% 3.41/1.87 thf(fact_206_curryE, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: ((((((((product_curry @ A) @ B) @ $o) @ F) @ A2) @ B2) => (F @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 3.41/1.87 thf(fact_207_curryD, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A2:A,B2:B]: ((((((((product_curry @ A) @ B) @ $o) @ F) @ A2) @ B2) => (F @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))).
% 3.41/1.87 thf(fact_208_product__swap, axiom, (! [B:$tType,A:$tType,A3:(set @ B),B3:(set @ A)]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu:B]: (B3)))) = ((((product_Sigma @ A) @ B) @ B3) @ (^ [Uu:A]: (A3))))))).
% 3.41/1.87 thf(fact_209_curry__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_curry @ A) @ B) @ C) = (^ [C5:(((product_prod @ A) @ B) > C),X2:A,Y4:B]: ((C5 @ ((((product_Pair @ A) @ B) @ X2) @ Y4)))))))).
% 3.41/1.87 thf(fact_210_inv__image__def, axiom, (! [A:$tType,B:$tType]: ((((inv_image @ B) @ A) = (^ [R3:(set @ ((product_prod @ B) @ B)),F2:(A > B)]: (((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X2:A,Y4:A]: ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F2 @ X2)) @ (F2 @ Y4))) @ R3))))))))))).
% 3.41/1.87 thf(fact_211_map__prod__surj__on, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F:(B > A),A3:(set @ B),A11:(set @ A),G3:(D > C),B3:(set @ D),B11:(set @ C)]: (((((((image @ B) @ A) @ F) @ A3) = A11) => ((((((image @ D) @ C) @ G3) @ B3) = B11) => (((((image @ ((product_prod @ B) @ D)) @ ((product_prod @ A) @ C)) @ ((((((product_map_prod @ B) @ A) @ D) @ C) @ F) @ G3)) @ ((((product_Sigma @ B) @ D) @ A3) @ (^ [Uu:B]: (B3)))) = ((((product_Sigma @ A) @ C) @ A11) @ (^ [Uu:A]: (B11))))))))).
% 3.41/1.87 thf(fact_212_map__prod__ident, axiom, (! [B:$tType,A:$tType]: ((((((((product_map_prod @ A) @ A) @ B) @ B) @ (^ [X2:A]: (X2))) @ (^ [Y4:B]: (Y4))) = (^ [Z4:((product_prod @ A) @ B)]: (Z4)))))).
% 3.41/1.87 thf(fact_213_map__prod__simp, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F:(C > A),G3:(D > B),A2:C,B2:D]: (((((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G3) @ ((((product_Pair @ C) @ D) @ A2) @ B2)) = ((((product_Pair @ A) @ B) @ (F @ A2)) @ (G3 @ B2)))))).
% 3.41/1.87 thf(fact_214_in__inv__image, axiom, (! [A:$tType,B:$tType,X:A,Y5:A,R2:(set @ ((product_prod @ B) @ B)),F:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y5)) @ ((((inv_image @ B) @ A) @ R2) @ F)) = (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F @ X)) @ (F @ Y5))) @ R2))))).
% 3.41/1.87 thf(fact_215_map__prod__imageI, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,A2:A,B2:B,R:(set @ ((product_prod @ A) @ B)),F:(A > C),G3:(B > D)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R) => (((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (F @ A2)) @ (G3 @ B2))) @ ((((image @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F) @ G3)) @ R)))))).
% 3.41/1.87 thf(fact_216_case__prod__map__prod, axiom, (! [C:$tType,A:$tType,B:$tType,E:$tType,D:$tType,H:(B > (C > A)),F:(D > B),G3:(E > C),X:((product_prod @ D) @ E)]: (((((((product_case_prod @ B) @ C) @ A) @ H) @ (((((((product_map_prod @ D) @ B) @ E) @ C) @ F) @ G3) @ X)) = (((((product_case_prod @ D) @ E) @ A) @ (^ [L:D,R3:E]: (((H @ (F @ L)) @ (G3 @ R3))))) @ X))))).
% 3.41/1.87 thf(fact_217_prod__fun__imageE, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,C2:((product_prod @ A) @ B),F:(C > A),G3:(D > B),R:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ A) @ B)) @ C2) @ ((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G3)) @ R)) => (~ ((! [X3:C,Y3:D]: (((C2 = ((((product_Pair @ A) @ B) @ (F @ X3)) @ (G3 @ Y3))) => (~ ((((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ X3) @ Y3)) @ R)))))))))))).
% 3.41/1.87 thf(fact_218_map__prod__def, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType]: ((((((product_map_prod @ A) @ C) @ B) @ D) = (^ [F2:(A > C),G4:(B > D)]: (((((product_case_prod @ A) @ B) @ ((product_prod @ C) @ D)) @ (^ [X2:A,Y4:B]: (((((product_Pair @ C) @ D) @ (F2 @ X2)) @ (G4 @ Y4))))))))))).
% 3.41/1.87 thf(fact_219_rp__inv__image__def, axiom, (! [B:$tType,A:$tType]: ((((fun_rp_inv_image @ A) @ B) = ((((product_case_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((B > A) > ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))))) @ (^ [R4:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ A) @ A)),F2:(B > A)]: (((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))) @ ((((inv_image @ A) @ B) @ R4) @ F2)) @ ((((inv_image @ A) @ B) @ S3) @ F2))))))))).
% 3.41/1.87 thf(fact_220_prod_Omap__ident, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: (((((((((product_map_prod @ A) @ A) @ B) @ B) @ (^ [X2:A]: (X2))) @ (^ [X2:B]: (X2))) @ T2) = T2)))).
% 3.41/1.87 thf(fact_221_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))))))).
% 3.41/1.87 thf(fact_222_The__split__eq, axiom, (! [A:$tType,B:$tType,X:A,Y5:B]: ((((the @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X6:A,Y9:B]: (((X = X6) & (Y5 = Y9)))))) = ((((product_Pair @ A) @ B) @ X) @ Y5))))).
% 3.41/1.87 thf(fact_223_UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 3.41/1.87 thf(fact_224_sup__top__right, axiom, (! [A:$tType]: ((((bounded_lattice_top @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ X) @ (top_top @ A)) = (top_top @ A)))))))).
% 3.41/1.87 thf(fact_225_sup__top__left, axiom, (! [A:$tType]: ((((bounded_lattice_top @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ (top_top @ A)) @ X) = (top_top @ A)))))))).
% 3.41/1.87 thf(fact_226_vimage__UNIV, axiom, (! [B:$tType,A:$tType,F:(A > B)]: ((((((vimage @ A) @ B) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A)))))).
% 3.41/1.87 thf(fact_227_UNIV__Plus__UNIV, axiom, (! [B:$tType,A:$tType]: ((((((sum_Plus @ A) @ B) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) = (top_top @ (set @ ((sum_sum @ A) @ B))))))).
% 3.41/1.87 thf(fact_228_UNIV__Times__UNIV, axiom, (! [B:$tType,A:$tType]: ((((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu:A]: ((top_top @ (set @ B))))) = (top_top @ (set @ ((product_prod @ A) @ B))))))).
% 3.41/1.87 thf(fact_229_UNIV__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (^ [X2:A]: ($true))))))).
% 3.41/1.87 thf(fact_230_rangeE, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))) => (~ ((! [X3:B]: ((B2 != (F @ X3)))))))))).
% 3.41/1.87 thf(fact_231_range__composition, axiom, (! [A:$tType,C:$tType,B:$tType,F:(C > A),G3:(B > C)]: ((((((image @ B) @ A) @ (^ [X2:B]: ((F @ (G3 @ X2))))) @ (top_top @ (set @ B))) = ((((image @ C) @ A) @ F) @ ((((image @ B) @ C) @ G3) @ (top_top @ (set @ B)))))))).
% 3.41/1.87 thf(fact_232_rangeI, axiom, (! [A:$tType,B:$tType,F:(B > A),X:B]: ((((member @ A) @ (F @ X)) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B))))))).
% 3.41/1.87 thf(fact_233_range__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),X:B]: (((B2 = (F @ X)) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))))))).
% 3.41/1.87 thf(fact_234_Un__UNIV__left, axiom, (! [A:$tType,B3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (top_top @ (set @ A))) @ B3) = (top_top @ (set @ A)))))).
% 3.41/1.87 thf(fact_235_Un__UNIV__right, axiom, (! [A:$tType,A3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ (top_top @ (set @ A))) = (top_top @ (set @ A)))))).
% 3.41/1.87 thf(fact_236_UNIV__eq__I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X3:A]: ((((member @ A) @ X3) @ A3))) => ((top_top @ (set @ A)) = A3))))).
% 3.41/1.87 thf(fact_237_UNIV__witness, axiom, (! [A:$tType]: ((? [X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))))).
% 3.41/1.87 thf(fact_238_UNIV__sum, axiom, (! [A:$tType,B:$tType]: (((top_top @ (set @ ((sum_sum @ A) @ B))) = (((sup_sup @ (set @ ((sum_sum @ A) @ B))) @ ((((image @ A) @ ((sum_sum @ A) @ B)) @ ((sum_Inl @ A) @ B)) @ (top_top @ (set @ A)))) @ ((((image @ B) @ ((sum_sum @ A) @ B)) @ ((sum_Inr @ B) @ A)) @ (top_top @ (set @ B)))))))).
% 3.41/1.87 thf(fact_239_map__prod__surj, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F:(A > B),G3:(C > D)]: (((((((image @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((((((image @ C) @ D) @ G3) @ (top_top @ (set @ C))) = (top_top @ (set @ D))) => (((((image @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)) @ ((((((product_map_prod @ A) @ B) @ C) @ D) @ F) @ G3)) @ (top_top @ (set @ ((product_prod @ A) @ C)))) = (top_top @ (set @ ((product_prod @ B) @ D))))))))).
% 3.41/1.87 thf(fact_240_old_Orec__sum__def, axiom, (! [B:$tType,T:$tType,A:$tType]: (((((sum_rec_sum @ A) @ T) @ B) = (^ [F12:(A > T),F23:(B > T),X2:((sum_sum @ A) @ B)]: (((the @ T) @ ((((((sum_rec_set_sum @ A) @ T) @ B) @ F12) @ F23) @ X2)))))))).
% 3.41/1.87 thf(fact_241_old_Orec__prod__def, axiom, (! [T:$tType,B:$tType,A:$tType]: (((((product_rec_prod @ A) @ B) @ T) = (^ [F12:(A > (B > T)),X2:((product_prod @ A) @ B)]: (((the @ T) @ (((((product_rec_set_prod @ A) @ B) @ T) @ F12) @ X2)))))))).
% 3.41/1.87 thf(fact_242_top__empty__eq2, axiom, (! [B:$tType,A:$tType]: (((top_top @ (A > (B > $o))) = (^ [X2:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y4)) @ (top_top @ (set @ ((product_prod @ A) @ B)))))))))).
% 3.41/1.87 thf(fact_243_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 3.41/1.87 thf(fact_244_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A))))))))).
% 3.41/1.87 thf(fact_245_the__sym__eq__trivial, axiom, (! [A:$tType,X:A]: ((((the @ A) @ ((^ [Y6:A,Z2:A]: ((Y6 = Z2))) @ X)) = X)))).
% 3.41/1.87 thf(fact_246_the__eq__trivial, axiom, (! [A:$tType,A2:A]: ((((the @ A) @ (^ [X2:A]: ((X2 = A2)))) = A2)))).
% 3.41/1.87 thf(fact_247_the__equality, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((P @ A2) => ((! [X3:A]: (((P @ X3) => (X3 = A2)))) => (((the @ A) @ P) = A2)))))).
% 3.41/1.87 thf(fact_248_top__conj_I2_J, axiom, (! [A:$tType,P:$o,X:A]: (((P & ((top_top @ (A > $o)) @ X)) = P)))).
% 3.41/1.87 thf(fact_249_top__conj_I1_J, axiom, (! [A:$tType,X:A,P:$o]: (((((top_top @ (A > $o)) @ X) & P) = P)))).
% 3.41/1.87 thf(fact_250_theI, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((P @ A2) => ((! [X3:A]: (((P @ X3) => (X3 = A2)))) => (P @ ((the @ A) @ P))))))).
% 3.41/1.87 thf(fact_251_theI_H, axiom, (! [A:$tType,P:(A > $o)]: (((? [X4:A]: (((P @ X4) & (! [Y3:A]: (((P @ Y3) => (Y3 = X4))))))) => (P @ ((the @ A) @ P)))))).
% 3.41/1.87 thf(fact_252_theI2, axiom, (! [A:$tType,P:(A > $o),A2:A,Q:(A > $o)]: (((P @ A2) => ((! [X3:A]: (((P @ X3) => (X3 = A2)))) => ((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (Q @ ((the @ A) @ P)))))))).
% 3.41/1.87 thf(fact_253_If__def, axiom, (! [A:$tType]: (((if @ A) = (^ [P4:$o,X2:A,Y4:A]: (((the @ A) @ (^ [Z4:A]: (((P4 => (Z4 = X2)) & ((~ (P4)) => (Z4 = Y4)))))))))))).
% 3.41/1.87 thf(fact_254_the1I2, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((? [X4:A]: (((P @ X4) & (! [Y3:A]: (((P @ Y3) => (Y3 = X4))))))) => ((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (Q @ ((the @ A) @ P))))))).
% 3.41/1.87 thf(fact_255_the1__equality, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((? [X4:A]: (((P @ X4) & (! [Y3:A]: (((P @ Y3) => (Y3 = X4))))))) => ((P @ A2) => (((the @ A) @ P) = A2)))))).
% 3.41/1.87 thf(tcon_HOL_Obool___Lattices_Obounded__lattice, axiom, ((bounded_lattice @ $o) @ (type2 @ $o))).
% 3.41/1.87 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1, axiom, (! [A7:$tType]: (((bounded_lattice @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 3.41/1.87 thf(tcon_fun___Lattices_Obounded__lattice_2, axiom, (! [A7:$tType,A12:$tType]: ((((bounded_lattice @ A12) @ (type2 @ A12)) => ((bounded_lattice @ (A7 > A12)) @ (type2 @ (A7 > A12))))))).
% 3.41/1.87 thf(tcon_fun___Lattices_Obounded__lattice__top, axiom, (! [A7:$tType,A12:$tType]: ((((bounded_lattice @ A12) @ (type2 @ A12)) => ((bounded_lattice_top @ (A7 > A12)) @ (type2 @ (A7 > A12))))))).
% 3.41/1.87 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A7:$tType,A12:$tType]: ((((semilattice_sup @ A12) @ (type2 @ A12)) => ((semilattice_sup @ (A7 > A12)) @ (type2 @ (A7 > A12))))))).
% 3.41/1.87 thf(tcon_fun___Lattices_Olattice, axiom, (! [A7:$tType,A12:$tType]: ((((lattice @ A12) @ (type2 @ A12)) => ((lattice @ (A7 > A12)) @ (type2 @ (A7 > A12))))))).
% 3.41/1.87 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_3, axiom, (! [A7:$tType]: (((bounded_lattice_top @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 3.41/1.87 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_4, axiom, (! [A7:$tType]: (((semilattice_sup @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 3.41/1.87 thf(tcon_Set_Oset___Lattices_Olattice_5, axiom, (! [A7:$tType]: (((lattice @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 3.41/1.87 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_6, axiom, ((bounded_lattice_top @ $o) @ (type2 @ $o))).
% 3.41/1.87 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_7, axiom, ((semilattice_sup @ $o) @ (type2 @ $o))).
% 3.41/1.87 thf(tcon_HOL_Obool___Lattices_Olattice_8, axiom, ((lattice @ $o) @ (type2 @ $o))).
% 3.41/1.87 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 3.41/1.87 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y5:A]: ((((((if @ A) @ $false) @ X) @ Y5) = Y5)))).
% 3.41/1.87 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y5:A]: ((((((if @ A) @ $true) @ X) @ Y5) = X)))).
% 3.41/1.87 thf(conj_0, conjecture, ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ t2)) @ (cont @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ tr1) @ tr2)))) = (((member @ dtree) @ t2) @ ((((image @ ((product_prod @ dtree) @ dtree)) @ dtree) @ parall1899940088le_par) @ ((((product_Sigma @ dtree) @ dtree) @ ((collect @ dtree) @ (^ [X2:dtree]: ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ X2)) @ (cont @ tr1)))))) @ (^ [Uu:dtree]: (((collect @ dtree) @ (^ [X2:dtree]: ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ X2)) @ (cont @ tr2)))))))))))).
% 3.41/1.87 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.ssUgw0micP/DTF2DTF15689.p
% 3.41/1.87 ------------------------
% 3.58/1.93 ---- Cleaned THF ---
% 3.58/1.93 thf(ty_t_Product__Type_Oprod,type,
% 3.58/1.93 product_prod: $tType > $tType > $tType ).
% 3.58/1.93
% 3.58/1.93 thf(ty_t_Sum__Type_Osum,type,
% 3.58/1.93 sum_sum: $tType > $tType > $tType ).
% 3.58/1.93
% 3.58/1.93 thf(ty_t_DTree_Odtree,type,
% 3.58/1.93 dtree: $tType ).
% 3.58/1.93
% 3.58/1.93 thf(ty_t_Set_Oset,type,
% 3.58/1.93 set: $tType > $tType ).
% 3.58/1.93
% 3.58/1.93 thf(ty_t_DTree_OT,type,
% 3.58/1.93 t: $tType ).
% 3.58/1.93
% 3.58/1.93 thf(ty_t_DTree_ON,type,
% 3.58/1.93 n: $tType ).
% 3.58/1.93
% 3.58/1.93 thf(ty_t_itself,type,
% 3.58/1.93 itself: $tType > $tType ).
% 3.58/1.93
% 3.58/1.93 thf(sy_cl_Lattices_Obounded__lattice,type,
% 3.58/1.93 bounded_lattice:
% 3.58/1.93 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_cl_HOL_Otype,type,
% 3.58/1.93 type:
% 3.58/1.93 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_cl_Lattices_Olattice,type,
% 3.58/1.93 lattice:
% 3.58/1.93 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 3.58/1.93 semilattice_sup:
% 3.58/1.93 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_cl_Lattices_Obounded__lattice__top,type,
% 3.58/1.93 bounded_lattice_top:
% 3.58/1.93 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 3.58/1.93 bNF_Greatest_image2:
% 3.58/1.93 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_DTree_Ocont,type,
% 3.58/1.93 cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_DTree_Ounfold,type,
% 3.58/1.93 unfold:
% 3.58/1.93 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Fun__Def_Orp__inv__image,type,
% 3.58/1.93 fun_rp_inv_image:
% 3.58/1.93 !>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_HOL_OThe,type,
% 3.58/1.93 the:
% 3.58/1.93 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_If,type,
% 3.58/1.93 if:
% 3.58/1.93 !>[A: $tType] : ( $o > A > A > A ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Lattices_Osup__class_Osup,type,
% 3.58/1.93 sup_sup:
% 3.58/1.93 !>[A: $tType] : ( A > A > A ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Orderings_Otop__class_Otop,type,
% 3.58/1.93 top_top:
% 3.58/1.93 !>[A: $tType] : A ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar,type,
% 3.58/1.93 parall1899940088le_par: ( product_prod @ dtree @ dtree ) > dtree ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c,type,
% 3.58/1.93 parall1914194347_par_c: ( product_prod @ dtree @ dtree ) > ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c__rel,type,
% 3.58/1.93 parall1418634510_c_rel: ( product_prod @ dtree @ dtree ) > ( product_prod @ dtree @ dtree ) > $o ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r,type,
% 3.58/1.93 parall1914194362_par_r: ( product_prod @ dtree @ dtree ) > n ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Product__Type_OPair,type,
% 3.58/1.93 product_Pair:
% 3.58/1.93 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Product__Type_OSigma,type,
% 3.58/1.93 product_Sigma:
% 3.58/1.93 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Product__Type_Ocurry,type,
% 3.58/1.93 product_curry:
% 3.58/1.93 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 3.58/1.93 produc2004651681e_prod:
% 3.58/1.93 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Product__Type_Omap__prod,type,
% 3.58/1.93 product_map_prod:
% 3.58/1.93 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 3.58/1.93 product_rec_prod:
% 3.58/1.93 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
% 3.58/1.93 product_rec_set_prod:
% 3.58/1.93 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 3.58/1.93 product_case_prod:
% 3.58/1.93 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 3.58/1.93 product_swap:
% 3.58/1.93 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Product__Type_Oproduct,type,
% 3.58/1.93 product_product:
% 3.58/1.93 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Pure_Otype,type,
% 3.58/1.93 type2:
% 3.58/1.93 !>[A: $tType] : ( itself @ A ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Relation_OId,type,
% 3.58/1.93 id:
% 3.58/1.93 !>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Relation_Oantisym,type,
% 3.58/1.93 antisym:
% 3.58/1.93 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Relation_Oinv__image,type,
% 3.58/1.93 inv_image:
% 3.58/1.93 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Set_OCollect,type,
% 3.58/1.93 collect:
% 3.58/1.93 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Set_Oimage,type,
% 3.58/1.93 image:
% 3.58/1.93 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Set_Ovimage,type,
% 3.58/1.93 vimage:
% 3.58/1.93 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Sum__Type_OInl,type,
% 3.58/1.93 sum_Inl:
% 3.58/1.93 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Sum__Type_OInr,type,
% 3.58/1.93 sum_Inr:
% 3.58/1.93 !>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Sum__Type_OPlus,type,
% 3.58/1.93 sum_Plus:
% 3.58/1.93 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Sum__Type_OSuml,type,
% 3.58/1.93 sum_Suml:
% 3.58/1.93 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( sum_sum @ A @ B ) > C ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Sum__Type_OSumr,type,
% 3.58/1.93 sum_Sumr:
% 3.58/1.93 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Sum__Type_Oold_Osum_Orec__set__sum,type,
% 3.58/1.93 sum_rec_set_sum:
% 3.58/1.93 !>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T > $o ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
% 3.58/1.93 sum_rec_sum:
% 3.58/1.93 !>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Wellfounded_Oaccp,type,
% 3.58/1.93 accp:
% 3.58/1.93 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Wellfounded_Olex__prod,type,
% 3.58/1.93 lex_prod:
% 3.58/1.93 !>[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 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_Wfrec_Osame__fst,type,
% 3.58/1.93 same_fst:
% 3.58/1.93 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_c_member,type,
% 3.58/1.93 member:
% 3.58/1.93 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.58/1.93
% 3.58/1.93 thf(sy_v_t,type,
% 3.58/1.93 t2: dtree ).
% 3.58/1.93
% 3.58/1.93 thf(sy_v_tr1,type,
% 3.58/1.93 tr1: dtree ).
% 3.58/1.93
% 3.58/1.93 thf(sy_v_tr2,type,
% 3.58/1.93 tr2: dtree ).
% 3.58/1.93
% 3.58/1.93 thf(fact_0_par__r_Ocases,axiom,
% 3.58/1.93 ! [X: product_prod @ dtree @ dtree] :
% 3.58/1.93 ~ ! [Tr1: dtree,Tr2: dtree] :
% 3.58/1.93 ( X
% 3.58/1.93 != ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_1_par__r_Oinduct,axiom,
% 3.58/1.93 ! [P: ( product_prod @ dtree @ dtree ) > $o,A0: product_prod @ dtree @ dtree] :
% 3.58/1.93 ( ! [Tr1: dtree,Tr2: dtree] : ( P @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 3.58/1.93 => ( P @ A0 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_2__092_060open_062Inr_A_N_096_Acont_A_Itr1_A_092_060parallel_062_Atr2_J_A_061_Apar_A_096_A_IInr_A_N_096_Acont_Atr1_A_092_060times_062_AInr_A_N_096_Acont_Atr2_J_092_060close_062,axiom,
% 3.58/1.93 ( ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) ) )
% 3.58/1.93 = ( image @ ( product_prod @ dtree @ dtree ) @ dtree @ parall1899940088le_par
% 3.58/1.93 @ ( product_Sigma @ dtree @ dtree @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ tr1 ) )
% 3.58/1.93 @ ^ [Uu: dtree] : ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ tr2 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_3_SigmaI,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,A2: A,A3: set @ A,B2: B,B3: A > ( set @ B )] :
% 3.58/1.93 ( ( member @ A @ A2 @ A3 )
% 3.58/1.93 => ( ( member @ B @ B2 @ ( B3 @ A2 ) )
% 3.58/1.93 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_4_mem__Sigma__iff,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 3.58/1.93 = ( ( member @ A @ A2 @ A3 )
% 3.58/1.93 & ( member @ B @ B2 @ ( B3 @ A2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_5_image__ident,axiom,
% 3.58/1.93 ! [A: $tType,Y: set @ A] :
% 3.58/1.93 ( ( image @ A @ A
% 3.58/1.93 @ ^ [X2: A] : X2
% 3.58/1.93 @ Y )
% 3.58/1.93 = Y ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_6_Inr__cont__par,axiom,
% 3.58/1.93 ! [Tr12: dtree,Tr22: dtree] :
% 3.58/1.93 ( ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) )
% 3.58/1.93 = ( image @ ( product_prod @ dtree @ dtree ) @ dtree @ parall1899940088le_par
% 3.58/1.93 @ ( product_Sigma @ dtree @ dtree @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr12 ) )
% 3.58/1.93 @ ^ [Uu: dtree] : ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr22 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_7_sum_Oinject_I2_J,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,X22: B,Y2: B] :
% 3.58/1.93 ( ( ( sum_Inr @ B @ A @ X22 )
% 3.58/1.93 = ( sum_Inr @ B @ A @ Y2 ) )
% 3.58/1.93 = ( X22 = Y2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_8_old_Osum_Oinject_I2_J,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,B2: B,B4: B] :
% 3.58/1.93 ( ( ( sum_Inr @ B @ A @ B2 )
% 3.58/1.93 = ( sum_Inr @ B @ A @ B4 ) )
% 3.58/1.93 = ( B2 = B4 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_9_image__eqI,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,B2: A,F: B > A,X: B,A3: set @ B] :
% 3.58/1.93 ( ( B2
% 3.58/1.93 = ( F @ X ) )
% 3.58/1.93 => ( ( member @ B @ X @ A3 )
% 3.58/1.93 => ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_10_prod_Oinject,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 3.58/1.93 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 3.58/1.93 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 3.58/1.93 = ( ( X1 = Y1 )
% 3.58/1.93 & ( X22 = Y2 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_11_old_Oprod_Oinject,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B4: B] :
% 3.58/1.93 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 3.58/1.93 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 3.58/1.93 = ( ( A2 = A4 )
% 3.58/1.93 & ( B2 = B4 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_12_SigmaE,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,C2: product_prod @ A @ B,A3: set @ A,B3: A > ( set @ B )] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 3.58/1.93 => ~ ! [X3: A] :
% 3.58/1.93 ( ( member @ A @ X3 @ A3 )
% 3.58/1.93 => ! [Y3: B] :
% 3.58/1.93 ( ( member @ B @ Y3 @ ( B3 @ X3 ) )
% 3.58/1.93 => ( C2
% 3.58/1.93 != ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_13_SigmaD1,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 3.58/1.93 => ( member @ A @ A2 @ A3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_14_vimageI,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,F: B > A,A2: B,B2: A,B3: set @ A] :
% 3.58/1.93 ( ( ( F @ A2 )
% 3.58/1.93 = B2 )
% 3.58/1.93 => ( ( member @ A @ B2 @ B3 )
% 3.58/1.93 => ( member @ B @ A2 @ ( vimage @ B @ A @ F @ B3 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_15_vimage__eq,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,A2: A,F: A > B,B3: set @ B] :
% 3.58/1.93 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B3 ) )
% 3.58/1.93 = ( member @ B @ ( F @ A2 ) @ B3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_16_vimage__ident,axiom,
% 3.58/1.93 ! [A: $tType,Y: set @ A] :
% 3.58/1.93 ( ( vimage @ A @ A
% 3.58/1.93 @ ^ [X2: A] : X2
% 3.58/1.93 @ Y )
% 3.58/1.93 = Y ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_17_vimage__Collect__eq,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,F: A > B,P: B > $o] :
% 3.58/1.93 ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
% 3.58/1.93 = ( collect @ A
% 3.58/1.93 @ ^ [Y4: A] : ( P @ ( F @ Y4 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_18_vimageD,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,A2: A,F: A > B,A3: set @ B] :
% 3.58/1.93 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ A3 ) )
% 3.58/1.93 => ( member @ B @ ( F @ A2 ) @ A3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_19_vimageE,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,A2: A,F: A > B,B3: set @ B] :
% 3.58/1.93 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B3 ) )
% 3.58/1.93 => ( member @ B @ ( F @ A2 ) @ B3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_20_vimageI2,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,F: B > A,A2: B,A3: set @ A] :
% 3.58/1.93 ( ( member @ A @ ( F @ A2 ) @ A3 )
% 3.58/1.93 => ( member @ B @ A2 @ ( vimage @ B @ A @ F @ A3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_21_vimage__def,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType] :
% 3.58/1.93 ( ( vimage @ A @ B )
% 3.58/1.93 = ( ^ [F2: A > B,B5: set @ B] :
% 3.58/1.93 ( collect @ A
% 3.58/1.93 @ ^ [X2: A] : ( member @ B @ ( F2 @ X2 ) @ B5 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_22_vimage__Collect,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,P: B > $o,F: A > B,Q: A > $o] :
% 3.58/1.93 ( ! [X3: A] :
% 3.58/1.93 ( ( P @ ( F @ X3 ) )
% 3.58/1.93 = ( Q @ X3 ) )
% 3.58/1.93 => ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
% 3.58/1.93 = ( collect @ A @ Q ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_23_old_Oprod_Oinducts,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 3.58/1.93 ( ! [A5: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B6 ) )
% 3.58/1.93 => ( P @ Prod ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_24_old_Oprod_Oexhaust,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,Y5: product_prod @ A @ B] :
% 3.58/1.93 ~ ! [A5: A,B6: B] :
% 3.58/1.93 ( Y5
% 3.58/1.93 != ( product_Pair @ A @ B @ A5 @ B6 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_25_prod__induct7,axiom,
% 3.58/1.93 ! [G: $tType,F3: $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 @ F3 @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) )] :
% 3.58/1.93 ( ! [A5: A,B6: B,C3: C,D2: D,E2: E,F4: F3,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G ) @ E2 @ ( product_Pair @ F3 @ G @ F4 @ G2 ) ) ) ) ) ) )
% 3.58/1.93 => ( P @ X ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_26_prod__induct6,axiom,
% 3.58/1.93 ! [F3: $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 @ F3 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 3.58/1.93 ( ! [A5: A,B6: B,C3: C,D2: D,E2: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
% 3.58/1.93 => ( P @ X ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_27_prod__induct5,axiom,
% 3.58/1.93 ! [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 ) ) )] :
% 3.58/1.93 ( ! [A5: A,B6: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 3.58/1.93 => ( P @ X ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_28_prod__induct4,axiom,
% 3.58/1.93 ! [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 ) )] :
% 3.58/1.93 ( ! [A5: A,B6: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B6 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 3.58/1.93 => ( P @ X ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_29_prod__induct3,axiom,
% 3.58/1.93 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.58/1.93 ( ! [A5: A,B6: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B6 @ C3 ) ) )
% 3.58/1.93 => ( P @ X ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_30_prod__cases7,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G: $tType,Y5: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) )] :
% 3.58/1.93 ~ ! [A5: A,B6: B,C3: C,D2: D,E2: E,F4: F3,G2: G] :
% 3.58/1.93 ( Y5
% 3.58/1.93 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G ) @ E2 @ ( product_Pair @ F3 @ G @ F4 @ G2 ) ) ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_31_prod__cases6,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y5: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 3.58/1.93 ~ ! [A5: A,B6: B,C3: C,D2: D,E2: E,F4: F3] :
% 3.58/1.93 ( Y5
% 3.58/1.93 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_32_prod__cases5,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y5: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 3.58/1.93 ~ ! [A5: A,B6: B,C3: C,D2: D,E2: E] :
% 3.58/1.93 ( Y5
% 3.58/1.93 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_33_prod__cases4,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y5: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 3.58/1.93 ~ ! [A5: A,B6: B,C3: C,D2: D] :
% 3.58/1.93 ( Y5
% 3.58/1.93 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B6 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_34_prod__cases3,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,C: $tType,Y5: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.58/1.93 ~ ! [A5: A,B6: B,C3: C] :
% 3.58/1.93 ( Y5
% 3.58/1.93 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B6 @ C3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_35_Pair__inject,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B4: B] :
% 3.58/1.93 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 3.58/1.93 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 3.58/1.93 => ~ ( ( A2 = A4 )
% 3.58/1.93 => ( B2 != B4 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_36_prod__cases,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 3.58/1.93 ( ! [A5: A,B6: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B6 ) )
% 3.58/1.93 => ( P @ P2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_37_surj__pair,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 3.58/1.93 ? [X3: A,Y3: B] :
% 3.58/1.93 ( P2
% 3.58/1.93 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_38_rev__image__eqI,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,X: A,A3: set @ A,B2: B,F: A > B] :
% 3.58/1.93 ( ( member @ A @ X @ A3 )
% 3.58/1.93 => ( ( B2
% 3.58/1.93 = ( F @ X ) )
% 3.58/1.93 => ( member @ B @ B2 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_39_ball__imageD,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
% 3.58/1.93 ( ! [X3: A] :
% 3.58/1.93 ( ( member @ A @ X3 @ ( image @ B @ A @ F @ A3 ) )
% 3.58/1.93 => ( P @ X3 ) )
% 3.58/1.93 => ! [X4: B] :
% 3.58/1.93 ( ( member @ B @ X4 @ A3 )
% 3.58/1.93 => ( P @ ( F @ X4 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_40_image__cong,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G3: A > B] :
% 3.58/1.93 ( ( M = N )
% 3.58/1.93 => ( ! [X3: A] :
% 3.58/1.93 ( ( member @ A @ X3 @ N )
% 3.58/1.93 => ( ( F @ X3 )
% 3.58/1.93 = ( G3 @ X3 ) ) )
% 3.58/1.93 => ( ( image @ A @ B @ F @ M )
% 3.58/1.93 = ( image @ A @ B @ G3 @ N ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_41_bex__imageD,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
% 3.58/1.93 ( ? [X4: A] :
% 3.58/1.93 ( ( member @ A @ X4 @ ( image @ B @ A @ F @ A3 ) )
% 3.58/1.93 & ( P @ X4 ) )
% 3.58/1.93 => ? [X3: B] :
% 3.58/1.93 ( ( member @ B @ X3 @ A3 )
% 3.58/1.93 & ( P @ ( F @ X3 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_42_image__iff,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,Z: A,F: B > A,A3: set @ B] :
% 3.58/1.93 ( ( member @ A @ Z @ ( image @ B @ A @ F @ A3 ) )
% 3.58/1.93 = ( ? [X2: B] :
% 3.58/1.93 ( ( member @ B @ X2 @ A3 )
% 3.58/1.93 & ( Z
% 3.58/1.93 = ( F @ X2 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_43_imageI,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,X: A,A3: set @ A,F: A > B] :
% 3.58/1.93 ( ( member @ A @ X @ A3 )
% 3.58/1.93 => ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_44_Inr__inject,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,X: B,Y5: B] :
% 3.58/1.93 ( ( ( sum_Inr @ B @ A @ X )
% 3.58/1.93 = ( sum_Inr @ B @ A @ Y5 ) )
% 3.58/1.93 => ( X = Y5 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_45_mem__Collect__eq,axiom,
% 3.58/1.93 ! [A: $tType,A2: A,P: A > $o] :
% 3.58/1.93 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 3.58/1.93 = ( P @ A2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_46_Collect__mem__eq,axiom,
% 3.58/1.93 ! [A: $tType,A3: set @ A] :
% 3.58/1.93 ( ( collect @ A
% 3.58/1.93 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 3.58/1.93 = A3 ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_47_Collect__cong,axiom,
% 3.58/1.93 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.58/1.93 ( ! [X3: A] :
% 3.58/1.93 ( ( P @ X3 )
% 3.58/1.93 = ( Q @ X3 ) )
% 3.58/1.93 => ( ( collect @ A @ P )
% 3.58/1.93 = ( collect @ A @ Q ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_48_ext,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,F: A > B,G3: A > B] :
% 3.58/1.93 ( ! [X3: A] :
% 3.58/1.93 ( ( F @ X3 )
% 3.58/1.93 = ( G3 @ X3 ) )
% 3.58/1.93 => ( F = G3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_49_Compr__image__eq,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
% 3.58/1.93 ( ( collect @ A
% 3.58/1.93 @ ^ [X2: A] :
% 3.58/1.93 ( ( member @ A @ X2 @ ( image @ B @ A @ F @ A3 ) )
% 3.58/1.93 & ( P @ X2 ) ) )
% 3.58/1.93 = ( image @ B @ A @ F
% 3.58/1.93 @ ( collect @ B
% 3.58/1.93 @ ^ [X2: B] :
% 3.58/1.93 ( ( member @ B @ X2 @ A3 )
% 3.58/1.93 & ( P @ ( F @ X2 ) ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_50_image__image,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,C: $tType,F: B > A,G3: C > B,A3: set @ C] :
% 3.58/1.93 ( ( image @ B @ A @ F @ ( image @ C @ B @ G3 @ A3 ) )
% 3.58/1.93 = ( image @ C @ A
% 3.58/1.93 @ ^ [X2: C] : ( F @ ( G3 @ X2 ) )
% 3.58/1.93 @ A3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_51_imageE,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,B2: A,F: B > A,A3: set @ B] :
% 3.58/1.93 ( ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) )
% 3.58/1.93 => ~ ! [X3: B] :
% 3.58/1.93 ( ( B2
% 3.58/1.93 = ( F @ X3 ) )
% 3.58/1.93 => ~ ( member @ B @ X3 @ A3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_52_Times__eq__cancel2,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,X: A,C4: set @ A,A3: set @ B,B3: set @ B] :
% 3.58/1.93 ( ( member @ A @ X @ C4 )
% 3.58/1.93 => ( ( ( product_Sigma @ B @ A @ A3
% 3.58/1.93 @ ^ [Uu: B] : C4 )
% 3.58/1.93 = ( product_Sigma @ B @ A @ B3
% 3.58/1.93 @ ^ [Uu: B] : C4 ) )
% 3.58/1.93 = ( A3 = B3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_53_Sigma__cong,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,C4: A > ( set @ B ),D3: A > ( set @ B )] :
% 3.58/1.93 ( ( A3 = B3 )
% 3.58/1.93 => ( ! [X3: A] :
% 3.58/1.93 ( ( member @ A @ X3 @ B3 )
% 3.58/1.93 => ( ( C4 @ X3 )
% 3.58/1.93 = ( D3 @ X3 ) ) )
% 3.58/1.93 => ( ( product_Sigma @ A @ B @ A3 @ C4 )
% 3.58/1.93 = ( product_Sigma @ A @ B @ B3 @ D3 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_54_SigmaE2,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 3.58/1.93 => ~ ( ( member @ A @ A2 @ A3 )
% 3.58/1.93 => ~ ( member @ B @ B2 @ ( B3 @ A2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_55_SigmaD2,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 3.58/1.93 => ( member @ B @ B2 @ ( B3 @ A2 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_56_old_Oprod_Orec,axiom,
% 3.58/1.93 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 3.58/1.93 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 3.58/1.93 = ( F1 @ A2 @ B2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_57_old_Osum_Osimps_I8_J,axiom,
% 3.58/1.93 ! [A: $tType,T: $tType,B: $tType,F1: A > T,F22: B > T,B2: B] :
% 3.58/1.93 ( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inr @ B @ A @ B2 ) )
% 3.58/1.93 = ( F22 @ B2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_58_member__product,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A3: set @ A,B3: set @ B] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A3 @ B3 ) )
% 3.58/1.93 = ( member @ ( product_prod @ A @ B ) @ X
% 3.58/1.93 @ ( product_Sigma @ A @ B @ A3
% 3.58/1.93 @ ^ [Uu: A] : B3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_59_Product__Type_Oproduct__def,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType] :
% 3.58/1.93 ( ( product_product @ A @ B )
% 3.58/1.93 = ( ^ [A6: set @ A,B5: set @ B] :
% 3.58/1.93 ( product_Sigma @ A @ B @ A6
% 3.58/1.93 @ ^ [Uu: A] : B5 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_60_Sup_OSUP__identity__eq,axiom,
% 3.58/1.93 ! [A: $tType,Sup: ( set @ A ) > A,A3: set @ A] :
% 3.58/1.93 ( ( Sup
% 3.58/1.93 @ ( image @ A @ A
% 3.58/1.93 @ ^ [X2: A] : X2
% 3.58/1.93 @ A3 ) )
% 3.58/1.93 = ( Sup @ A3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_61_Inf_OINF__identity__eq,axiom,
% 3.58/1.93 ! [A: $tType,Inf: ( set @ A ) > A,A3: set @ A] :
% 3.58/1.93 ( ( Inf
% 3.58/1.93 @ ( image @ A @ A
% 3.58/1.93 @ ^ [X2: A] : X2
% 3.58/1.93 @ A3 ) )
% 3.58/1.93 = ( Inf @ A3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_62_pred__equals__eq2,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 3.58/1.93 ( ( ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R ) )
% 3.58/1.93 = ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ S ) ) )
% 3.58/1.93 = ( R = S ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_63_internal__case__prod__conv,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
% 3.58/1.93 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 3.58/1.93 = ( C2 @ A2 @ B2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_64_Sumr_Osimps,axiom,
% 3.58/1.93 ! [A: $tType,C: $tType,B: $tType,F: B > C,X: B] :
% 3.58/1.93 ( ( sum_Sumr @ B @ C @ A @ F @ ( sum_Inr @ B @ A @ X ) )
% 3.58/1.93 = ( F @ X ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_65_not__arg__cong__Inr,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,X: A,Y5: A] :
% 3.58/1.93 ( ( X != Y5 )
% 3.58/1.93 => ( ( sum_Inr @ A @ B @ X )
% 3.58/1.93 != ( sum_Inr @ A @ B @ Y5 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_66_Inl__in__cont__par,axiom,
% 3.58/1.93 ! [T2: t,Tr12: dtree,Tr22: dtree] :
% 3.58/1.93 ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) )
% 3.58/1.93 = ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr12 ) )
% 3.58/1.93 | ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr22 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_67_sum_Oinject_I1_J,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,X1: A,Y1: A] :
% 3.58/1.93 ( ( ( sum_Inl @ A @ B @ X1 )
% 3.58/1.93 = ( sum_Inl @ A @ B @ Y1 ) )
% 3.58/1.93 = ( X1 = Y1 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_68_old_Osum_Oinject_I1_J,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,A2: A,A4: A] :
% 3.58/1.93 ( ( ( sum_Inl @ A @ B @ A2 )
% 3.58/1.93 = ( sum_Inl @ A @ B @ A4 ) )
% 3.58/1.93 = ( A2 = A4 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_69_old_Osum_Osimps_I7_J,axiom,
% 3.58/1.93 ! [B: $tType,T: $tType,A: $tType,F1: A > T,F22: B > T,A2: A] :
% 3.58/1.93 ( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inl @ A @ B @ A2 ) )
% 3.58/1.93 = ( F1 @ A2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_70_Inl__inject,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,X: A,Y5: A] :
% 3.58/1.93 ( ( ( sum_Inl @ A @ B @ X )
% 3.58/1.93 = ( sum_Inl @ A @ B @ Y5 ) )
% 3.58/1.93 => ( X = Y5 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_71_obj__sumE,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
% 3.58/1.93 ( ! [X3: A] :
% 3.58/1.93 ( S2
% 3.58/1.93 != ( sum_Inl @ A @ B @ X3 ) )
% 3.58/1.93 => ~ ! [X3: B] :
% 3.58/1.93 ( S2
% 3.58/1.93 != ( sum_Inr @ B @ A @ X3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_72_Sumr__inject,axiom,
% 3.58/1.93 ! [A: $tType,C: $tType,B: $tType,F: B > C,G3: B > C] :
% 3.58/1.93 ( ( ( sum_Sumr @ B @ C @ A @ F )
% 3.58/1.93 = ( sum_Sumr @ B @ C @ A @ G3 ) )
% 3.58/1.93 => ( F = G3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_73_sum_Odistinct_I1_J,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,X1: A,X22: B] :
% 3.58/1.93 ( ( sum_Inl @ A @ B @ X1 )
% 3.58/1.93 != ( sum_Inr @ B @ A @ X22 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_74_old_Osum_Odistinct_I2_J,axiom,
% 3.58/1.93 ! [B7: $tType,A7: $tType,B8: B7,A8: A7] :
% 3.58/1.93 ( ( sum_Inr @ B7 @ A7 @ B8 )
% 3.58/1.93 != ( sum_Inl @ A7 @ B7 @ A8 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_75_old_Osum_Odistinct_I1_J,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,A2: A,B4: B] :
% 3.58/1.93 ( ( sum_Inl @ A @ B @ A2 )
% 3.58/1.93 != ( sum_Inr @ B @ A @ B4 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_76_sumE,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
% 3.58/1.93 ( ! [X3: A] :
% 3.58/1.93 ( S2
% 3.58/1.93 != ( sum_Inl @ A @ B @ X3 ) )
% 3.58/1.93 => ~ ! [Y3: B] :
% 3.58/1.93 ( S2
% 3.58/1.93 != ( sum_Inr @ B @ A @ Y3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_77_Inr__not__Inl,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,B2: B,A2: A] :
% 3.58/1.93 ( ( sum_Inr @ B @ A @ B2 )
% 3.58/1.93 != ( sum_Inl @ A @ B @ A2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_78_split__sum__ex,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType] :
% 3.58/1.93 ( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 3.58/1.93 ? [X5: sum_sum @ A @ B] : ( P3 @ X5 ) )
% 3.58/1.93 = ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
% 3.58/1.93 ( ? [X2: A] : ( P4 @ ( sum_Inl @ A @ B @ X2 ) )
% 3.58/1.93 | ? [X2: B] : ( P4 @ ( sum_Inr @ B @ A @ X2 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_79_split__sum__all,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType] :
% 3.58/1.93 ( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 3.58/1.93 ! [X5: sum_sum @ A @ B] : ( P3 @ X5 ) )
% 3.58/1.93 = ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
% 3.58/1.93 ( ! [X2: A] : ( P4 @ ( sum_Inl @ A @ B @ X2 ) )
% 3.58/1.93 & ! [X2: B] : ( P4 @ ( sum_Inr @ B @ A @ X2 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_80_old_Osum_Oexhaust,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,Y5: sum_sum @ A @ B] :
% 3.58/1.93 ( ! [A5: A] :
% 3.58/1.93 ( Y5
% 3.58/1.93 != ( sum_Inl @ A @ B @ A5 ) )
% 3.58/1.93 => ~ ! [B6: B] :
% 3.58/1.93 ( Y5
% 3.58/1.93 != ( sum_Inr @ B @ A @ B6 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_81_old_Osum_Oinducts,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
% 3.58/1.93 ( ! [A5: A] : ( P @ ( sum_Inl @ A @ B @ A5 ) )
% 3.58/1.93 => ( ! [B6: B] : ( P @ ( sum_Inr @ B @ A @ B6 ) )
% 3.58/1.93 => ( P @ Sum ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_82_Inf_OINF__cong,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,A3: set @ B,B3: set @ B,C4: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 3.58/1.93 ( ( A3 = B3 )
% 3.58/1.93 => ( ! [X3: B] :
% 3.58/1.93 ( ( member @ B @ X3 @ B3 )
% 3.58/1.93 => ( ( C4 @ X3 )
% 3.58/1.93 = ( D3 @ X3 ) ) )
% 3.58/1.93 => ( ( Inf @ ( image @ B @ A @ C4 @ A3 ) )
% 3.58/1.93 = ( Inf @ ( image @ B @ A @ D3 @ B3 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_83_Sup_OSUP__cong,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,A3: set @ B,B3: set @ B,C4: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 3.58/1.93 ( ( A3 = B3 )
% 3.58/1.93 => ( ! [X3: B] :
% 3.58/1.93 ( ( member @ B @ X3 @ B3 )
% 3.58/1.93 => ( ( C4 @ X3 )
% 3.58/1.93 = ( D3 @ X3 ) ) )
% 3.58/1.93 => ( ( Sup @ ( image @ B @ A @ C4 @ A3 ) )
% 3.58/1.93 = ( Sup @ ( image @ B @ A @ D3 @ B3 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_84_Inl__Inr__False,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,X: A,Y5: B] :
% 3.58/1.93 ( ( sum_Inl @ A @ B @ X )
% 3.58/1.93 != ( sum_Inr @ B @ A @ Y5 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_85_Inr__Inl__False,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,X: B,Y5: A] :
% 3.58/1.93 ( ( sum_Inr @ B @ A @ X )
% 3.58/1.93 != ( sum_Inl @ A @ B @ Y5 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_86_Inl__Inr__image__cong,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,A3: set @ ( sum_sum @ A @ B ),B3: set @ ( sum_sum @ A @ B )] :
% 3.58/1.93 ( ( ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A3 )
% 3.58/1.93 = ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ B3 ) )
% 3.58/1.93 => ( ( ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ A3 )
% 3.58/1.93 = ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B3 ) )
% 3.58/1.93 => ( A3 = B3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_87_obj__sumE__f,axiom,
% 3.58/1.93 ! [A: $tType,C: $tType,B: $tType,S2: B,F: ( sum_sum @ A @ C ) > B,P: $o] :
% 3.58/1.93 ( ! [X3: A] :
% 3.58/1.93 ( ( S2
% 3.58/1.93 = ( F @ ( sum_Inl @ A @ C @ X3 ) ) )
% 3.58/1.93 => P )
% 3.58/1.93 => ( ! [X3: C] :
% 3.58/1.93 ( ( S2
% 3.58/1.93 = ( F @ ( sum_Inr @ C @ A @ X3 ) ) )
% 3.58/1.93 => P )
% 3.58/1.93 => ! [X4: sum_sum @ A @ C] :
% 3.58/1.93 ( ( S2
% 3.58/1.93 = ( F @ X4 ) )
% 3.58/1.93 => P ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_88_Inl__cont__par,axiom,
% 3.58/1.93 ! [Tr12: dtree,Tr22: dtree] :
% 3.58/1.93 ( ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) )
% 3.58/1.93 = ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( sup_sup @ ( set @ ( sum_sum @ t @ dtree ) ) @ ( cont @ Tr12 ) @ ( cont @ Tr22 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_89_Suml_Osimps,axiom,
% 3.58/1.93 ! [B: $tType,C: $tType,A: $tType,F: A > C,X: A] :
% 3.58/1.93 ( ( sum_Suml @ A @ C @ B @ F @ ( sum_Inl @ A @ B @ X ) )
% 3.58/1.93 = ( F @ X ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_90_image2__eqI,axiom,
% 3.58/1.93 ! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X: B,C2: C,G3: B > C,A3: set @ B] :
% 3.58/1.93 ( ( B2
% 3.58/1.93 = ( F @ X ) )
% 3.58/1.93 => ( ( C2
% 3.58/1.93 = ( G3 @ X ) )
% 3.58/1.93 => ( ( member @ B @ X @ A3 )
% 3.58/1.93 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A3 @ F @ G3 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_91_Un__iff,axiom,
% 3.58/1.93 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 3.58/1.93 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 3.58/1.93 = ( ( member @ A @ C2 @ A3 )
% 3.58/1.93 | ( member @ A @ C2 @ B3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_92_UnCI,axiom,
% 3.58/1.93 ! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
% 3.58/1.93 ( ( ~ ( member @ A @ C2 @ B3 )
% 3.58/1.93 => ( member @ A @ C2 @ A3 ) )
% 3.58/1.93 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_93_vimage__Un,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,F: A > B,A3: set @ B,B3: set @ B] :
% 3.58/1.93 ( ( vimage @ A @ B @ F @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
% 3.58/1.93 = ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F @ A3 ) @ ( vimage @ A @ B @ F @ B3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_94_Sigma__Un__distrib1,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,I: set @ A,J: set @ A,C4: A > ( set @ B )] :
% 3.58/1.93 ( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ I @ J ) @ C4 )
% 3.58/1.93 = ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ C4 ) @ ( product_Sigma @ A @ B @ J @ C4 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_95_Collect__disj__eq,axiom,
% 3.58/1.93 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.58/1.93 ( ( collect @ A
% 3.58/1.93 @ ^ [X2: A] :
% 3.58/1.93 ( ( P @ X2 )
% 3.58/1.93 | ( Q @ X2 ) ) )
% 3.58/1.93 = ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_96_Un__def,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( sup_sup @ ( set @ A ) )
% 3.58/1.93 = ( ^ [A6: set @ A,B5: set @ A] :
% 3.58/1.93 ( collect @ A
% 3.58/1.93 @ ^ [X2: A] :
% 3.58/1.93 ( ( member @ A @ X2 @ A6 )
% 3.58/1.93 | ( member @ A @ X2 @ B5 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_97_Sigma__Un__distrib2,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,I: set @ A,A3: A > ( set @ B ),B3: A > ( set @ B )] :
% 3.58/1.93 ( ( product_Sigma @ A @ B @ I
% 3.58/1.93 @ ^ [I2: A] : ( sup_sup @ ( set @ B ) @ ( A3 @ I2 ) @ ( B3 @ I2 ) ) )
% 3.58/1.93 = ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ A3 ) @ ( product_Sigma @ A @ B @ I @ B3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_98_Times__Un__distrib1,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,C4: set @ B] :
% 3.58/1.93 ( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A3 @ B3 )
% 3.58/1.93 @ ^ [Uu: A] : C4 )
% 3.58/1.93 = ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
% 3.58/1.93 @ ( product_Sigma @ A @ B @ A3
% 3.58/1.93 @ ^ [Uu: A] : C4 )
% 3.58/1.93 @ ( product_Sigma @ A @ B @ B3
% 3.58/1.93 @ ^ [Uu: A] : C4 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_99_Un__left__commute,axiom,
% 3.58/1.93 ! [A: $tType,A3: set @ A,B3: set @ A,C4: set @ A] :
% 3.58/1.93 ( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C4 ) )
% 3.58/1.93 = ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_100_Un__left__absorb,axiom,
% 3.58/1.93 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.58/1.93 ( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 3.58/1.93 = ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_101_Un__commute,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( sup_sup @ ( set @ A ) )
% 3.58/1.93 = ( ^ [A6: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_102_Un__absorb,axiom,
% 3.58/1.93 ! [A: $tType,A3: set @ A] :
% 3.58/1.93 ( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
% 3.58/1.93 = A3 ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_103_Un__assoc,axiom,
% 3.58/1.93 ! [A: $tType,A3: set @ A,B3: set @ A,C4: set @ A] :
% 3.58/1.93 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C4 )
% 3.58/1.93 = ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C4 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_104_ball__Un,axiom,
% 3.58/1.93 ! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
% 3.58/1.93 ( ( ! [X2: A] :
% 3.58/1.93 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 3.58/1.93 => ( P @ X2 ) ) )
% 3.58/1.93 = ( ! [X2: A] :
% 3.58/1.93 ( ( member @ A @ X2 @ A3 )
% 3.58/1.93 => ( P @ X2 ) )
% 3.58/1.93 & ! [X2: A] :
% 3.58/1.93 ( ( member @ A @ X2 @ B3 )
% 3.58/1.93 => ( P @ X2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_105_bex__Un,axiom,
% 3.58/1.93 ! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
% 3.58/1.93 ( ( ? [X2: A] :
% 3.58/1.93 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 3.58/1.93 & ( P @ X2 ) ) )
% 3.58/1.93 = ( ? [X2: A] :
% 3.58/1.93 ( ( member @ A @ X2 @ A3 )
% 3.58/1.93 & ( P @ X2 ) )
% 3.58/1.93 | ? [X2: A] :
% 3.58/1.93 ( ( member @ A @ X2 @ B3 )
% 3.58/1.93 & ( P @ X2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_106_UnI2,axiom,
% 3.58/1.93 ! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
% 3.58/1.93 ( ( member @ A @ C2 @ B3 )
% 3.58/1.93 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_107_UnI1,axiom,
% 3.58/1.93 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 3.58/1.93 ( ( member @ A @ C2 @ A3 )
% 3.58/1.93 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_108_UnE,axiom,
% 3.58/1.93 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 3.58/1.93 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 3.58/1.93 => ( ~ ( member @ A @ C2 @ A3 )
% 3.58/1.93 => ( member @ A @ C2 @ B3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_109_Suml__inject,axiom,
% 3.58/1.93 ! [B: $tType,C: $tType,A: $tType,F: A > C,G3: A > C] :
% 3.58/1.93 ( ( ( sum_Suml @ A @ C @ B @ F )
% 3.58/1.93 = ( sum_Suml @ A @ C @ B @ G3 ) )
% 3.58/1.93 => ( F = G3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_110_image__Un,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,B3: set @ B] :
% 3.58/1.93 ( ( image @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
% 3.58/1.93 = ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F @ A3 ) @ ( image @ B @ A @ F @ B3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_111_sup__apply,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType] :
% 3.58/1.93 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 3.58/1.93 => ( ( sup_sup @ ( A > B ) )
% 3.58/1.93 = ( ^ [F2: A > B,G4: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G4 @ X2 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_112_sup_Oidem,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.58/1.93 => ! [A2: A] :
% 3.58/1.93 ( ( sup_sup @ A @ A2 @ A2 )
% 3.58/1.93 = A2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_113_sup__idem,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.58/1.93 => ! [X: A] :
% 3.58/1.93 ( ( sup_sup @ A @ X @ X )
% 3.58/1.93 = X ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_114_sup_Oleft__idem,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.58/1.93 => ! [A2: A,B2: A] :
% 3.58/1.93 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
% 3.58/1.93 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_115_sup__left__idem,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.58/1.93 => ! [X: A,Y5: A] :
% 3.58/1.93 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y5 ) )
% 3.58/1.93 = ( sup_sup @ A @ X @ Y5 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_116_sup_Oright__idem,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.58/1.93 => ! [A2: A,B2: A] :
% 3.58/1.93 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
% 3.58/1.93 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_117_sup__Un__eq2,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 3.58/1.93 ( ( sup_sup @ ( A > B > $o )
% 3.58/1.93 @ ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R )
% 3.58/1.93 @ ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ S ) )
% 3.58/1.93 = ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_118_sup__set__def,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( sup_sup @ ( set @ A ) )
% 3.58/1.93 = ( ^ [A6: set @ A,B5: set @ A] :
% 3.58/1.93 ( collect @ A
% 3.58/1.93 @ ( sup_sup @ ( A > $o )
% 3.58/1.93 @ ^ [X2: A] : ( member @ A @ X2 @ A6 )
% 3.58/1.93 @ ^ [X2: A] : ( member @ A @ X2 @ B5 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_119_sup__Un__eq,axiom,
% 3.58/1.93 ! [A: $tType,R: set @ A,S: set @ A] :
% 3.58/1.93 ( ( sup_sup @ ( A > $o )
% 3.58/1.93 @ ^ [X2: A] : ( member @ A @ X2 @ R )
% 3.58/1.93 @ ^ [X2: A] : ( member @ A @ X2 @ S ) )
% 3.58/1.93 = ( ^ [X2: A] : ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ R @ S ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_120_sup__left__commute,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.58/1.93 => ! [X: A,Y5: A,Z: A] :
% 3.58/1.93 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y5 @ Z ) )
% 3.58/1.93 = ( sup_sup @ A @ Y5 @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_121_sup_Oleft__commute,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.58/1.93 => ! [B2: A,A2: A,C2: A] :
% 3.58/1.93 ( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C2 ) )
% 3.58/1.93 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_122_sup__commute,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.58/1.93 => ( ( sup_sup @ A )
% 3.58/1.93 = ( ^ [X2: A,Y4: A] : ( sup_sup @ A @ Y4 @ X2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_123_sup_Ocommute,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.58/1.93 => ( ( sup_sup @ A )
% 3.58/1.93 = ( ^ [A9: A,B9: A] : ( sup_sup @ A @ B9 @ A9 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_124_sup__assoc,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.58/1.93 => ! [X: A,Y5: A,Z: A] :
% 3.58/1.93 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y5 ) @ Z )
% 3.58/1.93 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y5 @ Z ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_125_sup_Oassoc,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.58/1.93 => ! [A2: A,B2: A,C2: A] :
% 3.58/1.93 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C2 )
% 3.58/1.93 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_126_sup__fun__def,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType] :
% 3.58/1.93 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 3.58/1.93 => ( ( sup_sup @ ( A > B ) )
% 3.58/1.93 = ( ^ [F2: A > B,G4: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G4 @ X2 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_127_inf__sup__aci_I5_J,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( lattice @ A @ ( type2 @ A ) )
% 3.58/1.93 => ( ( sup_sup @ A )
% 3.58/1.93 = ( ^ [X2: A,Y4: A] : ( sup_sup @ A @ Y4 @ X2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_128_inf__sup__aci_I6_J,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( lattice @ A @ ( type2 @ A ) )
% 3.58/1.93 => ! [X: A,Y5: A,Z: A] :
% 3.58/1.93 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y5 ) @ Z )
% 3.58/1.93 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y5 @ Z ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_129_inf__sup__aci_I7_J,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( lattice @ A @ ( type2 @ A ) )
% 3.58/1.93 => ! [X: A,Y5: A,Z: A] :
% 3.58/1.93 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y5 @ Z ) )
% 3.58/1.93 = ( sup_sup @ A @ Y5 @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_130_inf__sup__aci_I8_J,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( lattice @ A @ ( type2 @ A ) )
% 3.58/1.93 => ! [X: A,Y5: A] :
% 3.58/1.93 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y5 ) )
% 3.58/1.93 = ( sup_sup @ A @ X @ Y5 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_131_par__c_Oelims,axiom,
% 3.58/1.93 ! [X: product_prod @ dtree @ dtree,Y5: set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) )] :
% 3.58/1.93 ( ( ( parall1914194347_par_c @ X )
% 3.58/1.93 = Y5 )
% 3.58/1.93 => ~ ! [Tr1: dtree,Tr2: dtree] :
% 3.58/1.93 ( ( X
% 3.58/1.93 = ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 3.58/1.93 => ( Y5
% 3.58/1.93 != ( sup_sup @ ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) @ ( image @ t @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inl @ t @ ( product_prod @ dtree @ dtree ) ) @ ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( sup_sup @ ( set @ ( sum_sum @ t @ dtree ) ) @ ( cont @ Tr1 ) @ ( cont @ Tr2 ) ) ) )
% 3.58/1.93 @ ( image @ ( product_prod @ dtree @ dtree ) @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inr @ ( product_prod @ dtree @ dtree ) @ t )
% 3.58/1.93 @ ( product_Sigma @ dtree @ dtree @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr1 ) )
% 3.58/1.93 @ ^ [Uu: dtree] : ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr2 ) ) ) ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_132_par__c_Osimps,axiom,
% 3.58/1.93 ! [Tr12: dtree,Tr22: dtree] :
% 3.58/1.93 ( ( parall1914194347_par_c @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) )
% 3.58/1.93 = ( sup_sup @ ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) @ ( image @ t @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inl @ t @ ( product_prod @ dtree @ dtree ) ) @ ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( sup_sup @ ( set @ ( sum_sum @ t @ dtree ) ) @ ( cont @ Tr12 ) @ ( cont @ Tr22 ) ) ) )
% 3.58/1.93 @ ( image @ ( product_prod @ dtree @ dtree ) @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inr @ ( product_prod @ dtree @ dtree ) @ t )
% 3.58/1.93 @ ( product_Sigma @ dtree @ dtree @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr12 ) )
% 3.58/1.93 @ ^ [Uu: dtree] : ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr22 ) ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_133_par__c_Opelims,axiom,
% 3.58/1.93 ! [X: product_prod @ dtree @ dtree,Y5: set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) )] :
% 3.58/1.93 ( ( ( parall1914194347_par_c @ X )
% 3.58/1.93 = Y5 )
% 3.58/1.93 => ( ( accp @ ( product_prod @ dtree @ dtree ) @ parall1418634510_c_rel @ X )
% 3.58/1.93 => ~ ! [Tr1: dtree,Tr2: dtree] :
% 3.58/1.93 ( ( X
% 3.58/1.93 = ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 3.58/1.93 => ( ( Y5
% 3.58/1.93 = ( sup_sup @ ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) @ ( image @ t @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inl @ t @ ( product_prod @ dtree @ dtree ) ) @ ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( sup_sup @ ( set @ ( sum_sum @ t @ dtree ) ) @ ( cont @ Tr1 ) @ ( cont @ Tr2 ) ) ) )
% 3.58/1.93 @ ( image @ ( product_prod @ dtree @ dtree ) @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( sum_Inr @ ( product_prod @ dtree @ dtree ) @ t )
% 3.58/1.93 @ ( product_Sigma @ dtree @ dtree @ ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr1 ) )
% 3.58/1.93 @ ^ [Uu: dtree] : ( vimage @ dtree @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t ) @ ( cont @ Tr2 ) ) ) ) ) )
% 3.58/1.93 => ~ ( accp @ ( product_prod @ dtree @ dtree ) @ parall1418634510_c_rel @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_134_Plus__def,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType] :
% 3.58/1.93 ( ( sum_Plus @ A @ B )
% 3.58/1.93 = ( ^ [A6: set @ A,B5: set @ B] : ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A6 ) @ ( image @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B5 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_135_sup2CI,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,B3: A > B > $o,X: A,Y5: B,A3: A > B > $o] :
% 3.58/1.93 ( ( ~ ( B3 @ X @ Y5 )
% 3.58/1.93 => ( A3 @ X @ Y5 ) )
% 3.58/1.93 => ( sup_sup @ ( A > B > $o ) @ A3 @ B3 @ X @ Y5 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_136_sup1CI,axiom,
% 3.58/1.93 ! [A: $tType,B3: A > $o,X: A,A3: A > $o] :
% 3.58/1.93 ( ( ~ ( B3 @ X )
% 3.58/1.93 => ( A3 @ X ) )
% 3.58/1.93 => ( sup_sup @ ( A > $o ) @ A3 @ B3 @ X ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_137_InrI,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,B2: A,B3: set @ A,A3: set @ B] :
% 3.58/1.93 ( ( member @ A @ B2 @ B3 )
% 3.58/1.93 => ( member @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B @ B2 ) @ ( sum_Plus @ B @ A @ A3 @ B3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_138_InlI,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,A2: A,A3: set @ A,B3: set @ B] :
% 3.58/1.93 ( ( member @ A @ A2 @ A3 )
% 3.58/1.93 => ( member @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ A2 ) @ ( sum_Plus @ A @ B @ A3 @ B3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_139_sup1E,axiom,
% 3.58/1.93 ! [A: $tType,A3: A > $o,B3: A > $o,X: A] :
% 3.58/1.93 ( ( sup_sup @ ( A > $o ) @ A3 @ B3 @ X )
% 3.58/1.93 => ( ~ ( A3 @ X )
% 3.58/1.93 => ( B3 @ X ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_140_sup2E,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,A3: A > B > $o,B3: A > B > $o,X: A,Y5: B] :
% 3.58/1.93 ( ( sup_sup @ ( A > B > $o ) @ A3 @ B3 @ X @ Y5 )
% 3.58/1.93 => ( ~ ( A3 @ X @ Y5 )
% 3.58/1.93 => ( B3 @ X @ Y5 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_141_sup1I1,axiom,
% 3.58/1.93 ! [A: $tType,A3: A > $o,X: A,B3: A > $o] :
% 3.58/1.93 ( ( A3 @ X )
% 3.58/1.93 => ( sup_sup @ ( A > $o ) @ A3 @ B3 @ X ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_142_sup1I2,axiom,
% 3.58/1.93 ! [A: $tType,B3: A > $o,X: A,A3: A > $o] :
% 3.58/1.93 ( ( B3 @ X )
% 3.58/1.93 => ( sup_sup @ ( A > $o ) @ A3 @ B3 @ X ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_143_sup2I1,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,A3: A > B > $o,X: A,Y5: B,B3: A > B > $o] :
% 3.58/1.93 ( ( A3 @ X @ Y5 )
% 3.58/1.93 => ( sup_sup @ ( A > B > $o ) @ A3 @ B3 @ X @ Y5 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_144_sup2I2,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,B3: A > B > $o,X: A,Y5: B,A3: A > B > $o] :
% 3.58/1.93 ( ( B3 @ X @ Y5 )
% 3.58/1.93 => ( sup_sup @ ( A > B > $o ) @ A3 @ B3 @ X @ Y5 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_145_PlusE,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,U: sum_sum @ A @ B,A3: set @ A,B3: set @ B] :
% 3.58/1.93 ( ( member @ ( sum_sum @ A @ B ) @ U @ ( sum_Plus @ A @ B @ A3 @ B3 ) )
% 3.58/1.93 => ( ! [X3: A] :
% 3.58/1.93 ( ( member @ A @ X3 @ A3 )
% 3.58/1.93 => ( U
% 3.58/1.93 != ( sum_Inl @ A @ B @ X3 ) ) )
% 3.58/1.93 => ~ ! [Y3: B] :
% 3.58/1.93 ( ( member @ B @ Y3 @ B3 )
% 3.58/1.93 => ( U
% 3.58/1.93 != ( sum_Inr @ B @ A @ Y3 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_146_reflclp__idemp,axiom,
% 3.58/1.93 ! [A: $tType,P: A > A > $o] :
% 3.58/1.93 ( ( sup_sup @ ( A > A > $o )
% 3.58/1.93 @ ( sup_sup @ ( A > A > $o ) @ P
% 3.58/1.93 @ ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) )
% 3.58/1.93 @ ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) )
% 3.58/1.93 = ( sup_sup @ ( A > A > $o ) @ P
% 3.58/1.93 @ ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_147_accp_Ocases,axiom,
% 3.58/1.93 ! [A: $tType,R2: A > A > $o,A2: A] :
% 3.58/1.93 ( ( accp @ A @ R2 @ A2 )
% 3.58/1.93 => ! [Y7: A] :
% 3.58/1.93 ( ( R2 @ Y7 @ A2 )
% 3.58/1.93 => ( accp @ A @ R2 @ Y7 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_148_accp_Osimps,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( accp @ A )
% 3.58/1.93 = ( ^ [R3: A > A > $o,A9: A] :
% 3.58/1.93 ? [X2: A] :
% 3.58/1.93 ( ( A9 = X2 )
% 3.58/1.93 & ! [Y4: A] :
% 3.58/1.93 ( ( R3 @ Y4 @ X2 )
% 3.58/1.93 => ( accp @ A @ R3 @ Y4 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_149_accp_Ointros,axiom,
% 3.58/1.93 ! [A: $tType,R2: A > A > $o,X: A] :
% 3.58/1.93 ( ! [Y3: A] :
% 3.58/1.93 ( ( R2 @ Y3 @ X )
% 3.58/1.93 => ( accp @ A @ R2 @ Y3 ) )
% 3.58/1.93 => ( accp @ A @ R2 @ X ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_150_accp__induct__rule,axiom,
% 3.58/1.93 ! [A: $tType,R2: A > A > $o,A2: A,P: A > $o] :
% 3.58/1.93 ( ( accp @ A @ R2 @ A2 )
% 3.58/1.93 => ( ! [X3: A] :
% 3.58/1.93 ( ( accp @ A @ R2 @ X3 )
% 3.58/1.93 => ( ! [Y7: A] :
% 3.58/1.93 ( ( R2 @ Y7 @ X3 )
% 3.58/1.93 => ( P @ Y7 ) )
% 3.58/1.93 => ( P @ X3 ) ) )
% 3.58/1.93 => ( P @ A2 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_151_not__accp__down,axiom,
% 3.58/1.93 ! [A: $tType,R: A > A > $o,X: A] :
% 3.58/1.93 ( ~ ( accp @ A @ R @ X )
% 3.58/1.93 => ~ ! [Z3: A] :
% 3.58/1.93 ( ( R @ Z3 @ X )
% 3.58/1.93 => ( accp @ A @ R @ Z3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_152_accp__downward,axiom,
% 3.58/1.93 ! [A: $tType,R2: A > A > $o,B2: A,A2: A] :
% 3.58/1.93 ( ( accp @ A @ R2 @ B2 )
% 3.58/1.93 => ( ( R2 @ A2 @ B2 )
% 3.58/1.93 => ( accp @ A @ R2 @ A2 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_153_accp_Oinducts,axiom,
% 3.58/1.93 ! [A: $tType,R2: A > A > $o,X: A,P: A > $o] :
% 3.58/1.93 ( ( accp @ A @ R2 @ X )
% 3.58/1.93 => ( ! [X3: A] :
% 3.58/1.93 ( ! [Y7: A] :
% 3.58/1.93 ( ( R2 @ Y7 @ X3 )
% 3.58/1.93 => ( accp @ A @ R2 @ Y7 ) )
% 3.58/1.93 => ( ! [Y7: A] :
% 3.58/1.93 ( ( R2 @ Y7 @ X3 )
% 3.58/1.93 => ( P @ Y7 ) )
% 3.58/1.93 => ( P @ X3 ) ) )
% 3.58/1.93 => ( P @ X ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_154_accp__induct,axiom,
% 3.58/1.93 ! [A: $tType,R2: A > A > $o,A2: A,P: A > $o] :
% 3.58/1.93 ( ( accp @ A @ R2 @ A2 )
% 3.58/1.93 => ( ! [X3: A] :
% 3.58/1.93 ( ( accp @ A @ R2 @ X3 )
% 3.58/1.93 => ( ! [Y7: A] :
% 3.58/1.93 ( ( R2 @ Y7 @ X3 )
% 3.58/1.93 => ( P @ Y7 ) )
% 3.58/1.93 => ( P @ X3 ) ) )
% 3.58/1.93 => ( P @ A2 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_155_in__lex__prod,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B4: B,R2: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B )] :
% 3.58/1.93 ( ( 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 @ A4 @ B4 ) ) @ ( lex_prod @ A @ B @ R2 @ S2 ) )
% 3.58/1.93 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A4 ) @ R2 )
% 3.58/1.93 | ( ( A2 = A4 )
% 3.58/1.93 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B4 ) @ S2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_156_reflcl__set__eq,axiom,
% 3.58/1.93 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 3.58/1.93 ( ( sup_sup @ ( A > A > $o )
% 3.58/1.93 @ ^ [X2: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R2 )
% 3.58/1.93 @ ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) )
% 3.58/1.93 = ( ^ [X2: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id @ A ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_157_par__def,axiom,
% 3.58/1.93 ( parall1899940088le_par
% 3.58/1.93 = ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_158_IdI,axiom,
% 3.58/1.93 ! [A: $tType,A2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id @ A ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_159_pair__in__Id__conv,axiom,
% 3.58/1.93 ! [A: $tType,A2: A,B2: A] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id @ A ) )
% 3.58/1.93 = ( A2 = B2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_160_IdE,axiom,
% 3.58/1.93 ! [A: $tType,P2: product_prod @ A @ A] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ A ) @ P2 @ ( id @ A ) )
% 3.58/1.93 => ~ ! [X3: A] :
% 3.58/1.93 ( P2
% 3.58/1.93 != ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_161_IdD,axiom,
% 3.58/1.93 ! [A: $tType,A2: A,B2: A] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id @ A ) )
% 3.58/1.93 => ( A2 = B2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_162_same__fstI,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,P: A > $o,X: A,Y8: B,Y5: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
% 3.58/1.93 ( ( P @ X )
% 3.58/1.93 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y5 ) @ ( R @ X ) )
% 3.58/1.93 => ( 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 @ Y8 ) @ ( product_Pair @ A @ B @ X @ Y5 ) ) @ ( same_fst @ A @ B @ P @ R ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_163_antisym__reflcl,axiom,
% 3.58/1.93 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 3.58/1.93 ( ( antisym @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id @ A ) ) )
% 3.58/1.93 = ( antisym @ A @ R2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_164_pair__imageI,axiom,
% 3.58/1.93 ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set @ ( product_prod @ A @ B ),F: A > B > C] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ A3 )
% 3.58/1.93 => ( member @ C @ ( F @ A2 @ B2 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ A3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_165_case__prodI,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
% 3.58/1.93 ( ( F @ A2 @ B2 )
% 3.58/1.93 => ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_166_case__prodI2,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,C2: A > B > $o] :
% 3.58/1.93 ( ! [A5: A,B6: B] :
% 3.58/1.93 ( ( P2
% 3.58/1.93 = ( product_Pair @ A @ B @ A5 @ B6 ) )
% 3.58/1.93 => ( C2 @ A5 @ B6 ) )
% 3.58/1.93 => ( product_case_prod @ A @ B @ $o @ C2 @ P2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_167_mem__case__prodI,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),A2: B,B2: C] :
% 3.58/1.93 ( ( member @ A @ Z @ ( C2 @ A2 @ B2 ) )
% 3.58/1.93 => ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_168_mem__case__prodI2,axiom,
% 3.58/1.93 ! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z: C,C2: A > B > ( set @ C )] :
% 3.58/1.93 ( ! [A5: A,B6: B] :
% 3.58/1.93 ( ( P2
% 3.58/1.93 = ( product_Pair @ A @ B @ A5 @ B6 ) )
% 3.58/1.93 => ( member @ C @ Z @ ( C2 @ A5 @ B6 ) ) )
% 3.58/1.93 => ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P2 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_169_case__prodI2_H,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
% 3.58/1.93 ( ! [A5: A,B6: B] :
% 3.58/1.93 ( ( ( product_Pair @ A @ B @ A5 @ B6 )
% 3.58/1.93 = P2 )
% 3.58/1.93 => ( C2 @ A5 @ B6 @ X ) )
% 3.58/1.93 => ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P2 @ X ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_170_Collect__case__prod,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,P: A > $o,Q: B > $o] :
% 3.58/1.93 ( ( collect @ ( product_prod @ A @ B )
% 3.58/1.93 @ ( product_case_prod @ A @ B @ $o
% 3.58/1.93 @ ^ [A9: A,B9: B] :
% 3.58/1.93 ( ( P @ A9 )
% 3.58/1.93 & ( Q @ B9 ) ) ) )
% 3.58/1.93 = ( product_Sigma @ A @ B @ ( collect @ A @ P )
% 3.58/1.93 @ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_171_case__prod__conv,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B2: C] :
% 3.58/1.93 ( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 3.58/1.93 = ( F @ A2 @ B2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_172_lex__prod__def,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType] :
% 3.58/1.93 ( ( lex_prod @ A @ B )
% 3.58/1.93 = ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
% 3.58/1.93 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 3.58/1.93 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 3.58/1.93 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 3.58/1.93 @ ^ [A9: A,B9: B] :
% 3.58/1.93 ( product_case_prod @ A @ B @ $o
% 3.58/1.93 @ ^ [A10: A,B10: B] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A9 @ A10 ) @ Ra )
% 3.58/1.93 | ( ( A9 = A10 )
% 3.58/1.93 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B9 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_173_case__prodD_H,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A2: A,B2: B,C2: C] :
% 3.58/1.93 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 )
% 3.58/1.93 => ( R @ A2 @ B2 @ C2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_174_case__prodE_H,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P2: product_prod @ A @ B,Z: C] :
% 3.58/1.93 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P2 @ Z )
% 3.58/1.93 => ~ ! [X3: A,Y3: B] :
% 3.58/1.93 ( ( P2
% 3.58/1.93 = ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 3.58/1.93 => ~ ( C2 @ X3 @ Y3 @ Z ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_175_case__prodE2,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z: product_prod @ B @ C] :
% 3.58/1.93 ( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z ) )
% 3.58/1.93 => ~ ! [X3: B,Y3: C] :
% 3.58/1.93 ( ( Z
% 3.58/1.93 = ( product_Pair @ B @ C @ X3 @ Y3 ) )
% 3.58/1.93 => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_176_case__prod__eta,axiom,
% 3.58/1.93 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 3.58/1.93 ( ( product_case_prod @ A @ B @ C
% 3.58/1.93 @ ^ [X2: A,Y4: B] : ( F @ ( product_Pair @ A @ B @ X2 @ Y4 ) ) )
% 3.58/1.93 = F ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_177_cond__case__prod__eta,axiom,
% 3.58/1.93 ! [C: $tType,B: $tType,A: $tType,F: A > B > C,G3: ( product_prod @ A @ B ) > C] :
% 3.58/1.93 ( ! [X3: A,Y3: B] :
% 3.58/1.93 ( ( F @ X3 @ Y3 )
% 3.58/1.93 = ( G3 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
% 3.58/1.93 => ( ( product_case_prod @ A @ B @ C @ F )
% 3.58/1.93 = G3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_178_mem__case__prodE,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 3.58/1.93 ( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P2 ) )
% 3.58/1.93 => ~ ! [X3: B,Y3: C] :
% 3.58/1.93 ( ( P2
% 3.58/1.93 = ( product_Pair @ B @ C @ X3 @ Y3 ) )
% 3.58/1.93 => ~ ( member @ A @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_179_old_Oprod_Ocase,axiom,
% 3.58/1.93 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
% 3.58/1.93 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 3.58/1.93 = ( F @ X1 @ X22 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_180_same__fst__def,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType] :
% 3.58/1.93 ( ( same_fst @ A @ B )
% 3.58/1.93 = ( ^ [P4: A > $o,R4: A > ( set @ ( product_prod @ B @ B ) )] :
% 3.58/1.93 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 3.58/1.93 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 3.58/1.93 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 3.58/1.93 @ ^ [X6: A,Y9: B] :
% 3.58/1.93 ( product_case_prod @ A @ B @ $o
% 3.58/1.93 @ ^ [X2: A,Y4: B] :
% 3.58/1.93 ( ( X6 = X2 )
% 3.58/1.93 & ( P4 @ X2 )
% 3.58/1.93 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y9 @ Y4 ) @ ( R4 @ X2 ) ) ) ) ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_181_antisym__Id,axiom,
% 3.58/1.93 ! [A: $tType] : ( antisym @ A @ ( id @ A ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_182_antisymD,axiom,
% 3.58/1.93 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
% 3.58/1.93 ( ( antisym @ A @ R2 )
% 3.58/1.93 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
% 3.58/1.93 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A2 ) @ R2 )
% 3.58/1.93 => ( A2 = B2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_183_antisymI,axiom,
% 3.58/1.93 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 3.58/1.93 ( ! [X3: A,Y3: A] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
% 3.58/1.93 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R2 )
% 3.58/1.93 => ( X3 = Y3 ) ) )
% 3.58/1.93 => ( antisym @ A @ R2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_184_antisym__def,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( antisym @ A )
% 3.58/1.93 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 3.58/1.93 ! [X2: A,Y4: A] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R3 )
% 3.58/1.93 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ R3 )
% 3.58/1.93 => ( X2 = Y4 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_185_case__prod__Pair__iden,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 3.58/1.93 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
% 3.58/1.93 = P2 ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_186_case__prodD,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
% 3.58/1.93 ( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 3.58/1.93 => ( F @ A2 @ B2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_187_case__prodE,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,C2: A > B > $o,P2: product_prod @ A @ B] :
% 3.58/1.93 ( ( product_case_prod @ A @ B @ $o @ C2 @ P2 )
% 3.58/1.93 => ~ ! [X3: A,Y3: B] :
% 3.58/1.93 ( ( P2
% 3.58/1.93 = ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 3.58/1.93 => ~ ( C2 @ X3 @ Y3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_188_Collect__case__prod__Sigma,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] :
% 3.58/1.93 ( ( collect @ ( product_prod @ A @ B )
% 3.58/1.93 @ ( product_case_prod @ A @ B @ $o
% 3.58/1.93 @ ^ [X2: A,Y4: B] :
% 3.58/1.93 ( ( P @ X2 )
% 3.58/1.93 & ( Q @ X2 @ Y4 ) ) ) )
% 3.58/1.93 = ( product_Sigma @ A @ B @ ( collect @ A @ P )
% 3.58/1.93 @ ^ [X2: A] : ( collect @ B @ ( Q @ X2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_189_case__prod__app,axiom,
% 3.58/1.93 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 3.58/1.93 ( ( product_case_prod @ B @ C @ ( D > A ) )
% 3.58/1.93 = ( ^ [F2: B > C > D > A,X2: product_prod @ B @ C,Y4: D] :
% 3.58/1.93 ( product_case_prod @ B @ C @ A
% 3.58/1.93 @ ^ [L: B,R3: C] : ( F2 @ L @ R3 @ Y4 )
% 3.58/1.93 @ X2 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_190_prod_Ocase__distrib,axiom,
% 3.58/1.93 ! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
% 3.58/1.93 ( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 3.58/1.93 = ( product_case_prod @ A @ B @ D
% 3.58/1.93 @ ^ [X12: A,X23: B] : ( H @ ( F @ X12 @ X23 ) )
% 3.58/1.93 @ Prod ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_191_antisymP__equality,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( antisym @ A
% 3.58/1.93 @ ( collect @ ( product_prod @ A @ A )
% 3.58/1.93 @ ( product_case_prod @ A @ A @ $o
% 3.58/1.93 @ ^ [Y6: A,Z2: A] : ( Y6 = Z2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_192_internal__case__prod__def,axiom,
% 3.58/1.93 ! [C: $tType,B: $tType,A: $tType] :
% 3.58/1.93 ( ( produc2004651681e_prod @ A @ B @ C )
% 3.58/1.93 = ( product_case_prod @ A @ B @ C ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_193_swap__product,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,A3: set @ B,B3: set @ A] :
% 3.58/1.93 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
% 3.58/1.93 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
% 3.58/1.93 @ ^ [I2: B,J2: A] : ( product_Pair @ A @ B @ J2 @ I2 ) )
% 3.58/1.93 @ ( product_Sigma @ B @ A @ A3
% 3.58/1.93 @ ^ [Uu: B] : B3 ) )
% 3.58/1.93 = ( product_Sigma @ A @ B @ B3
% 3.58/1.93 @ ^ [Uu: A] : A3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_194_split__cong,axiom,
% 3.58/1.93 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
% 3.58/1.93 ( ! [X3: A,Y3: B] :
% 3.58/1.93 ( ( ( product_Pair @ A @ B @ X3 @ Y3 )
% 3.58/1.93 = Q2 )
% 3.58/1.93 => ( ( F @ X3 @ Y3 )
% 3.58/1.93 = ( G3 @ X3 @ Y3 ) ) )
% 3.58/1.93 => ( ( P2 = Q2 )
% 3.58/1.93 => ( ( product_case_prod @ A @ B @ C @ F @ P2 )
% 3.58/1.93 = ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_195_case__swap,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,C: $tType,F: C > B > A,P2: product_prod @ C @ B] :
% 3.58/1.93 ( ( product_case_prod @ B @ C @ A
% 3.58/1.93 @ ^ [Y4: B,X2: C] : ( F @ X2 @ Y4 )
% 3.58/1.93 @ ( product_swap @ C @ B @ P2 ) )
% 3.58/1.93 = ( product_case_prod @ C @ B @ A @ F @ P2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_196_case__prod__curry,axiom,
% 3.58/1.93 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 3.58/1.93 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 3.58/1.93 = F ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_197_curryI,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 3.58/1.93 ( ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 3.58/1.93 => ( product_curry @ A @ B @ $o @ F @ A2 @ B2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_198_swap__swap,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 3.58/1.93 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 3.58/1.93 = P2 ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_199_split__part,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
% 3.58/1.93 ( ( product_case_prod @ A @ B @ $o
% 3.58/1.93 @ ^ [A9: A,B9: B] :
% 3.58/1.93 ( P
% 3.58/1.93 & ( Q @ A9 @ B9 ) ) )
% 3.58/1.93 = ( ^ [Ab: product_prod @ A @ B] :
% 3.58/1.93 ( P
% 3.58/1.93 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_200_swap__simp,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,X: B,Y5: A] :
% 3.58/1.93 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y5 ) )
% 3.58/1.93 = ( product_Pair @ A @ B @ Y5 @ X ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_201_curry__conv,axiom,
% 3.58/1.93 ! [A: $tType,C: $tType,B: $tType] :
% 3.58/1.93 ( ( product_curry @ B @ C @ A )
% 3.58/1.93 = ( ^ [F2: ( product_prod @ B @ C ) > A,A9: B,B9: C] : ( F2 @ ( product_Pair @ B @ C @ A9 @ B9 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_202_curry__case__prod,axiom,
% 3.58/1.93 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 3.58/1.93 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
% 3.58/1.93 = F ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_203_pair__in__swap__image,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,Y5: A,X: B,A3: set @ ( product_prod @ B @ A )] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y5 @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A3 ) )
% 3.58/1.93 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y5 ) @ A3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_204_prod_Odisc__eq__case,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 3.58/1.93 ( product_case_prod @ A @ B @ $o
% 3.58/1.93 @ ^ [Uu: A,Uv: B] : $true
% 3.58/1.93 @ Prod ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_205_curry__K,axiom,
% 3.58/1.93 ! [B: $tType,C: $tType,A: $tType,C2: C] :
% 3.58/1.93 ( ( product_curry @ A @ B @ C
% 3.58/1.93 @ ^ [X2: product_prod @ A @ B] : C2 )
% 3.58/1.93 = ( ^ [X2: A,Y4: B] : C2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_206_curryE,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 3.58/1.93 ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
% 3.58/1.93 => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_207_curryD,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
% 3.58/1.93 ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
% 3.58/1.93 => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_208_product__swap,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,A3: set @ B,B3: set @ A] :
% 3.58/1.93 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
% 3.58/1.93 @ ( product_Sigma @ B @ A @ A3
% 3.58/1.93 @ ^ [Uu: B] : B3 ) )
% 3.58/1.93 = ( product_Sigma @ A @ B @ B3
% 3.58/1.93 @ ^ [Uu: A] : A3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_209_curry__def,axiom,
% 3.58/1.93 ! [C: $tType,B: $tType,A: $tType] :
% 3.58/1.93 ( ( product_curry @ A @ B @ C )
% 3.58/1.93 = ( ^ [C5: ( product_prod @ A @ B ) > C,X2: A,Y4: B] : ( C5 @ ( product_Pair @ A @ B @ X2 @ Y4 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_210_inv__image__def,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType] :
% 3.58/1.93 ( ( inv_image @ B @ A )
% 3.58/1.93 = ( ^ [R3: set @ ( product_prod @ B @ B ),F2: A > B] :
% 3.58/1.93 ( collect @ ( product_prod @ A @ A )
% 3.58/1.93 @ ( product_case_prod @ A @ A @ $o
% 3.58/1.93 @ ^ [X2: A,Y4: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) @ R3 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_211_map__prod__surj__on,axiom,
% 3.58/1.93 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: B > A,A3: set @ B,A11: set @ A,G3: D > C,B3: set @ D,B11: set @ C] :
% 3.58/1.93 ( ( ( image @ B @ A @ F @ A3 )
% 3.58/1.93 = A11 )
% 3.58/1.93 => ( ( ( image @ D @ C @ G3 @ B3 )
% 3.58/1.93 = B11 )
% 3.58/1.93 => ( ( image @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F @ G3 )
% 3.58/1.93 @ ( product_Sigma @ B @ D @ A3
% 3.58/1.93 @ ^ [Uu: B] : B3 ) )
% 3.58/1.93 = ( product_Sigma @ A @ C @ A11
% 3.58/1.93 @ ^ [Uu: A] : B11 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_212_map__prod__ident,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType] :
% 3.58/1.93 ( ( product_map_prod @ A @ A @ B @ B
% 3.58/1.93 @ ^ [X2: A] : X2
% 3.58/1.93 @ ^ [Y4: B] : Y4 )
% 3.58/1.93 = ( ^ [Z4: product_prod @ A @ B] : Z4 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_213_map__prod__simp,axiom,
% 3.58/1.93 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G3: D > B,A2: C,B2: D] :
% 3.58/1.93 ( ( product_map_prod @ C @ A @ D @ B @ F @ G3 @ ( product_Pair @ C @ D @ A2 @ B2 ) )
% 3.58/1.93 = ( product_Pair @ A @ B @ ( F @ A2 ) @ ( G3 @ B2 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_214_in__inv__image,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,X: A,Y5: A,R2: set @ ( product_prod @ B @ B ),F: A > B] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ ( inv_image @ B @ A @ R2 @ F ) )
% 3.58/1.93 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y5 ) ) @ R2 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_215_map__prod__imageI,axiom,
% 3.58/1.93 ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R: set @ ( product_prod @ A @ B ),F: A > C,G3: B > D] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R )
% 3.58/1.93 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A2 ) @ ( G3 @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G3 ) @ R ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_216_case__prod__map__prod,axiom,
% 3.58/1.93 ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F: D > B,G3: E > C,X: product_prod @ D @ E] :
% 3.58/1.93 ( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F @ G3 @ X ) )
% 3.58/1.93 = ( product_case_prod @ D @ E @ A
% 3.58/1.93 @ ^ [L: D,R3: E] : ( H @ ( F @ L ) @ ( G3 @ R3 ) )
% 3.58/1.93 @ X ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_217_prod__fun__imageE,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F: C > A,G3: D > B,R: set @ ( product_prod @ C @ D )] :
% 3.58/1.93 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G3 ) @ R ) )
% 3.58/1.93 => ~ ! [X3: C,Y3: D] :
% 3.58/1.93 ( ( C2
% 3.58/1.93 = ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G3 @ Y3 ) ) )
% 3.58/1.93 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_218_map__prod__def,axiom,
% 3.58/1.93 ! [B: $tType,D: $tType,C: $tType,A: $tType] :
% 3.58/1.93 ( ( product_map_prod @ A @ C @ B @ D )
% 3.58/1.93 = ( ^ [F2: A > C,G4: B > D] :
% 3.58/1.93 ( product_case_prod @ A @ B @ ( product_prod @ C @ D )
% 3.58/1.93 @ ^ [X2: A,Y4: B] : ( product_Pair @ C @ D @ ( F2 @ X2 ) @ ( G4 @ Y4 ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_219_rp__inv__image__def,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType] :
% 3.58/1.93 ( ( fun_rp_inv_image @ A @ B )
% 3.58/1.93 = ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
% 3.58/1.93 @ ^ [R4: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A ),F2: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R4 @ F2 ) @ ( inv_image @ A @ B @ S3 @ F2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_220_prod_Omap__ident,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 3.58/1.93 ( ( product_map_prod @ A @ A @ B @ B
% 3.58/1.93 @ ^ [X2: A] : X2
% 3.58/1.93 @ ^ [X2: B] : X2
% 3.58/1.93 @ T2 )
% 3.58/1.93 = T2 ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_221_surj__swap,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType] :
% 3.58/1.93 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 3.58/1.93 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_222_The__split__eq,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,X: A,Y5: B] :
% 3.58/1.93 ( ( the @ ( product_prod @ A @ B )
% 3.58/1.93 @ ( product_case_prod @ A @ B @ $o
% 3.58/1.93 @ ^ [X6: A,Y9: B] :
% 3.58/1.93 ( ( X = X6 )
% 3.58/1.93 & ( Y5 = Y9 ) ) ) )
% 3.58/1.93 = ( product_Pair @ A @ B @ X @ Y5 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_223_UNIV__I,axiom,
% 3.58/1.93 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_224_sup__top__right,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 3.58/1.93 => ! [X: A] :
% 3.58/1.93 ( ( sup_sup @ A @ X @ ( top_top @ A ) )
% 3.58/1.93 = ( top_top @ A ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_225_sup__top__left,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 3.58/1.93 => ! [X: A] :
% 3.58/1.93 ( ( sup_sup @ A @ ( top_top @ A ) @ X )
% 3.58/1.93 = ( top_top @ A ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_226_vimage__UNIV,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType,F: A > B] :
% 3.58/1.93 ( ( vimage @ A @ B @ F @ ( top_top @ ( set @ B ) ) )
% 3.58/1.93 = ( top_top @ ( set @ A ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_227_UNIV__Plus__UNIV,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType] :
% 3.58/1.93 ( ( sum_Plus @ A @ B @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.58/1.93 = ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_228_UNIV__Times__UNIV,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType] :
% 3.58/1.93 ( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 3.58/1.93 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
% 3.58/1.93 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_229_UNIV__def,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( top_top @ ( set @ A ) )
% 3.58/1.93 = ( collect @ A
% 3.58/1.93 @ ^ [X2: A] : $true ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_230_rangeE,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,B2: A,F: B > A] :
% 3.58/1.93 ( ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
% 3.58/1.93 => ~ ! [X3: B] :
% 3.58/1.93 ( B2
% 3.58/1.93 != ( F @ X3 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_231_range__composition,axiom,
% 3.58/1.93 ! [A: $tType,C: $tType,B: $tType,F: C > A,G3: B > C] :
% 3.58/1.93 ( ( image @ B @ A
% 3.58/1.93 @ ^ [X2: B] : ( F @ ( G3 @ X2 ) )
% 3.58/1.93 @ ( top_top @ ( set @ B ) ) )
% 3.58/1.93 = ( image @ C @ A @ F @ ( image @ B @ C @ G3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_232_rangeI,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_233_range__eqI,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType,B2: A,F: B > A,X: B] :
% 3.58/1.93 ( ( B2
% 3.58/1.93 = ( F @ X ) )
% 3.58/1.93 => ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_234_Un__UNIV__left,axiom,
% 3.58/1.93 ! [A: $tType,B3: set @ A] :
% 3.58/1.93 ( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 )
% 3.58/1.93 = ( top_top @ ( set @ A ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_235_Un__UNIV__right,axiom,
% 3.58/1.93 ! [A: $tType,A3: set @ A] :
% 3.58/1.93 ( ( sup_sup @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
% 3.58/1.93 = ( top_top @ ( set @ A ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_236_UNIV__eq__I,axiom,
% 3.58/1.93 ! [A: $tType,A3: set @ A] :
% 3.58/1.93 ( ! [X3: A] : ( member @ A @ X3 @ A3 )
% 3.58/1.93 => ( ( top_top @ ( set @ A ) )
% 3.58/1.93 = A3 ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_237_UNIV__witness,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_238_UNIV__sum,axiom,
% 3.58/1.93 ! [A: $tType,B: $tType] :
% 3.58/1.93 ( ( top_top @ ( set @ ( sum_sum @ A @ B ) ) )
% 3.58/1.93 = ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ ( top_top @ ( set @ A ) ) ) @ ( image @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_239_map__prod__surj,axiom,
% 3.58/1.93 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G3: C > D] :
% 3.58/1.93 ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 3.58/1.93 = ( top_top @ ( set @ B ) ) )
% 3.58/1.93 => ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
% 3.58/1.93 = ( top_top @ ( set @ D ) ) )
% 3.58/1.93 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 3.58/1.93 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_240_old_Orec__sum__def,axiom,
% 3.58/1.93 ! [B: $tType,T: $tType,A: $tType] :
% 3.58/1.93 ( ( sum_rec_sum @ A @ T @ B )
% 3.58/1.93 = ( ^ [F12: A > T,F23: B > T,X2: sum_sum @ A @ B] : ( the @ T @ ( sum_rec_set_sum @ A @ T @ B @ F12 @ F23 @ X2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_241_old_Orec__prod__def,axiom,
% 3.58/1.93 ! [T: $tType,B: $tType,A: $tType] :
% 3.58/1.93 ( ( product_rec_prod @ A @ B @ T )
% 3.58/1.93 = ( ^ [F12: A > B > T,X2: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X2 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_242_top__empty__eq2,axiom,
% 3.58/1.93 ! [B: $tType,A: $tType] :
% 3.58/1.93 ( ( top_top @ ( A > B > $o ) )
% 3.58/1.93 = ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_243_top__set__def,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( top_top @ ( set @ A ) )
% 3.58/1.93 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_244_top__empty__eq,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( top_top @ ( A > $o ) )
% 3.58/1.93 = ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_245_the__sym__eq__trivial,axiom,
% 3.58/1.93 ! [A: $tType,X: A] :
% 3.58/1.93 ( ( the @ A
% 3.58/1.93 @ ( ^ [Y6: A,Z2: A] : ( Y6 = Z2 )
% 3.58/1.93 @ X ) )
% 3.58/1.93 = X ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_246_the__eq__trivial,axiom,
% 3.58/1.93 ! [A: $tType,A2: A] :
% 3.58/1.93 ( ( the @ A
% 3.58/1.93 @ ^ [X2: A] : ( X2 = A2 ) )
% 3.58/1.93 = A2 ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_247_the__equality,axiom,
% 3.58/1.93 ! [A: $tType,P: A > $o,A2: A] :
% 3.58/1.93 ( ( P @ A2 )
% 3.58/1.93 => ( ! [X3: A] :
% 3.58/1.93 ( ( P @ X3 )
% 3.58/1.93 => ( X3 = A2 ) )
% 3.58/1.93 => ( ( the @ A @ P )
% 3.58/1.93 = A2 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_248_top__conj_I2_J,axiom,
% 3.58/1.93 ! [A: $tType,P: $o,X: A] :
% 3.58/1.93 ( ( P
% 3.58/1.93 & ( top_top @ ( A > $o ) @ X ) )
% 3.58/1.93 = P ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_249_top__conj_I1_J,axiom,
% 3.58/1.93 ! [A: $tType,X: A,P: $o] :
% 3.58/1.93 ( ( ( top_top @ ( A > $o ) @ X )
% 3.58/1.93 & P )
% 3.58/1.93 = P ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_250_theI,axiom,
% 3.58/1.93 ! [A: $tType,P: A > $o,A2: A] :
% 3.58/1.93 ( ( P @ A2 )
% 3.58/1.93 => ( ! [X3: A] :
% 3.58/1.93 ( ( P @ X3 )
% 3.58/1.93 => ( X3 = A2 ) )
% 3.58/1.93 => ( P @ ( the @ A @ P ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_251_theI_H,axiom,
% 3.58/1.93 ! [A: $tType,P: A > $o] :
% 3.58/1.93 ( ? [X4: A] :
% 3.58/1.93 ( ( P @ X4 )
% 3.58/1.93 & ! [Y3: A] :
% 3.58/1.93 ( ( P @ Y3 )
% 3.58/1.93 => ( Y3 = X4 ) ) )
% 3.58/1.93 => ( P @ ( the @ A @ P ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_252_theI2,axiom,
% 3.58/1.93 ! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
% 3.58/1.93 ( ( P @ A2 )
% 3.58/1.93 => ( ! [X3: A] :
% 3.58/1.93 ( ( P @ X3 )
% 3.58/1.93 => ( X3 = A2 ) )
% 3.58/1.93 => ( ! [X3: A] :
% 3.58/1.93 ( ( P @ X3 )
% 3.58/1.93 => ( Q @ X3 ) )
% 3.58/1.93 => ( Q @ ( the @ A @ P ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_253_If__def,axiom,
% 3.58/1.93 ! [A: $tType] :
% 3.58/1.93 ( ( if @ A )
% 3.58/1.93 = ( ^ [P4: $o,X2: A,Y4: A] :
% 3.58/1.93 ( the @ A
% 3.58/1.93 @ ^ [Z4: A] :
% 3.58/1.93 ( ( P4
% 3.58/1.93 => ( Z4 = X2 ) )
% 3.58/1.93 & ( ~ P4
% 3.58/1.93 => ( Z4 = Y4 ) ) ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_254_the1I2,axiom,
% 3.58/1.93 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.58/1.93 ( ? [X4: A] :
% 3.58/1.93 ( ( P @ X4 )
% 3.58/1.93 & ! [Y3: A] :
% 3.58/1.93 ( ( P @ Y3 )
% 3.58/1.93 => ( Y3 = X4 ) ) )
% 3.58/1.93 => ( ! [X3: A] :
% 3.58/1.93 ( ( P @ X3 )
% 3.58/1.93 => ( Q @ X3 ) )
% 3.58/1.93 => ( Q @ ( the @ A @ P ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(fact_255_the1__equality,axiom,
% 3.58/1.93 ! [A: $tType,P: A > $o,A2: A] :
% 3.58/1.93 ( ? [X4: A] :
% 3.58/1.93 ( ( P @ X4 )
% 3.58/1.93 & ! [Y3: A] :
% 3.58/1.93 ( ( P @ Y3 )
% 3.58/1.93 => ( Y3 = X4 ) ) )
% 3.58/1.93 => ( ( P @ A2 )
% 3.58/1.93 => ( ( the @ A @ P )
% 3.58/1.93 = A2 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 3.58/1.93 bounded_lattice @ $o @ ( type2 @ $o ) ).
% 3.58/1.93
% 3.58/1.93 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 3.58/1.93 ! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 3.58/1.93 ! [A7: $tType,A12: $tType] :
% 3.58/1.93 ( ( bounded_lattice @ A12 @ ( type2 @ A12 ) )
% 3.58/1.93 => ( bounded_lattice @ ( A7 > A12 ) @ ( type2 @ ( A7 > A12 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
% 3.58/1.93 ! [A7: $tType,A12: $tType] :
% 3.58/1.93 ( ( bounded_lattice @ A12 @ ( type2 @ A12 ) )
% 3.58/1.93 => ( bounded_lattice_top @ ( A7 > A12 ) @ ( type2 @ ( A7 > A12 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 3.58/1.93 ! [A7: $tType,A12: $tType] :
% 3.58/1.93 ( ( semilattice_sup @ A12 @ ( type2 @ A12 ) )
% 3.58/1.93 => ( semilattice_sup @ ( A7 > A12 ) @ ( type2 @ ( A7 > A12 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(tcon_fun___Lattices_Olattice,axiom,
% 3.58/1.93 ! [A7: $tType,A12: $tType] :
% 3.58/1.93 ( ( lattice @ A12 @ ( type2 @ A12 ) )
% 3.58/1.93 => ( lattice @ ( A7 > A12 ) @ ( type2 @ ( A7 > A12 ) ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_3,axiom,
% 3.58/1.93 ! [A7: $tType] : ( bounded_lattice_top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_4,axiom,
% 3.58/1.93 ! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(tcon_Set_Oset___Lattices_Olattice_5,axiom,
% 3.58/1.93 ! [A7: $tType] : ( lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_6,axiom,
% 3.58/1.93 bounded_lattice_top @ $o @ ( type2 @ $o ) ).
% 3.58/1.93
% 3.58/1.93 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_7,axiom,
% 3.58/1.93 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 3.58/1.93
% 3.58/1.93 thf(tcon_HOL_Obool___Lattices_Olattice_8,axiom,
% 3.58/1.93 lattice @ $o @ ( type2 @ $o ) ).
% 3.58/1.93
% 3.58/1.93 thf(help_If_3_1_T,axiom,
% 3.58/1.93 ! [P: $o] :
% 3.58/1.93 ( ( P = $true )
% 3.58/1.93 | ( P = $false ) ) ).
% 3.58/1.93
% 3.58/1.93 thf(help_If_2_1_T,axiom,
% 3.58/1.93 ! [A: $tType,X: A,Y5: A] :
% 3.58/1.93 ( ( if @ A @ $false @ X @ Y5 )
% 3.58/1.93 = Y5 ) ).
% 3.58/1.93
% 3.58/1.93 thf(help_If_1_1_T,axiom,
% 3.58/1.93 ! [A: $tType,X: A,Y5: A] :
% 3.58/1.93 ( ( if @ A @ $true @ X @ Y5 )
% 3.58/1.93 = X ) ).
% 3.58/1.93
% 3.58/1.93 thf(conj_0,conjecture,
% 3.58/1.93 ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ t2 ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) ) )
% 3.58/1.93 = ( member @ dtree @ t2
% 3.58/1.93 @ ( image @ ( product_prod @ dtree @ dtree ) @ dtree @ parall1899940088le_par
% 3.58/1.93 @ ( product_Sigma @ dtree @ dtree
% 3.58/1.93 @ ( collect @ dtree
% 3.58/1.93 @ ^ [X2: dtree] : ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ X2 ) @ ( cont @ tr1 ) ) )
% 3.58/1.93 @ ^ [Uu: dtree] :
% 3.58/1.93 ( collect @ dtree
% 3.58/1.93 @ ^ [X2: dtree] : ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ X2 ) @ ( cont @ tr2 ) ) ) ) ) ) ) ).
% 3.58/1.93 ------------------------
% 3.58/1.97 % (15804)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_15689 for (2999ds/2Mi)
% 3.58/1.97 % (15801)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_15689 for (2999ds/183Mi)
% 3.58/1.97 % (15803)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_15689 for (2999ds/27Mi)
% 3.58/1.97 % (15802)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_15689 for (2999ds/4Mi)
% 3.58/1.97 % (15805)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_15689 for (2999ds/2Mi)
% 3.58/1.97 % (15807)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_15689 for (2999ds/18Mi)
% 3.58/1.97 % (15806)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_15689 for (2999ds/275Mi)
% 3.58/1.97 % (15804)Instruction limit reached!
% 3.58/1.97 % (15804)------------------------------
% 3.58/1.97 % (15804)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.58/1.97 % (15804)Termination reason: Unknown
% 3.58/1.97 % (15804)Termination phase: shuffling
% 3.58/1.97
% 3.58/1.97 % (15805)Instruction limit reached!
% 3.58/1.97 % (15805)------------------------------
% 3.58/1.97 % (15805)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.58/1.97 % (15805)Termination reason: Unknown
% 3.58/1.97 % (15805)Termination phase: shuffling
% 3.58/1.97
% 3.58/1.97 % (15805)Memory used [KB]: 1407
% 3.58/1.97 % (15805)Time elapsed: 0.003 s
% 3.58/1.97 % (15805)Instructions burned: 3 (million)
% 3.58/1.97 % (15805)------------------------------
% 3.58/1.97 % (15805)------------------------------
% 3.58/1.97 % (15804)Memory used [KB]: 1407
% 3.58/1.97 % (15804)Time elapsed: 0.003 s
% 3.58/1.97 % (15804)Instructions burned: 3 (million)
% 3.58/1.97 % (15804)------------------------------
% 3.58/1.97 % (15804)------------------------------
% 3.58/1.97 % (15802)Instruction limit reached!
% 3.58/1.97 % (15802)------------------------------
% 3.58/1.97 % (15802)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.58/1.97 % (15802)Termination reason: Unknown
% 3.58/1.97 % (15802)Termination phase: shuffling
% 3.58/1.97
% 3.58/1.97 % (15802)Memory used [KB]: 1407
% 3.58/1.97 % (15802)Time elapsed: 0.004 s
% 3.58/1.97 % (15802)Instructions burned: 4 (million)
% 3.58/1.97 % (15802)------------------------------
% 3.58/1.97 % (15802)------------------------------
% 3.58/1.98 % (15807)Instruction limit reached!
% 3.58/1.98 % (15807)------------------------------
% 3.58/1.98 % (15807)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.58/1.98 % (15807)Termination reason: Unknown
% 3.58/1.98 % (15807)Termination phase: shuffling
% 3.58/1.98
% 3.58/1.98 % (15807)Memory used [KB]: 1791
% 3.58/1.98 % (15807)Time elapsed: 0.011 s
% 3.58/1.98 % (15807)Instructions burned: 18 (million)
% 3.58/1.98 % (15807)------------------------------
% 3.58/1.98 % (15807)------------------------------
% 3.58/1.98 % (15803)Instruction limit reached!
% 3.58/1.98 % (15803)------------------------------
% 3.58/1.98 % (15803)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.58/1.98 % (15803)Termination reason: Unknown
% 3.58/1.98 % (15803)Termination phase: Property scanning
% 3.58/1.98
% 3.58/1.98 % (15803)Memory used [KB]: 1918
% 3.58/1.98 % (15803)Time elapsed: 0.016 s
% 3.58/1.98 % (15803)Instructions burned: 28 (million)
% 3.58/1.98 % (15803)------------------------------
% 3.58/1.98 % (15803)------------------------------
% 3.58/1.99 % (15808)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_15689 for (2999ds/3Mi)
% 3.58/1.99 % (15809)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_15689 for (2999ds/37Mi)
% 3.58/1.99 % (15810)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_15689 for (2999ds/15Mi)
% 3.58/1.99 % (15808)Instruction limit reached!
% 3.58/1.99 % (15808)------------------------------
% 3.58/1.99 % (15808)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.58/1.99 % (15808)Termination reason: Unknown
% 3.58/1.99 % (15808)Termination phase: shuffling
% 3.58/1.99
% 3.58/1.99 % (15808)Memory used [KB]: 1407
% 3.58/1.99 % (15808)Time elapsed: 0.003 s
% 3.58/1.99 % (15808)Instructions burned: 3 (million)
% 3.58/1.99 % (15808)------------------------------
% 3.58/1.99 % (15808)------------------------------
% 3.58/1.99 % (15811)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_15689 for (2999ds/3Mi)
% 3.58/1.99 % (15810)Instruction limit reached!
% 3.58/1.99 % (15810)------------------------------
% 3.58/1.99 % (15810)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.58/1.99 % (15810)Termination reason: Unknown
% 3.58/1.99 % (15810)Termination phase: shuffling
% 3.58/1.99
% 3.58/1.99 % (15810)Memory used [KB]: 1791
% 3.58/2.00 % (15810)Time elapsed: 0.010 s
% 3.58/2.00 % (15810)Instructions burned: 16 (million)
% 3.58/2.00 % (15810)------------------------------
% 3.58/2.00 % (15810)------------------------------
% 3.58/2.00 % (15811)Instruction limit reached!
% 3.58/2.00 % (15811)------------------------------
% 3.58/2.00 % (15811)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.58/2.00 % (15811)Termination reason: Unknown
% 3.58/2.00 % (15811)Termination phase: shuffling
% 3.58/2.00
% 3.58/2.00 % (15811)Memory used [KB]: 1407
% 3.58/2.00 % (15811)Time elapsed: 0.003 s
% 3.58/2.00 % (15811)Instructions burned: 3 (million)
% 3.58/2.00 % (15811)------------------------------
% 3.58/2.00 % (15811)------------------------------
% 3.58/2.00 % (15812)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_15689 for (2999ds/1041Mi)
% 3.58/2.00 % (15813)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_15689 for (2999ds/7Mi)
% 3.58/2.00 % (15809)Instruction limit reached!
% 3.58/2.00 % (15809)------------------------------
% 3.58/2.00 % (15809)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.58/2.00 % (15809)Termination reason: Unknown
% 3.58/2.00 % (15809)Termination phase: shuffling
% 3.58/2.00
% 3.58/2.00 % (15809)Memory used [KB]: 2174
% 3.58/2.00 % (15809)Time elapsed: 0.020 s
% 3.58/2.00 % (15809)Instructions burned: 39 (million)
% 3.58/2.00 % (15809)------------------------------
% 3.58/2.00 % (15809)------------------------------
% 3.58/2.01 % (15813)Instruction limit reached!
% 3.58/2.01 % (15813)------------------------------
% 3.58/2.01 % (15813)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.58/2.01 % (15813)Termination reason: Unknown
% 3.58/2.01 % (15813)Termination phase: shuffling
% 3.58/2.01
% 3.58/2.01 % (15813)Memory used [KB]: 1535
% 3.58/2.01 % (15813)Time elapsed: 0.005 s
% 3.58/2.01 % (15813)Instructions burned: 7 (million)
% 3.58/2.01 % (15813)------------------------------
% 3.58/2.01 % (15813)------------------------------
% 3.58/2.01 % (15814)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_15689 for (2999ds/16Mi)
% 3.58/2.01 % (15815)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_15689 for (2999ds/3Mi)
% 3.58/2.01 % (15815)Instruction limit reached!
% 3.58/2.01 % (15815)------------------------------
% 3.58/2.01 % (15815)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.58/2.01 % (15815)Termination reason: Unknown
% 3.58/2.01 % (15815)Termination phase: shuffling
% 3.58/2.01
% 3.58/2.01 % (15815)Memory used [KB]: 1407
% 3.58/2.01 % (15815)Time elapsed: 0.003 s
% 3.58/2.01 % (15815)Instructions burned: 3 (million)
% 3.58/2.01 % (15815)------------------------------
% 3.58/2.01 % (15815)------------------------------
% 3.58/2.02 % (15814)Instruction limit reached!
% 3.58/2.02 % (15814)------------------------------
% 3.58/2.02 % (15814)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.58/2.02 % (15814)Termination reason: Unknown
% 3.58/2.02 % (15814)Termination phase: shuffling
% 3.58/2.02
% 3.58/2.02 % (15814)Memory used [KB]: 1791
% 3.58/2.02 % (15814)Time elapsed: 0.010 s
% 3.58/2.02 % (15814)Instructions burned: 16 (million)
% 3.58/2.02 % (15814)------------------------------
% 3.58/2.02 % (15814)------------------------------
% 3.58/2.02 % (15816)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_15689 for (2999ds/3Mi)
% 3.58/2.02 % (15816)Instruction limit reached!
% 3.58/2.02 % (15816)------------------------------
% 3.58/2.02 % (15816)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.58/2.02 % (15816)Termination reason: Unknown
% 3.58/2.02 % (15816)Termination phase: shuffling
% 3.58/2.02
% 3.58/2.02 % (15816)Memory used [KB]: 1407
% 3.58/2.02 % (15816)Time elapsed: 0.003 s
% 3.58/2.02 % (15816)Instructions burned: 3 (million)
% 3.58/2.02 % (15816)------------------------------
% 3.58/2.02 % (15816)------------------------------
% 3.58/2.02 % (15817)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_15689 for (2999ds/7Mi)
% 4.07/2.03 % (15817)Instruction limit reached!
% 4.07/2.03 % (15817)------------------------------
% 4.07/2.03 % (15817)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.03 % (15817)Termination reason: Unknown
% 4.07/2.03 % (15817)Termination phase: shuffling
% 4.07/2.03
% 4.07/2.03 % (15817)Memory used [KB]: 1535
% 4.07/2.03 % (15817)Time elapsed: 0.005 s
% 4.07/2.03 % (15817)Instructions burned: 7 (million)
% 4.07/2.03 % (15817)------------------------------
% 4.07/2.03 % (15817)------------------------------
% 4.07/2.03 % (15818)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_15689 for (2999ds/3Mi)
% 4.07/2.03 % (15818)Instruction limit reached!
% 4.07/2.03 % (15818)------------------------------
% 4.07/2.03 % (15818)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.03 % (15818)Termination reason: Unknown
% 4.07/2.03 % (15818)Termination phase: shuffling
% 4.07/2.03
% 4.07/2.03 % (15818)Memory used [KB]: 1407
% 4.07/2.03 % (15818)Time elapsed: 0.003 s
% 4.07/2.03 % (15818)Instructions burned: 3 (million)
% 4.07/2.03 % (15818)------------------------------
% 4.07/2.03 % (15818)------------------------------
% 4.07/2.04 % (15819)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_15689 for (2999ds/4Mi)
% 4.07/2.04 % (15819)Instruction limit reached!
% 4.07/2.04 % (15819)------------------------------
% 4.07/2.04 % (15819)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.04 % (15819)Termination reason: Unknown
% 4.07/2.04 % (15819)Termination phase: shuffling
% 4.07/2.04
% 4.07/2.04 % (15819)Memory used [KB]: 1535
% 4.07/2.04 % (15819)Time elapsed: 0.004 s
% 4.07/2.04 % (15819)Instructions burned: 5 (million)
% 4.07/2.04 % (15819)------------------------------
% 4.07/2.04 % (15819)------------------------------
% 4.07/2.04 % (15821)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_15689 for (2999ds/18Mi)
% 4.07/2.04 % (15822)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_15689 for (2999ds/710Mi)
% 4.07/2.05 % (15823)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_15689 for (2999ds/6Mi)
% 4.07/2.05 % (15821)Instruction limit reached!
% 4.07/2.05 % (15821)------------------------------
% 4.07/2.05 % (15821)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.05 % (15821)Termination reason: Unknown
% 4.07/2.05 % (15821)Termination phase: shuffling
% 4.07/2.05
% 4.07/2.05 % (15821)Memory used [KB]: 1791
% 4.07/2.05 % (15821)Time elapsed: 0.011 s
% 4.07/2.05 % (15821)Instructions burned: 18 (million)
% 4.07/2.05 % (15821)------------------------------
% 4.07/2.05 % (15821)------------------------------
% 4.07/2.05 % (15823)Instruction limit reached!
% 4.07/2.05 % (15823)------------------------------
% 4.07/2.05 % (15823)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.05 % (15823)Termination reason: Unknown
% 4.07/2.05 % (15823)Termination phase: shuffling
% 4.07/2.05
% 4.07/2.05 % (15823)Memory used [KB]: 1535
% 4.07/2.05 % (15823)Time elapsed: 0.005 s
% 4.07/2.05 % (15823)Instructions burned: 7 (million)
% 4.07/2.05 % (15823)------------------------------
% 4.07/2.05 % (15823)------------------------------
% 4.07/2.05 % (15824)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_15689 for (2999ds/902Mi)
% 4.07/2.06 % (15801)Instruction limit reached!
% 4.07/2.06 % (15801)------------------------------
% 4.07/2.06 % (15801)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.06 % (15801)Termination reason: Unknown
% 4.07/2.06 % (15801)Termination phase: Saturation
% 4.07/2.06
% 4.07/2.06 % (15801)Memory used [KB]: 7419
% 4.07/2.06 % (15801)Time elapsed: 0.093 s
% 4.07/2.06 % (15801)Instructions burned: 184 (million)
% 4.07/2.06 % (15801)------------------------------
% 4.07/2.06 % (15801)------------------------------
% 4.07/2.06 % (15825)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_15689 for (2998ds/21Mi)
% 4.07/2.07 % (15826)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_15689 for (2998ds/5Mi)
% 4.07/2.07 % (15826)Instruction limit reached!
% 4.07/2.07 % (15826)------------------------------
% 4.07/2.07 % (15826)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.07 % (15826)Termination reason: Unknown
% 4.07/2.07 % (15826)Termination phase: shuffling
% 4.07/2.07
% 4.07/2.07 % (15826)Memory used [KB]: 1535
% 4.07/2.07 % (15826)Time elapsed: 0.004 s
% 4.07/2.07 % (15826)Instructions burned: 5 (million)
% 4.07/2.07 % (15826)------------------------------
% 4.07/2.07 % (15826)------------------------------
% 4.07/2.08 % (15825)Instruction limit reached!
% 4.07/2.08 % (15825)------------------------------
% 4.07/2.08 % (15825)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.08 % (15825)Termination reason: Unknown
% 4.07/2.08 % (15825)Termination phase: shuffling
% 4.07/2.08
% 4.07/2.08 % (15825)Memory used [KB]: 1918
% 4.07/2.08 % (15825)Time elapsed: 0.013 s
% 4.07/2.08 % (15825)Instructions burned: 22 (million)
% 4.07/2.08 % (15825)------------------------------
% 4.07/2.08 % (15825)------------------------------
% 4.07/2.08 % (15827)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_15689 for (2998ds/6Mi)
% 4.07/2.08 % (15827)Instruction limit reached!
% 4.07/2.08 % (15827)------------------------------
% 4.07/2.08 % (15827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.08 % (15827)Termination reason: Unknown
% 4.07/2.08 % (15827)Termination phase: shuffling
% 4.07/2.08
% 4.07/2.08 % (15827)Memory used [KB]: 1535
% 4.07/2.08 % (15827)Time elapsed: 0.005 s
% 4.07/2.08 % (15827)Instructions burned: 7 (million)
% 4.07/2.08 % (15827)------------------------------
% 4.07/2.08 % (15827)------------------------------
% 4.07/2.08 % (15828)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_15689 for (2998ds/377Mi)
% 4.07/2.09 % (15829)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_15689 for (2998ds/779Mi)
% 4.07/2.10 % (15830)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_15689 for (2998ds/19Mi)
% 4.07/2.10 % (15830)Instruction limit reached!
% 4.07/2.10 % (15830)------------------------------
% 4.07/2.10 % (15830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.10 % (15830)Termination reason: Unknown
% 4.07/2.10 % (15830)Termination phase: shuffling
% 4.07/2.10
% 4.07/2.10 % (15830)Memory used [KB]: 1791
% 4.07/2.10 % (15830)Time elapsed: 0.011 s
% 4.07/2.10 % (15830)Instructions burned: 20 (million)
% 4.07/2.10 % (15830)------------------------------
% 4.07/2.10 % (15830)------------------------------
% 4.07/2.11 % (15806)Instruction limit reached!
% 4.07/2.11 % (15806)------------------------------
% 4.07/2.11 % (15806)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.07/2.11 % (15806)Termination reason: Unknown
% 4.07/2.11 % (15806)Termination phase: Saturation
% 4.07/2.11
% 4.07/2.11 % (15806)Memory used [KB]: 7675
% 4.07/2.11 % (15806)Time elapsed: 0.141 s
% 4.07/2.11 % (15806)Instructions burned: 275 (million)
% 4.07/2.11 % (15806)------------------------------
% 4.07/2.11 % (15806)------------------------------
% 4.07/2.12 % (15831)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_15689 for (2998ds/879Mi)
% 4.07/2.13 % (15832)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_15689 for (2998ds/17Mi)
% 4.52/2.13 % (15832)Instruction limit reached!
% 4.52/2.13 % (15832)------------------------------
% 4.52/2.13 % (15832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.52/2.13 % (15832)Termination reason: Unknown
% 4.52/2.13 % (15832)Termination phase: shuffling
% 4.52/2.13
% 4.52/2.13 % (15832)Memory used [KB]: 1791
% 4.52/2.13 % (15832)Time elapsed: 0.011 s
% 4.52/2.13 % (15832)Instructions burned: 18 (million)
% 4.52/2.13 % (15832)------------------------------
% 4.52/2.13 % (15832)------------------------------
% 4.52/2.15 % (15833)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_15689 for (2998ds/3Mi)
% 4.52/2.15 % (15833)Instruction limit reached!
% 4.52/2.15 % (15833)------------------------------
% 4.52/2.15 % (15833)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.52/2.15 % (15833)Termination reason: Unknown
% 4.52/2.15 % (15833)Termination phase: shuffling
% 4.52/2.15
% 4.52/2.15 % (15833)Memory used [KB]: 1407
% 4.52/2.15 % (15833)Time elapsed: 0.004 s
% 4.52/2.15 % (15833)Instructions burned: 4 (million)
% 4.52/2.15 % (15833)------------------------------
% 4.52/2.15 % (15833)------------------------------
% 4.52/2.17 % (15834)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_15689 for (2997ds/30Mi)
% 4.52/2.18 % (15834)Instruction limit reached!
% 4.52/2.18 % (15834)------------------------------
% 4.52/2.18 % (15834)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.52/2.18 % (15834)Termination reason: Unknown
% 4.52/2.18 % (15834)Termination phase: shuffling
% 4.52/2.18
% 4.52/2.18 % (15834)Memory used [KB]: 2046
% 4.52/2.18 % (15834)Time elapsed: 0.017 s
% 4.52/2.18 % (15834)Instructions burned: 30 (million)
% 4.52/2.18 % (15834)------------------------------
% 4.52/2.18 % (15834)------------------------------
% 4.66/2.20 % (15835)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_15689 for (2997ds/127Mi)
% 4.66/2.26 % (15835)Instruction limit reached!
% 4.66/2.26 % (15835)------------------------------
% 4.66/2.26 % (15835)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.66/2.26 % (15835)Termination reason: Unknown
% 4.66/2.26 % (15835)Termination phase: Saturation
% 4.66/2.26
% 4.66/2.26 % (15835)Memory used [KB]: 7164
% 4.66/2.26 % (15835)Time elapsed: 0.064 s
% 4.66/2.26 % (15835)Instructions burned: 128 (million)
% 4.66/2.26 % (15835)------------------------------
% 4.66/2.26 % (15835)------------------------------
% 4.66/2.28 % (15828)Instruction limit reached!
% 4.66/2.28 % (15828)------------------------------
% 4.66/2.28 % (15828)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.66/2.28 % (15828)Termination reason: Unknown
% 4.66/2.28 % (15828)Termination phase: Saturation
% 4.66/2.28
% 4.66/2.28 % (15828)Memory used [KB]: 8827
% 4.66/2.28 % (15828)Time elapsed: 0.194 s
% 4.66/2.28 % (15828)Instructions burned: 378 (million)
% 4.66/2.28 % (15828)------------------------------
% 4.66/2.28 % (15828)------------------------------
% 4.66/2.28 % (15836)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_15689 for (2996ds/100Mi)
% 4.66/2.29 % (15837)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_15689 for (2996ds/3Mi)
% 4.66/2.30 % (15837)Instruction limit reached!
% 4.66/2.30 % (15837)------------------------------
% 4.66/2.30 % (15837)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.66/2.30 % (15837)Termination reason: Unknown
% 4.66/2.30 % (15837)Termination phase: shuffling
% 4.66/2.30
% 4.66/2.30 % (15837)Memory used [KB]: 1407
% 4.66/2.30 % (15837)Time elapsed: 0.004 s
% 4.66/2.30 % (15837)Instructions burned: 4 (million)
% 4.66/2.30 % (15837)------------------------------
% 4.66/2.30 % (15837)------------------------------
% 5.05/2.31 % (15838)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_15689 for (2996ds/20Mi)
% 5.05/2.32 % (15838)Instruction limit reached!
% 5.05/2.32 % (15838)------------------------------
% 5.05/2.32 % (15838)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.05/2.32 % (15838)Termination reason: Unknown
% 5.05/2.32 % (15838)Termination phase: shuffling
% 5.05/2.32
% 5.05/2.32 % (15838)Memory used [KB]: 1791
% 5.05/2.32 % (15838)Time elapsed: 0.013 s
% 5.05/2.32 % (15838)Instructions burned: 20 (million)
% 5.05/2.32 % (15838)------------------------------
% 5.05/2.32 % (15838)------------------------------
% 5.05/2.33 % (15836)Instruction limit reached!
% 5.05/2.33 % (15836)------------------------------
% 5.05/2.33 % (15836)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.05/2.33 % (15836)Termination reason: Unknown
% 5.05/2.33 % (15836)Termination phase: Property scanning
% 5.05/2.33
% 5.05/2.33 % (15836)Memory used [KB]: 2814
% 5.05/2.33 % (15836)Time elapsed: 0.049 s
% 5.05/2.33 % (15836)Instructions burned: 100 (million)
% 5.05/2.33 % (15836)------------------------------
% 5.05/2.33 % (15836)------------------------------
% 5.05/2.34 % (15839)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_15689 for (2996ds/86Mi)
% 5.05/2.34 % (15840)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_15689 for (2996ds/107Mi)
% 5.05/2.38 % (15839)Instruction limit reached!
% 5.05/2.38 % (15839)------------------------------
% 5.05/2.38 % (15839)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.05/2.38 % (15839)Termination reason: Unknown
% 5.05/2.38 % (15839)Termination phase: Property scanning
% 5.05/2.38
% 5.05/2.38 % (15839)Memory used [KB]: 2430
% 5.05/2.38 % (15839)Time elapsed: 0.043 s
% 5.05/2.38 % (15839)Instructions burned: 86 (million)
% 5.05/2.38 % (15839)------------------------------
% 5.05/2.38 % (15839)------------------------------
% 5.05/2.40 % (15840)Instruction limit reached!
% 5.05/2.40 % (15840)------------------------------
% 5.05/2.40 % (15840)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.05/2.40 % (15840)Termination reason: Unknown
% 5.05/2.40 % (15840)Termination phase: Saturation
% 5.05/2.40
% 5.05/2.40 % (15840)Memory used [KB]: 7036
% 5.05/2.40 % (15840)Time elapsed: 0.054 s
% 5.05/2.40 % (15840)Instructions burned: 108 (million)
% 5.05/2.40 % (15840)------------------------------
% 5.05/2.40 % (15840)------------------------------
% 5.32/2.40 % (15841)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_15689 for (2995ds/39Mi)
% 5.32/2.41 % (15822)Instruction limit reached!
% 5.32/2.41 % (15822)------------------------------
% 5.32/2.41 % (15822)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.32/2.41 % (15822)Termination reason: Unknown
% 5.32/2.41 % (15822)Termination phase: Saturation
% 5.32/2.41
% 5.32/2.41 % (15822)Memory used [KB]: 8699
% 5.32/2.41 % (15822)Time elapsed: 0.364 s
% 5.32/2.41 % (15822)Instructions burned: 710 (million)
% 5.32/2.41 % (15822)------------------------------
% 5.32/2.41 % (15822)------------------------------
% 5.32/2.41 % (15842)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_15689 for (2995ds/448Mi)
% 5.32/2.42 % (15841)Instruction limit reached!
% 5.32/2.42 % (15841)------------------------------
% 5.32/2.42 % (15841)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.32/2.42 % (15841)Termination reason: Unknown
% 5.32/2.42 % (15841)Termination phase: shuffling
% 5.32/2.42
% 5.32/2.42 % (15841)Memory used [KB]: 2174
% 5.32/2.42 % (15841)Time elapsed: 0.020 s
% 5.32/2.42 % (15841)Instructions burned: 39 (million)
% 5.32/2.42 % (15841)------------------------------
% 5.32/2.42 % (15841)------------------------------
% 5.32/2.42 % (15843)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_15689 for (2995ds/46Mi)
% 5.32/2.43 % (15844)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_15689 for (2995ds/98Mi)
% 5.32/2.44 % (15843)Instruction limit reached!
% 5.32/2.44 % (15843)------------------------------
% 5.32/2.44 % (15843)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.32/2.44 % (15843)Termination reason: Unknown
% 5.32/2.44 % (15843)Termination phase: Preprocessing 1
% 5.32/2.44
% 5.32/2.44 % (15843)Memory used [KB]: 2046
% 5.32/2.44 % (15843)Time elapsed: 0.025 s
% 5.32/2.44 % (15843)Instructions burned: 47 (million)
% 5.32/2.44 % (15843)------------------------------
% 5.32/2.44 % (15843)------------------------------
% 5.32/2.46 % (15845)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_15689 for (2994ds/507Mi)
% 5.32/2.47 % (15831)First to succeed.
% 5.32/2.48 % (15829)Instruction limit reached!
% 5.32/2.48 % (15829)------------------------------
% 5.32/2.48 % (15829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.32/2.48 % (15829)Termination reason: Unknown
% 5.32/2.48 % (15829)Termination phase: Saturation
% 5.32/2.48
% 5.32/2.48 % (15829)Memory used [KB]: 9210
% 5.32/2.48 % (15829)Time elapsed: 0.388 s
% 5.32/2.48 % (15829)Instructions burned: 779 (million)
% 5.32/2.48 % (15829)------------------------------
% 5.32/2.48 % (15829)------------------------------
% 5.32/2.48 % (15831)Refutation found. Thanks to Tanya!
% 5.32/2.48 % SZS status Theorem for DTF2THF_15689
% 5.32/2.48 % SZS output start Proof for DTF2THF_15689
% 5.32/2.48 thf(type_def_5, type, itself: ($tType) > $tType).
% 5.32/2.48 thf(type_def_7, type, set: ($tType) > $tType).
% 5.32/2.48 thf(type_def_8, type, product_prod: ($tType * $tType) > $tType).
% 5.32/2.48 thf(type_def_9, type, dtree: $tType).
% 5.32/2.48 thf(type_def_10, type, t: $tType).
% 5.32/2.48 thf(type_def_11, type, sum_sum: ($tType * $tType) > $tType).
% 5.32/2.48 thf(type_def_12, type, n: $tType).
% 5.32/2.48 thf(func_def_0, type, product_prod: ($tType * $tType) > $tType).
% 5.32/2.48 thf(func_def_1, type, sum_sum: ($tType * $tType) > $tType).
% 5.32/2.48 thf(func_def_2, type, dtree: $tType).
% 5.32/2.48 thf(func_def_3, type, set: ($tType) > $tType).
% 5.32/2.48 thf(func_def_4, type, t: $tType).
% 5.32/2.48 thf(func_def_5, type, n: $tType).
% 5.32/2.48 thf(func_def_6, type, itself: ($tType) > $tType).
% 5.32/2.48 thf(func_def_7, type, bounded_lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.32/2.48 thf(func_def_8, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.32/2.48 thf(func_def_9, type, lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.32/2.48 thf(func_def_10, type, semilattice_sup: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.32/2.48 thf(func_def_11, type, bounded_lattice_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.32/2.48 thf(func_def_12, type, bNF_Greatest_image2: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ X0 > (X0 > X1) > (X0 > X2) > set @ product_prod @ X1 @ X2)).
% 5.32/2.48 thf(func_def_13, type, cont: dtree > set @ sum_sum @ t @ dtree).
% 5.32/2.48 thf(func_def_14, type, unfold: !>[X0: $tType]:((X0 > n) > (X0 > set @ sum_sum @ t @ X0) > X0 > dtree)).
% 5.32/2.48 thf(func_def_15, type, fun_rp_inv_image: !>[X0: $tType, X1: $tType]:(product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X0 @ X0 > (X1 > X0) > product_prod @ set @ product_prod @ X1 @ X1 @ set @ product_prod @ X1 @ X1)).
% 5.32/2.48 thf(func_def_16, type, the: !>[X0: $tType]:((X0 > $o) > X0)).
% 5.32/2.48 thf(func_def_17, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 5.32/2.48 thf(func_def_18, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 5.32/2.48 thf(func_def_19, type, top_top: !>[X0: $tType]:(X0)).
% 5.32/2.48 thf(func_def_20, type, parall1899940088le_par: product_prod @ dtree @ dtree > dtree).
% 5.32/2.48 thf(func_def_21, type, parall1914194347_par_c: product_prod @ dtree @ dtree > set @ sum_sum @ t @ product_prod @ dtree @ dtree).
% 5.32/2.48 thf(func_def_22, type, parall1418634510_c_rel: product_prod @ dtree @ dtree > product_prod @ dtree @ dtree > $o).
% 5.32/2.48 thf(func_def_23, type, parall1914194362_par_r: product_prod @ dtree @ dtree > n).
% 5.32/2.48 thf(func_def_24, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 5.32/2.48 thf(func_def_25, type, product_Sigma: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > set @ X1) > set @ product_prod @ X0 @ X1)).
% 5.32/2.48 thf(func_def_26, type, product_curry: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X0 @ X1 > X2) > X0 > X1 > X2)).
% 5.32/2.48 thf(func_def_27, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 5.32/2.48 thf(func_def_28, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 5.32/2.48 thf(func_def_29, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 5.32/2.48 thf(func_def_30, type, product_rec_set_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2 > $o)).
% 5.32/2.48 thf(func_def_31, type, product_case_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 5.32/2.48 thf(func_def_32, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 5.32/2.48 thf(func_def_33, type, product_product: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > set @ product_prod @ X0 @ X1)).
% 5.32/2.48 thf(func_def_34, type, type2: !>[X0: $tType]:(itself @ X0)).
% 5.32/2.48 thf(func_def_35, type, id: !>[X0: $tType]:(set @ product_prod @ X0 @ X0)).
% 5.32/2.48 thf(func_def_36, type, antisym: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 5.32/2.48 thf(func_def_37, type, inv_image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X1 > X0) > set @ product_prod @ X1 @ X1)).
% 5.32/2.48 thf(func_def_38, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 5.32/2.48 thf(func_def_39, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 5.32/2.48 thf(func_def_40, type, vimage: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X1 > set @ X0)).
% 5.32/2.48 thf(func_def_41, type, sum_Inl: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X0 @ X1)).
% 5.32/2.48 thf(func_def_42, type, sum_Inr: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X1 @ X0)).
% 5.32/2.48 thf(func_def_43, type, sum_Plus: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > set @ sum_sum @ X0 @ X1)).
% 5.32/2.48 thf(func_def_44, type, sum_Suml: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > sum_sum @ X0 @ X2 > X1)).
% 5.32/2.48 thf(func_def_45, type, sum_Sumr: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > sum_sum @ X2 @ X0 > X1)).
% 5.32/2.48 thf(func_def_46, type, sum_rec_set_sum: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X1) > sum_sum @ X0 @ X2 > X1 > $o)).
% 5.32/2.48 thf(func_def_47, type, sum_rec_sum: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X1) > sum_sum @ X0 @ X2 > X1)).
% 5.32/2.48 thf(func_def_48, type, accp: !>[X0: $tType]:((X0 > X0 > $o) > X0 > $o)).
% 5.32/2.48 thf(func_def_49, 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)).
% 5.32/2.48 thf(func_def_50, 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)).
% 5.32/2.48 thf(func_def_51, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 5.32/2.48 thf(func_def_52, type, t2: dtree).
% 5.32/2.48 thf(func_def_53, type, tr1: dtree).
% 5.32/2.48 thf(func_def_54, type, tr2: dtree).
% 5.32/2.48 thf(func_def_72, type, sK0: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 5.32/2.48 thf(func_def_73, type, sK1: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 5.32/2.48 thf(func_def_74, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 5.32/2.48 thf(func_def_75, type, sK3: (product_prod @ dtree @ dtree > $o) > dtree).
% 5.32/2.48 thf(func_def_76, type, sK4: (product_prod @ dtree @ dtree > $o) > dtree).
% 5.32/2.48 thf(func_def_77, type, sK5: !>[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)).
% 5.32/2.48 thf(func_def_78, type, sK6: !>[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)).
% 5.32/2.48 thf(func_def_79, type, sK7: !>[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)).
% 5.32/2.48 thf(func_def_80, type, sK8: !>[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)).
% 5.32/2.48 thf(func_def_81, type, sK9: !>[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)).
% 5.32/2.48 thf(func_def_82, type, sK10: !>[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)).
% 5.32/2.48 thf(func_def_83, type, sK11: !>[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)).
% 5.32/2.48 thf(func_def_84, type, sK12: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 5.32/2.48 thf(func_def_85, type, sK13: !>[X0: $tType, X1: $tType, X2: $tType]:($o > (sum_sum @ X0 @ X1 > X2) > X2 > X1)).
% 5.32/2.48 thf(func_def_86, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType]:($o > (sum_sum @ X0 @ X1 > X2) > X2 > X0)).
% 5.32/2.48 thf(func_def_87, type, sK15: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X0)).
% 5.32/2.48 thf(func_def_88, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 5.32/2.48 thf(func_def_89, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 5.32/2.48 thf(func_def_90, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 5.32/2.48 thf(func_def_91, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 5.32/2.48 thf(func_def_92, type, sK20: set @ sum_sum @ t @ product_prod @ dtree @ dtree > product_prod @ dtree @ dtree > dtree).
% 5.32/2.48 thf(func_def_93, type, sK21: set @ sum_sum @ t @ product_prod @ dtree @ dtree > product_prod @ dtree @ dtree > dtree).
% 5.32/2.48 thf(func_def_94, type, sK22: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 5.32/2.48 thf(func_def_95, type, sK23: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 5.32/2.48 thf(func_def_96, type, sK24: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 5.32/2.48 thf(func_def_97, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 5.32/2.48 thf(func_def_98, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 5.32/2.48 thf(func_def_99, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 5.32/2.48 thf(func_def_100, type, sK28: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 5.32/2.48 thf(func_def_101, type, sK29: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 5.32/2.48 thf(func_def_102, type, sK30: product_prod @ dtree @ dtree > dtree).
% 5.32/2.48 thf(func_def_103, type, sK31: product_prod @ dtree @ dtree > dtree).
% 5.32/2.48 thf(func_def_104, type, sK32: !>[X0: $tType, X1: $tType]:((X1 > X0) > X0 > X1)).
% 5.32/2.48 thf(func_def_105, type, sK33: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 5.32/2.48 thf(func_def_106, type, sK34: !>[X0: $tType, X1: $tType]:((X0 > set @ X1) > set @ X0 > product_prod @ X0 @ X1 > X0)).
% 5.32/2.48 thf(func_def_107, type, sK35: !>[X0: $tType, X1: $tType]:((X0 > set @ X1) > set @ X0 > product_prod @ X0 @ X1 > X1)).
% 5.32/2.48 thf(func_def_108, type, sK36: !>[X0: $tType, X1: $tType]:((sum_sum @ X1 @ X0 > $o) > X1)).
% 5.32/2.48 thf(func_def_109, type, sK37: !>[X0: $tType, X1: $tType]:((sum_sum @ X1 @ X0 > $o) > X0)).
% 5.32/2.48 thf(func_def_110, type, sK38: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X1)).
% 5.32/2.48 thf(func_def_111, type, sK39: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X0)).
% 5.32/2.48 thf(func_def_112, type, sK40: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 5.32/2.48 thf(func_def_113, type, sK41: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 5.32/2.48 thf(func_def_114, type, sK42: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 5.32/2.48 thf(func_def_115, type, sK43: set @ sum_sum @ t @ product_prod @ dtree @ dtree > product_prod @ dtree @ dtree > dtree).
% 5.32/2.48 thf(func_def_116, type, sK44: set @ sum_sum @ t @ product_prod @ dtree @ dtree > product_prod @ dtree @ dtree > dtree).
% 5.32/2.48 thf(func_def_117, type, sK45: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 5.32/2.48 thf(func_def_118, type, sK46: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 5.32/2.48 thf(func_def_119, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 5.32/2.48 thf(func_def_120, type, sK48: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 5.32/2.48 thf(func_def_121, type, sK49: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 5.32/2.48 thf(func_def_122, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 5.32/2.48 thf(func_def_123, type, sK51: !>[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)).
% 5.32/2.48 thf(func_def_124, type, sK52: !>[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)).
% 5.32/2.48 thf(func_def_125, type, sK53: !>[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)).
% 5.32/2.48 thf(func_def_126, type, sK54: !>[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)).
% 5.32/2.48 thf(func_def_127, type, sK55: !>[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)).
% 5.32/2.48 thf(func_def_128, type, sK56: !>[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)).
% 5.32/2.48 thf(func_def_129, type, sK57: !>[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)).
% 5.32/2.48 thf(func_def_130, type, sK58: !>[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)).
% 5.32/2.48 thf(func_def_131, type, sK59: !>[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)).
% 5.32/2.48 thf(func_def_132, type, sK60: !>[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)).
% 5.32/2.48 thf(func_def_133, type, sK61: !>[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)).
% 5.32/2.48 thf(func_def_134, type, sK62: !>[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)).
% 5.32/2.48 thf(func_def_135, type, sK63: !>[X0: $tType, X1: $tType]:((X1 > $o) > (X1 > X0) > (X0 > $o) > X1)).
% 5.32/2.48 thf(func_def_136, type, sK64: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X0)).
% 5.32/2.48 thf(func_def_137, type, sK65: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X1)).
% 5.32/2.48 thf(func_def_138, type, sK66: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(set @ product_prod @ X3 @ X2 > (X2 > X0) > (X3 > X1) > product_prod @ X1 @ X0 > X3)).
% 5.32/2.48 thf(func_def_139, type, sK67: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(set @ product_prod @ X3 @ X2 > (X2 > X0) > (X3 > X1) > product_prod @ X1 @ X0 > X2)).
% 5.32/2.48 thf(func_def_140, type, sK68: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > X0 > X1)).
% 5.32/2.48 thf(func_def_141, type, sK69: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 5.32/2.48 thf(func_def_142, type, sK70: !>[X0: $tType, X1: $tType]:((X1 > set @ X0) > (X1 > set @ X0) > set @ X1 > X1)).
% 5.32/2.48 thf(func_def_143, type, sK71: !>[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)).
% 5.32/2.48 thf(func_def_144, type, sK72: !>[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)).
% 5.32/2.48 thf(func_def_145, type, sK73: !>[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)).
% 5.32/2.48 thf(func_def_146, type, sK74: !>[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)).
% 5.32/2.48 thf(func_def_147, type, sK75: !>[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)).
% 5.32/2.48 thf(func_def_148, type, sK76: !>[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)).
% 5.32/2.48 thf(func_def_149, type, sK77: !>[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)).
% 5.32/2.48 thf(func_def_150, type, sK78: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 5.32/2.48 thf(func_def_151, type, sK79: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 5.32/2.48 thf(func_def_152, type, sK80: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X0)).
% 5.32/2.48 thf(func_def_153, type, sK81: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X1)).
% 5.32/2.48 thf(func_def_154, type, sK82: !>[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)).
% 5.32/2.48 thf(func_def_155, type, sK83: !>[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)).
% 5.32/2.48 thf(func_def_156, type, sK84: !>[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)).
% 5.32/2.48 thf(func_def_157, type, sK85: !>[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)).
% 5.32/2.48 thf(func_def_158, type, sK86: !>[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)).
% 5.32/2.48 thf(func_def_159, type, sK87: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X1)).
% 5.32/2.48 thf(func_def_160, type, sK88: !>[X0: $tType, X1: $tType]:(set @ X0 > sum_sum @ X0 @ X1 > X0)).
% 5.32/2.48 thf(func_def_161, type, sK89: !>[X0: $tType, X1: $tType]:(set @ X1 > sum_sum @ X0 @ X1 > X1)).
% 5.32/2.48 thf(func_def_162, type, sK90: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 5.32/2.48 thf(func_def_163, type, sK91: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 5.32/2.48 thf(func_def_164, type, sK92: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 5.32/2.48 thf(func_def_166, type, sK94: !>[X0: $tType, X1: $tType]:((X1 > X0) > set @ X1 > X0 > X1)).
% 5.32/2.48 thf(func_def_167, type, ph95: !>[X0: $tType]:(X0)).
% 5.32/2.48 thf(f1991,plain,(
% 5.32/2.48 $false),
% 5.32/2.48 inference(avatar_sat_refutation,[status(thm)],[f1166,f1175,f1948,f1951,f1990])).
% 5.32/2.48 thf(f1990,plain,(
% 5.32/2.48 ~spl93_1 | ~spl93_4),
% 5.32/2.48 inference(avatar_contradiction_clause,[status(thm)],[f1989])).
% 5.32/2.48 thf(f1989,plain,(
% 5.32/2.48 $false | (~spl93_1 | ~spl93_4)),
% 5.32/2.48 inference(trivial_inequality_removal,[status(thm)],[f1988])).
% 5.32/2.48 thf(f1988,plain,(
% 5.32/2.48 ($false = $true) | (~spl93_1 | ~spl93_4)),
% 5.32/2.48 inference(forward_demodulation,[status(thm)],[f1982,f1174])).
% 5.32/2.48 thf(f1174,plain,(
% 5.32/2.48 ((member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ t2) @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) = $false) | ~spl93_4),
% 5.32/2.48 inference(avatar_component_clause,[status(thm)],[f1172])).
% 5.32/2.48 thf(f1172,definition,(
% 5.32/2.48 spl93_4 <=> ((member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ t2) @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) = $false)),
% 5.32/2.48 introduced(definition,[new_symbols(naming,[spl93_4])],[avatar_definition])).
% 5.32/2.48 thf(f1982,plain,(
% 5.32/2.48 ((member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ t2) @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) = $true) | ~spl93_1),
% 5.32/2.48 inference(beta_eta_normalization,[status(thm)],[f1981])).
% 5.32/2.48 thf(f1981,plain,(
% 5.32/2.48 (((^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2))))) @ t2) = $true) | ~spl93_1),
% 5.32/2.48 inference(trivial_inequality_removal,[status(thm)],[f1980])).
% 5.32/2.48 thf(f1980,plain,(
% 5.32/2.48 (((^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2))))) @ t2) = $true) | ($false = $true) | ~spl93_1),
% 5.32/2.48 inference(superposition,[status(thm)],[f1113,f1953])).
% 5.32/2.48 thf(f1953,plain,(
% 5.32/2.48 ((member @ dtree @ t2 @ (collect @ dtree @ (^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2))))))) = $true) | ~spl93_1),
% 5.32/2.48 inference(forward_demodulation,[status(thm)],[f1161,f1086])).
% 5.32/2.48 thf(f1086,plain,(
% 5.32/2.48 ( ! [X0 : dtree,X1 : dtree] : (((collect @ dtree @ (^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ X0 @ X1)))))) = (image @ product_prod @ dtree @ dtree @ dtree @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c) @ (product_Sigma @ dtree @ dtree @ (collect @ dtree @ (^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ X0)))) @ (^[Y0 : dtree]: (collect @ dtree @ (^[Y1 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y1) @ (cont @ X1))))))))) )),
% 5.32/2.48 inference(beta_eta_normalization,[status(thm)],[f1044])).
% 5.32/2.48 thf(f1044,plain,(
% 5.32/2.48 ( ! [X0 : dtree,X1 : dtree] : ((((^[Y0 : dtree > sum_sum @ t @ dtree]: ((^[Y1 : set @ sum_sum @ t @ dtree]: (collect @ dtree @ (^[Y2 : dtree]: (member @ sum_sum @ t @ dtree @ (Y0 @ Y2) @ Y1)))))) @ sum_Inr @ dtree @ t @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ X0 @ X1)))) = (image @ product_prod @ dtree @ dtree @ dtree @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c) @ (product_Sigma @ dtree @ dtree @ ((^[Y0 : dtree > sum_sum @ t @ dtree]: ((^[Y1 : set @ sum_sum @ t @ dtree]: (collect @ dtree @ (^[Y2 : dtree]: (member @ sum_sum @ t @ dtree @ (Y0 @ Y2) @ Y1)))))) @ sum_Inr @ dtree @ t @ (cont @ X0)) @ (^[Y0 : dtree]: ((^[Y1 : dtree > sum_sum @ t @ dtree]: ((^[Y2 : set @ sum_sum @ t @ dtree]: (collect @ dtree @ (^[Y3 : dtree]: (member @ sum_sum @ t @ dtree @ (Y1 @ Y3) @ Y2)))))) @ sum_Inr @ dtree @ t @ (cont @ X1))))))) )),
% 5.32/2.48 inference(definition_unfolding,[status(thm)],[f1008,f897,f925,f925,f925,f897])).
% 5.32/2.48 thf(f925,plain,(
% 5.32/2.48 ( ! [X1 : $tType,X0 : $tType] : ((vimage @ X1 @ X0 = (^[Y0 : X1 > X0]: ((^[Y1 : set @ X0]: (collect @ X1 @ (^[Y2 : X1]: (member @ X0 @ (Y0 @ Y2) @ Y1)))))))) )),
% 5.32/2.48 inference(cnf_transformation,[status(thm)],[f668])).
% 5.32/2.48 thf(f668,plain,(
% 5.32/2.48 ! [X0 : $tType,X1 : $tType] : (vimage @ X1 @ X0 = (^[Y0 : X1 > X0]: ((^[Y1 : set @ X0]: (collect @ X1 @ (^[Y2 : X1]: (member @ X0 @ (Y0 @ Y2) @ Y1)))))))),
% 5.32/2.48 inference(fool_elimination,[status(thm)],[f667])).
% 5.32/2.48 thf(f667,plain,(
% 5.32/2.48 ! [X0 : $tType,X1 : $tType] : ((^[X2 : X1 > X0, X3 : set @ X0] : (collect @ X1 @ (^[X4 : X1] : (member @ X0 @ (X2 @ X4) @ X3)))) = vimage @ X1 @ X0)),
% 5.32/2.48 inference(rectify,[status(thm)],[f22])).
% 5.32/2.48 thf(f22,axiom,(
% 5.32/2.48 ! [X2 : $tType,X0 : $tType] : ((^[X30 : X0 > X2, X31 : set @ X2] : (collect @ X0 @ (^[X16 : X0] : (member @ X2 @ (X30 @ X16) @ X31)))) = vimage @ X0 @ X2)),
% 5.32/2.48 file('/export/starexec/sandbox/tmp/tmp.ssUgw0micP/DTF2THF_15689.p',fact_21_vimage__def)).
% 5.32/2.48 thf(f897,plain,(
% 5.32/2.48 (parall1899940088le_par = (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c))),
% 5.32/2.48 inference(cnf_transformation,[status(thm)],[f158])).
% 5.32/2.48 thf(f158,axiom,(
% 5.32/2.48 (parall1899940088le_par = (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c))),
% 5.32/2.48 file('/export/starexec/sandbox/tmp/tmp.ssUgw0micP/DTF2THF_15689.p',fact_157_par__def)).
% 5.32/2.48 thf(f1008,plain,(
% 5.32/2.48 ( ! [X0 : dtree,X1 : dtree] : (((image @ product_prod @ dtree @ dtree @ dtree @ parall1899940088le_par @ (product_Sigma @ dtree @ dtree @ (vimage @ dtree @ sum_sum @ t @ dtree @ sum_Inr @ dtree @ t @ (cont @ X0)) @ (^[Y0 : dtree]: (vimage @ dtree @ sum_sum @ t @ dtree @ sum_Inr @ dtree @ t @ (cont @ X1))))) = (vimage @ dtree @ sum_sum @ t @ dtree @ sum_Inr @ dtree @ t @ (cont @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ X1)))))) )),
% 5.32/2.48 inference(cnf_transformation,[status(thm)],[f338])).
% 5.32/2.48 thf(f338,plain,(
% 5.32/2.48 ! [X0 : dtree,X1 : dtree] : ((image @ product_prod @ dtree @ dtree @ dtree @ parall1899940088le_par @ (product_Sigma @ dtree @ dtree @ (vimage @ dtree @ sum_sum @ t @ dtree @ sum_Inr @ dtree @ t @ (cont @ X0)) @ (^[Y0 : dtree]: (vimage @ dtree @ sum_sum @ t @ dtree @ sum_Inr @ dtree @ t @ (cont @ X1))))) = (vimage @ dtree @ sum_sum @ t @ dtree @ sum_Inr @ dtree @ t @ (cont @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ X1)))))),
% 5.32/2.48 inference(fool_elimination,[status(thm)],[f337])).
% 5.32/2.48 thf(f337,plain,(
% 5.32/2.48 ! [X0 : dtree,X1 : dtree] : ((image @ product_prod @ dtree @ dtree @ dtree @ parall1899940088le_par @ (product_Sigma @ dtree @ dtree @ (vimage @ dtree @ sum_sum @ t @ dtree @ sum_Inr @ dtree @ t @ (cont @ X0)) @ (^[X2 : dtree] : (vimage @ dtree @ sum_sum @ t @ dtree @ sum_Inr @ dtree @ t @ (cont @ X1))))) = (vimage @ dtree @ sum_sum @ t @ dtree @ sum_Inr @ dtree @ t @ (cont @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X0 @ X1)))))),
% 5.32/2.48 inference(rectify,[status(thm)],[f7])).
% 5.32/2.48 thf(f7,axiom,(
% 5.32/2.48 ! [X17 : dtree,X18 : dtree] : ((image @ product_prod @ dtree @ dtree @ dtree @ parall1899940088le_par @ (product_Sigma @ dtree @ dtree @ (vimage @ dtree @ sum_sum @ t @ dtree @ sum_Inr @ dtree @ t @ (cont @ X17)) @ (^[X10 : dtree] : (vimage @ dtree @ sum_sum @ t @ dtree @ sum_Inr @ dtree @ t @ (cont @ X18))))) = (vimage @ dtree @ sum_sum @ t @ dtree @ sum_Inr @ dtree @ t @ (cont @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ X17 @ X18)))))),
% 5.32/2.48 file('/export/starexec/sandbox/tmp/tmp.ssUgw0micP/DTF2THF_15689.p',fact_6_Inr__cont__par)).
% 5.32/2.48 thf(f1161,plain,(
% 5.32/2.48 ((member @ dtree @ t2 @ (image @ product_prod @ dtree @ dtree @ dtree @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c) @ (product_Sigma @ dtree @ dtree @ (collect @ dtree @ (^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ tr1)))) @ (^[Y0 : dtree]: (collect @ dtree @ (^[Y1 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y1) @ (cont @ tr2)))))))) = $true) | ~spl93_1),
% 5.32/2.48 inference(avatar_component_clause,[status(thm)],[f1159])).
% 5.32/2.48 thf(f1159,definition,(
% 5.32/2.48 spl93_1 <=> ((member @ dtree @ t2 @ (image @ product_prod @ dtree @ dtree @ dtree @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c) @ (product_Sigma @ dtree @ dtree @ (collect @ dtree @ (^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ tr1)))) @ (^[Y0 : dtree]: (collect @ dtree @ (^[Y1 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y1) @ (cont @ tr2)))))))) = $true)),
% 5.32/2.48 introduced(definition,[new_symbols(naming,[spl93_1])],[avatar_definition])).
% 5.32/2.48 thf(f1113,plain,(
% 5.32/2.48 ( ! [X0 : $tType,X2 : X0 > $o,X1 : X0] : (((member @ X0 @ X1 @ (collect @ X0 @ X2)) = $false) | ((X2 @ X1) = $true)) )),
% 5.32/2.48 inference(binary_proxy_clausification,[status(thm)],[f1017])).
% 5.32/2.48 thf(f1017,plain,(
% 5.32/2.48 ( ! [X0 : $tType,X2 : X0 > $o,X1 : X0] : (((member @ X0 @ X1 @ (collect @ X0 @ X2)) = (X2 @ X1))) )),
% 5.32/2.48 inference(cnf_transformation,[status(thm)],[f722])).
% 5.32/2.48 thf(f722,plain,(
% 5.32/2.48 ! [X0 : $tType,X1 : X0,X2 : X0 > $o] : ((member @ X0 @ X1 @ (collect @ X0 @ X2)) = (X2 @ X1))),
% 5.32/2.48 inference(rectify,[status(thm)],[f46])).
% 5.32/2.48 thf(f46,axiom,(
% 5.32/2.48 ! [X0 : $tType,X11 : X0,X8 : X0 > $o] : ((member @ X0 @ X11 @ (collect @ X0 @ X8)) = (X8 @ X11))),
% 5.32/2.48 file('/export/starexec/sandbox/tmp/tmp.ssUgw0micP/DTF2THF_15689.p',fact_45_mem__Collect__eq)).
% 5.32/2.48 thf(f1951,plain,(
% 5.32/2.48 ~spl93_2 | ~spl93_4),
% 5.32/2.48 inference(avatar_contradiction_clause,[status(thm)],[f1950])).
% 5.32/2.48 thf(f1950,plain,(
% 5.32/2.48 $false | (~spl93_2 | ~spl93_4)),
% 5.32/2.48 inference(trivial_inequality_removal,[status(thm)],[f1949])).
% 5.32/2.48 thf(f1949,plain,(
% 5.32/2.48 ($false = $true) | (~spl93_2 | ~spl93_4)),
% 5.32/2.48 inference(backward_demodulation,[status(thm)],[f1165,f1174])).
% 5.32/2.48 thf(f1165,plain,(
% 5.32/2.48 ((member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ t2) @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) = $true) | ~spl93_2),
% 5.32/2.48 inference(avatar_component_clause,[status(thm)],[f1163])).
% 5.32/2.48 thf(f1163,definition,(
% 5.32/2.48 spl93_2 <=> ((member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ t2) @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) = $true)),
% 5.32/2.48 introduced(definition,[new_symbols(naming,[spl93_2])],[avatar_definition])).
% 5.32/2.48 thf(f1948,plain,(
% 5.32/2.48 spl93_4 | ~spl93_3),
% 5.32/2.48 inference(avatar_split_clause,[status(thm)],[f1939,f1168,f1172])).
% 5.32/2.48 thf(f1168,definition,(
% 5.32/2.48 spl93_3 <=> ($false = (member @ dtree @ t2 @ (image @ product_prod @ dtree @ dtree @ dtree @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c) @ (product_Sigma @ dtree @ dtree @ (collect @ dtree @ (^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ tr1)))) @ (^[Y0 : dtree]: (collect @ dtree @ (^[Y1 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y1) @ (cont @ tr2)))))))))),
% 5.32/2.48 introduced(definition,[new_symbols(naming,[spl93_3])],[avatar_definition])).
% 5.32/2.48 thf(f1939,plain,(
% 5.32/2.48 ((member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ t2) @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) = $false) | ~spl93_3),
% 5.32/2.48 inference(beta_eta_normalization,[status(thm)],[f1938])).
% 5.32/2.48 thf(f1938,plain,(
% 5.32/2.48 (((^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2))))) @ t2) = $false) | ~spl93_3),
% 5.32/2.48 inference(trivial_inequality_removal,[status(thm)],[f1936])).
% 5.32/2.48 thf(f1936,plain,(
% 5.32/2.48 (((^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2))))) @ t2) = $false) | ($false = $true) | ~spl93_3),
% 5.32/2.48 inference(superposition,[status(thm)],[f1178,f1112])).
% 5.32/2.48 thf(f1112,plain,(
% 5.32/2.48 ( ! [X0 : $tType,X2 : X0 > $o,X1 : X0] : (((member @ X0 @ X1 @ (collect @ X0 @ X2)) = $true) | ((X2 @ X1) = $false)) )),
% 5.32/2.48 inference(binary_proxy_clausification,[status(thm)],[f1017])).
% 5.32/2.48 thf(f1178,plain,(
% 5.32/2.48 ((member @ dtree @ t2 @ (collect @ dtree @ (^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2))))))) = $false) | ~spl93_3),
% 5.32/2.48 inference(forward_demodulation,[status(thm)],[f1170,f1086])).
% 5.32/2.48 thf(f1170,plain,(
% 5.32/2.48 ($false = (member @ dtree @ t2 @ (image @ product_prod @ dtree @ dtree @ dtree @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c) @ (product_Sigma @ dtree @ dtree @ (collect @ dtree @ (^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ tr1)))) @ (^[Y0 : dtree]: (collect @ dtree @ (^[Y1 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y1) @ (cont @ tr2))))))))) | ~spl93_3),
% 5.32/2.48 inference(avatar_component_clause,[status(thm)],[f1168])).
% 5.32/2.48 thf(f1175,plain,(
% 5.32/2.48 spl93_3 | spl93_4),
% 5.32/2.48 inference(avatar_split_clause,[status(thm)],[f1148,f1172,f1168])).
% 5.32/2.48 thf(f1148,plain,(
% 5.32/2.48 ($false = (member @ dtree @ t2 @ (image @ product_prod @ dtree @ dtree @ dtree @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c) @ (product_Sigma @ dtree @ dtree @ (collect @ dtree @ (^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ tr1)))) @ (^[Y0 : dtree]: (collect @ dtree @ (^[Y1 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y1) @ (cont @ tr2))))))))) | ((member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ t2) @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) = $false)),
% 5.32/2.48 inference(binary_proxy_clausification,[status(thm)],[f1039])).
% 5.32/2.48 thf(f1039,plain,(
% 5.32/2.48 ((member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ t2) @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) != (member @ dtree @ t2 @ (image @ product_prod @ dtree @ dtree @ dtree @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c) @ (product_Sigma @ dtree @ dtree @ (collect @ dtree @ (^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ tr1)))) @ (^[Y0 : dtree]: (collect @ dtree @ (^[Y1 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y1) @ (cont @ tr2)))))))))),
% 5.32/2.48 inference(definition_unfolding,[status(thm)],[f969,f897,f897])).
% 5.32/2.48 thf(f969,plain,(
% 5.32/2.48 ((member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ t2) @ (cont @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) != (member @ dtree @ t2 @ (image @ product_prod @ dtree @ dtree @ dtree @ parall1899940088le_par @ (product_Sigma @ dtree @ dtree @ (collect @ dtree @ (^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ tr1)))) @ (^[Y0 : dtree]: (collect @ dtree @ (^[Y1 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y1) @ (cont @ tr2)))))))))),
% 5.32/2.48 inference(cnf_transformation,[status(thm)],[f687])).
% 5.32/2.48 thf(f687,plain,(
% 5.32/2.48 ((member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ t2) @ (cont @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) != (member @ dtree @ t2 @ (image @ product_prod @ dtree @ dtree @ dtree @ parall1899940088le_par @ (product_Sigma @ dtree @ dtree @ (collect @ dtree @ (^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ tr1)))) @ (^[Y0 : dtree]: (collect @ dtree @ (^[Y1 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y1) @ (cont @ tr2)))))))))),
% 5.32/2.48 inference(flattening,[status(thm)],[f642])).
% 5.32/2.48 thf(f642,plain,(
% 5.32/2.48 ~((member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ t2) @ (cont @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) = (member @ dtree @ t2 @ (image @ product_prod @ dtree @ dtree @ dtree @ parall1899940088le_par @ (product_Sigma @ dtree @ dtree @ (collect @ dtree @ (^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ tr1)))) @ (^[Y0 : dtree]: (collect @ dtree @ (^[Y1 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y1) @ (cont @ tr2)))))))))),
% 5.32/2.48 inference(fool_elimination,[status(thm)],[f641])).
% 5.32/2.48 thf(f641,plain,(
% 5.32/2.48 ~((member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ t2) @ (cont @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) = (member @ dtree @ t2 @ (image @ product_prod @ dtree @ dtree @ dtree @ parall1899940088le_par @ (product_Sigma @ dtree @ dtree @ (collect @ dtree @ (^[X0 : dtree] : (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ X0) @ (cont @ tr1)))) @ (^[X1 : dtree] : (collect @ dtree @ (^[X2 : dtree] : (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ X2) @ (cont @ tr2)))))))))),
% 5.32/2.48 inference(rectify,[status(thm)],[f273])).
% 5.32/2.48 thf(f273,negated_conjecture,(
% 5.32/2.48 ~((member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ t2) @ (cont @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) = (member @ dtree @ t2 @ (image @ product_prod @ dtree @ dtree @ dtree @ parall1899940088le_par @ (product_Sigma @ dtree @ dtree @ (collect @ dtree @ (^[X16 : dtree] : (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ X16) @ (cont @ tr1)))) @ (^[X10 : dtree] : (collect @ dtree @ (^[X16 : dtree] : (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ X16) @ (cont @ tr2)))))))))),
% 5.32/2.48 inference(negated_conjecture,[status(cth)],[f272])).
% 5.32/2.48 thf(f272,conjecture,(
% 5.32/2.48 ((member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ t2) @ (cont @ (parall1899940088le_par @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) = (member @ dtree @ t2 @ (image @ product_prod @ dtree @ dtree @ dtree @ parall1899940088le_par @ (product_Sigma @ dtree @ dtree @ (collect @ dtree @ (^[X16 : dtree] : (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ X16) @ (cont @ tr1)))) @ (^[X10 : dtree] : (collect @ dtree @ (^[X16 : dtree] : (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ X16) @ (cont @ tr2)))))))))),
% 5.32/2.48 file('/export/starexec/sandbox/tmp/tmp.ssUgw0micP/DTF2THF_15689.p',conj_0)).
% 5.32/2.48 thf(f1166,plain,(
% 5.32/2.48 spl93_1 | spl93_2),
% 5.32/2.48 inference(avatar_split_clause,[status(thm)],[f1147,f1163,f1159])).
% 5.32/2.48 thf(f1147,plain,(
% 5.32/2.48 ((member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ t2) @ (cont @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)))) = $true) | ((member @ dtree @ t2 @ (image @ product_prod @ dtree @ dtree @ dtree @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c) @ (product_Sigma @ dtree @ dtree @ (collect @ dtree @ (^[Y0 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y0) @ (cont @ tr1)))) @ (^[Y0 : dtree]: (collect @ dtree @ (^[Y1 : dtree]: (member @ sum_sum @ t @ dtree @ (sum_Inr @ dtree @ t @ Y1) @ (cont @ tr2)))))))) = $true)),
% 5.32/2.48 inference(binary_proxy_clausification,[status(thm)],[f1039])).
% 5.32/2.48 % SZS output end Proof for DTF2THF_15689
% 5.32/2.48 % (15831)------------------------------
% 5.32/2.48 % (15831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.32/2.48 % (15831)Termination reason: Refutation
% 5.32/2.48
% 5.32/2.48 % (15831)Memory used [KB]: 8443
% 5.32/2.48 % (15831)Time elapsed: 0.378 s
% 5.32/2.48 % (15831)Instructions burned: 689 (million)
% 5.32/2.48 % (15831)------------------------------
% 5.32/2.48 % (15831)------------------------------
% 5.32/2.48 % (15800)Success in time 0.543 s
%------------------------------------------------------------------------------