%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP128^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n020.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:46 AM UTC 2026
% Result : Theorem 2.98s 1.64s
% Output : Refutation 2.98s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP128^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n020.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:27:27 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.30/0.45 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : ITP128^2 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.46 % Domain : Interactive Theorem Proving
% 0.30/0.46 % Problem : Sledgehammer Monomorphic_Monad problem prob_47__7041882_1
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English :
% 0.30/0.46
% 0.30/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.46 % Source : [Des21]
% 0.30/0.46 % Names : Monomorphic_Monad/prob_47__7041882_1 [Des21]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.33 v8.1.0, 0.25 v7.5.0
% 0.30/0.46 % Syntax : Number of formulae : 346 ( 95 unt; 62 typ; 0 def)
% 0.30/0.46 % Number of atoms : 1001 ( 298 equ; 10 cnn)
% 0.30/0.46 % Maximal formula atoms : 20 ( 3 avg)
% 0.30/0.46 % Number of connectives : 4591 ( 18 ~; 1 |; 32 &;4155 @)
% 0.30/0.46 % ( 0 <=>; 385 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 25 ( 11 avg)
% 0.30/0.46 % Number of types : 5 ( 4 usr)
% 0.30/0.46 % Number of type conns : 1318 (1318 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 62 ( 58 usr; 3 con; 0-8 aty)
% 0.30/0.46 % Number of variables : 1767 ( 254 ^;1412 !; 19 ?;1767 :)
% 0.30/0.46 % ( 82 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.46 % SPC : TH1_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:15:22.560
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % Could-be-implicit typings (7)
% 0.30/0.46 thf(ty_t_Multiset_Omultiset,type,
% 0.30/0.46 multiset: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_FSet_Ofset,type,
% 0.30/0.46 fset: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Set_Oset,type,
% 0.30/0.46 set: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_tf_d,type,
% 0.30/0.46 d: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_tf_c,type,
% 0.30/0.46 c: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_tf_b,type,
% 0.30/0.46 b: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_tf_a,type,
% 0.30/0.46 a: $tType ).
% 0.30/0.46
% 0.30/0.46 % Explicit typings (55)
% 0.30/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.30/0.46 preorder:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_HOL_Otype,type,
% 0.30/0.46 type:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oplus,type,
% 0.30/0.46 plus:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.30/0.46 order:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.30/0.46 group_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.30/0.46 semigroup_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.30/0.46 comm_monoid_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.30/0.46 ab_semigroup_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Complete__Partial__Order_Occpo,type,
% 0.30/0.46 comple1141879883l_ccpo:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.30/0.46 cancel_semigroup_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.30/0.46 ordere779506340up_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.30/0.46 comple187826305attice:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.30/0.46 canoni770627133id_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.30/0.46 ordere236663937imp_le:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_BNF__Def_Orel__fun,type,
% 0.30/0.46 bNF_rel_fun:
% 0.30/0.46 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp,type,
% 0.30/0.46 comple939513234o_fixp:
% 0.30/0.46 !>[A: $tType] : ( ( A > A ) > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Complete__Partial__Order_Omonotone,type,
% 0.30/0.46 comple1396247847notone:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_FSet_Offold,type,
% 0.30/0.46 ffold:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( fset @ A ) > B ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
% 0.30/0.46 finite100568337ommute:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Finite__Set_Ocomp__fun__idem,type,
% 0.30/0.46 finite_comp_fun_idem:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Finite__Set_Ocomp__fun__idem__axioms,type,
% 0.30/0.46 finite852775215axioms:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Finite__Set_Ofold__graph,type,
% 0.30/0.46 finite_fold_graph:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Finite__Set_Ofolding,type,
% 0.30/0.46 finite_folding:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Finite__Set_Ofolding__idem,type,
% 0.30/0.46 finite_folding_idem:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Finite__Set_Ofolding__idem__axioms,type,
% 0.30/0.46 finite1921348288axioms:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Fun_Ocomp,type,
% 0.30/0.46 comp:
% 0.30/0.46 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Fun_Ofun__upd,type,
% 0.30/0.46 fun_upd:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Fun_Omap__fun,type,
% 0.30/0.46 map_fun:
% 0.30/0.46 !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.30/0.46 plus_plus:
% 0.30/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_If,type,
% 0.30/0.46 if:
% 0.30/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Multiset_Ofold__mset,type,
% 0.30/0.46 fold_mset:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( multiset @ A ) > B ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.46 ord_less_eq:
% 0.30/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oorder__class_Omono,type,
% 0.30/0.46 order_mono:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Partial__Function_Ofun__ord,type,
% 0.30/0.46 partial_fun_ord:
% 0.30/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Quotient_OBex1__rel,type,
% 0.30/0.46 bex1_rel:
% 0.30/0.46 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Quotient_OQuotient3,type,
% 0.30/0.46 quotient3:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Relation_ODomainp,type,
% 0.30/0.46 domainp:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Relation_Oreflp,type,
% 0.30/0.46 reflp:
% 0.30/0.46 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Relation_Orelcompp,type,
% 0.30/0.46 relcompp:
% 0.30/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_OCollect,type,
% 0.30/0.46 collect:
% 0.30/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Transfer_Obi__total,type,
% 0.30/0.46 bi_total:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Transfer_Obi__unique,type,
% 0.30/0.46 bi_unique:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Transfer_Oleft__total,type,
% 0.30/0.46 left_total:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Transfer_Oleft__unique,type,
% 0.30/0.46 left_unique:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Transfer_Orev__implies,type,
% 0.30/0.46 rev_implies: $o > $o > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Transfer_Oright__total,type,
% 0.30/0.46 right_total:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Transfer_Oright__unique,type,
% 0.30/0.46 right_unique:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Transfer_Otransfer__bforall,type,
% 0.30/0.46 transfer_bforall:
% 0.30/0.46 !>[A: $tType] : ( ( A > $o ) > ( A > $o ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Transfer_Otransfer__forall,type,
% 0.30/0.46 transfer_forall:
% 0.30/0.46 !>[A: $tType] : ( ( A > $o ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Transitive__Closure_Ortranclp,type,
% 0.30/0.46 transitive_rtranclp:
% 0.30/0.46 !>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_member,type,
% 0.30/0.46 member:
% 0.30/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_A,type,
% 0.30/0.46 a2: a > b > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_B,type,
% 0.30/0.46 b2: c > d > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_f1,type,
% 0.30/0.46 f1: a > c > c ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_f2,type,
% 0.30/0.46 f2: b > d > d ).
% 0.30/0.46
% 0.30/0.46 % Relevant facts (256)
% 0.30/0.46 thf(fact_0_assms_I2_J,axiom,
% 0.30/0.46 finite100568337ommute @ a @ c @ f1 ).
% 0.30/0.46
% 0.30/0.46 % assms(2)
% 0.30/0.46 thf(fact_1_comp__fun__commute_Ofun__left__comm,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: A > B > B,Y: A,X: A,Z: B] :
% 0.30/0.46 ( ( finite100568337ommute @ A @ B @ F )
% 0.30/0.46 => ( ( F @ Y @ ( F @ X @ Z ) )
% 0.30/0.46 = ( F @ X @ ( F @ Y @ Z ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_commute.fun_left_comm
% 0.30/0.46 thf(fact_2_assms_I3_J,axiom,
% 0.30/0.46 finite100568337ommute @ b @ d @ f2 ).
% 0.30/0.46
% 0.30/0.46 % assms(3)
% 0.30/0.46 thf(fact_3__C12_C,axiom,
% 0.30/0.46 bNF_rel_fun @ a @ b @ ( c > c ) @ ( d > d ) @ a2 @ ( bNF_rel_fun @ c @ d @ c @ d @ b2 @ b2 ) @ f1 @ f2 ).
% 0.30/0.46
% 0.30/0.46 % "12"
% 0.30/0.46 thf(fact_4_comp__fun__commute_Ofold__mset__fusion,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,F: A > B > B,G: A > C > C,H: C > B,W: C,A2: multiset @ A] :
% 0.30/0.46 ( ( finite100568337ommute @ A @ B @ F )
% 0.30/0.46 => ( ( finite100568337ommute @ A @ C @ G )
% 0.30/0.46 => ( ! [X2: A,Y2: C] :
% 0.30/0.46 ( ( H @ ( G @ X2 @ Y2 ) )
% 0.30/0.46 = ( F @ X2 @ ( H @ Y2 ) ) )
% 0.30/0.46 => ( ( H @ ( fold_mset @ A @ C @ G @ W @ A2 ) )
% 0.30/0.46 = ( fold_mset @ A @ B @ F @ ( H @ W ) @ A2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_commute.fold_mset_fusion
% 0.30/0.46 thf(fact_5_comp__fun__commute_Ofold__mset__fun__left__comm,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B,X: A,S: B,M: multiset @ A] :
% 0.30/0.46 ( ( finite100568337ommute @ A @ B @ F )
% 0.30/0.46 => ( ( F @ X @ ( fold_mset @ A @ B @ F @ S @ M ) )
% 0.30/0.46 = ( fold_mset @ A @ B @ F @ ( F @ X @ S ) @ M ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_commute.fold_mset_fun_left_comm
% 0.30/0.46 thf(fact_6_comp__fun__idem_Oaxioms_I1_J,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B] :
% 0.30/0.46 ( ( finite_comp_fun_idem @ A @ B @ F )
% 0.30/0.46 => ( finite100568337ommute @ A @ B @ F ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_idem.axioms(1)
% 0.30/0.46 thf(fact_7_comp__fun__commute_Ofold__graph__determ,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: A > B > B,Z: B,A2: set @ A,X: B,Y: B] :
% 0.30/0.46 ( ( finite100568337ommute @ A @ B @ F )
% 0.30/0.46 => ( ( finite_fold_graph @ A @ B @ F @ Z @ A2 @ X )
% 0.30/0.46 => ( ( finite_fold_graph @ A @ B @ F @ Z @ A2 @ Y )
% 0.30/0.46 => ( Y = X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_commute.fold_graph_determ
% 0.30/0.46 thf(fact_8_comp__fun__commute_Offold__fun__left__comm,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B,X: A,Z: B,A2: fset @ A] :
% 0.30/0.46 ( ( finite100568337ommute @ A @ B @ F )
% 0.30/0.46 => ( ( F @ X @ ( ffold @ A @ B @ F @ Z @ A2 ) )
% 0.30/0.46 = ( ffold @ A @ B @ F @ ( F @ X @ Z ) @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_commute.ffold_fun_left_comm
% 0.30/0.46 thf(fact_9_comp__fun__commute_Ocomp__comp__fun__commute,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,F: A > B > B,G: C > A] :
% 0.30/0.46 ( ( finite100568337ommute @ A @ B @ F )
% 0.30/0.46 => ( finite100568337ommute @ C @ B @ ( comp @ A @ ( B > B ) @ C @ F @ G ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_commute.comp_comp_fun_commute
% 0.30/0.46 thf(fact_10_comp__fun__commute__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( finite100568337ommute @ A @ B )
% 0.30/0.46 = ( ^ [F2: A > B > B] :
% 0.30/0.46 ! [Y3: A,X3: A] :
% 0.30/0.46 ( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) )
% 0.30/0.46 = ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_commute_def
% 0.30/0.46 thf(fact_11_comp__fun__commute_Ointro,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B] :
% 0.30/0.46 ( ! [Y2: A,X2: A] :
% 0.30/0.46 ( ( comp @ B @ B @ B @ ( F @ Y2 ) @ ( F @ X2 ) )
% 0.30/0.46 = ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( finite100568337ommute @ A @ B @ F ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_commute.intro
% 0.30/0.46 thf(fact_12_comp__fun__commute_Ocomp__fun__commute,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B,Y: A,X: A] :
% 0.30/0.46 ( ( finite100568337ommute @ A @ B @ F )
% 0.30/0.46 => ( ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) )
% 0.30/0.46 = ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_commute.comp_fun_commute
% 0.30/0.46 thf(fact_13_fold__graph__closed__eq,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: set @ A,B2: set @ B,F: A > B > B,G: A > B > B,Z: B] :
% 0.30/0.46 ( ! [A3: A,B3: B] :
% 0.30/0.46 ( ( member @ A @ A3 @ A2 )
% 0.30/0.46 => ( ( member @ B @ B3 @ B2 )
% 0.30/0.46 => ( ( F @ A3 @ B3 )
% 0.30/0.46 = ( G @ A3 @ B3 ) ) ) )
% 0.30/0.46 => ( ! [A3: A,B3: B] :
% 0.30/0.46 ( ( member @ A @ A3 @ A2 )
% 0.30/0.46 => ( ( member @ B @ B3 @ B2 )
% 0.30/0.46 => ( member @ B @ ( G @ A3 @ B3 ) @ B2 ) ) )
% 0.30/0.46 => ( ( member @ B @ Z @ B2 )
% 0.30/0.46 => ( ( finite_fold_graph @ A @ B @ F @ Z @ A2 )
% 0.30/0.46 = ( finite_fold_graph @ A @ B @ G @ Z @ A2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fold_graph_closed_eq
% 0.30/0.46 thf(fact_14_fold__graph__closed__lemma,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,G: A > B > B,Z: B,A2: set @ A,X: B,B2: set @ B,F: A > B > B] :
% 0.30/0.46 ( ( finite_fold_graph @ A @ B @ G @ Z @ A2 @ X )
% 0.30/0.46 => ( ! [A3: A,B3: B] :
% 0.30/0.46 ( ( member @ A @ A3 @ A2 )
% 0.30/0.46 => ( ( member @ B @ B3 @ B2 )
% 0.30/0.46 => ( ( F @ A3 @ B3 )
% 0.30/0.46 = ( G @ A3 @ B3 ) ) ) )
% 0.30/0.46 => ( ! [A3: A,B3: B] :
% 0.30/0.46 ( ( member @ A @ A3 @ A2 )
% 0.30/0.46 => ( ( member @ B @ B3 @ B2 )
% 0.30/0.46 => ( member @ B @ ( G @ A3 @ B3 ) @ B2 ) ) )
% 0.30/0.46 => ( ( member @ B @ Z @ B2 )
% 0.30/0.46 => ( ( finite_fold_graph @ A @ B @ F @ Z @ A2 @ X )
% 0.30/0.46 & ( member @ B @ X @ B2 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fold_graph_closed_lemma
% 0.30/0.46 thf(fact_15_comp__fun__idem_Ocomp__fun__idem,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B,X: A] :
% 0.30/0.46 ( ( finite_comp_fun_idem @ A @ B @ F )
% 0.30/0.46 => ( ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ X ) )
% 0.30/0.46 = ( F @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_idem.comp_fun_idem
% 0.30/0.46 thf(fact_16_comp__fun__idem_Ofun__left__idem,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: A > B > B,X: A,Z: B] :
% 0.30/0.46 ( ( finite_comp_fun_idem @ A @ B @ F )
% 0.30/0.46 => ( ( F @ X @ ( F @ X @ Z ) )
% 0.30/0.46 = ( F @ X @ Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_idem.fun_left_idem
% 0.30/0.46 thf(fact_17_comp__fun__idem_Ocomp__comp__fun__idem,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,F: A > B > B,G: C > A] :
% 0.30/0.46 ( ( finite_comp_fun_idem @ A @ B @ F )
% 0.30/0.46 => ( finite_comp_fun_idem @ C @ B @ ( comp @ A @ ( B > B ) @ C @ F @ G ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_idem.comp_comp_fun_idem
% 0.30/0.46 thf(fact_18_comp__fun__commute_Ocommute__left__comp,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,F: A > B > B,Y: A,X: A,G: C > B] :
% 0.30/0.46 ( ( finite100568337ommute @ A @ B @ F )
% 0.30/0.46 => ( ( comp @ B @ B @ C @ ( F @ Y ) @ ( comp @ B @ B @ C @ ( F @ X ) @ G ) )
% 0.30/0.46 = ( comp @ B @ B @ C @ ( F @ X ) @ ( comp @ B @ B @ C @ ( F @ Y ) @ G ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_commute.commute_left_comp
% 0.30/0.46 thf(fact_19_comp__apply,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType] :
% 0.30/0.46 ( ( comp @ B @ A @ C )
% 0.30/0.46 = ( ^ [F2: B > A,G2: C > B,X3: C] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_apply
% 0.30/0.46 thf(fact_20_rel__funI,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F: A > C,G: B > D] :
% 0.30/0.46 ( ! [X2: A,Y2: B] :
% 0.30/0.46 ( ( A2 @ X2 @ Y2 )
% 0.30/0.46 => ( B2 @ ( F @ X2 ) @ ( G @ Y2 ) ) )
% 0.30/0.46 => ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G ) ) ).
% 0.30/0.46
% 0.30/0.46 % rel_funI
% 0.30/0.46 thf(fact_21_If__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( B > B > B )
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A2 @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) )
% 0.30/0.46 @ ( if @ A )
% 0.30/0.46 @ ( if @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % If_transfer
% 0.30/0.46 thf(fact_22_rel__fun__def__butlast,axiom,
% 0.30/0.46 ! [B: $tType,D: $tType,C: $tType,E: $tType,F3: $tType,A: $tType,R: A > B > $o,S2: C > E > $o,T: D > F3 > $o,F: A > C > D,G: B > E > F3] :
% 0.30/0.46 ( ( bNF_rel_fun @ A @ B @ ( C > D ) @ ( E > F3 ) @ R @ ( bNF_rel_fun @ C @ E @ D @ F3 @ S2 @ T ) @ F @ G )
% 0.30/0.46 = ( ! [X3: A,Y3: B] :
% 0.30/0.46 ( ( R @ X3 @ Y3 )
% 0.30/0.46 => ( bNF_rel_fun @ C @ E @ D @ F3 @ S2 @ T @ ( F @ X3 ) @ ( G @ Y3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rel_fun_def_butlast
% 0.30/0.46 thf(fact_23_o__rsp_I2_J,axiom,
% 0.30/0.46 ! [E: $tType,F3: $tType,H2: $tType,G3: $tType,R1: E > F3 > $o] :
% 0.30/0.46 ( bNF_rel_fun @ ( G3 > H2 ) @ ( G3 > H2 ) @ ( ( E > G3 ) > E > H2 ) @ ( ( F3 > G3 ) > F3 > H2 )
% 0.30/0.46 @ ^ [Y4: G3 > H2,Z2: G3 > H2] : ( Y4 = Z2 )
% 0.30/0.46 @ ( bNF_rel_fun @ ( E > G3 ) @ ( F3 > G3 ) @ ( E > H2 ) @ ( F3 > H2 )
% 0.30/0.46 @ ( bNF_rel_fun @ E @ F3 @ G3 @ G3 @ R1
% 0.30/0.46 @ ^ [Y4: G3,Z2: G3] : ( Y4 = Z2 ) )
% 0.30/0.46 @ ( bNF_rel_fun @ E @ F3 @ H2 @ H2 @ R1
% 0.30/0.46 @ ^ [Y4: H2,Z2: H2] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ( comp @ G3 @ H2 @ E )
% 0.30/0.46 @ ( comp @ G3 @ H2 @ F3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % o_rsp(2)
% 0.30/0.46 thf(fact_24_o__rsp_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,R2: A > C > $o,R3: B > D > $o,R1: E > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F3 > C ) > F3 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R2 @ R3 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F3 > C ) @ ( E > B ) @ ( F3 > D ) @ ( bNF_rel_fun @ E @ F3 @ A @ C @ R1 @ R2 ) @ ( bNF_rel_fun @ E @ F3 @ B @ D @ R1 @ R3 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % o_rsp(1)
% 0.30/0.46 thf(fact_25_fun_Omap__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,D: $tType,G3: $tType,F3: $tType,Rb: A > F3 > $o,Sd: B > G3 > $o] :
% 0.30/0.46 ( bNF_rel_fun @ ( A > B ) @ ( F3 > G3 ) @ ( ( D > A ) > D > B ) @ ( ( D > F3 ) > D > G3 ) @ ( bNF_rel_fun @ A @ F3 @ B @ G3 @ Rb @ Sd )
% 0.30/0.46 @ ( bNF_rel_fun @ ( D > A ) @ ( D > F3 ) @ ( D > B ) @ ( D > G3 )
% 0.30/0.46 @ ( bNF_rel_fun @ D @ D @ A @ F3
% 0.30/0.46 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 0.30/0.46 @ Rb )
% 0.30/0.46 @ ( bNF_rel_fun @ D @ D @ B @ G3
% 0.30/0.46 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 0.30/0.46 @ Sd ) )
% 0.30/0.46 @ ( comp @ A @ B @ D )
% 0.30/0.46 @ ( comp @ F3 @ G3 @ D ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.map_transfer
% 0.30/0.46 thf(fact_26_comp__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,B2: A > C > $o,C2: B > D > $o,A2: E > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F3 > C ) > F3 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ B2 @ C2 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F3 > C ) @ ( E > B ) @ ( F3 > D ) @ ( bNF_rel_fun @ E @ F3 @ A @ C @ A2 @ B2 ) @ ( bNF_rel_fun @ E @ F3 @ B @ D @ A2 @ C2 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_transfer
% 0.30/0.46 thf(fact_27_comp__fun__idem_Ointro,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B] :
% 0.30/0.46 ( ( finite100568337ommute @ A @ B @ F )
% 0.30/0.46 => ( ( finite852775215axioms @ A @ B @ F )
% 0.30/0.46 => ( finite_comp_fun_idem @ A @ B @ F ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_idem.intro
% 0.30/0.46 thf(fact_28_comp__fun__idem__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( finite_comp_fun_idem @ A @ B )
% 0.30/0.46 = ( ^ [F2: A > B > B] :
% 0.30/0.46 ( ( finite100568337ommute @ A @ B @ F2 )
% 0.30/0.46 & ( finite852775215axioms @ A @ B @ F2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_idem_def
% 0.30/0.46 thf(fact_29_comp__fun__idem_Oaxioms_I2_J,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B] :
% 0.30/0.46 ( ( finite_comp_fun_idem @ A @ B @ F )
% 0.30/0.46 => ( finite852775215axioms @ A @ B @ F ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_idem.axioms(2)
% 0.30/0.46 thf(fact_30_comp__fun__idem__axioms_Ointro,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B] :
% 0.30/0.46 ( ! [X2: A] :
% 0.30/0.46 ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
% 0.30/0.46 = ( F @ X2 ) )
% 0.30/0.46 => ( finite852775215axioms @ A @ B @ F ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_idem_axioms.intro
% 0.30/0.46 thf(fact_31_comp__fun__idem__axioms__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( finite852775215axioms @ A @ B )
% 0.30/0.46 = ( ^ [F2: A > B > B] :
% 0.30/0.46 ! [X3: A] :
% 0.30/0.46 ( ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
% 0.30/0.46 = ( F2 @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_idem_axioms_def
% 0.30/0.46 thf(fact_32_fun_Orel__transfer,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,E: $tType,D: $tType,Sa: A > C > $o,Sc: B > E > $o] :
% 0.30/0.46 ( bNF_rel_fun @ ( A > B > $o ) @ ( C > E > $o ) @ ( ( D > A ) > ( D > B ) > $o ) @ ( ( D > C ) > ( D > E ) > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( E > $o ) @ Sa
% 0.30/0.46 @ ( bNF_rel_fun @ B @ E @ $o @ $o @ Sc
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ ( D > A ) @ ( D > C ) @ ( ( D > B ) > $o ) @ ( ( D > E ) > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ D @ D @ A @ C
% 0.30/0.46 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 0.30/0.46 @ Sa )
% 0.30/0.46 @ ( bNF_rel_fun @ ( D > B ) @ ( D > E ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ D @ D @ B @ E
% 0.30/0.46 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 0.30/0.46 @ Sc )
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ D @ D @ A @ B
% 0.30/0.46 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 ) )
% 0.30/0.46 @ ( bNF_rel_fun @ D @ D @ C @ E
% 0.30/0.46 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.rel_transfer
% 0.30/0.46 thf(fact_33_fun_Orel__refl,axiom,
% 0.30/0.46 ! [B: $tType,D: $tType,Ra: B > B > $o,X: D > B] :
% 0.30/0.46 ( ! [X2: B] : ( Ra @ X2 @ X2 )
% 0.30/0.46 => ( bNF_rel_fun @ D @ D @ B @ B
% 0.30/0.46 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 0.30/0.46 @ Ra
% 0.30/0.46 @ X
% 0.30/0.46 @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.rel_refl
% 0.30/0.46 thf(fact_34_fun_Orel__eq,axiom,
% 0.30/0.46 ! [A: $tType,D: $tType] :
% 0.30/0.46 ( ( bNF_rel_fun @ D @ D @ A @ A
% 0.30/0.46 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.30/0.46 = ( ^ [Y4: D > A,Z2: D > A] : ( Y4 = Z2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.rel_eq
% 0.30/0.46 thf(fact_35_rel__fun__mono_H,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,Y5: A > B > $o,X4: A > B > $o,A2: C > D > $o,B2: C > D > $o,F: A > C,G: B > D] :
% 0.30/0.46 ( ! [X2: A,Y2: B] :
% 0.30/0.46 ( ( Y5 @ X2 @ Y2 )
% 0.30/0.46 => ( X4 @ X2 @ Y2 ) )
% 0.30/0.46 => ( ! [X2: C,Y2: D] :
% 0.30/0.46 ( ( A2 @ X2 @ Y2 )
% 0.30/0.46 => ( B2 @ X2 @ Y2 ) )
% 0.30/0.46 => ( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A2 @ F @ G )
% 0.30/0.46 => ( bNF_rel_fun @ A @ B @ C @ D @ Y5 @ B2 @ F @ G ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rel_fun_mono'
% 0.30/0.46 thf(fact_36_rel__fun__mono,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,X4: A > B > $o,A2: C > D > $o,F: A > C,G: B > D,Y5: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A2 @ F @ G )
% 0.30/0.46 => ( ! [X2: A,Y2: B] :
% 0.30/0.46 ( ( Y5 @ X2 @ Y2 )
% 0.30/0.46 => ( X4 @ X2 @ Y2 ) )
% 0.30/0.46 => ( ! [X2: C,Y2: D] :
% 0.30/0.46 ( ( A2 @ X2 @ Y2 )
% 0.30/0.46 => ( B2 @ X2 @ Y2 ) )
% 0.30/0.46 => ( bNF_rel_fun @ A @ B @ C @ D @ Y5 @ B2 @ F @ G ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rel_fun_mono
% 0.30/0.46 thf(fact_37_let__rsp,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o] :
% 0.30/0.46 ( bNF_rel_fun @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ R1 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ C @ D @ ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 ) @ R2 )
% 0.30/0.46 @ ^ [S3: A,F2: A > C] : ( F2 @ S3 )
% 0.30/0.46 @ ^ [S3: B,F2: B > D] : ( F2 @ S3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % let_rsp
% 0.30/0.46 thf(fact_38_rel__funD,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F: A > C,G: B > D,X: A,Y: B] :
% 0.30/0.46 ( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G )
% 0.30/0.46 => ( ( A2 @ X @ Y )
% 0.30/0.46 => ( B2 @ ( F @ X ) @ ( G @ Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rel_funD
% 0.30/0.46 thf(fact_39_rewriteR__comp__comp2,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H: A > C,R12: D > B,R22: A > D,F: B > E,L: D > E] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ G @ H )
% 0.30/0.46 = ( comp @ D @ B @ A @ R12 @ R22 ) )
% 0.30/0.46 => ( ( ( comp @ B @ E @ D @ F @ R12 )
% 0.30/0.46 = L )
% 0.30/0.46 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H )
% 0.30/0.46 = ( comp @ D @ E @ A @ L @ R22 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rewriteR_comp_comp2
% 0.30/0.46 thf(fact_40_rewriteL__comp__comp2,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L2: A > D,H: E > A,R4: E > D] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ F @ G )
% 0.30/0.46 = ( comp @ D @ B @ A @ L1 @ L2 ) )
% 0.30/0.46 => ( ( ( comp @ A @ D @ E @ L2 @ H )
% 0.30/0.46 = R4 )
% 0.30/0.46 => ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H ) )
% 0.30/0.46 = ( comp @ D @ B @ E @ L1 @ R4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rewriteL_comp_comp2
% 0.30/0.46 thf(fact_41_rewriteR__comp__comp,axiom,
% 0.30/0.46 ! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R4: A > B,F: B > D] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ G @ H )
% 0.30/0.46 = R4 )
% 0.30/0.46 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H )
% 0.30/0.46 = ( comp @ B @ D @ A @ F @ R4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rewriteR_comp_comp
% 0.30/0.46 thf(fact_42_rewriteL__comp__comp,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L: A > B,H: D > A] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ F @ G )
% 0.30/0.46 = L )
% 0.30/0.46 => ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H ) )
% 0.30/0.46 = ( comp @ A @ B @ D @ L @ H ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rewriteL_comp_comp
% 0.30/0.46 thf(fact_43_fun_Omap__comp,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
% 0.30/0.46 ( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
% 0.30/0.46 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.map_comp
% 0.30/0.46 thf(fact_44_comp__apply__eq,axiom,
% 0.30/0.46 ! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H: D > A,K: C > D] :
% 0.30/0.46 ( ( ( F @ ( G @ X ) )
% 0.30/0.46 = ( H @ ( K @ X ) ) )
% 0.30/0.46 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 0.30/0.46 = ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_apply_eq
% 0.30/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.46 ! [A: $tType,A4: A,P: A > $o] :
% 0.30/0.46 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 0.30/0.46 = ( P @ A4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Collect_eq
% 0.30/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ( collect @ A
% 0.30/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
% 0.30/0.46 = A2 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mem_eq
% 0.30/0.46 thf(fact_47_Collect__cong,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X2: A] :
% 0.30/0.46 ( ( P @ X2 )
% 0.30/0.46 = ( Q @ X2 ) )
% 0.30/0.46 => ( ( collect @ A @ P )
% 0.30/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_cong
% 0.30/0.46 thf(fact_48_ext,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/0.46 ( ! [X2: A] :
% 0.30/0.46 ( ( F @ X2 )
% 0.30/0.46 = ( G @ X2 ) )
% 0.30/0.46 => ( F = G ) ) ).
% 0.30/0.46
% 0.30/0.46 % ext
% 0.30/0.46 thf(fact_49_comp__eq__dest__lhs,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,A4: C > B,B4: A > C,C3: A > B,V: A] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ A4 @ B4 )
% 0.30/0.46 = C3 )
% 0.30/0.46 => ( ( A4 @ ( B4 @ V ) )
% 0.30/0.46 = ( C3 @ V ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_eq_dest_lhs
% 0.30/0.46 thf(fact_50_comp__eq__elim,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,D: $tType,A: $tType,A4: C > B,B4: A > C,C3: D > B,D2: A > D] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ A4 @ B4 )
% 0.30/0.46 = ( comp @ D @ B @ A @ C3 @ D2 ) )
% 0.30/0.46 => ! [V2: A] :
% 0.30/0.46 ( ( A4 @ ( B4 @ V2 ) )
% 0.30/0.46 = ( C3 @ ( D2 @ V2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_eq_elim
% 0.30/0.46 thf(fact_51_comp__eq__dest,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,D: $tType,A: $tType,A4: C > B,B4: A > C,C3: D > B,D2: A > D,V: A] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ A4 @ B4 )
% 0.30/0.46 = ( comp @ D @ B @ A @ C3 @ D2 ) )
% 0.30/0.46 => ( ( A4 @ ( B4 @ V ) )
% 0.30/0.46 = ( C3 @ ( D2 @ V ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_eq_dest
% 0.30/0.46 thf(fact_52_comp__assoc,axiom,
% 0.30/0.46 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
% 0.30/0.46 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
% 0.30/0.46 = ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_assoc
% 0.30/0.46 thf(fact_53_comp__def,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.46 ( ( comp @ B @ C @ A )
% 0.30/0.46 = ( ^ [F2: B > C,G2: A > B,X3: A] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_def
% 0.30/0.46 thf(fact_54_Let__transfer,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( bNF_rel_fun @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ A2 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ C @ D @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) @ B2 )
% 0.30/0.46 @ ^ [S3: A,F2: A > C] : ( F2 @ S3 )
% 0.30/0.46 @ ^ [S3: B,F2: B > D] : ( F2 @ S3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Let_transfer
% 0.30/0.46 thf(fact_55_type__copy__map__cong0,axiom,
% 0.30/0.46 ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X: C,N: D > A,H: C > D,F: A > E] :
% 0.30/0.46 ( ( ( M @ ( G @ X ) )
% 0.30/0.46 = ( N @ ( H @ X ) ) )
% 0.30/0.46 => ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M ) @ G @ X )
% 0.30/0.46 = ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N ) @ H @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_copy_map_cong0
% 0.30/0.46 thf(fact_56_function__factors__right,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,G: B > C,F: A > C] :
% 0.30/0.46 ( ( ! [X3: A] :
% 0.30/0.46 ? [Y3: B] :
% 0.30/0.46 ( ( G @ Y3 )
% 0.30/0.46 = ( F @ X3 ) ) )
% 0.30/0.46 = ( ? [H3: A > B] :
% 0.30/0.46 ( F
% 0.30/0.46 = ( comp @ B @ C @ A @ G @ H3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % function_factors_right
% 0.30/0.46 thf(fact_57_function__factors__left,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,G: A > B,F: A > C] :
% 0.30/0.46 ( ( ! [X3: A,Y3: A] :
% 0.30/0.46 ( ( ( G @ X3 )
% 0.30/0.46 = ( G @ Y3 ) )
% 0.30/0.46 => ( ( F @ X3 )
% 0.30/0.46 = ( F @ Y3 ) ) ) )
% 0.30/0.46 = ( ? [H3: B > C] :
% 0.30/0.46 ( F
% 0.30/0.46 = ( comp @ B @ C @ A @ H3 @ G ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % function_factors_left
% 0.30/0.46 thf(fact_58_comp__cong,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X: C,F4: D > A,G4: E > D,X5: E] :
% 0.30/0.46 ( ( ( F @ ( G @ X ) )
% 0.30/0.46 = ( F4 @ ( G4 @ X5 ) ) )
% 0.30/0.46 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 0.30/0.46 = ( comp @ D @ A @ E @ F4 @ G4 @ X5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_cong
% 0.30/0.46 thf(fact_59_if__rsp,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.30/0.46 => ( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( A > A > A )
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ A @ ( A > A ) @ ( A > A ) @ R @ ( bNF_rel_fun @ A @ A @ A @ A @ R @ R ) )
% 0.30/0.46 @ ( if @ A )
% 0.30/0.46 @ ( if @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % if_rsp
% 0.30/0.46 thf(fact_60_rel__funD2,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,A2: A > A > $o,B2: B > C > $o,F: A > B,G: A > C,X: A] :
% 0.30/0.46 ( ( bNF_rel_fun @ A @ A @ B @ C @ A2 @ B2 @ F @ G )
% 0.30/0.46 => ( ( A2 @ X @ X )
% 0.30/0.46 => ( B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rel_funD2
% 0.30/0.46 thf(fact_61_apply__rsp_H,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o,F: A > C,G: B > D,X: A,Y: B] :
% 0.30/0.46 ( ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 @ F @ G )
% 0.30/0.46 => ( ( R1 @ X @ Y )
% 0.30/0.46 => ( R2 @ ( F @ X ) @ ( G @ Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % apply_rsp'
% 0.30/0.46 thf(fact_62_rel__funE,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F: A > C,G: B > D,X: A,Y: B] :
% 0.30/0.46 ( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G )
% 0.30/0.46 => ( ( A2 @ X @ Y )
% 0.30/0.46 => ( B2 @ ( F @ X ) @ ( G @ Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rel_funE
% 0.30/0.46 thf(fact_63_quot__rel__rsp,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.30/0.46 => ( bNF_rel_fun @ A @ A @ ( A > $o ) @ ( A > $o ) @ R
% 0.30/0.46 @ ( bNF_rel_fun @ A @ A @ $o @ $o @ R
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 0.30/0.46 @ R
% 0.30/0.46 @ R ) ) ).
% 0.30/0.46
% 0.30/0.46 % quot_rel_rsp
% 0.30/0.46 thf(fact_64_Quotient3__rep__reflp,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A4: B] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.30/0.46 => ( R @ ( Rep @ A4 ) @ ( Rep @ A4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Quotient3_rep_reflp
% 0.30/0.46 thf(fact_65_Quotient3__rep__abs,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.30/0.46 => ( ( R @ R4 @ R4 )
% 0.30/0.46 => ( R @ ( Rep @ ( Abs @ R4 ) ) @ R4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Quotient3_rep_abs
% 0.30/0.46 thf(fact_66_Quotient3__rel__rep,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A4: B,B4: B] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.30/0.46 => ( ( R @ ( Rep @ A4 ) @ ( Rep @ B4 ) )
% 0.30/0.46 = ( A4 = B4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Quotient3_rel_rep
% 0.30/0.46 thf(fact_67_Quotient3__rel__abs,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S: A] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.30/0.46 => ( ( R @ R4 @ S )
% 0.30/0.46 => ( ( Abs @ R4 )
% 0.30/0.46 = ( Abs @ S ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Quotient3_rel_abs
% 0.30/0.46 thf(fact_68_Quotient3__abs__rep,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A4: B] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.30/0.46 => ( ( Abs @ ( Rep @ A4 ) )
% 0.30/0.46 = A4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Quotient3_abs_rep
% 0.30/0.46 thf(fact_69_rep__abs__rsp__left,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.30/0.46 => ( ( R @ X1 @ X22 )
% 0.30/0.46 => ( R @ ( Rep @ ( Abs @ X1 ) ) @ X22 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rep_abs_rsp_left
% 0.30/0.46 thf(fact_70_Quotient3__refl2,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S: A] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.30/0.46 => ( ( R @ R4 @ S )
% 0.30/0.46 => ( R @ S @ S ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Quotient3_refl2
% 0.30/0.46 thf(fact_71_Quotient3__refl1,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S: A] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.30/0.46 => ( ( R @ R4 @ S )
% 0.30/0.46 => ( R @ R4 @ R4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Quotient3_refl1
% 0.30/0.46 thf(fact_72_Quotient3__rel,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S: A] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.30/0.46 => ( ( ( R @ R4 @ R4 )
% 0.30/0.46 & ( R @ S @ S )
% 0.30/0.46 & ( ( Abs @ R4 )
% 0.30/0.46 = ( Abs @ S ) ) )
% 0.30/0.46 = ( R @ R4 @ S ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Quotient3_rel
% 0.30/0.46 thf(fact_73_Quotient3__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( quotient3 @ A @ B )
% 0.30/0.46 = ( ^ [R5: A > A > $o,Abs2: A > B,Rep2: B > A] :
% 0.30/0.46 ( ! [A5: B] :
% 0.30/0.46 ( ( Abs2 @ ( Rep2 @ A5 ) )
% 0.30/0.46 = A5 )
% 0.30/0.46 & ! [A5: B] : ( R5 @ ( Rep2 @ A5 ) @ ( Rep2 @ A5 ) )
% 0.30/0.46 & ! [R6: A,S3: A] :
% 0.30/0.46 ( ( R5 @ R6 @ S3 )
% 0.30/0.46 = ( ( R5 @ R6 @ R6 )
% 0.30/0.46 & ( R5 @ S3 @ S3 )
% 0.30/0.46 & ( ( Abs2 @ R6 )
% 0.30/0.46 = ( Abs2 @ S3 ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Quotient3_def
% 0.30/0.46 thf(fact_74_rep__abs__rsp,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.30/0.46 => ( ( R @ X1 @ X22 )
% 0.30/0.46 => ( R @ X1 @ ( Rep @ ( Abs @ X22 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rep_abs_rsp
% 0.30/0.46 thf(fact_75_equals__rsp,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,Xa: A,Xb: A,Ya: A,Yb: A] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.30/0.46 => ( ( R @ Xa @ Xb )
% 0.30/0.46 => ( ( R @ Ya @ Yb )
% 0.30/0.46 => ( ( R @ Xa @ Ya )
% 0.30/0.46 = ( R @ Xb @ Yb ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % equals_rsp
% 0.30/0.46 thf(fact_76_Quotient3I,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Abs: B > A,Rep: A > B,R: B > B > $o] :
% 0.30/0.46 ( ! [A3: A] :
% 0.30/0.46 ( ( Abs @ ( Rep @ A3 ) )
% 0.30/0.46 = A3 )
% 0.30/0.46 => ( ! [A3: A] : ( R @ ( Rep @ A3 ) @ ( Rep @ A3 ) )
% 0.30/0.46 => ( ! [R7: B,S4: B] :
% 0.30/0.46 ( ( R @ R7 @ S4 )
% 0.30/0.46 = ( ( R @ R7 @ R7 )
% 0.30/0.46 & ( R @ S4 @ S4 )
% 0.30/0.46 & ( ( Abs @ R7 )
% 0.30/0.46 = ( Abs @ S4 ) ) ) )
% 0.30/0.46 => ( quotient3 @ B @ A @ R @ Abs @ Rep ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Quotient3I
% 0.30/0.46 thf(fact_77_cond__prs,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: A > A > $o,Absf: A > B,Repf: B > A,A4: $o,B4: B,C3: B] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Absf @ Repf )
% 0.30/0.46 => ( ( A4
% 0.30/0.46 => ( ( Absf @ ( if @ A @ A4 @ ( Repf @ B4 ) @ ( Repf @ C3 ) ) )
% 0.30/0.46 = B4 ) )
% 0.30/0.46 & ( ~ A4
% 0.30/0.46 => ( ( Absf @ ( if @ A @ A4 @ ( Repf @ B4 ) @ ( Repf @ C3 ) ) )
% 0.30/0.46 = C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % cond_prs
% 0.30/0.46 thf(fact_78_apply__rspQ3,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R2: C > C > $o,F: A > C,G: A > C,X: A,Y: A] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
% 0.30/0.46 => ( ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 @ F @ G )
% 0.30/0.46 => ( ( R1 @ X @ Y )
% 0.30/0.46 => ( R2 @ ( F @ X ) @ ( G @ Y ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % apply_rspQ3
% 0.30/0.46 thf(fact_79_apply__rspQ3_H_H,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,S2: C > C > $o,F: A > C,X: B] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 0.30/0.46 => ( ( bNF_rel_fun @ A @ A @ C @ C @ R @ S2 @ F @ F )
% 0.30/0.46 => ( S2 @ ( F @ ( Rep @ X ) ) @ ( F @ ( Rep @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % apply_rspQ3''
% 0.30/0.46 thf(fact_80_bex1__rel__rsp,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > A > $o,Absf: A > B,Repf: B > A] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R @ Absf @ Repf )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( A > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ A @ $o @ $o @ R
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 0.30/0.46 @ ( bex1_rel @ A @ R )
% 0.30/0.46 @ ( bex1_rel @ A @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bex1_rel_rsp
% 0.30/0.46 thf(fact_81_map__fun__parametric,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,E: $tType,F3: $tType,H2: $tType,G3: $tType,D: $tType,C: $tType,A2: A > C > $o,B2: B > D > $o,C2: E > G3 > $o,D3: F3 > H2 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > F3 ) > ( B > E ) > A > F3 ) @ ( ( G3 > H2 ) > ( D > G3 ) > C > H2 ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ ( E > F3 ) @ ( G3 > H2 ) @ ( ( B > E ) > A > F3 ) @ ( ( D > G3 ) > C > H2 ) @ ( bNF_rel_fun @ E @ G3 @ F3 @ H2 @ C2 @ D3 ) @ ( bNF_rel_fun @ ( B > E ) @ ( D > G3 ) @ ( A > F3 ) @ ( C > H2 ) @ ( bNF_rel_fun @ B @ D @ E @ G3 @ B2 @ C2 ) @ ( bNF_rel_fun @ A @ C @ F3 @ H2 @ A2 @ D3 ) ) ) @ ( map_fun @ A @ B @ E @ F3 ) @ ( map_fun @ C @ D @ G3 @ H2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % map_fun_parametric
% 0.30/0.46 thf(fact_82_folding__idem__axioms__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( finite1921348288axioms @ A @ B )
% 0.30/0.46 = ( ^ [F2: A > B > B] :
% 0.30/0.46 ! [X3: A] :
% 0.30/0.46 ( ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
% 0.30/0.46 = ( F2 @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % folding_idem_axioms_def
% 0.30/0.46 thf(fact_83_folding__idem__axioms_Ointro,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B] :
% 0.30/0.46 ( ! [X2: A] :
% 0.30/0.46 ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
% 0.30/0.46 = ( F @ X2 ) )
% 0.30/0.46 => ( finite1921348288axioms @ A @ B @ F ) ) ).
% 0.30/0.46
% 0.30/0.46 % folding_idem_axioms.intro
% 0.30/0.46 thf(fact_84_fun__quotient3,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs22: C > D,Rep22: D > C] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 0.30/0.46 => ( ( quotient3 @ C @ D @ R2 @ Abs22 @ Rep22 )
% 0.30/0.46 => ( quotient3 @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 ) @ ( map_fun @ B @ A @ C @ D @ Rep12 @ Abs22 ) @ ( map_fun @ A @ B @ D @ C @ Abs12 @ Rep22 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_quotient3
% 0.30/0.46 thf(fact_85_fun__ord__parametric,axiom,
% 0.30/0.46 ! [C: $tType,D: $tType,A: $tType,B: $tType,F3: $tType,E: $tType,C2: A > B > $o,A2: C > E > $o,B2: D > F3 > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ C2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( C > D > $o ) @ ( E > F3 > $o ) @ ( ( A > C ) > ( A > D ) > $o ) @ ( ( B > E ) > ( B > F3 ) > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ E @ ( D > $o ) @ ( F3 > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ D @ F3 @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ ( A > C ) @ ( B > E ) @ ( ( A > D ) > $o ) @ ( ( B > F3 ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ E @ C2 @ A2 )
% 0.30/0.46 @ ( bNF_rel_fun @ ( A > D ) @ ( B > F3 ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ D @ F3 @ C2 @ B2 )
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ( partial_fun_ord @ C @ D @ A )
% 0.30/0.46 @ ( partial_fun_ord @ E @ F3 @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_ord_parametric
% 0.30/0.46 thf(fact_86_comp__fun__commute_Ofold__mset__union,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B,S: B,M: multiset @ A,N: multiset @ A] :
% 0.30/0.46 ( ( finite100568337ommute @ A @ B @ F )
% 0.30/0.46 => ( ( fold_mset @ A @ B @ F @ S @ ( plus_plus @ ( multiset @ A ) @ M @ N ) )
% 0.30/0.46 = ( fold_mset @ A @ B @ F @ ( fold_mset @ A @ B @ F @ S @ M ) @ N ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_commute.fold_mset_union
% 0.30/0.46 thf(fact_87_map__fun__apply,axiom,
% 0.30/0.46 ! [D: $tType,A: $tType,C: $tType,B: $tType] :
% 0.30/0.46 ( ( map_fun @ B @ C @ D @ A )
% 0.30/0.46 = ( ^ [F2: B > C,G2: D > A,H3: C > D,X3: B] : ( G2 @ ( H3 @ ( F2 @ X3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % map_fun_apply
% 0.30/0.46 thf(fact_88_o__prs_I1_J,axiom,
% 0.30/0.46 ! [C: $tType,E: $tType,A: $tType,B: $tType,F3: $tType,D: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R2: C > C > $o,Abs23: C > D,Rep23: D > C,R3: E > E > $o,Abs3: E > F3,Rep3: F3 > E] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
% 0.30/0.46 => ( ( quotient3 @ C @ D @ R2 @ Abs23 @ Rep23 )
% 0.30/0.46 => ( ( quotient3 @ E @ F3 @ R3 @ Abs3 @ Rep3 )
% 0.30/0.46 => ( ( map_fun @ ( D > F3 ) @ ( C > E ) @ ( ( A > C ) > A > E ) @ ( ( B > D ) > B > F3 ) @ ( map_fun @ C @ D @ F3 @ E @ Abs23 @ Rep3 ) @ ( map_fun @ ( B > D ) @ ( A > C ) @ ( A > E ) @ ( B > F3 ) @ ( map_fun @ A @ B @ D @ C @ Abs1 @ Rep23 ) @ ( map_fun @ B @ A @ E @ F3 @ Rep1 @ Abs3 ) ) @ ( comp @ C @ E @ A ) )
% 0.30/0.46 = ( comp @ D @ F3 @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % o_prs(1)
% 0.30/0.46 thf(fact_89_let__prs,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType,D: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R2: C > C > $o,Abs23: C > D,Rep23: D > C] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
% 0.30/0.46 => ( ( quotient3 @ C @ D @ R2 @ Abs23 @ Rep23 )
% 0.30/0.46 => ( ( map_fun @ D @ C @ ( ( C > A ) > A ) @ ( ( D > B ) > B ) @ Rep23 @ ( map_fun @ ( D > B ) @ ( C > A ) @ A @ B @ ( map_fun @ C @ D @ B @ A @ Abs23 @ Rep1 ) @ Abs1 )
% 0.30/0.46 @ ^ [S3: C,F2: C > A] : ( F2 @ S3 ) )
% 0.30/0.46 = ( ^ [S3: D,F2: D > B] : ( F2 @ S3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % let_prs
% 0.30/0.46 thf(fact_90_bi__total__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( bi_total @ A @ A
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_total_eq
% 0.30/0.46 thf(fact_91_bi__total__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( bi_total @ A @ B )
% 0.30/0.46 = ( ^ [R5: A > B > $o] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ? [X6: B] : ( R5 @ X3 @ X6 )
% 0.30/0.46 & ! [Y3: B] :
% 0.30/0.46 ? [X3: A] : ( R5 @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_total_def
% 0.30/0.46 thf(fact_92_union__assoc,axiom,
% 0.30/0.46 ! [A: $tType,M: multiset @ A,N: multiset @ A,K2: multiset @ A] :
% 0.30/0.46 ( ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M @ N ) @ K2 )
% 0.30/0.46 = ( plus_plus @ ( multiset @ A ) @ M @ ( plus_plus @ ( multiset @ A ) @ N @ K2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % union_assoc
% 0.30/0.46 thf(fact_93_union__lcomm,axiom,
% 0.30/0.46 ! [A: $tType,M: multiset @ A,N: multiset @ A,K2: multiset @ A] :
% 0.30/0.46 ( ( plus_plus @ ( multiset @ A ) @ M @ ( plus_plus @ ( multiset @ A ) @ N @ K2 ) )
% 0.30/0.46 = ( plus_plus @ ( multiset @ A ) @ N @ ( plus_plus @ ( multiset @ A ) @ M @ K2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % union_lcomm
% 0.30/0.46 thf(fact_94_union__commute,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( plus_plus @ ( multiset @ A ) )
% 0.30/0.46 = ( ^ [M2: multiset @ A,N2: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ N2 @ M2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % union_commute
% 0.30/0.46 thf(fact_95_union__left__cancel,axiom,
% 0.30/0.46 ! [A: $tType,K2: multiset @ A,M: multiset @ A,N: multiset @ A] :
% 0.30/0.46 ( ( ( plus_plus @ ( multiset @ A ) @ K2 @ M )
% 0.30/0.46 = ( plus_plus @ ( multiset @ A ) @ K2 @ N ) )
% 0.30/0.46 = ( M = N ) ) ).
% 0.30/0.46
% 0.30/0.46 % union_left_cancel
% 0.30/0.46 thf(fact_96_union__right__cancel,axiom,
% 0.30/0.46 ! [A: $tType,M: multiset @ A,K2: multiset @ A,N: multiset @ A] :
% 0.30/0.46 ( ( ( plus_plus @ ( multiset @ A ) @ M @ K2 )
% 0.30/0.46 = ( plus_plus @ ( multiset @ A ) @ N @ K2 ) )
% 0.30/0.46 = ( M = N ) ) ).
% 0.30/0.46
% 0.30/0.46 % union_right_cancel
% 0.30/0.46 thf(fact_97_multi__union__self__other__eq,axiom,
% 0.30/0.46 ! [A: $tType,A2: multiset @ A,X4: multiset @ A,Y5: multiset @ A] :
% 0.30/0.46 ( ( ( plus_plus @ ( multiset @ A ) @ A2 @ X4 )
% 0.30/0.46 = ( plus_plus @ ( multiset @ A ) @ A2 @ Y5 ) )
% 0.30/0.46 => ( X4 = Y5 ) ) ).
% 0.30/0.46
% 0.30/0.46 % multi_union_self_other_eq
% 0.30/0.46 thf(fact_98_comp__fun__commute__plus__mset,axiom,
% 0.30/0.46 ! [A: $tType] : ( finite100568337ommute @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_fun_commute_plus_mset
% 0.30/0.46 thf(fact_99_bex1__rel__aux,axiom,
% 0.30/0.46 ! [A: $tType,R: A > A > $o,X: A > $o,Y: A > $o] :
% 0.30/0.46 ( ! [Xa2: A,Ya2: A] :
% 0.30/0.46 ( ( R @ Xa2 @ Ya2 )
% 0.30/0.46 => ( ( X @ Xa2 )
% 0.30/0.46 = ( Y @ Ya2 ) ) )
% 0.30/0.46 => ( ( bex1_rel @ A @ R @ X )
% 0.30/0.46 => ( bex1_rel @ A @ R @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bex1_rel_aux
% 0.30/0.46 thf(fact_100_bex1__rel__aux2,axiom,
% 0.30/0.46 ! [A: $tType,R: A > A > $o,X: A > $o,Y: A > $o] :
% 0.30/0.46 ( ! [Xa2: A,Ya2: A] :
% 0.30/0.46 ( ( R @ Xa2 @ Ya2 )
% 0.30/0.46 => ( ( X @ Xa2 )
% 0.30/0.46 = ( Y @ Ya2 ) ) )
% 0.30/0.46 => ( ( bex1_rel @ A @ R @ Y )
% 0.30/0.46 => ( bex1_rel @ A @ R @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bex1_rel_aux2
% 0.30/0.46 thf(fact_101_map__fun_Ocompositionality,axiom,
% 0.30/0.46 ! [D: $tType,F3: $tType,C: $tType,E: $tType,B: $tType,A: $tType,F: E > C,G: D > F3,H: C > A,I: B > D,Fun: A > B] :
% 0.30/0.46 ( ( map_fun @ E @ C @ D @ F3 @ F @ G @ ( map_fun @ C @ A @ B @ D @ H @ I @ Fun ) )
% 0.30/0.46 = ( map_fun @ E @ A @ B @ F3 @ ( comp @ C @ A @ E @ H @ F ) @ ( comp @ D @ F3 @ B @ G @ I ) @ Fun ) ) ).
% 0.30/0.46
% 0.30/0.46 % map_fun.compositionality
% 0.30/0.46 thf(fact_102_map__fun_Ocomp,axiom,
% 0.30/0.46 ! [E: $tType,C: $tType,A: $tType,F3: $tType,D: $tType,B: $tType,F: E > C,G: D > F3,H: C > A,I: B > D] :
% 0.30/0.46 ( ( comp @ ( C > D ) @ ( E > F3 ) @ ( A > B ) @ ( map_fun @ E @ C @ D @ F3 @ F @ G ) @ ( map_fun @ C @ A @ B @ D @ H @ I ) )
% 0.30/0.46 = ( map_fun @ E @ A @ B @ F3 @ ( comp @ C @ A @ E @ H @ F ) @ ( comp @ D @ F3 @ B @ G @ I ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % map_fun.comp
% 0.30/0.46 thf(fact_103_map__fun__def,axiom,
% 0.30/0.46 ! [B: $tType,D: $tType,A: $tType,C: $tType] :
% 0.30/0.46 ( ( map_fun @ C @ A @ B @ D )
% 0.30/0.46 = ( ^ [F2: C > A,G2: B > D,H3: A > B] : ( comp @ A @ D @ C @ ( comp @ B @ D @ A @ G2 @ H3 ) @ F2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % map_fun_def
% 0.30/0.46 thf(fact_104_add__right__cancel,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( cancel_semigroup_add @ A )
% 0.30/0.46 => ! [B4: A,A4: A,C3: A] :
% 0.30/0.46 ( ( ( plus_plus @ A @ B4 @ A4 )
% 0.30/0.46 = ( plus_plus @ A @ C3 @ A4 ) )
% 0.30/0.46 = ( B4 = C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_right_cancel
% 0.30/0.46 thf(fact_105_add__left__cancel,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( cancel_semigroup_add @ A )
% 0.30/0.46 => ! [A4: A,B4: A,C3: A] :
% 0.30/0.46 ( ( ( plus_plus @ A @ A4 @ B4 )
% 0.30/0.46 = ( plus_plus @ A @ A4 @ C3 ) )
% 0.30/0.46 = ( B4 = C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_left_cancel
% 0.30/0.46 thf(fact_106_set__plus__intro,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( plus @ A )
% 0.30/0.46 => ! [A4: A,C2: set @ A,B4: A,D3: set @ A] :
% 0.30/0.46 ( ( member @ A @ A4 @ C2 )
% 0.30/0.46 => ( ( member @ A @ B4 @ D3 )
% 0.30/0.46 => ( member @ A @ ( plus_plus @ A @ A4 @ B4 ) @ ( plus_plus @ ( set @ A ) @ C2 @ D3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_plus_intro
% 0.30/0.46 thf(fact_107_transfer__forall__transfer_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 0.30/0.46 @ ( transfer_forall @ A )
% 0.30/0.46 @ ( transfer_forall @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % transfer_forall_transfer(1)
% 0.30/0.46 thf(fact_108_monotone__parametric,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( ( C > C > $o ) > ( A > C ) > $o ) @ ( ( D > D > $o ) > ( B > D ) > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ ( C > C > $o ) @ ( D > D > $o ) @ ( ( A > C ) > $o ) @ ( ( B > D ) > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B2
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 )
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ( comple1396247847notone @ A @ C )
% 0.30/0.46 @ ( comple1396247847notone @ B @ D ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % monotone_parametric
% 0.30/0.46 thf(fact_109_relcompp__transfer,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,E: $tType,F3: $tType,B: $tType,D: $tType,B2: A > B > $o,A2: C > D > $o,C2: E > F3 > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ B2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( ( A > E > $o ) > C > E > $o ) @ ( ( B > F3 > $o ) > D > F3 > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ ( A > E > $o ) @ ( B > F3 > $o ) @ ( C > E > $o ) @ ( D > F3 > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( E > $o ) @ ( F3 > $o ) @ B2
% 0.30/0.46 @ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ ( E > $o ) @ ( F3 > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) ) )
% 0.30/0.46 @ ( relcompp @ C @ A @ E )
% 0.30/0.46 @ ( relcompp @ D @ B @ F3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcompp_transfer
% 0.30/0.46 thf(fact_110_fun_Orel__compp,axiom,
% 0.30/0.46 ! [A: $tType,D: $tType,C: $tType,B: $tType,R: A > B > $o,S2: B > C > $o] :
% 0.30/0.46 ( ( bNF_rel_fun @ D @ D @ A @ C
% 0.30/0.46 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 0.30/0.46 @ ( relcompp @ A @ B @ C @ R @ S2 ) )
% 0.30/0.46 = ( relcompp @ ( D > A ) @ ( D > B ) @ ( D > C )
% 0.30/0.46 @ ( bNF_rel_fun @ D @ D @ A @ B
% 0.30/0.46 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 0.30/0.46 @ R )
% 0.30/0.46 @ ( bNF_rel_fun @ D @ D @ B @ C
% 0.30/0.46 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 0.30/0.46 @ S2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.rel_compp
% 0.30/0.46 thf(fact_111_eq__comp__r,axiom,
% 0.30/0.46 ! [A: $tType,R: A > A > $o] :
% 0.30/0.46 ( ( relcompp @ A @ A @ A
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 0.30/0.46 @ ( relcompp @ A @ A @ A @ R
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) )
% 0.30/0.46 = R ) ).
% 0.30/0.46
% 0.30/0.46 % eq_comp_r
% 0.30/0.46 thf(fact_112_bi__total__OO,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,A2: A > B > $o,B2: B > C > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( ( bi_total @ B @ C @ B2 )
% 0.30/0.46 => ( bi_total @ A @ C @ ( relcompp @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_total_OO
% 0.30/0.46 thf(fact_113_OOO__quotient3,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R2: B > B > $o,Abs23: B > C,Rep23: C > B,R23: A > A > $o] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
% 0.30/0.46 => ( ( quotient3 @ B @ C @ R2 @ Abs23 @ Rep23 )
% 0.30/0.46 => ( ! [X2: A,Y2: A] :
% 0.30/0.46 ( ( R23 @ X2 @ Y2 )
% 0.30/0.46 => ( ( R1 @ X2 @ X2 )
% 0.30/0.46 => ( ( R1 @ Y2 @ Y2 )
% 0.30/0.46 => ( R2 @ ( Abs1 @ X2 ) @ ( Abs1 @ Y2 ) ) ) ) )
% 0.30/0.46 => ( ! [X2: B,Y2: B] :
% 0.30/0.46 ( ( R2 @ X2 @ Y2 )
% 0.30/0.46 => ( R23 @ ( Rep1 @ X2 ) @ ( Rep1 @ Y2 ) ) )
% 0.30/0.46 => ( quotient3 @ A @ C @ ( relcompp @ A @ A @ A @ R1 @ ( relcompp @ A @ A @ A @ R23 @ R1 ) ) @ ( comp @ B @ C @ A @ Abs23 @ Abs1 ) @ ( comp @ B @ A @ C @ Rep1 @ Rep23 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % OOO_quotient3
% 0.30/0.46 thf(fact_114_OOO__eq__quotient3,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,Abs23: B > C,Rep23: C > B] :
% 0.30/0.46 ( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
% 0.30/0.46 => ( ( quotient3 @ B @ C
% 0.30/0.46 @ ^ [Y4: B,Z2: B] : ( Y4 = Z2 )
% 0.30/0.46 @ Abs23
% 0.30/0.46 @ Rep23 )
% 0.30/0.46 => ( quotient3 @ A @ C
% 0.30/0.46 @ ( relcompp @ A @ A @ A @ R1
% 0.30/0.46 @ ( relcompp @ A @ A @ A
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 0.30/0.46 @ R1 ) )
% 0.30/0.46 @ ( comp @ B @ C @ A @ Abs23 @ Abs1 )
% 0.30/0.46 @ ( comp @ B @ A @ C @ Rep1 @ Rep23 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % OOO_eq_quotient3
% 0.30/0.46 thf(fact_115_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ab_semigroup_add @ A )
% 0.30/0.46 => ! [A4: A,B4: A,C3: A] :
% 0.30/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B4 ) @ C3 )
% 0.30/0.46 = ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B4 @ C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.30/0.46 thf(fact_116_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere779506340up_add @ A )
% 0.30/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.46 ( ( ( I = J )
% 0.30/0.46 & ( K = L ) )
% 0.30/0.46 => ( ( plus_plus @ A @ I @ K )
% 0.30/0.46 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_semiring(4)
% 0.30/0.46 thf(fact_117_group__cancel_Oadd1,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comm_monoid_add @ A )
% 0.30/0.46 => ! [A2: A,K: A,A4: A,B4: A] :
% 0.30/0.46 ( ( A2
% 0.30/0.46 = ( plus_plus @ A @ K @ A4 ) )
% 0.30/0.46 => ( ( plus_plus @ A @ A2 @ B4 )
% 0.30/0.46 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A4 @ B4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group_cancel.add1
% 0.30/0.46 thf(fact_118_group__cancel_Oadd2,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comm_monoid_add @ A )
% 0.30/0.46 => ! [B2: A,K: A,B4: A,A4: A] :
% 0.30/0.46 ( ( B2
% 0.30/0.46 = ( plus_plus @ A @ K @ B4 ) )
% 0.30/0.46 => ( ( plus_plus @ A @ A4 @ B2 )
% 0.30/0.46 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A4 @ B4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group_cancel.add2
% 0.30/0.46 thf(fact_119_set__plus__elim,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( plus @ A )
% 0.30/0.46 => ! [X: A,A2: set @ A,B2: set @ A] :
% 0.30/0.46 ( ( member @ A @ X @ ( plus_plus @ ( set @ A ) @ A2 @ B2 ) )
% 0.30/0.46 => ~ ! [A3: A,B3: A] :
% 0.30/0.46 ( ( X
% 0.30/0.46 = ( plus_plus @ A @ A3 @ B3 ) )
% 0.30/0.46 => ( ( member @ A @ A3 @ A2 )
% 0.30/0.46 => ~ ( member @ A @ B3 @ B2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_plus_elim
% 0.30/0.46 thf(fact_120_add_Oassoc,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( semigroup_add @ A )
% 0.30/0.46 => ! [A4: A,B4: A,C3: A] :
% 0.30/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B4 ) @ C3 )
% 0.30/0.46 = ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B4 @ C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.assoc
% 0.30/0.46 thf(fact_121_add_Oleft__cancel,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( group_add @ A )
% 0.30/0.46 => ! [A4: A,B4: A,C3: A] :
% 0.30/0.46 ( ( ( plus_plus @ A @ A4 @ B4 )
% 0.30/0.46 = ( plus_plus @ A @ A4 @ C3 ) )
% 0.30/0.46 = ( B4 = C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_cancel
% 0.30/0.46 thf(fact_122_add_Oright__cancel,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( group_add @ A )
% 0.30/0.46 => ! [B4: A,A4: A,C3: A] :
% 0.30/0.46 ( ( ( plus_plus @ A @ B4 @ A4 )
% 0.30/0.46 = ( plus_plus @ A @ C3 @ A4 ) )
% 0.30/0.46 = ( B4 = C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.right_cancel
% 0.30/0.46 thf(fact_123_add_Ocommute,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ab_semigroup_add @ A )
% 0.30/0.46 => ( ( plus_plus @ A )
% 0.30/0.46 = ( ^ [A5: A,B5: A] : ( plus_plus @ A @ B5 @ A5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.commute
% 0.30/0.46 thf(fact_124_add_Oleft__commute,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ab_semigroup_add @ A )
% 0.30/0.46 => ! [B4: A,A4: A,C3: A] :
% 0.30/0.46 ( ( plus_plus @ A @ B4 @ ( plus_plus @ A @ A4 @ C3 ) )
% 0.30/0.46 = ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B4 @ C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_commute
% 0.30/0.46 thf(fact_125_add__left__imp__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( cancel_semigroup_add @ A )
% 0.30/0.46 => ! [A4: A,B4: A,C3: A] :
% 0.30/0.46 ( ( ( plus_plus @ A @ A4 @ B4 )
% 0.30/0.46 = ( plus_plus @ A @ A4 @ C3 ) )
% 0.30/0.46 => ( B4 = C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_left_imp_eq
% 0.30/0.46 thf(fact_126_add__right__imp__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( cancel_semigroup_add @ A )
% 0.30/0.46 => ! [B4: A,A4: A,C3: A] :
% 0.30/0.46 ( ( ( plus_plus @ A @ B4 @ A4 )
% 0.30/0.46 = ( plus_plus @ A @ C3 @ A4 ) )
% 0.30/0.46 => ( B4 = C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_right_imp_eq
% 0.30/0.46 thf(fact_127_transfer__forall__transfer_I4_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 0.30/0.46 @ rev_implies
% 0.30/0.46 @ ( transfer_forall @ A )
% 0.30/0.46 @ ( transfer_forall @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % transfer_forall_transfer(4)
% 0.30/0.46 thf(fact_128_transfer__forall__transfer_I5_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ rev_implies ) @ rev_implies @ ( transfer_forall @ A ) @ ( transfer_forall @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % transfer_forall_transfer(5)
% 0.30/0.46 thf(fact_129_Domainp__transfer,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType,D: $tType,B2: A > B > $o,A2: C > D > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ B2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( C > $o ) @ ( D > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 0.30/0.46 @ ( domainp @ C @ A )
% 0.30/0.46 @ ( domainp @ D @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Domainp_transfer
% 0.30/0.46 thf(fact_130_folding__idem_Oaxioms_I2_J,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B] :
% 0.30/0.46 ( ( finite_folding_idem @ A @ B @ F )
% 0.30/0.46 => ( finite1921348288axioms @ A @ B @ F ) ) ).
% 0.30/0.46
% 0.30/0.46 % folding_idem.axioms(2)
% 0.30/0.46 thf(fact_131_OO__eq,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( relcompp @ A @ B @ B @ R
% 0.30/0.46 @ ^ [Y4: B,Z2: B] : ( Y4 = Z2 ) )
% 0.30/0.46 = R ) ).
% 0.30/0.46
% 0.30/0.46 % OO_eq
% 0.30/0.46 thf(fact_132_Domainp__iff,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( domainp @ A @ B )
% 0.30/0.46 = ( ^ [T2: A > B > $o,X3: A] :
% 0.30/0.46 ? [X6: B] : ( T2 @ X3 @ X6 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Domainp_iff
% 0.30/0.46 thf(fact_133_Domainp__refl,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( domainp @ A @ B )
% 0.30/0.46 = ( domainp @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % Domainp_refl
% 0.30/0.46 thf(fact_134_rev__implies__def,axiom,
% 0.30/0.46 ( rev_implies
% 0.30/0.46 = ( ^ [X3: $o,Y3: $o] :
% 0.30/0.46 ( Y3
% 0.30/0.46 => X3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rev_implies_def
% 0.30/0.46 thf(fact_135_folding__idem_Ocomp__fun__idem,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B,X: A] :
% 0.30/0.46 ( ( finite_folding_idem @ A @ B @ F )
% 0.30/0.46 => ( ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ X ) )
% 0.30/0.46 = ( F @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % folding_idem.comp_fun_idem
% 0.30/0.46 thf(fact_136_relcompp_OrelcompI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,R4: A > B > $o,A4: A,B4: B,S: B > C > $o,C3: C] :
% 0.30/0.46 ( ( R4 @ A4 @ B4 )
% 0.30/0.46 => ( ( S @ B4 @ C3 )
% 0.30/0.46 => ( relcompp @ A @ B @ C @ R4 @ S @ A4 @ C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcompp.relcompI
% 0.30/0.46 thf(fact_137_relcompp_Oinducts,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,R4: A > B > $o,S: B > C > $o,X1: A,X22: C,P: A > C > $o] :
% 0.30/0.46 ( ( relcompp @ A @ B @ C @ R4 @ S @ X1 @ X22 )
% 0.30/0.46 => ( ! [A3: A,B3: B,C4: C] :
% 0.30/0.46 ( ( R4 @ A3 @ B3 )
% 0.30/0.46 => ( ( S @ B3 @ C4 )
% 0.30/0.46 => ( P @ A3 @ C4 ) ) )
% 0.30/0.46 => ( P @ X1 @ X22 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcompp.inducts
% 0.30/0.46 thf(fact_138_relcompp__assoc,axiom,
% 0.30/0.46 ! [A: $tType,D: $tType,B: $tType,C: $tType,R4: A > D > $o,S: D > C > $o,T3: C > B > $o] :
% 0.30/0.46 ( ( relcompp @ A @ C @ B @ ( relcompp @ A @ D @ C @ R4 @ S ) @ T3 )
% 0.30/0.46 = ( relcompp @ A @ D @ B @ R4 @ ( relcompp @ D @ C @ B @ S @ T3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcompp_assoc
% 0.30/0.46 thf(fact_139_relcompp__apply,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( relcompp @ A @ B @ C )
% 0.30/0.46 = ( ^ [R5: A > B > $o,S5: B > C > $o,A5: A,C5: C] :
% 0.30/0.46 ? [B5: B] :
% 0.30/0.46 ( ( R5 @ A5 @ B5 )
% 0.30/0.46 & ( S5 @ B5 @ C5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcompp_apply
% 0.30/0.46 thf(fact_140_relcompp_Osimps,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( relcompp @ A @ B @ C )
% 0.30/0.46 = ( ^ [R6: A > B > $o,S3: B > C > $o,A1: A,A22: C] :
% 0.30/0.46 ? [A5: A,B5: B,C5: C] :
% 0.30/0.46 ( ( A1 = A5 )
% 0.30/0.46 & ( A22 = C5 )
% 0.30/0.46 & ( R6 @ A5 @ B5 )
% 0.30/0.46 & ( S3 @ B5 @ C5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcompp.simps
% 0.30/0.46 thf(fact_141_relcompp_Ocases,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,R4: A > B > $o,S: B > C > $o,A12: A,A23: C] :
% 0.30/0.46 ( ( relcompp @ A @ B @ C @ R4 @ S @ A12 @ A23 )
% 0.30/0.46 => ~ ! [B3: B] :
% 0.30/0.46 ( ( R4 @ A12 @ B3 )
% 0.30/0.46 => ~ ( S @ B3 @ A23 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcompp.cases
% 0.30/0.46 thf(fact_142_relcomppE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,R4: A > B > $o,S: B > C > $o,A4: A,C3: C] :
% 0.30/0.46 ( ( relcompp @ A @ B @ C @ R4 @ S @ A4 @ C3 )
% 0.30/0.46 => ~ ! [B3: B] :
% 0.30/0.46 ( ( R4 @ A4 @ B3 )
% 0.30/0.46 => ~ ( S @ B3 @ C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcomppE
% 0.30/0.46 thf(fact_143_eq__OO,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( relcompp @ A @ A @ B
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 0.30/0.46 @ R )
% 0.30/0.46 = R ) ).
% 0.30/0.46
% 0.30/0.46 % eq_OO
% 0.30/0.46 thf(fact_144_folding__idem__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( finite_folding_idem @ A @ B )
% 0.30/0.46 = ( ^ [F2: A > B > B] :
% 0.30/0.46 ( ( finite_folding @ A @ B @ F2 )
% 0.30/0.46 & ( finite1921348288axioms @ A @ B @ F2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % folding_idem_def
% 0.30/0.46 thf(fact_145_folding__idem_Ointro,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B] :
% 0.30/0.46 ( ( finite_folding @ A @ B @ F )
% 0.30/0.46 => ( ( finite1921348288axioms @ A @ B @ F )
% 0.30/0.46 => ( finite_folding_idem @ A @ B @ F ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % folding_idem.intro
% 0.30/0.46 thf(fact_146_reflp__transfer_I5_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ rev_implies
% 0.30/0.46 @ ( reflp @ A )
% 0.30/0.46 @ ( reflp @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % reflp_transfer(5)
% 0.30/0.46 thf(fact_147_reflp__transfer_I4_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ rev_implies ) ) @ rev_implies @ ( reflp @ A ) @ ( reflp @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % reflp_transfer(4)
% 0.30/0.46 thf(fact_148_fun_Orel__reflp,axiom,
% 0.30/0.46 ! [D: $tType,A: $tType,R: A > A > $o] :
% 0.30/0.46 ( ( reflp @ A @ R )
% 0.30/0.46 => ( reflp @ ( D > A )
% 0.30/0.46 @ ( bNF_rel_fun @ D @ D @ A @ A
% 0.30/0.46 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 0.30/0.46 @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.rel_reflp
% 0.30/0.46 thf(fact_149_DEADID_Orel__reflp,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( reflp @ A
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % DEADID.rel_reflp
% 0.30/0.46 thf(fact_150_folding__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( finite_folding @ A @ B )
% 0.30/0.46 = ( ^ [F2: A > B > B] :
% 0.30/0.46 ! [Y3: A,X3: A] :
% 0.30/0.46 ( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) )
% 0.30/0.46 = ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % folding_def
% 0.30/0.46 thf(fact_151_folding_Ointro,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B] :
% 0.30/0.46 ( ! [Y2: A,X2: A] :
% 0.30/0.46 ( ( comp @ B @ B @ B @ ( F @ Y2 ) @ ( F @ X2 ) )
% 0.30/0.46 = ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( finite_folding @ A @ B @ F ) ) ).
% 0.30/0.46
% 0.30/0.46 % folding.intro
% 0.30/0.46 thf(fact_152_folding_Ocomp__fun__commute,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B,Y: A,X: A] :
% 0.30/0.46 ( ( finite_folding @ A @ B @ F )
% 0.30/0.46 => ( ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) )
% 0.30/0.46 = ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % folding.comp_fun_commute
% 0.30/0.46 thf(fact_153_folding__idem_Oaxioms_I1_J,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B > B] :
% 0.30/0.46 ( ( finite_folding_idem @ A @ B @ F )
% 0.30/0.46 => ( finite_folding @ A @ B @ F ) ) ).
% 0.30/0.46
% 0.30/0.46 % folding_idem.axioms(1)
% 0.30/0.46 thf(fact_154_reflp__transfer_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 0.30/0.46 @ ( reflp @ A )
% 0.30/0.46 @ ( reflp @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % reflp_transfer(1)
% 0.30/0.46 thf(fact_155_left__total__parametric,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( ( bi_total @ C @ D @ B2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 0.30/0.46 @ ( left_total @ A @ C )
% 0.30/0.46 @ ( left_total @ B @ D ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_total_parametric
% 0.30/0.46 thf(fact_156_reflp__transfer_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ (=>) ) ) @ (=>) @ ( reflp @ A ) @ ( reflp @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % reflp_transfer(2)
% 0.30/0.46 thf(fact_157_reflp__transfer_I3_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ (=>)
% 0.30/0.46 @ ( reflp @ A )
% 0.30/0.46 @ ( reflp @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % reflp_transfer(3)
% 0.30/0.46 thf(fact_158_left__totalE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( left_total @ A @ B @ R )
% 0.30/0.46 => ! [X7: A] :
% 0.30/0.46 ? [X_1: B] : ( R @ X7 @ X_1 ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_totalE
% 0.30/0.46 thf(fact_159_left__totalI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ! [X2: A] :
% 0.30/0.46 ? [X_12: B] : ( R @ X2 @ X_12 )
% 0.30/0.46 => ( left_total @ A @ B @ R ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_totalI
% 0.30/0.46 thf(fact_160_right__totalE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o,Y: B] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ~ ! [X2: A] :
% 0.30/0.46 ~ ( A2 @ X2 @ Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_totalE
% 0.30/0.46 thf(fact_161_right__totalI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: B > A > $o] :
% 0.30/0.46 ( ! [Y2: A] :
% 0.30/0.46 ? [X7: B] : ( A2 @ X7 @ Y2 )
% 0.30/0.46 => ( right_total @ B @ A @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_totalI
% 0.30/0.46 thf(fact_162_left__total__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( left_total @ A @ A
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_total_eq
% 0.30/0.46 thf(fact_163_left__total__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( left_total @ A @ B )
% 0.30/0.46 = ( ^ [R5: A > B > $o] :
% 0.30/0.46 ! [X3: A] :
% 0.30/0.46 ? [X6: B] : ( R5 @ X3 @ X6 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_total_def
% 0.30/0.46 thf(fact_164_right__total__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( right_total @ A @ A
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_total_eq
% 0.30/0.46 thf(fact_165_right__total__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( right_total @ A @ B )
% 0.30/0.46 = ( ^ [R5: A > B > $o] :
% 0.30/0.46 ! [Y3: B] :
% 0.30/0.46 ? [X3: A] : ( R5 @ X3 @ Y3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_total_def
% 0.30/0.46 thf(fact_166_bi__total__alt__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( bi_total @ A @ B )
% 0.30/0.46 = ( ^ [A6: A > B > $o] :
% 0.30/0.46 ( ( left_total @ A @ B @ A6 )
% 0.30/0.46 & ( right_total @ A @ B @ A6 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_total_alt_def
% 0.30/0.46 thf(fact_167_bi__totalI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( left_total @ A @ B @ R )
% 0.30/0.46 => ( ( right_total @ A @ B @ R )
% 0.30/0.46 => ( bi_total @ A @ B @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_totalI
% 0.30/0.46 thf(fact_168_left__total__OO,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,R: A > B > $o,S2: B > C > $o] :
% 0.30/0.46 ( ( left_total @ A @ B @ R )
% 0.30/0.46 => ( ( left_total @ B @ C @ S2 )
% 0.30/0.46 => ( left_total @ A @ C @ ( relcompp @ A @ B @ C @ R @ S2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_total_OO
% 0.30/0.46 thf(fact_169_right__total__OO,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,A2: A > B > $o,B2: B > C > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( ( right_total @ B @ C @ B2 )
% 0.30/0.46 => ( right_total @ A @ C @ ( relcompp @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_total_OO
% 0.30/0.46 thf(fact_170_pcr__Domainp__total,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,B2: A > B > $o,A2: C > A > $o,P: C > $o] :
% 0.30/0.46 ( ( left_total @ A @ B @ B2 )
% 0.30/0.46 => ( ( ( domainp @ C @ A @ A2 )
% 0.30/0.46 = P )
% 0.30/0.46 => ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
% 0.30/0.46 = P ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pcr_Domainp_total
% 0.30/0.46 thf(fact_171_pcr__Domainp__par__left__total,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,B2: A > B > $o,P: A > $o,A2: C > A > $o,P2: C > $o] :
% 0.30/0.46 ( ( ( domainp @ A @ B @ B2 )
% 0.30/0.46 = P )
% 0.30/0.46 => ( ( left_total @ C @ A @ A2 )
% 0.30/0.46 => ( ( bNF_rel_fun @ C @ A @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 0.30/0.46 @ P2
% 0.30/0.46 @ P )
% 0.30/0.46 => ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
% 0.30/0.46 = P2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pcr_Domainp_par_left_total
% 0.30/0.46 thf(fact_172_transfer__forall__transfer_I3_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ (=>) ) @ (=>) @ ( transfer_forall @ A ) @ ( transfer_forall @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % transfer_forall_transfer(3)
% 0.30/0.46 thf(fact_173_transfer__forall__transfer_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 0.30/0.46 @ (=>)
% 0.30/0.46 @ ( transfer_forall @ A )
% 0.30/0.46 @ ( transfer_forall @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % transfer_forall_transfer(2)
% 0.30/0.46 thf(fact_174_right__total__parametric,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( ( bi_total @ C @ D @ B2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 0.30/0.46 @ ( right_total @ A @ C )
% 0.30/0.46 @ ( right_total @ B @ D ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_total_parametric
% 0.30/0.46 thf(fact_175_Domainp__forall__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 0.30/0.46 @ ( transfer_bforall @ A @ ( domainp @ A @ B @ A2 ) )
% 0.30/0.46 @ ( transfer_forall @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Domainp_forall_transfer
% 0.30/0.46 thf(fact_176_nchotomy__relcomppE,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,D: $tType,F: B > A,R4: C > A > $o,S: A > D > $o,A4: C,C3: D] :
% 0.30/0.46 ( ! [Y2: A] :
% 0.30/0.46 ? [X7: B] :
% 0.30/0.46 ( Y2
% 0.30/0.46 = ( F @ X7 ) )
% 0.30/0.46 => ( ( relcompp @ C @ A @ D @ R4 @ S @ A4 @ C3 )
% 0.30/0.46 => ~ ! [B3: B] :
% 0.30/0.46 ( ( R4 @ A4 @ ( F @ B3 ) )
% 0.30/0.46 => ~ ( S @ ( F @ B3 ) @ C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nchotomy_relcomppE
% 0.30/0.46 thf(fact_177_pos__fun__distr,axiom,
% 0.30/0.46 ! [E: $tType,C: $tType,A: $tType,B: $tType,D: $tType,F3: $tType,R: A > E > $o,S2: B > F3 > $o,R8: E > C > $o,S6: F3 > D > $o] : ( ord_less_eq @ ( ( A > B ) > ( C > D ) > $o ) @ ( relcompp @ ( A > B ) @ ( E > F3 ) @ ( C > D ) @ ( bNF_rel_fun @ A @ E @ B @ F3 @ R @ S2 ) @ ( bNF_rel_fun @ E @ C @ F3 @ D @ R8 @ S6 ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ ( relcompp @ A @ E @ C @ R @ R8 ) @ ( relcompp @ B @ F3 @ D @ S2 @ S6 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pos_fun_distr
% 0.30/0.46 thf(fact_178_add__le__cancel__left,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere236663937imp_le @ A )
% 0.30/0.46 => ! [C3: A,A4: A,B4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A4 ) @ ( plus_plus @ A @ C3 @ B4 ) )
% 0.30/0.46 = ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_le_cancel_left
% 0.30/0.46 thf(fact_179_add__le__cancel__right,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere236663937imp_le @ A )
% 0.30/0.46 => ! [A4: A,C3: A,B4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B4 @ C3 ) )
% 0.30/0.46 = ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_le_cancel_right
% 0.30/0.46 thf(fact_180_reflp__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( reflp @ A )
% 0.30/0.46 = ( ord_less_eq @ ( A > A > $o )
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % reflp_eq
% 0.30/0.46 thf(fact_181_reflp__ge__eq,axiom,
% 0.30/0.46 ! [A: $tType,R: A > A > $o] :
% 0.30/0.46 ( ( reflp @ A @ R )
% 0.30/0.46 => ( ord_less_eq @ ( A > A > $o )
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 0.30/0.46 @ R ) ) ).
% 0.30/0.46
% 0.30/0.46 % reflp_ge_eq
% 0.30/0.46 thf(fact_182_leq__OOI,axiom,
% 0.30/0.46 ! [A: $tType,R: A > A > $o] :
% 0.30/0.46 ( ( R
% 0.30/0.46 = ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) )
% 0.30/0.46 => ( ord_less_eq @ ( A > A > $o ) @ R @ ( relcompp @ A @ A @ A @ R @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % leq_OOI
% 0.30/0.46 thf(fact_183_relcompp__mono,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,R9: A > B > $o,R4: A > B > $o,S7: B > C > $o,S: B > C > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > B > $o ) @ R9 @ R4 )
% 0.30/0.46 => ( ( ord_less_eq @ ( B > C > $o ) @ S7 @ S )
% 0.30/0.46 => ( ord_less_eq @ ( A > C > $o ) @ ( relcompp @ A @ B @ C @ R9 @ S7 ) @ ( relcompp @ A @ B @ C @ R4 @ S ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % relcompp_mono
% 0.30/0.46 thf(fact_184_predicate2D__conj,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X: A,Y: B] :
% 0.30/0.46 ( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.30/0.46 & R )
% 0.30/0.46 => ( R
% 0.30/0.46 & ( ( P @ X @ Y )
% 0.30/0.46 => ( Q @ X @ Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % predicate2D_conj
% 0.30/0.46 thf(fact_185_refl__ge__eq,axiom,
% 0.30/0.46 ! [A: $tType,R: A > A > $o] :
% 0.30/0.46 ( ! [X2: A] : ( R @ X2 @ X2 )
% 0.30/0.46 => ( ord_less_eq @ ( A > A > $o )
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 0.30/0.46 @ R ) ) ).
% 0.30/0.46
% 0.30/0.46 % refl_ge_eq
% 0.30/0.46 thf(fact_186_ge__eq__refl,axiom,
% 0.30/0.46 ! [A: $tType,R: A > A > $o,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > A > $o )
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 0.30/0.46 @ R )
% 0.30/0.46 => ( R @ X @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % ge_eq_refl
% 0.30/0.46 thf(fact_187_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere779506340up_add @ A )
% 0.30/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.46 ( ( ( ord_less_eq @ A @ I @ J )
% 0.30/0.46 & ( K = L ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_semiring(3)
% 0.30/0.46 thf(fact_188_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere779506340up_add @ A )
% 0.30/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.46 ( ( ( I = J )
% 0.30/0.46 & ( ord_less_eq @ A @ K @ L ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_semiring(2)
% 0.30/0.46 thf(fact_189_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere779506340up_add @ A )
% 0.30/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.30/0.46 ( ( ( ord_less_eq @ A @ I @ J )
% 0.30/0.46 & ( ord_less_eq @ A @ K @ L ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_semiring(1)
% 0.30/0.46 thf(fact_190_add__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere779506340up_add @ A )
% 0.30/0.46 => ! [A4: A,B4: A,C3: A,D2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ C3 @ D2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B4 @ D2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono
% 0.30/0.46 thf(fact_191_add__left__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere779506340up_add @ A )
% 0.30/0.46 => ! [A4: A,B4: A,C3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A4 ) @ ( plus_plus @ A @ C3 @ B4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_left_mono
% 0.30/0.46 thf(fact_192_less__eqE,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( canoni770627133id_add @ A )
% 0.30/0.46 => ! [A4: A,B4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.46 => ~ ! [C4: A] :
% 0.30/0.46 ( B4
% 0.30/0.46 != ( plus_plus @ A @ A4 @ C4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_eqE
% 0.30/0.46 thf(fact_193_add__right__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere779506340up_add @ A )
% 0.30/0.46 => ! [A4: A,B4: A,C3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B4 @ C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_right_mono
% 0.30/0.46 thf(fact_194_le__iff__add,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( canoni770627133id_add @ A )
% 0.30/0.46 => ( ( ord_less_eq @ A )
% 0.30/0.46 = ( ^ [A5: A,B5: A] :
% 0.30/0.46 ? [C5: A] :
% 0.30/0.46 ( B5
% 0.30/0.46 = ( plus_plus @ A @ A5 @ C5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_iff_add
% 0.30/0.46 thf(fact_195_add__le__imp__le__left,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere236663937imp_le @ A )
% 0.30/0.46 => ! [C3: A,A4: A,B4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A4 ) @ ( plus_plus @ A @ C3 @ B4 ) )
% 0.30/0.46 => ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_le_imp_le_left
% 0.30/0.46 thf(fact_196_add__le__imp__le__right,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere236663937imp_le @ A )
% 0.30/0.46 => ! [A4: A,C3: A,B4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B4 @ C3 ) )
% 0.30/0.46 => ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_le_imp_le_right
% 0.30/0.46 thf(fact_197_fun_Orel__mono,axiom,
% 0.30/0.46 ! [D: $tType,B: $tType,A: $tType,R: A > B > $o,Ra: A > B > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra )
% 0.30/0.46 => ( ord_less_eq @ ( ( D > A ) > ( D > B ) > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ D @ D @ A @ B
% 0.30/0.46 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 0.30/0.46 @ R )
% 0.30/0.46 @ ( bNF_rel_fun @ D @ D @ A @ B
% 0.30/0.46 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 0.30/0.46 @ Ra ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.rel_mono
% 0.30/0.46 thf(fact_198_fun__mono,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,C2: A > B > $o,A2: A > B > $o,B2: C > D > $o,D3: C > D > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( A > B > $o ) @ C2 @ A2 )
% 0.30/0.46 => ( ( ord_less_eq @ ( C > D > $o ) @ B2 @ D3 )
% 0.30/0.46 => ( ord_less_eq @ ( ( A > C ) > ( B > D ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ A @ B @ C @ D @ C2 @ D3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_mono
% 0.30/0.46 thf(fact_199_lfp_Omonotone__if__bot,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A )
% 0.30/0.46 => ! [Bound: A,G: A > B,Bot: B,F: A > B,Ord: B > B > $o] :
% 0.30/0.46 ( ! [X2: A] :
% 0.30/0.46 ( ( ( ord_less_eq @ A @ X2 @ Bound )
% 0.30/0.46 => ( ( G @ X2 )
% 0.30/0.46 = Bot ) )
% 0.30/0.46 & ( ~ ( ord_less_eq @ A @ X2 @ Bound )
% 0.30/0.46 => ( ( G @ X2 )
% 0.30/0.46 = ( F @ X2 ) ) ) )
% 0.30/0.46 => ( ! [X2: A,Y2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.30/0.46 => ( ~ ( ord_less_eq @ A @ X2 @ Bound )
% 0.30/0.46 => ( Ord @ ( F @ X2 ) @ ( F @ Y2 ) ) ) )
% 0.30/0.46 => ( ! [X2: A] :
% 0.30/0.46 ( ~ ( ord_less_eq @ A @ X2 @ Bound )
% 0.30/0.46 => ( Ord @ Bot @ ( F @ X2 ) ) )
% 0.30/0.46 => ( ( Ord @ Bot @ Bot )
% 0.30/0.46 => ( comple1396247847notone @ A @ B @ ( ord_less_eq @ A ) @ Ord @ G ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % lfp.monotone_if_bot
% 0.30/0.46 thf(fact_200_mono__transfer,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType] :
% 0.30/0.46 ( ( ( order @ B )
% 0.30/0.46 & ( order @ D )
% 0.30/0.46 & ( order @ C )
% 0.30/0.46 & ( order @ A ) )
% 0.30/0.46 => ! [A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 0.30/0.46 @ ( ord_less_eq @ A )
% 0.30/0.46 @ ( ord_less_eq @ B ) )
% 0.30/0.46 => ( ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B2
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 0.30/0.46 @ ( ord_less_eq @ C )
% 0.30/0.46 @ ( ord_less_eq @ D ) )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 )
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 0.30/0.46 @ ( order_mono @ A @ C )
% 0.30/0.46 @ ( order_mono @ B @ D ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mono_transfer
% 0.30/0.46 thf(fact_201_neg__fun__distr1,axiom,
% 0.30/0.46 ! [D: $tType,A: $tType,B: $tType,C: $tType,E: $tType,F3: $tType,R: A > B > $o,R8: B > C > $o,S2: D > F3 > $o,S6: F3 > E > $o] :
% 0.30/0.46 ( ( left_unique @ A @ B @ R )
% 0.30/0.46 => ( ( right_total @ A @ B @ R )
% 0.30/0.46 => ( ( right_unique @ B @ C @ R8 )
% 0.30/0.46 => ( ( left_total @ B @ C @ R8 )
% 0.30/0.46 => ( ord_less_eq @ ( ( A > D ) > ( C > E ) > $o ) @ ( bNF_rel_fun @ A @ C @ D @ E @ ( relcompp @ A @ B @ C @ R @ R8 ) @ ( relcompp @ D @ F3 @ E @ S2 @ S6 ) ) @ ( relcompp @ ( A > D ) @ ( B > F3 ) @ ( C > E ) @ ( bNF_rel_fun @ A @ B @ D @ F3 @ R @ S2 ) @ ( bNF_rel_fun @ B @ C @ F3 @ E @ R8 @ S6 ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % neg_fun_distr1
% 0.30/0.46 thf(fact_202_set__plus__mono2,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( plus @ A )
% 0.30/0.46 => ! [C2: set @ A,D3: set @ A,E2: set @ A,F5: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ C2 @ D3 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ E2 @ F5 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( plus_plus @ ( set @ A ) @ C2 @ E2 ) @ ( plus_plus @ ( set @ A ) @ D3 @ F5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_plus_mono2
% 0.30/0.46 thf(fact_203_set__plus__mono2__b,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( plus @ A )
% 0.30/0.46 => ! [C2: set @ A,D3: set @ A,E2: set @ A,F5: set @ A,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ C2 @ D3 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ E2 @ F5 )
% 0.30/0.46 => ( ( member @ A @ X @ ( plus_plus @ ( set @ A ) @ C2 @ E2 ) )
% 0.30/0.46 => ( member @ A @ X @ ( plus_plus @ ( set @ A ) @ D3 @ F5 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_plus_mono2_b
% 0.30/0.46 thf(fact_204_left__uniqueD,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: A > B > $o,X: A,Z: B,Y: A] :
% 0.30/0.46 ( ( left_unique @ A @ B @ A2 )
% 0.30/0.46 => ( ( A2 @ X @ Z )
% 0.30/0.46 => ( ( A2 @ Y @ Z )
% 0.30/0.46 => ( X = Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_uniqueD
% 0.30/0.46 thf(fact_205_left__uniqueI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ! [X2: A,Y2: A,Z3: B] :
% 0.30/0.46 ( ( A2 @ X2 @ Z3 )
% 0.30/0.46 => ( ( A2 @ Y2 @ Z3 )
% 0.30/0.46 => ( X2 = Y2 ) ) )
% 0.30/0.46 => ( left_unique @ A @ B @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_uniqueI
% 0.30/0.46 thf(fact_206_right__uniqueD,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o,X: A,Y: B,Z: B] :
% 0.30/0.46 ( ( right_unique @ A @ B @ A2 )
% 0.30/0.46 => ( ( A2 @ X @ Y )
% 0.30/0.46 => ( ( A2 @ X @ Z )
% 0.30/0.46 => ( Y = Z ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_uniqueD
% 0.30/0.46 thf(fact_207_right__uniqueI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ! [X2: A,Y2: B,Z3: B] :
% 0.30/0.46 ( ( A2 @ X2 @ Y2 )
% 0.30/0.46 => ( ( A2 @ X2 @ Z3 )
% 0.30/0.46 => ( Y2 = Z3 ) ) )
% 0.30/0.46 => ( right_unique @ A @ B @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_uniqueI
% 0.30/0.46 thf(fact_208_left__unique__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( left_unique @ A @ A
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_unique_eq
% 0.30/0.46 thf(fact_209_left__unique__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( left_unique @ A @ B )
% 0.30/0.46 = ( ^ [R5: A > B > $o] :
% 0.30/0.46 ! [X3: A,Y3: A,Z4: B] :
% 0.30/0.46 ( ( R5 @ X3 @ Z4 )
% 0.30/0.46 => ( ( R5 @ Y3 @ Z4 )
% 0.30/0.46 => ( X3 = Y3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_unique_def
% 0.30/0.46 thf(fact_210_right__unique__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( right_unique @ A @ A
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_unique_eq
% 0.30/0.46 thf(fact_211_right__unique__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( right_unique @ A @ B )
% 0.30/0.46 = ( ^ [R5: A > B > $o] :
% 0.30/0.46 ! [X3: A,Y3: B,Z4: B] :
% 0.30/0.46 ( ( R5 @ X3 @ Y3 )
% 0.30/0.46 => ( ( R5 @ X3 @ Z4 )
% 0.30/0.46 => ( Y3 = Z4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_unique_def
% 0.30/0.46 thf(fact_212_left__unique__OO,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,R: A > B > $o,S2: B > C > $o] :
% 0.30/0.46 ( ( left_unique @ A @ B @ R )
% 0.30/0.46 => ( ( left_unique @ B @ C @ S2 )
% 0.30/0.46 => ( left_unique @ A @ C @ ( relcompp @ A @ B @ C @ R @ S2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_unique_OO
% 0.30/0.46 thf(fact_213_right__unique__OO,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,A2: A > B > $o,B2: B > C > $o] :
% 0.30/0.46 ( ( right_unique @ A @ B @ A2 )
% 0.30/0.46 => ( ( right_unique @ B @ C @ B2 )
% 0.30/0.46 => ( right_unique @ A @ C @ ( relcompp @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_unique_OO
% 0.30/0.46 thf(fact_214_functional__converse__relation,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( left_unique @ A @ B @ R )
% 0.30/0.46 => ( ( right_total @ A @ B @ R )
% 0.30/0.46 => ! [Y6: B] :
% 0.30/0.46 ? [X2: A] :
% 0.30/0.46 ( ( R @ X2 @ Y6 )
% 0.30/0.46 & ! [Ya3: A] :
% 0.30/0.46 ( ( R @ Ya3 @ Y6 )
% 0.30/0.46 => ( Ya3 = X2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % functional_converse_relation
% 0.30/0.46 thf(fact_215_functional__relation,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( right_unique @ A @ B @ R )
% 0.30/0.46 => ( ( left_total @ A @ B @ R )
% 0.30/0.46 => ! [X7: A] :
% 0.30/0.46 ? [Xa2: B] :
% 0.30/0.46 ( ( R @ X7 @ Xa2 )
% 0.30/0.46 & ! [Y6: B] :
% 0.30/0.46 ( ( R @ X7 @ Y6 )
% 0.30/0.46 => ( Y6 = Xa2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % functional_relation
% 0.30/0.46 thf(fact_216_right__unique__fun,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( ( right_unique @ C @ D @ B2 )
% 0.30/0.46 => ( right_unique @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_unique_fun
% 0.30/0.46 thf(fact_217_left__unique__fun,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( left_total @ A @ B @ A2 )
% 0.30/0.46 => ( ( left_unique @ C @ D @ B2 )
% 0.30/0.46 => ( left_unique @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_unique_fun
% 0.30/0.46 thf(fact_218_right__total__fun,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( right_unique @ A @ B @ A2 )
% 0.30/0.46 => ( ( right_total @ C @ D @ B2 )
% 0.30/0.46 => ( right_total @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_total_fun
% 0.30/0.46 thf(fact_219_left__total__fun,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( left_unique @ A @ B @ A2 )
% 0.30/0.46 => ( ( left_total @ C @ D @ B2 )
% 0.30/0.46 => ( left_total @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_total_fun
% 0.30/0.46 thf(fact_220_right__unique__alt__def2,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( right_unique @ A @ B )
% 0.30/0.46 = ( ^ [R5: A > B > $o] :
% 0.30/0.46 ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ R5 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5 @ (=>) )
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 0.30/0.46 @ ^ [Y4: B,Z2: B] : ( Y4 = Z2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_unique_alt_def2
% 0.30/0.46 thf(fact_221_eq__imp__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( right_unique @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ (=>) )
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 0.30/0.46 @ ^ [Y4: B,Z2: B] : ( Y4 = Z2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_imp_transfer
% 0.30/0.46 thf(fact_222_neg__fun__distr2,axiom,
% 0.30/0.46 ! [F3: $tType,E: $tType,A: $tType,B: $tType,D: $tType,C: $tType,R8: A > B > $o,S6: C > D > $o,R: E > A > $o,S2: F3 > C > $o] :
% 0.30/0.46 ( ( right_unique @ A @ B @ R8 )
% 0.30/0.46 => ( ( left_total @ A @ B @ R8 )
% 0.30/0.46 => ( ( left_unique @ C @ D @ S6 )
% 0.30/0.46 => ( ( right_total @ C @ D @ S6 )
% 0.30/0.46 => ( ord_less_eq @ ( ( E > F3 ) > ( B > D ) > $o ) @ ( bNF_rel_fun @ E @ B @ F3 @ D @ ( relcompp @ E @ A @ B @ R @ R8 ) @ ( relcompp @ F3 @ C @ D @ S2 @ S6 ) ) @ ( relcompp @ ( E > F3 ) @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ E @ A @ F3 @ C @ R @ S2 ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R8 @ S6 ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % neg_fun_distr2
% 0.30/0.46 thf(fact_223_fixp__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comple1141879883l_ccpo @ A )
% 0.30/0.46 => ! [F: A > A,G: A > A] :
% 0.30/0.46 ( ( partial_fun_ord @ A @ A @ A @ ( ord_less_eq @ A ) @ F @ G )
% 0.30/0.46 => ( ( comple1396247847notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 0.30/0.46 => ( ( comple1396247847notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ G )
% 0.30/0.46 => ( ord_less_eq @ A @ ( comple939513234o_fixp @ A @ F ) @ ( comple939513234o_fixp @ A @ G ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fixp_mono
% 0.30/0.46 thf(fact_224_left__unique__parametric,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( bi_unique @ A @ B @ A2 )
% 0.30/0.46 => ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( ( bi_total @ C @ D @ B2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 0.30/0.46 @ ( left_unique @ A @ C )
% 0.30/0.46 @ ( left_unique @ B @ D ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_unique_parametric
% 0.30/0.46 thf(fact_225_bi__uniqueI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R: A > B > $o] :
% 0.30/0.46 ( ( left_unique @ A @ B @ R )
% 0.30/0.46 => ( ( right_unique @ A @ B @ R )
% 0.30/0.46 => ( bi_unique @ A @ B @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_uniqueI
% 0.30/0.46 thf(fact_226_bi__unique__alt__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( bi_unique @ A @ B )
% 0.30/0.46 = ( ^ [A6: A > B > $o] :
% 0.30/0.46 ( ( left_unique @ A @ B @ A6 )
% 0.30/0.46 & ( right_unique @ A @ B @ A6 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_unique_alt_def
% 0.30/0.46 thf(fact_227_bi__unique__OO,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,A2: A > B > $o,B2: B > C > $o] :
% 0.30/0.46 ( ( bi_unique @ A @ B @ A2 )
% 0.30/0.46 => ( ( bi_unique @ B @ C @ B2 )
% 0.30/0.46 => ( bi_unique @ A @ C @ ( relcompp @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_unique_OO
% 0.30/0.46 thf(fact_228_bi__unique__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( bi_unique @ A @ B )
% 0.30/0.46 = ( ^ [R5: A > B > $o] :
% 0.30/0.46 ( ! [X3: A,Y3: B,Z4: B] :
% 0.30/0.46 ( ( R5 @ X3 @ Y3 )
% 0.30/0.46 => ( ( R5 @ X3 @ Z4 )
% 0.30/0.46 => ( Y3 = Z4 ) ) )
% 0.30/0.46 & ! [X3: A,Y3: A,Z4: B] :
% 0.30/0.46 ( ( R5 @ X3 @ Z4 )
% 0.30/0.46 => ( ( R5 @ Y3 @ Z4 )
% 0.30/0.46 => ( X3 = Y3 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_unique_def
% 0.30/0.46 thf(fact_229_bi__unique__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( bi_unique @ A @ A
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_unique_eq
% 0.30/0.46 thf(fact_230_bi__uniqueDr,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o,X: A,Y: B,Z: B] :
% 0.30/0.46 ( ( bi_unique @ A @ B @ A2 )
% 0.30/0.46 => ( ( A2 @ X @ Y )
% 0.30/0.46 => ( ( A2 @ X @ Z )
% 0.30/0.46 => ( Y = Z ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_uniqueDr
% 0.30/0.46 thf(fact_231_bi__uniqueDl,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: A > B > $o,X: A,Y: B,Z: A] :
% 0.30/0.46 ( ( bi_unique @ A @ B @ A2 )
% 0.30/0.46 => ( ( A2 @ X @ Y )
% 0.30/0.46 => ( ( A2 @ Z @ Y )
% 0.30/0.46 => ( X = Z ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_uniqueDl
% 0.30/0.46 thf(fact_232_bi__unique__fun,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( ( bi_unique @ C @ D @ B2 )
% 0.30/0.46 => ( bi_unique @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_unique_fun
% 0.30/0.46 thf(fact_233_bi__unique__alt__def2,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( bi_unique @ A @ B )
% 0.30/0.46 = ( ^ [R5: A > B > $o] :
% 0.30/0.46 ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ R5
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 0.30/0.46 @ ^ [Y4: B,Z2: B] : ( Y4 = Z2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_unique_alt_def2
% 0.30/0.46 thf(fact_234_eq__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( bi_unique @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 0.30/0.46 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 0.30/0.46 @ ^ [Y4: B,Z2: B] : ( Y4 = Z2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_transfer
% 0.30/0.46 thf(fact_235_bi__total__fun,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( bi_unique @ A @ B @ A2 )
% 0.30/0.46 => ( ( bi_total @ C @ D @ B2 )
% 0.30/0.46 => ( bi_total @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bi_total_fun
% 0.30/0.46 thf(fact_236_right__unique__transfer,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( ( right_total @ C @ D @ B2 )
% 0.30/0.46 => ( ( bi_unique @ C @ D @ B2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ (=>)
% 0.30/0.46 @ ( right_unique @ A @ C )
% 0.30/0.46 @ ( right_unique @ B @ D ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_unique_transfer
% 0.30/0.46 thf(fact_237_left__unique__transfer,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( right_total @ A @ B @ A2 )
% 0.30/0.46 => ( ( right_total @ C @ D @ B2 )
% 0.30/0.46 => ( ( bi_unique @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ (=>)
% 0.30/0.46 @ ( left_unique @ A @ C )
% 0.30/0.46 @ ( left_unique @ B @ D ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % left_unique_transfer
% 0.30/0.46 thf(fact_238_right__unique__parametric,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( ( bi_unique @ C @ D @ B2 )
% 0.30/0.46 => ( ( bi_total @ C @ D @ B2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 0.30/0.46 @ ( right_unique @ A @ C )
% 0.30/0.46 @ ( right_unique @ B @ D ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % right_unique_parametric
% 0.30/0.46 thf(fact_239_fixp__lowerbound,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comple1141879883l_ccpo @ A )
% 0.30/0.46 => ! [F: A > A,Z: A] :
% 0.30/0.46 ( ( comple1396247847notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 0.30/0.46 => ( ( ord_less_eq @ A @ ( F @ Z ) @ Z )
% 0.30/0.46 => ( ord_less_eq @ A @ ( comple939513234o_fixp @ A @ F ) @ Z ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fixp_lowerbound
% 0.30/0.46 thf(fact_240_fixp__unfold,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comple1141879883l_ccpo @ A )
% 0.30/0.46 => ! [F: A > A] :
% 0.30/0.46 ( ( comple1396247847notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 0.30/0.46 => ( ( comple939513234o_fixp @ A @ F )
% 0.30/0.46 = ( F @ ( comple939513234o_fixp @ A @ F ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fixp_unfold
% 0.30/0.46 thf(fact_241_monotone__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( comple1396247847notone @ A @ B )
% 0.30/0.46 = ( ^ [Orda: A > A > $o,Ordb: B > B > $o,F2: A > B] :
% 0.30/0.46 ! [X3: A,Y3: A] :
% 0.30/0.46 ( ( Orda @ X3 @ Y3 )
% 0.30/0.46 => ( Ordb @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % monotone_def
% 0.30/0.46 thf(fact_242_monotoneI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Orda2: A > A > $o,Ordb2: B > B > $o,F: A > B] :
% 0.30/0.46 ( ! [X2: A,Y2: A] :
% 0.30/0.46 ( ( Orda2 @ X2 @ Y2 )
% 0.30/0.46 => ( Ordb2 @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( comple1396247847notone @ A @ B @ Orda2 @ Ordb2 @ F ) ) ).
% 0.30/0.46
% 0.30/0.46 % monotoneI
% 0.30/0.46 thf(fact_243_monotoneD,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Orda2: A > A > $o,Ordb2: B > B > $o,F: A > B,X: A,Y: A] :
% 0.30/0.46 ( ( comple1396247847notone @ A @ B @ Orda2 @ Ordb2 @ F )
% 0.30/0.46 => ( ( Orda2 @ X @ Y )
% 0.30/0.46 => ( Ordb2 @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % monotoneD
% 0.30/0.46 thf(fact_244_fun__upd__transfer,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 0.30/0.46 ( ( bi_unique @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( A > C > A > C ) @ ( B > D > B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ A @ B @ ( C > A > C ) @ ( D > B > D ) @ A2 @ ( bNF_rel_fun @ C @ D @ ( A > C ) @ ( B > D ) @ B2 @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) @ ( fun_upd @ A @ C ) @ ( fun_upd @ B @ D ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_transfer
% 0.30/0.46 thf(fact_245_rtranclp__parametric,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 0.30/0.46 ( ( bi_unique @ A @ B @ A2 )
% 0.30/0.46 => ( ( bi_total @ A @ B @ A2 )
% 0.30/0.46 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( A > A > $o ) @ ( B > B > $o )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 0.30/0.46 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 0.30/0.46 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 0.30/0.46 @ ( transitive_rtranclp @ A )
% 0.30/0.46 @ ( transitive_rtranclp @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rtranclp_parametric
% 0.30/0.46 thf(fact_246_fun__upd__apply,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( fun_upd @ B @ A )
% 0.30/0.46 = ( ^ [F2: B > A,X3: B,Y3: A,Z4: B] : ( if @ A @ ( Z4 = X3 ) @ Y3 @ ( F2 @ Z4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_apply
% 0.30/0.46 thf(fact_247_fun__upd__triv,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 0.30/0.46 ( ( fun_upd @ A @ B @ F @ X @ ( F @ X ) )
% 0.30/0.46 = F ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_triv
% 0.30/0.46 thf(fact_248_fun__upd__upd,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,Z: B] :
% 0.30/0.46 ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ X @ Z )
% 0.30/0.46 = ( fun_upd @ A @ B @ F @ X @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_upd
% 0.30/0.46 thf(fact_249_fun__upd__idem__iff,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B] :
% 0.30/0.46 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 0.30/0.46 = F )
% 0.30/0.46 = ( ( F @ X )
% 0.30/0.46 = Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_idem_iff
% 0.30/0.46 thf(fact_250_fun__upd__twist,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A4: A,C3: A,M3: A > B,B4: B,D2: B] :
% 0.30/0.46 ( ( A4 != C3 )
% 0.30/0.46 => ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M3 @ A4 @ B4 ) @ C3 @ D2 )
% 0.30/0.46 = ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M3 @ C3 @ D2 ) @ A4 @ B4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_twist
% 0.30/0.46 thf(fact_251_fun__upd__other,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Z: A,X: A,F: A > B,Y: B] :
% 0.30/0.46 ( ( Z != X )
% 0.30/0.46 => ( ( fun_upd @ A @ B @ F @ X @ Y @ Z )
% 0.30/0.46 = ( F @ Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_other
% 0.30/0.46 thf(fact_252_fun__upd__same,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: B > A,X: B,Y: A] :
% 0.30/0.46 ( ( fun_upd @ B @ A @ F @ X @ Y @ X )
% 0.30/0.46 = Y ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_same
% 0.30/0.46 thf(fact_253_fun__upd__idem,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: B > A,X: B,Y: A] :
% 0.30/0.46 ( ( ( F @ X )
% 0.30/0.46 = Y )
% 0.30/0.46 => ( ( fun_upd @ B @ A @ F @ X @ Y )
% 0.30/0.46 = F ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_idem
% 0.30/0.46 thf(fact_254_fun__upd__eqD,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,G: A > B,Z: B] :
% 0.30/0.46 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 0.30/0.46 = ( fun_upd @ A @ B @ G @ X @ Z ) )
% 0.30/0.46 => ( Y = Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_eqD
% 0.30/0.46 thf(fact_255_fun__upd__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( fun_upd @ A @ B )
% 0.30/0.46 = ( ^ [F2: A > B,A5: A,B5: B,X3: A] : ( if @ B @ ( X3 = A5 ) @ B5 @ ( F2 @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_def
% 0.30/0.46
% 0.30/0.46 % Type constructors (24)
% 0.30/0.46 thf(tcon_Multiset_Omultiset___Orderings_Opreorder,axiom,
% 0.30/0.46 ! [A7: $tType] :
% 0.30/0.46 ( ( preorder @ A7 )
% 0.30/0.46 => ( preorder @ ( multiset @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_1,axiom,
% 0.30/0.46 preorder @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
% 0.30/0.46 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Opreorder_3,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( preorder @ A8 )
% 0.30/0.46 => ( preorder @ ( A7 > A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A8 )
% 0.30/0.46 => ( comple187826305attice @ ( A7 > A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( comple187826305attice @ A8 )
% 0.30/0.46 => ( comple1141879883l_ccpo @ ( A7 > A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/0.46 ! [A7: $tType,A8: $tType] :
% 0.30/0.46 ( ( order @ A8 )
% 0.30/0.46 => ( order @ ( A7 > A8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_4,axiom,
% 0.30/0.46 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_5,axiom,
% 0.30/0.46 ! [A7: $tType] : ( comple1141879883l_ccpo @ ( set @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Groups_Oab__semigroup__add,axiom,
% 0.30/0.46 ! [A7: $tType] :
% 0.30/0.46 ( ( ab_semigroup_add @ A7 )
% 0.30/0.46 => ( ab_semigroup_add @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Groups_Ocomm__monoid__add,axiom,
% 0.30/0.46 ! [A7: $tType] :
% 0.30/0.46 ( ( comm_monoid_add @ A7 )
% 0.30/0.46 => ( comm_monoid_add @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Groups_Osemigroup__add,axiom,
% 0.30/0.46 ! [A7: $tType] :
% 0.30/0.46 ( ( semigroup_add @ A7 )
% 0.30/0.46 => ( semigroup_add @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 0.30/0.46 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Groups_Oplus,axiom,
% 0.30/0.46 ! [A7: $tType] :
% 0.30/0.46 ( ( plus @ A7 )
% 0.30/0.46 => ( plus @ ( set @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_7,axiom,
% 0.30/0.46 comple187826305attice @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_8,axiom,
% 0.30/0.46 comple1141879883l_ccpo @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 0.30/0.46 order @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add,axiom,
% 0.30/0.46 ! [A7: $tType] :
% 0.30/0.46 ( ( preorder @ A7 )
% 0.30/0.46 => ( ordere779506340up_add @ ( multiset @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add,axiom,
% 0.30/0.46 ! [A7: $tType] : ( cancel_semigroup_add @ ( multiset @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_10,axiom,
% 0.30/0.46 ! [A7: $tType] : ( ab_semigroup_add @ ( multiset @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_11,axiom,
% 0.30/0.46 ! [A7: $tType] : ( comm_monoid_add @ ( multiset @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_12,axiom,
% 0.30/0.46 ! [A7: $tType] : ( semigroup_add @ ( multiset @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Multiset_Omultiset___Orderings_Oorder_13,axiom,
% 0.30/0.46 ! [A7: $tType] :
% 0.30/0.46 ( ( preorder @ A7 )
% 0.30/0.46 => ( order @ ( multiset @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Multiset_Omultiset___Groups_Oplus_14,axiom,
% 0.30/0.46 ! [A7: $tType] : ( plus @ ( multiset @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Helper facts (3)
% 0.30/0.46 thf(help_If_3_1_T,axiom,
% 0.30/0.46 ! [P: $o] :
% 0.30/0.46 ( ( P = $true )
% 0.30/0.46 | ( P = $false ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(help_If_2_1_T,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Y: A] :
% 0.30/0.46 ( ( if @ A @ $false @ X @ Y )
% 0.30/0.46 = Y ) ).
% 0.30/0.46
% 0.30/0.46 thf(help_If_1_1_T,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Y: A] :
% 0.30/0.46 ( ( if @ A @ $true @ X @ Y )
% 0.30/0.46 = X ) ).
% 0.30/0.46
% 0.30/0.46 % Conjectures (1)
% 0.30/0.46 thf(conj_0,conjecture,
% 0.30/0.46 finite100568337ommute @ a @ c @ f1 ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 2.89/1.53 ---- Embedded in THF ---
% 2.89/1.53 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.89/1.53 %%% Generated on Tue Feb 24 08:27:28 EST 2026
% 2.89/1.53 %%% using '$$dhol' embedding, version 1.3.0.
% 2.89/1.53 %%% Logic specification used:
% 2.89/1.53 %%% thf(spec, logic, $$dhol).
% 2.89/1.53
% 2.89/1.53 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.XftS7e7A06/DTF2DTF9803.p
% 2.89/1.53 thf(ty_t_Multiset_Omultiset, type, multiset: ($tType > $tType)).
% 2.89/1.53 thf(ty_t_FSet_Ofset, type, fset: ($tType > $tType)).
% 2.89/1.53 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.89/1.53 thf(ty_tf_d, type, d: $tType).
% 2.89/1.53 thf(ty_tf_c, type, c: $tType).
% 2.89/1.53 thf(ty_tf_b, type, b: $tType).
% 2.89/1.53 thf(ty_tf_a, type, a: $tType).
% 2.89/1.53 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.89/1.53 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.89/1.53 thf(sy_cl_Groups_Oplus, type, plus: (!> [A:$tType]: ($o))).
% 2.89/1.53 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.89/1.53 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.89/1.53 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))).
% 2.89/1.53 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 2.89/1.53 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))).
% 2.89/1.53 thf(sy_cl_Complete__Partial__Order_Occpo, type, comple1141879883l_ccpo: (!> [A:$tType]: ($o))).
% 2.89/1.53 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 2.89/1.53 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: ($o))).
% 2.89/1.53 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple187826305attice: (!> [A:$tType]: ($o))).
% 2.89/1.53 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.89/1.53 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: ($o))).
% 2.89/1.53 thf(sy_c_BNF__Def_Orel__fun, type, bNF_rel_fun: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > (C > $o)) > ((B > (D > $o)) > ((A > B) > ((C > D) > $o))))))).
% 2.89/1.53 thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp, type, comple939513234o_fixp: (!> [A:$tType]: (((A > A) > A)))).
% 2.89/1.53 thf(sy_c_Complete__Partial__Order_Omonotone, type, comple1396247847notone: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((B > (B > $o)) > ((A > B) > $o)))))).
% 2.89/1.53 thf(sy_c_FSet_Offold, type, ffold: (!> [A:$tType,B:$tType]: (((A > (B > B)) > (B > ((fset @ A) > B)))))).
% 2.89/1.53 thf(sy_c_Finite__Set_Ocomp__fun__commute, type, finite100568337ommute: (!> [A:$tType,B:$tType]: (((A > (B > B)) > $o)))).
% 2.89/1.53 thf(sy_c_Finite__Set_Ocomp__fun__idem, type, finite_comp_fun_idem: (!> [A:$tType,B:$tType]: (((A > (B > B)) > $o)))).
% 2.89/1.53 thf(sy_c_Finite__Set_Ocomp__fun__idem__axioms, type, finite852775215axioms: (!> [A:$tType,B:$tType]: (((A > (B > B)) > $o)))).
% 2.89/1.53 thf(sy_c_Finite__Set_Ofold__graph, type, finite_fold_graph: (!> [A:$tType,B:$tType]: (((A > (B > B)) > (B > ((set @ A) > (B > $o))))))).
% 2.89/1.53 thf(sy_c_Finite__Set_Ofolding, type, finite_folding: (!> [A:$tType,B:$tType]: (((A > (B > B)) > $o)))).
% 2.89/1.53 thf(sy_c_Finite__Set_Ofolding__idem, type, finite_folding_idem: (!> [A:$tType,B:$tType]: (((A > (B > B)) > $o)))).
% 2.89/1.53 thf(sy_c_Finite__Set_Ofolding__idem__axioms, type, finite1921348288axioms: (!> [A:$tType,B:$tType]: (((A > (B > B)) > $o)))).
% 2.89/1.53 thf(sy_c_Fun_Ocomp, type, comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > B) > (A > C)))))).
% 2.89/1.53 thf(sy_c_Fun_Ofun__upd, type, fun_upd: (!> [A:$tType,B:$tType]: (((A > B) > (A > (B > (A > B))))))).
% 2.89/1.53 thf(sy_c_Fun_Omap__fun, type, map_fun: (!> [C:$tType,A:$tType,B:$tType,D:$tType]: (((C > A) > ((B > D) > ((A > B) > (C > D))))))).
% 2.89/1.53 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.89/1.53 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.89/1.53 thf(sy_c_Multiset_Ofold__mset, type, fold_mset: (!> [A:$tType,B:$tType]: (((A > (B > B)) > (B > ((multiset @ A) > B)))))).
% 2.89/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.89/1.53 thf(sy_c_Orderings_Oorder__class_Omono, type, order_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.89/1.53 thf(sy_c_Partial__Function_Ofun__ord, type, partial_fun_ord: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((C > A) > ((C > B) > $o)))))).
% 2.89/1.53 thf(sy_c_Quotient_OBex1__rel, type, bex1_rel: (!> [A:$tType]: (((A > (A > $o)) > ((A > $o) > $o))))).
% 2.89/1.53 thf(sy_c_Quotient_OQuotient3, type, quotient3: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > ((B > A) > $o)))))).
% 2.89/1.53 thf(sy_c_Relation_ODomainp, type, domainp: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > (A > $o))))).
% 2.89/1.53 thf(sy_c_Relation_Oreflp, type, reflp: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 2.89/1.53 thf(sy_c_Relation_Orelcompp, type, relcompp: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((B > (C > $o)) > (A > (C > $o))))))).
% 2.89/1.53 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.89/1.53 thf(sy_c_Transfer_Obi__total, type, bi_total: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > $o)))).
% 2.89/1.53 thf(sy_c_Transfer_Obi__unique, type, bi_unique: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > $o)))).
% 2.89/1.53 thf(sy_c_Transfer_Oleft__total, type, left_total: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > $o)))).
% 2.89/1.53 thf(sy_c_Transfer_Oleft__unique, type, left_unique: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > $o)))).
% 2.89/1.53 thf(sy_c_Transfer_Orev__implies, type, rev_implies: ($o > ($o > $o))).
% 2.89/1.53 thf(sy_c_Transfer_Oright__total, type, right_total: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > $o)))).
% 2.89/1.53 thf(sy_c_Transfer_Oright__unique, type, right_unique: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > $o)))).
% 2.89/1.53 thf(sy_c_Transfer_Otransfer__bforall, type, transfer_bforall: (!> [A:$tType]: (((A > $o) > ((A > $o) > $o))))).
% 2.89/1.53 thf(sy_c_Transfer_Otransfer__forall, type, transfer_forall: (!> [A:$tType]: (((A > $o) > $o)))).
% 2.89/1.53 thf(sy_c_Transitive__Closure_Ortranclp, type, transitive_rtranclp: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > $o)))))).
% 2.89/1.53 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.89/1.53 thf(sy_v_A, type, a2: (a > (b > $o))).
% 2.89/1.53 thf(sy_v_B, type, b2: (c > (d > $o))).
% 2.89/1.53 thf(sy_v_f1, type, f1: (a > (c > c))).
% 2.89/1.53 thf(sy_v_f2, type, f2: (b > (d > d))).
% 2.89/1.53 thf(fact_0_assms_I2_J, axiom, (((finite100568337ommute @ a) @ c) @ f1)).
% 2.89/1.53 thf(fact_1_comp__fun__commute_Ofun__left__comm, axiom, (! [A:$tType,B:$tType,F:(A > (B > B)),Y:A,X:A,Z:B]: (((((finite100568337ommute @ A) @ B) @ F) => (((F @ Y) @ ((F @ X) @ Z)) = ((F @ X) @ ((F @ Y) @ Z))))))).
% 2.89/1.53 thf(fact_2_assms_I3_J, axiom, (((finite100568337ommute @ b) @ d) @ f2)).
% 2.89/1.53 thf(fact_3__C12_C, axiom, ((((((((bNF_rel_fun @ a) @ b) @ (c > c)) @ (d > d)) @ a2) @ ((((((bNF_rel_fun @ c) @ d) @ c) @ d) @ b2) @ b2)) @ f1) @ f2)).
% 2.89/1.53 thf(fact_4_comp__fun__commute_Ofold__mset__fusion, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > B)),G:(A > (C > C)),H:(C > B),W:C,A2:(multiset @ A)]: (((((finite100568337ommute @ A) @ B) @ F) => ((((finite100568337ommute @ A) @ C) @ G) => ((! [X2:A,Y2:C]: (((H @ ((G @ X2) @ Y2)) = ((F @ X2) @ (H @ Y2))))) => ((H @ (((((fold_mset @ A) @ C) @ G) @ W) @ A2)) = (((((fold_mset @ A) @ B) @ F) @ (H @ W)) @ A2)))))))).
% 2.89/1.53 thf(fact_5_comp__fun__commute_Ofold__mset__fun__left__comm, axiom, (! [B:$tType,A:$tType,F:(A > (B > B)),X:A,S:B,M:(multiset @ A)]: (((((finite100568337ommute @ A) @ B) @ F) => (((F @ X) @ (((((fold_mset @ A) @ B) @ F) @ S) @ M)) = (((((fold_mset @ A) @ B) @ F) @ ((F @ X) @ S)) @ M)))))).
% 2.89/1.53 thf(fact_6_comp__fun__idem_Oaxioms_I1_J, axiom, (! [B:$tType,A:$tType,F:(A > (B > B))]: (((((finite_comp_fun_idem @ A) @ B) @ F) => (((finite100568337ommute @ A) @ B) @ F))))).
% 2.89/1.53 thf(fact_7_comp__fun__commute_Ofold__graph__determ, axiom, (! [A:$tType,B:$tType,F:(A > (B > B)),Z:B,A2:(set @ A),X:B,Y:B]: (((((finite100568337ommute @ A) @ B) @ F) => (((((((finite_fold_graph @ A) @ B) @ F) @ Z) @ A2) @ X) => (((((((finite_fold_graph @ A) @ B) @ F) @ Z) @ A2) @ Y) => (Y = X))))))).
% 2.89/1.53 thf(fact_8_comp__fun__commute_Offold__fun__left__comm, axiom, (! [B:$tType,A:$tType,F:(A > (B > B)),X:A,Z:B,A2:(fset @ A)]: (((((finite100568337ommute @ A) @ B) @ F) => (((F @ X) @ (((((ffold @ A) @ B) @ F) @ Z) @ A2)) = (((((ffold @ A) @ B) @ F) @ ((F @ X) @ Z)) @ A2)))))).
% 2.89/1.53 thf(fact_9_comp__fun__commute_Ocomp__comp__fun__commute, axiom, (! [B:$tType,A:$tType,C:$tType,F:(A > (B > B)),G:(C > A)]: (((((finite100568337ommute @ A) @ B) @ F) => (((finite100568337ommute @ C) @ B) @ (((((comp @ A) @ (B > B)) @ C) @ F) @ G)))))).
% 2.89/1.53 thf(fact_10_comp__fun__commute__def, axiom, (! [B:$tType,A:$tType]: ((((finite100568337ommute @ A) @ B) = (^ [F2:(A > (B > B))]: ((! [Y3:A,X3:A]: (((((((comp @ B) @ B) @ B) @ (F2 @ Y3)) @ (F2 @ X3)) = (((((comp @ B) @ B) @ B) @ (F2 @ X3)) @ (F2 @ Y3))))))))))).
% 2.89/1.53 thf(fact_11_comp__fun__commute_Ointro, axiom, (! [B:$tType,A:$tType,F:(A > (B > B))]: (((! [Y2:A,X2:A]: (((((((comp @ B) @ B) @ B) @ (F @ Y2)) @ (F @ X2)) = (((((comp @ B) @ B) @ B) @ (F @ X2)) @ (F @ Y2))))) => (((finite100568337ommute @ A) @ B) @ F))))).
% 2.89/1.53 thf(fact_12_comp__fun__commute_Ocomp__fun__commute, axiom, (! [B:$tType,A:$tType,F:(A > (B > B)),Y:A,X:A]: (((((finite100568337ommute @ A) @ B) @ F) => ((((((comp @ B) @ B) @ B) @ (F @ Y)) @ (F @ X)) = (((((comp @ B) @ B) @ B) @ (F @ X)) @ (F @ Y))))))).
% 2.89/1.53 thf(fact_13_fold__graph__closed__eq, axiom, (! [B:$tType,A:$tType,A2:(set @ A),B2:(set @ B),F:(A > (B > B)),G:(A > (B > B)),Z:B]: (((! [A3:A,B3:B]: (((((member @ A) @ A3) @ A2) => ((((member @ B) @ B3) @ B2) => (((F @ A3) @ B3) = ((G @ A3) @ B3)))))) => ((! [A3:A,B3:B]: (((((member @ A) @ A3) @ A2) => ((((member @ B) @ B3) @ B2) => (((member @ B) @ ((G @ A3) @ B3)) @ B2))))) => ((((member @ B) @ Z) @ B2) => ((((((finite_fold_graph @ A) @ B) @ F) @ Z) @ A2) = (((((finite_fold_graph @ A) @ B) @ G) @ Z) @ A2)))))))).
% 2.89/1.53 thf(fact_14_fold__graph__closed__lemma, axiom, (! [A:$tType,B:$tType,G:(A > (B > B)),Z:B,A2:(set @ A),X:B,B2:(set @ B),F:(A > (B > B))]: ((((((((finite_fold_graph @ A) @ B) @ G) @ Z) @ A2) @ X) => ((! [A3:A,B3:B]: (((((member @ A) @ A3) @ A2) => ((((member @ B) @ B3) @ B2) => (((F @ A3) @ B3) = ((G @ A3) @ B3)))))) => ((! [A3:A,B3:B]: (((((member @ A) @ A3) @ A2) => ((((member @ B) @ B3) @ B2) => (((member @ B) @ ((G @ A3) @ B3)) @ B2))))) => ((((member @ B) @ Z) @ B2) => (((((((finite_fold_graph @ A) @ B) @ F) @ Z) @ A2) @ X) & (((member @ B) @ X) @ B2))))))))).
% 2.89/1.53 thf(fact_15_comp__fun__idem_Ocomp__fun__idem, axiom, (! [B:$tType,A:$tType,F:(A > (B > B)),X:A]: (((((finite_comp_fun_idem @ A) @ B) @ F) => ((((((comp @ B) @ B) @ B) @ (F @ X)) @ (F @ X)) = (F @ X)))))).
% 2.89/1.53 thf(fact_16_comp__fun__idem_Ofun__left__idem, axiom, (! [A:$tType,B:$tType,F:(A > (B > B)),X:A,Z:B]: (((((finite_comp_fun_idem @ A) @ B) @ F) => (((F @ X) @ ((F @ X) @ Z)) = ((F @ X) @ Z)))))).
% 2.89/1.53 thf(fact_17_comp__fun__idem_Ocomp__comp__fun__idem, axiom, (! [B:$tType,A:$tType,C:$tType,F:(A > (B > B)),G:(C > A)]: (((((finite_comp_fun_idem @ A) @ B) @ F) => (((finite_comp_fun_idem @ C) @ B) @ (((((comp @ A) @ (B > B)) @ C) @ F) @ G)))))).
% 2.89/1.53 thf(fact_18_comp__fun__commute_Ocommute__left__comp, axiom, (! [A:$tType,B:$tType,C:$tType,F:(A > (B > B)),Y:A,X:A,G:(C > B)]: (((((finite100568337ommute @ A) @ B) @ F) => ((((((comp @ B) @ B) @ C) @ (F @ Y)) @ (((((comp @ B) @ B) @ C) @ (F @ X)) @ G)) = (((((comp @ B) @ B) @ C) @ (F @ X)) @ (((((comp @ B) @ B) @ C) @ (F @ Y)) @ G))))))).
% 2.89/1.53 thf(fact_19_comp__apply, axiom, (! [C:$tType,A:$tType,B:$tType]: (((((comp @ B) @ A) @ C) = (^ [F2:(B > A),G2:(C > B),X3:C]: ((F2 @ (G2 @ X3)))))))).
% 2.89/1.53 thf(fact_20_rel__funI, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o)),F:(A > C),G:(B > D)]: (((! [X2:A,Y2:B]: ((((A2 @ X2) @ Y2) => ((B2 @ (F @ X2)) @ (G @ Y2))))) => ((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2) @ F) @ G))))).
% 2.89/1.53 thf(fact_21_If__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((((((bNF_rel_fun @ $o) @ $o) @ (A > (A > A))) @ (B > (B > B))) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))) @ ((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A2) @ A2))) @ (if @ A)) @ (if @ B))))).
% 2.89/1.53 thf(fact_22_rel__fun__def__butlast, axiom, (! [B:$tType,D:$tType,C:$tType,E:$tType,F3:$tType,A:$tType,R:(A > (B > $o)),S2:(C > (E > $o)),T:(D > (F3 > $o)),F:(A > (C > D)),G:(B > (E > F3))]: ((((((((((bNF_rel_fun @ A) @ B) @ (C > D)) @ (E > F3)) @ R) @ ((((((bNF_rel_fun @ C) @ E) @ D) @ F3) @ S2) @ T)) @ F) @ G) = (! [X3:A,Y3:B]: ((((R @ X3) @ Y3) => ((((((((bNF_rel_fun @ C) @ E) @ D) @ F3) @ S2) @ T) @ (F @ X3)) @ (G @ Y3))))))))).
% 2.89/1.53 thf(fact_23_o__rsp_I2_J, axiom, (! [E:$tType,F3:$tType,H2:$tType,G3:$tType,R1:(E > (F3 > $o))]: (((((((((bNF_rel_fun @ (G3 > H2)) @ (G3 > H2)) @ ((E > G3) > (E > H2))) @ ((F3 > G3) > (F3 > H2))) @ (^ [Y4:(G3 > H2),Z2:(G3 > H2)]: ((Y4 = Z2)))) @ ((((((bNF_rel_fun @ (E > G3)) @ (F3 > G3)) @ (E > H2)) @ (F3 > H2)) @ ((((((bNF_rel_fun @ E) @ F3) @ G3) @ G3) @ R1) @ (^ [Y4:G3,Z2:G3]: ((Y4 = Z2))))) @ ((((((bNF_rel_fun @ E) @ F3) @ H2) @ H2) @ R1) @ (^ [Y4:H2,Z2:H2]: ((Y4 = Z2)))))) @ (((comp @ G3) @ H2) @ E)) @ (((comp @ G3) @ H2) @ F3))))).
% 2.89/1.53 thf(fact_24_o__rsp_I1_J, axiom, (! [A:$tType,B:$tType,E:$tType,F3:$tType,D:$tType,C:$tType,R2:(A > (C > $o)),R3:(B > (D > $o)),R1:(E > (F3 > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (C > D)) @ ((E > A) > (E > B))) @ ((F3 > C) > (F3 > D))) @ ((((((bNF_rel_fun @ A) @ C) @ B) @ D) @ R2) @ R3)) @ ((((((bNF_rel_fun @ (E > A)) @ (F3 > C)) @ (E > B)) @ (F3 > D)) @ ((((((bNF_rel_fun @ E) @ F3) @ A) @ C) @ R1) @ R2)) @ ((((((bNF_rel_fun @ E) @ F3) @ B) @ D) @ R1) @ R3))) @ (((comp @ A) @ B) @ E)) @ (((comp @ C) @ D) @ F3))))).
% 2.89/1.53 thf(fact_25_fun_Omap__transfer, axiom, (! [A:$tType,B:$tType,D:$tType,G3:$tType,F3:$tType,Rb:(A > (F3 > $o)),Sd:(B > (G3 > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (F3 > G3)) @ ((D > A) > (D > B))) @ ((D > F3) > (D > G3))) @ ((((((bNF_rel_fun @ A) @ F3) @ B) @ G3) @ Rb) @ Sd)) @ ((((((bNF_rel_fun @ (D > A)) @ (D > F3)) @ (D > B)) @ (D > G3)) @ ((((((bNF_rel_fun @ D) @ D) @ A) @ F3) @ (^ [Y4:D,Z2:D]: ((Y4 = Z2)))) @ Rb)) @ ((((((bNF_rel_fun @ D) @ D) @ B) @ G3) @ (^ [Y4:D,Z2:D]: ((Y4 = Z2)))) @ Sd))) @ (((comp @ A) @ B) @ D)) @ (((comp @ F3) @ G3) @ D))))).
% 2.89/1.53 thf(fact_26_comp__transfer, axiom, (! [A:$tType,B:$tType,E:$tType,F3:$tType,D:$tType,C:$tType,B2:(A > (C > $o)),C2:(B > (D > $o)),A2:(E > (F3 > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (C > D)) @ ((E > A) > (E > B))) @ ((F3 > C) > (F3 > D))) @ ((((((bNF_rel_fun @ A) @ C) @ B) @ D) @ B2) @ C2)) @ ((((((bNF_rel_fun @ (E > A)) @ (F3 > C)) @ (E > B)) @ (F3 > D)) @ ((((((bNF_rel_fun @ E) @ F3) @ A) @ C) @ A2) @ B2)) @ ((((((bNF_rel_fun @ E) @ F3) @ B) @ D) @ A2) @ C2))) @ (((comp @ A) @ B) @ E)) @ (((comp @ C) @ D) @ F3))))).
% 2.89/1.53 thf(fact_27_comp__fun__idem_Ointro, axiom, (! [B:$tType,A:$tType,F:(A > (B > B))]: (((((finite100568337ommute @ A) @ B) @ F) => ((((finite852775215axioms @ A) @ B) @ F) => (((finite_comp_fun_idem @ A) @ B) @ F)))))).
% 2.89/1.53 thf(fact_28_comp__fun__idem__def, axiom, (! [B:$tType,A:$tType]: ((((finite_comp_fun_idem @ A) @ B) = (^ [F2:(A > (B > B))]: (((((finite100568337ommute @ A) @ B) @ F2) & (((finite852775215axioms @ A) @ B) @ F2)))))))).
% 2.89/1.53 thf(fact_29_comp__fun__idem_Oaxioms_I2_J, axiom, (! [B:$tType,A:$tType,F:(A > (B > B))]: (((((finite_comp_fun_idem @ A) @ B) @ F) => (((finite852775215axioms @ A) @ B) @ F))))).
% 2.89/1.53 thf(fact_30_comp__fun__idem__axioms_Ointro, axiom, (! [B:$tType,A:$tType,F:(A > (B > B))]: (((! [X2:A]: (((((((comp @ B) @ B) @ B) @ (F @ X2)) @ (F @ X2)) = (F @ X2)))) => (((finite852775215axioms @ A) @ B) @ F))))).
% 2.89/1.53 thf(fact_31_comp__fun__idem__axioms__def, axiom, (! [B:$tType,A:$tType]: ((((finite852775215axioms @ A) @ B) = (^ [F2:(A > (B > B))]: ((! [X3:A]: (((((((comp @ B) @ B) @ B) @ (F2 @ X3)) @ (F2 @ X3)) = (F2 @ X3)))))))))).
% 2.89/1.53 thf(fact_32_fun_Orel__transfer, axiom, (! [B:$tType,A:$tType,C:$tType,E:$tType,D:$tType,Sa:(A > (C > $o)),Sc:(B > (E > $o))]: (((((((((bNF_rel_fun @ (A > (B > $o))) @ (C > (E > $o))) @ ((D > A) > ((D > B) > $o))) @ ((D > C) > ((D > E) > $o))) @ ((((((bNF_rel_fun @ A) @ C) @ (B > $o)) @ (E > $o)) @ Sa) @ ((((((bNF_rel_fun @ B) @ E) @ $o) @ $o) @ Sc) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ ((((((bNF_rel_fun @ (D > A)) @ (D > C)) @ ((D > B) > $o)) @ ((D > E) > $o)) @ ((((((bNF_rel_fun @ D) @ D) @ A) @ C) @ (^ [Y4:D,Z2:D]: ((Y4 = Z2)))) @ Sa)) @ ((((((bNF_rel_fun @ (D > B)) @ (D > E)) @ $o) @ $o) @ ((((((bNF_rel_fun @ D) @ D) @ B) @ E) @ (^ [Y4:D,Z2:D]: ((Y4 = Z2)))) @ Sc)) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ (((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y4:D,Z2:D]: ((Y4 = Z2))))) @ (((((bNF_rel_fun @ D) @ D) @ C) @ E) @ (^ [Y4:D,Z2:D]: ((Y4 = Z2)))))))).
% 2.89/1.53 thf(fact_33_fun_Orel__refl, axiom, (! [B:$tType,D:$tType,Ra:(B > (B > $o)),X:(D > B)]: (((! [X2:B]: (((Ra @ X2) @ X2))) => ((((((((bNF_rel_fun @ D) @ D) @ B) @ B) @ (^ [Y4:D,Z2:D]: ((Y4 = Z2)))) @ Ra) @ X) @ X))))).
% 2.89/1.53 thf(fact_34_fun_Orel__eq, axiom, (! [A:$tType,D:$tType]: ((((((((bNF_rel_fun @ D) @ D) @ A) @ A) @ (^ [Y4:D,Z2:D]: ((Y4 = Z2)))) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2)))) = (^ [Y4:(D > A),Z2:(D > A)]: ((Y4 = Z2))))))).
% 2.89/1.53 thf(fact_35_rel__fun__mono_H, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,Y5:(A > (B > $o)),X4:(A > (B > $o)),A2:(C > (D > $o)),B2:(C > (D > $o)),F:(A > C),G:(B > D)]: (((! [X2:A,Y2:B]: ((((Y5 @ X2) @ Y2) => ((X4 @ X2) @ Y2)))) => ((! [X2:C,Y2:D]: ((((A2 @ X2) @ Y2) => ((B2 @ X2) @ Y2)))) => (((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ X4) @ A2) @ F) @ G) => ((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ Y5) @ B2) @ F) @ G))))))).
% 2.89/1.53 thf(fact_36_rel__fun__mono, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,X4:(A > (B > $o)),A2:(C > (D > $o)),F:(A > C),G:(B > D),Y5:(A > (B > $o)),B2:(C > (D > $o))]: ((((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ X4) @ A2) @ F) @ G) => ((! [X2:A,Y2:B]: ((((Y5 @ X2) @ Y2) => ((X4 @ X2) @ Y2)))) => ((! [X2:C,Y2:D]: ((((A2 @ X2) @ Y2) => ((B2 @ X2) @ Y2)))) => ((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ Y5) @ B2) @ F) @ G))))))).
% 2.89/1.53 thf(fact_37_let__rsp, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,R1:(A > (B > $o)),R2:(C > (D > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ ((A > C) > C)) @ ((B > D) > D)) @ R1) @ ((((((bNF_rel_fun @ (A > C)) @ (B > D)) @ C) @ D) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ R1) @ R2)) @ R2)) @ (^ [S3:A,F2:(A > C)]: ((F2 @ S3)))) @ (^ [S3:B,F2:(B > D)]: ((F2 @ S3))))))).
% 2.89/1.53 thf(fact_38_rel__funD, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o)),F:(A > C),G:(B > D),X:A,Y:B]: ((((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2) @ F) @ G) => (((A2 @ X) @ Y) => ((B2 @ (F @ X)) @ (G @ Y))))))).
% 2.89/1.53 thf(fact_39_rewriteR__comp__comp2, axiom, (! [C:$tType,B:$tType,E:$tType,D:$tType,A:$tType,G:(C > B),H:(A > C),R12:(D > B),R22:(A > D),F:(B > E),L:(D > E)]: ((((((((comp @ C) @ B) @ A) @ G) @ H) = (((((comp @ D) @ B) @ A) @ R12) @ R22)) => (((((((comp @ B) @ E) @ D) @ F) @ R12) = L) => ((((((comp @ C) @ E) @ A) @ (((((comp @ B) @ E) @ C) @ F) @ G)) @ H) = (((((comp @ D) @ E) @ A) @ L) @ R22))))))).
% 2.89/1.53 thf(fact_40_rewriteL__comp__comp2, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,E:$tType,F:(C > B),G:(A > C),L1:(D > B),L2:(A > D),H:(E > A),R4:(E > D)]: ((((((((comp @ C) @ B) @ A) @ F) @ G) = (((((comp @ D) @ B) @ A) @ L1) @ L2)) => (((((((comp @ A) @ D) @ E) @ L2) @ H) = R4) => ((((((comp @ C) @ B) @ E) @ F) @ (((((comp @ A) @ C) @ E) @ G) @ H)) = (((((comp @ D) @ B) @ E) @ L1) @ R4))))))).
% 2.89/1.53 thf(fact_41_rewriteR__comp__comp, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,G:(C > B),H:(A > C),R4:(A > B),F:(B > D)]: ((((((((comp @ C) @ B) @ A) @ G) @ H) = R4) => ((((((comp @ C) @ D) @ A) @ (((((comp @ B) @ D) @ C) @ F) @ G)) @ H) = (((((comp @ B) @ D) @ A) @ F) @ R4)))))).
% 2.89/1.53 thf(fact_42_rewriteL__comp__comp, axiom, (! [C:$tType,B:$tType,A:$tType,D:$tType,F:(C > B),G:(A > C),L:(A > B),H:(D > A)]: ((((((((comp @ C) @ B) @ A) @ F) @ G) = L) => ((((((comp @ C) @ B) @ D) @ F) @ (((((comp @ A) @ C) @ D) @ G) @ H)) = (((((comp @ A) @ B) @ D) @ L) @ H)))))).
% 2.89/1.53 thf(fact_43_fun_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,D:$tType,G:(B > C),F:(A > B),V:(D > A)]: (((((((comp @ B) @ C) @ D) @ G) @ (((((comp @ A) @ B) @ D) @ F) @ V)) = (((((comp @ A) @ C) @ D) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ V))))).
% 2.89/1.53 thf(fact_44_comp__apply__eq, axiom, (! [B:$tType,D:$tType,A:$tType,C:$tType,F:(B > A),G:(C > B),X:C,H:(D > A),K:(C > D)]: ((((F @ (G @ X)) = (H @ (K @ X))) => (((((((comp @ B) @ A) @ C) @ F) @ G) @ X) = ((((((comp @ D) @ A) @ C) @ H) @ K) @ X)))))).
% 2.89/1.53 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A4:A,P:(A > $o)]: (((((member @ A) @ A4) @ ((collect @ A) @ P)) = (P @ A4))))).
% 2.89/1.53 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A2)))) = A2)))).
% 2.89/1.53 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.89/1.53 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 2.89/1.53 thf(fact_49_comp__eq__dest__lhs, axiom, (! [C:$tType,B:$tType,A:$tType,A4:(C > B),B4:(A > C),C3:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A4) @ B4) = C3) => ((A4 @ (B4 @ V)) = (C3 @ V)))))).
% 2.89/1.53 thf(fact_50_comp__eq__elim, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A4:(C > B),B4:(A > C),C3:(D > B),D2:(A > D)]: ((((((((comp @ C) @ B) @ A) @ A4) @ B4) = (((((comp @ D) @ B) @ A) @ C3) @ D2)) => (! [V2:A]: (((A4 @ (B4 @ V2)) = (C3 @ (D2 @ V2))))))))).
% 2.89/1.53 thf(fact_51_comp__eq__dest, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A4:(C > B),B4:(A > C),C3:(D > B),D2:(A > D),V:A]: ((((((((comp @ C) @ B) @ A) @ A4) @ B4) = (((((comp @ D) @ B) @ A) @ C3) @ D2)) => ((A4 @ (B4 @ V)) = (C3 @ (D2 @ V))))))).
% 2.89/1.53 thf(fact_52_comp__assoc, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F:(D > B),G:(C > D),H:(A > C)]: (((((((comp @ C) @ B) @ A) @ (((((comp @ D) @ B) @ C) @ F) @ G)) @ H) = (((((comp @ D) @ B) @ A) @ F) @ (((((comp @ C) @ D) @ A) @ G) @ H)))))).
% 2.89/1.53 thf(fact_53_comp__def, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((comp @ B) @ C) @ A) = (^ [F2:(B > C),G2:(A > B),X3:A]: ((F2 @ (G2 @ X3)))))))).
% 2.89/1.53 thf(fact_54_Let__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ ((A > C) > C)) @ ((B > D) > D)) @ A2) @ ((((((bNF_rel_fun @ (A > C)) @ (B > D)) @ C) @ D) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2)) @ B2)) @ (^ [S3:A,F2:(A > C)]: ((F2 @ S3)))) @ (^ [S3:B,F2:(B > D)]: ((F2 @ S3))))))).
% 2.89/1.53 thf(fact_55_type__copy__map__cong0, axiom, (! [B:$tType,D:$tType,E:$tType,A:$tType,C:$tType,M:(B > A),G:(C > B),X:C,N:(D > A),H:(C > D),F:(A > E)]: ((((M @ (G @ X)) = (N @ (H @ X))) => (((((((comp @ B) @ E) @ C) @ (((((comp @ A) @ E) @ B) @ F) @ M)) @ G) @ X) = ((((((comp @ D) @ E) @ C) @ (((((comp @ A) @ E) @ D) @ F) @ N)) @ H) @ X)))))).
% 2.89/1.53 thf(fact_56_function__factors__right, axiom, (! [C:$tType,B:$tType,A:$tType,G:(B > C),F:(A > C)]: (((! [X3:A]: ((? [Y3:B]: (((G @ Y3) = (F @ X3)))))) = (? [H3:(A > B)]: ((F = (((((comp @ B) @ C) @ A) @ G) @ H3)))))))).
% 2.89/1.53 thf(fact_57_function__factors__left, axiom, (! [A:$tType,C:$tType,B:$tType,G:(A > B),F:(A > C)]: (((! [X3:A,Y3:A]: ((((G @ X3) = (G @ Y3)) => ((F @ X3) = (F @ Y3))))) = (? [H3:(B > C)]: ((F = (((((comp @ B) @ C) @ A) @ H3) @ G)))))))).
% 2.89/1.53 thf(fact_58_comp__cong, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,E:$tType,F:(B > A),G:(C > B),X:C,F4:(D > A),G4:(E > D),X5:E]: ((((F @ (G @ X)) = (F4 @ (G4 @ X5))) => (((((((comp @ B) @ A) @ C) @ F) @ G) @ X) = ((((((comp @ D) @ A) @ E) @ F4) @ G4) @ X5)))))).
% 2.89/1.53 thf(fact_59_if__rsp, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A)]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((((((((bNF_rel_fun @ $o) @ $o) @ (A > (A > A))) @ (A > (A > A))) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))) @ ((((((bNF_rel_fun @ A) @ A) @ (A > A)) @ (A > A)) @ R) @ ((((((bNF_rel_fun @ A) @ A) @ A) @ A) @ R) @ R))) @ (if @ A)) @ (if @ A)))))).
% 2.89/1.53 thf(fact_60_rel__funD2, axiom, (! [B:$tType,C:$tType,A:$tType,A2:(A > (A > $o)),B2:(B > (C > $o)),F:(A > B),G:(A > C),X:A]: ((((((((((bNF_rel_fun @ A) @ A) @ B) @ C) @ A2) @ B2) @ F) @ G) => (((A2 @ X) @ X) => ((B2 @ (F @ X)) @ (G @ X))))))).
% 2.89/1.53 thf(fact_61_apply__rsp_H, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,R1:(A > (B > $o)),R2:(C > (D > $o)),F:(A > C),G:(B > D),X:A,Y:B]: ((((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ R1) @ R2) @ F) @ G) => (((R1 @ X) @ Y) => ((R2 @ (F @ X)) @ (G @ Y))))))).
% 2.89/1.53 thf(fact_62_rel__funE, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o)),F:(A > C),G:(B > D),X:A,Y:B]: ((((((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2) @ F) @ G) => (((A2 @ X) @ Y) => ((B2 @ (F @ X)) @ (G @ Y))))))).
% 2.89/1.53 thf(fact_63_quot__rel__rsp, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A)]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((((((((bNF_rel_fun @ A) @ A) @ (A > $o)) @ (A > $o)) @ R) @ ((((((bNF_rel_fun @ A) @ A) @ $o) @ $o) @ R) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2))))) @ R) @ R))))).
% 2.89/1.53 thf(fact_64_Quotient3__rep__reflp, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),A4:B]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((R @ (Rep @ A4)) @ (Rep @ A4)))))).
% 2.89/1.53 thf(fact_65_Quotient3__rep__abs, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R4:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R4) @ R4) => ((R @ (Rep @ (Abs @ R4))) @ R4)))))).
% 2.89/1.53 thf(fact_66_Quotient3__rel__rep, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),A4:B,B4:B]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ (Rep @ A4)) @ (Rep @ B4)) = (A4 = B4)))))).
% 2.89/1.53 thf(fact_67_Quotient3__rel__abs, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R4:A,S:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R4) @ S) => ((Abs @ R4) = (Abs @ S))))))).
% 2.89/1.53 thf(fact_68_Quotient3__abs__rep, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),A4:B]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((Abs @ (Rep @ A4)) = A4))))).
% 2.89/1.53 thf(fact_69_rep__abs__rsp__left, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),X1:A,X22:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ X1) @ X22) => ((R @ (Rep @ (Abs @ X1))) @ X22)))))).
% 2.89/1.53 thf(fact_70_Quotient3__refl2, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R4:A,S:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R4) @ S) => ((R @ S) @ S)))))).
% 2.89/1.53 thf(fact_71_Quotient3__refl1, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R4:A,S:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ R4) @ S) => ((R @ R4) @ R4)))))).
% 2.89/1.53 thf(fact_72_Quotient3__rel, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),R4:A,S:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => ((((R @ R4) @ R4) & (((R @ S) @ S) & ((Abs @ R4) = (Abs @ S)))) = ((R @ R4) @ S)))))).
% 2.89/1.53 thf(fact_73_Quotient3__def, axiom, (! [B:$tType,A:$tType]: ((((quotient3 @ A) @ B) = (^ [R5:(A > (A > $o)),Abs2:(A > B),Rep2:(B > A)]: (((! [A5:B]: (((Abs2 @ (Rep2 @ A5)) = A5))) & ((! [A5:B]: (((R5 @ (Rep2 @ A5)) @ (Rep2 @ A5)))) & (! [R6:A,S3:A]: ((((R5 @ R6) @ S3) = (((R5 @ R6) @ R6) & (((R5 @ S3) @ S3) & ((Abs2 @ R6) = (Abs2 @ S3))))))))))))))).
% 2.89/1.53 thf(fact_74_rep__abs__rsp, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),X1:A,X22:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ X1) @ X22) => ((R @ X1) @ (Rep @ (Abs @ X22)))))))).
% 2.89/1.53 thf(fact_75_equals__rsp, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),Xa:A,Xb:A,Ya:A,Yb:A]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((R @ Xa) @ Xb) => (((R @ Ya) @ Yb) => (((R @ Xa) @ Ya) = ((R @ Xb) @ Yb)))))))).
% 2.89/1.53 thf(fact_76_Quotient3I, axiom, (! [B:$tType,A:$tType,Abs:(B > A),Rep:(A > B),R:(B > (B > $o))]: (((! [A3:A]: (((Abs @ (Rep @ A3)) = A3))) => ((! [A3:A]: (((R @ (Rep @ A3)) @ (Rep @ A3)))) => ((! [R7:B,S4:B]: ((((R @ R7) @ S4) = (((R @ R7) @ R7) & (((R @ S4) @ S4) & ((Abs @ R7) = (Abs @ S4))))))) => (((((quotient3 @ B) @ A) @ R) @ Abs) @ Rep))))))).
% 2.89/1.53 thf(fact_77_cond__prs, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),Absf:(A > B),Repf:(B > A),A4:$o,B4:B,C3:B]: (((((((quotient3 @ A) @ B) @ R) @ Absf) @ Repf) => ((A4 => ((Absf @ ((((if @ A) @ A4) @ (Repf @ B4)) @ (Repf @ C3))) = B4)) & ((~ (A4)) => ((Absf @ ((((if @ A) @ A4) @ (Repf @ B4)) @ (Repf @ C3))) = C3))))))).
% 2.89/1.53 thf(fact_78_apply__rspQ3, axiom, (! [B:$tType,C:$tType,A:$tType,R1:(A > (A > $o)),Abs1:(A > B),Rep1:(B > A),R2:(C > (C > $o)),F:(A > C),G:(A > C),X:A,Y:A]: (((((((quotient3 @ A) @ B) @ R1) @ Abs1) @ Rep1) => (((((((((bNF_rel_fun @ A) @ A) @ C) @ C) @ R1) @ R2) @ F) @ G) => (((R1 @ X) @ Y) => ((R2 @ (F @ X)) @ (G @ Y)))))))).
% 2.89/1.53 thf(fact_79_apply__rspQ3_H_H, axiom, (! [C:$tType,A:$tType,B:$tType,R:(A > (A > $o)),Abs:(A > B),Rep:(B > A),S2:(C > (C > $o)),F:(A > C),X:B]: (((((((quotient3 @ A) @ B) @ R) @ Abs) @ Rep) => (((((((((bNF_rel_fun @ A) @ A) @ C) @ C) @ R) @ S2) @ F) @ F) => ((S2 @ (F @ (Rep @ X))) @ (F @ (Rep @ X)))))))).
% 2.89/1.53 thf(fact_80_bex1__rel__rsp, axiom, (! [B:$tType,A:$tType,R:(A > (A > $o)),Absf:(A > B),Repf:(B > A)]: (((((((quotient3 @ A) @ B) @ R) @ Absf) @ Repf) => ((((((((bNF_rel_fun @ (A > $o)) @ (A > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ A) @ $o) @ $o) @ R) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2))))) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))) @ ((bex1_rel @ A) @ R)) @ ((bex1_rel @ A) @ R)))))).
% 2.89/1.53 thf(fact_81_map__fun__parametric, axiom, (! [A:$tType,B:$tType,E:$tType,F3:$tType,H2:$tType,G3:$tType,D:$tType,C:$tType,A2:(A > (C > $o)),B2:(B > (D > $o)),C2:(E > (G3 > $o)),D3:(F3 > (H2 > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (C > D)) @ ((E > F3) > ((B > E) > (A > F3)))) @ ((G3 > H2) > ((D > G3) > (C > H2)))) @ ((((((bNF_rel_fun @ A) @ C) @ B) @ D) @ A2) @ B2)) @ ((((((bNF_rel_fun @ (E > F3)) @ (G3 > H2)) @ ((B > E) > (A > F3))) @ ((D > G3) > (C > H2))) @ ((((((bNF_rel_fun @ E) @ G3) @ F3) @ H2) @ C2) @ D3)) @ ((((((bNF_rel_fun @ (B > E)) @ (D > G3)) @ (A > F3)) @ (C > H2)) @ ((((((bNF_rel_fun @ B) @ D) @ E) @ G3) @ B2) @ C2)) @ ((((((bNF_rel_fun @ A) @ C) @ F3) @ H2) @ A2) @ D3)))) @ ((((map_fun @ A) @ B) @ E) @ F3)) @ ((((map_fun @ C) @ D) @ G3) @ H2))))).
% 2.89/1.53 thf(fact_82_folding__idem__axioms__def, axiom, (! [B:$tType,A:$tType]: ((((finite1921348288axioms @ A) @ B) = (^ [F2:(A > (B > B))]: ((! [X3:A]: (((((((comp @ B) @ B) @ B) @ (F2 @ X3)) @ (F2 @ X3)) = (F2 @ X3)))))))))).
% 2.89/1.53 thf(fact_83_folding__idem__axioms_Ointro, axiom, (! [B:$tType,A:$tType,F:(A > (B > B))]: (((! [X2:A]: (((((((comp @ B) @ B) @ B) @ (F @ X2)) @ (F @ X2)) = (F @ X2)))) => (((finite1921348288axioms @ A) @ B) @ F))))).
% 2.89/1.53 thf(fact_84_fun__quotient3, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,R1:(A > (A > $o)),Abs12:(A > B),Rep12:(B > A),R2:(C > (C > $o)),Abs22:(C > D),Rep22:(D > C)]: (((((((quotient3 @ A) @ B) @ R1) @ Abs12) @ Rep12) => ((((((quotient3 @ C) @ D) @ R2) @ Abs22) @ Rep22) => (((((quotient3 @ (A > C)) @ (B > D)) @ ((((((bNF_rel_fun @ A) @ A) @ C) @ C) @ R1) @ R2)) @ ((((((map_fun @ B) @ A) @ C) @ D) @ Rep12) @ Abs22)) @ ((((((map_fun @ A) @ B) @ D) @ C) @ Abs12) @ Rep22))))))).
% 2.89/1.53 thf(fact_85_fun__ord__parametric, axiom, (! [C:$tType,D:$tType,A:$tType,B:$tType,F3:$tType,E:$tType,C2:(A > (B > $o)),A2:(C > (E > $o)),B2:(D > (F3 > $o))]: (((((bi_total @ A) @ B) @ C2) => ((((((((bNF_rel_fun @ (C > (D > $o))) @ (E > (F3 > $o))) @ ((A > C) > ((A > D) > $o))) @ ((B > E) > ((B > F3) > $o))) @ ((((((bNF_rel_fun @ C) @ E) @ (D > $o)) @ (F3 > $o)) @ A2) @ ((((((bNF_rel_fun @ D) @ F3) @ $o) @ $o) @ B2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ ((((((bNF_rel_fun @ (A > C)) @ (B > E)) @ ((A > D) > $o)) @ ((B > F3) > $o)) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ E) @ C2) @ A2)) @ ((((((bNF_rel_fun @ (A > D)) @ (B > F3)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ D) @ F3) @ C2) @ B2)) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ (((partial_fun_ord @ C) @ D) @ A)) @ (((partial_fun_ord @ E) @ F3) @ B)))))).
% 2.89/1.53 thf(fact_86_comp__fun__commute_Ofold__mset__union, axiom, (! [B:$tType,A:$tType,F:(A > (B > B)),S:B,M:(multiset @ A),N:(multiset @ A)]: (((((finite100568337ommute @ A) @ B) @ F) => ((((((fold_mset @ A) @ B) @ F) @ S) @ (((plus_plus @ (multiset @ A)) @ M) @ N)) = (((((fold_mset @ A) @ B) @ F) @ (((((fold_mset @ A) @ B) @ F) @ S) @ M)) @ N)))))).
% 2.89/1.53 thf(fact_87_map__fun__apply, axiom, (! [D:$tType,A:$tType,C:$tType,B:$tType]: ((((((map_fun @ B) @ C) @ D) @ A) = (^ [F2:(B > C),G2:(D > A),H3:(C > D),X3:B]: ((G2 @ (H3 @ (F2 @ X3))))))))).
% 2.89/1.53 thf(fact_88_o__prs_I1_J, axiom, (! [C:$tType,E:$tType,A:$tType,B:$tType,F3:$tType,D:$tType,R1:(A > (A > $o)),Abs1:(A > B),Rep1:(B > A),R2:(C > (C > $o)),Abs23:(C > D),Rep23:(D > C),R3:(E > (E > $o)),Abs3:(E > F3),Rep3:(F3 > E)]: (((((((quotient3 @ A) @ B) @ R1) @ Abs1) @ Rep1) => ((((((quotient3 @ C) @ D) @ R2) @ Abs23) @ Rep23) => ((((((quotient3 @ E) @ F3) @ R3) @ Abs3) @ Rep3) => ((((((((map_fun @ (D > F3)) @ (C > E)) @ ((A > C) > (A > E))) @ ((B > D) > (B > F3))) @ ((((((map_fun @ C) @ D) @ F3) @ E) @ Abs23) @ Rep3)) @ ((((((map_fun @ (B > D)) @ (A > C)) @ (A > E)) @ (B > F3)) @ ((((((map_fun @ A) @ B) @ D) @ C) @ Abs1) @ Rep23)) @ ((((((map_fun @ B) @ A) @ E) @ F3) @ Rep1) @ Abs3))) @ (((comp @ C) @ E) @ A)) = (((comp @ D) @ F3) @ B)))))))).
% 2.89/1.53 thf(fact_89_let__prs, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,R1:(A > (A > $o)),Abs1:(A > B),Rep1:(B > A),R2:(C > (C > $o)),Abs23:(C > D),Rep23:(D > C)]: (((((((quotient3 @ A) @ B) @ R1) @ Abs1) @ Rep1) => ((((((quotient3 @ C) @ D) @ R2) @ Abs23) @ Rep23) => ((((((((map_fun @ D) @ C) @ ((C > A) > A)) @ ((D > B) > B)) @ Rep23) @ ((((((map_fun @ (D > B)) @ (C > A)) @ A) @ B) @ ((((((map_fun @ C) @ D) @ B) @ A) @ Abs23) @ Rep1)) @ Abs1)) @ (^ [S3:C,F2:(C > A)]: ((F2 @ S3)))) = (^ [S3:D,F2:(D > B)]: ((F2 @ S3))))))))).
% 2.89/1.53 thf(fact_90_bi__total__eq, axiom, (! [A:$tType]: ((((bi_total @ A) @ A) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2))))))).
% 2.89/1.53 thf(fact_91_bi__total__def, axiom, (! [B:$tType,A:$tType]: ((((bi_total @ A) @ B) = (^ [R5:(A > (B > $o))]: (((! [X3:A]: ((? [X6:B]: (((R5 @ X3) @ X6))))) & (! [Y3:B]: ((? [X3:A]: (((R5 @ X3) @ Y3)))))))))))).
% 2.89/1.53 thf(fact_92_union__assoc, axiom, (! [A:$tType,M:(multiset @ A),N:(multiset @ A),K2:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ M) @ N)) @ K2) = (((plus_plus @ (multiset @ A)) @ M) @ (((plus_plus @ (multiset @ A)) @ N) @ K2)))))).
% 2.89/1.53 thf(fact_93_union__lcomm, axiom, (! [A:$tType,M:(multiset @ A),N:(multiset @ A),K2:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ M) @ (((plus_plus @ (multiset @ A)) @ N) @ K2)) = (((plus_plus @ (multiset @ A)) @ N) @ (((plus_plus @ (multiset @ A)) @ M) @ K2)))))).
% 2.89/1.53 thf(fact_94_union__commute, axiom, (! [A:$tType]: (((plus_plus @ (multiset @ A)) = (^ [M2:(multiset @ A),N2:(multiset @ A)]: ((((plus_plus @ (multiset @ A)) @ N2) @ M2))))))).
% 2.89/1.53 thf(fact_95_union__left__cancel, axiom, (! [A:$tType,K2:(multiset @ A),M:(multiset @ A),N:(multiset @ A)]: ((((((plus_plus @ (multiset @ A)) @ K2) @ M) = (((plus_plus @ (multiset @ A)) @ K2) @ N)) = (M = N))))).
% 2.89/1.53 thf(fact_96_union__right__cancel, axiom, (! [A:$tType,M:(multiset @ A),K2:(multiset @ A),N:(multiset @ A)]: ((((((plus_plus @ (multiset @ A)) @ M) @ K2) = (((plus_plus @ (multiset @ A)) @ N) @ K2)) = (M = N))))).
% 2.89/1.53 thf(fact_97_multi__union__self__other__eq, axiom, (! [A:$tType,A2:(multiset @ A),X4:(multiset @ A),Y5:(multiset @ A)]: ((((((plus_plus @ (multiset @ A)) @ A2) @ X4) = (((plus_plus @ (multiset @ A)) @ A2) @ Y5)) => (X4 = Y5))))).
% 2.89/1.53 thf(fact_98_comp__fun__commute__plus__mset, axiom, (! [A:$tType]: ((((finite100568337ommute @ (multiset @ A)) @ (multiset @ A)) @ (plus_plus @ (multiset @ A)))))).
% 2.89/1.53 thf(fact_99_bex1__rel__aux, axiom, (! [A:$tType,R:(A > (A > $o)),X:(A > $o),Y:(A > $o)]: (((! [Xa2:A,Ya2:A]: ((((R @ Xa2) @ Ya2) => ((X @ Xa2) = (Y @ Ya2))))) => ((((bex1_rel @ A) @ R) @ X) => (((bex1_rel @ A) @ R) @ Y)))))).
% 2.89/1.53 thf(fact_100_bex1__rel__aux2, axiom, (! [A:$tType,R:(A > (A > $o)),X:(A > $o),Y:(A > $o)]: (((! [Xa2:A,Ya2:A]: ((((R @ Xa2) @ Ya2) => ((X @ Xa2) = (Y @ Ya2))))) => ((((bex1_rel @ A) @ R) @ Y) => (((bex1_rel @ A) @ R) @ X)))))).
% 2.89/1.53 thf(fact_101_map__fun_Ocompositionality, axiom, (! [D:$tType,F3:$tType,C:$tType,E:$tType,B:$tType,A:$tType,F:(E > C),G:(D > F3),H:(C > A),I:(B > D),Fun:(A > B)]: (((((((((map_fun @ E) @ C) @ D) @ F3) @ F) @ G) @ (((((((map_fun @ C) @ A) @ B) @ D) @ H) @ I) @ Fun)) = (((((((map_fun @ E) @ A) @ B) @ F3) @ (((((comp @ C) @ A) @ E) @ H) @ F)) @ (((((comp @ D) @ F3) @ B) @ G) @ I)) @ Fun))))).
% 2.89/1.53 thf(fact_102_map__fun_Ocomp, axiom, (! [E:$tType,C:$tType,A:$tType,F3:$tType,D:$tType,B:$tType,F:(E > C),G:(D > F3),H:(C > A),I:(B > D)]: (((((((comp @ (C > D)) @ (E > F3)) @ (A > B)) @ ((((((map_fun @ E) @ C) @ D) @ F3) @ F) @ G)) @ ((((((map_fun @ C) @ A) @ B) @ D) @ H) @ I)) = ((((((map_fun @ E) @ A) @ B) @ F3) @ (((((comp @ C) @ A) @ E) @ H) @ F)) @ (((((comp @ D) @ F3) @ B) @ G) @ I)))))).
% 2.89/1.53 thf(fact_103_map__fun__def, axiom, (! [B:$tType,D:$tType,A:$tType,C:$tType]: ((((((map_fun @ C) @ A) @ B) @ D) = (^ [F2:(C > A),G2:(B > D),H3:(A > B)]: ((((((comp @ A) @ D) @ C) @ (((((comp @ B) @ D) @ A) @ G2) @ H3)) @ F2))))))).
% 2.89/1.53 thf(fact_104_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B4:A,A4:A,C3:A]: ((((((plus_plus @ A) @ B4) @ A4) = (((plus_plus @ A) @ C3) @ A4)) = (B4 = C3)))))))).
% 2.89/1.53 thf(fact_105_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A4:A,B4:A,C3:A]: ((((((plus_plus @ A) @ A4) @ B4) = (((plus_plus @ A) @ A4) @ C3)) = (B4 = C3)))))))).
% 2.89/1.53 thf(fact_106_set__plus__intro, axiom, (! [A:$tType]: (((plus @ A) => (! [A4:A,C2:(set @ A),B4:A,D3:(set @ A)]: (((((member @ A) @ A4) @ C2) => ((((member @ A) @ B4) @ D3) => (((member @ A) @ (((plus_plus @ A) @ A4) @ B4)) @ (((plus_plus @ (set @ A)) @ C2) @ D3)))))))))).
% 2.89/1.53 thf(fact_107_transfer__forall__transfer_I1_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2))))) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))) @ (transfer_forall @ A)) @ (transfer_forall @ B)))))).
% 2.89/1.53 thf(fact_108_monotone__parametric, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > (A > $o))) @ (B > (B > $o))) @ ((C > (C > $o)) > ((A > C) > $o))) @ ((D > (D > $o)) > ((B > D) > $o))) @ ((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ ((((((bNF_rel_fun @ (C > (C > $o))) @ (D > (D > $o))) @ ((A > C) > $o)) @ ((B > D) > $o)) @ ((((((bNF_rel_fun @ C) @ D) @ (C > $o)) @ (D > $o)) @ B2) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ B2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ ((((((bNF_rel_fun @ (A > C)) @ (B > D)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2)) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ ((comple1396247847notone @ A) @ C)) @ ((comple1396247847notone @ B) @ D)))))).
% 2.89/1.53 thf(fact_109_relcompp__transfer, axiom, (! [C:$tType,A:$tType,E:$tType,F3:$tType,B:$tType,D:$tType,B2:(A > (B > $o)),A2:(C > (D > $o)),C2:(E > (F3 > $o))]: (((((bi_total @ A) @ B) @ B2) => ((((((((bNF_rel_fun @ (C > (A > $o))) @ (D > (B > $o))) @ ((A > (E > $o)) > (C > (E > $o)))) @ ((B > (F3 > $o)) > (D > (F3 > $o)))) @ ((((((bNF_rel_fun @ C) @ D) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ B2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ ((((((bNF_rel_fun @ (A > (E > $o))) @ (B > (F3 > $o))) @ (C > (E > $o))) @ (D > (F3 > $o))) @ ((((((bNF_rel_fun @ A) @ B) @ (E > $o)) @ (F3 > $o)) @ B2) @ ((((((bNF_rel_fun @ E) @ F3) @ $o) @ $o) @ C2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ ((((((bNF_rel_fun @ C) @ D) @ (E > $o)) @ (F3 > $o)) @ A2) @ ((((((bNF_rel_fun @ E) @ F3) @ $o) @ $o) @ C2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2))))))) @ (((relcompp @ C) @ A) @ E)) @ (((relcompp @ D) @ B) @ F3)))))).
% 2.89/1.53 thf(fact_110_fun_Orel__compp, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType,R:(A > (B > $o)),S2:(B > (C > $o))]: ((((((((bNF_rel_fun @ D) @ D) @ A) @ C) @ (^ [Y4:D,Z2:D]: ((Y4 = Z2)))) @ (((((relcompp @ A) @ B) @ C) @ R) @ S2)) = (((((relcompp @ (D > A)) @ (D > B)) @ (D > C)) @ ((((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y4:D,Z2:D]: ((Y4 = Z2)))) @ R)) @ ((((((bNF_rel_fun @ D) @ D) @ B) @ C) @ (^ [Y4:D,Z2:D]: ((Y4 = Z2)))) @ S2)))))).
% 2.89/1.53 thf(fact_111_eq__comp__r, axiom, (! [A:$tType,R:(A > (A > $o))]: (((((((relcompp @ A) @ A) @ A) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2)))) @ (((((relcompp @ A) @ A) @ A) @ R) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2))))) = R)))).
% 2.89/1.53 thf(fact_112_bi__total__OO, axiom, (! [A:$tType,C:$tType,B:$tType,A2:(A > (B > $o)),B2:(B > (C > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((bi_total @ B) @ C) @ B2) => (((bi_total @ A) @ C) @ (((((relcompp @ A) @ B) @ C) @ A2) @ B2))))))).
% 2.89/1.53 thf(fact_113_OOO__quotient3, axiom, (! [A:$tType,B:$tType,C:$tType,R1:(A > (A > $o)),Abs1:(A > B),Rep1:(B > A),R2:(B > (B > $o)),Abs23:(B > C),Rep23:(C > B),R23:(A > (A > $o))]: (((((((quotient3 @ A) @ B) @ R1) @ Abs1) @ Rep1) => ((((((quotient3 @ B) @ C) @ R2) @ Abs23) @ Rep23) => ((! [X2:A,Y2:A]: ((((R23 @ X2) @ Y2) => (((R1 @ X2) @ X2) => (((R1 @ Y2) @ Y2) => ((R2 @ (Abs1 @ X2)) @ (Abs1 @ Y2))))))) => ((! [X2:B,Y2:B]: ((((R2 @ X2) @ Y2) => ((R23 @ (Rep1 @ X2)) @ (Rep1 @ Y2))))) => (((((quotient3 @ A) @ C) @ (((((relcompp @ A) @ A) @ A) @ R1) @ (((((relcompp @ A) @ A) @ A) @ R23) @ R1))) @ (((((comp @ B) @ C) @ A) @ Abs23) @ Abs1)) @ (((((comp @ B) @ A) @ C) @ Rep1) @ Rep23))))))))).
% 2.89/1.53 thf(fact_114_OOO__eq__quotient3, axiom, (! [A:$tType,B:$tType,C:$tType,R1:(A > (A > $o)),Abs1:(A > B),Rep1:(B > A),Abs23:(B > C),Rep23:(C > B)]: (((((((quotient3 @ A) @ B) @ R1) @ Abs1) @ Rep1) => ((((((quotient3 @ B) @ C) @ (^ [Y4:B,Z2:B]: ((Y4 = Z2)))) @ Abs23) @ Rep23) => (((((quotient3 @ A) @ C) @ (((((relcompp @ A) @ A) @ A) @ R1) @ (((((relcompp @ A) @ A) @ A) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2)))) @ R1))) @ (((((comp @ B) @ C) @ A) @ Abs23) @ Abs1)) @ (((((comp @ B) @ A) @ C) @ Rep1) @ Rep23))))))).
% 2.89/1.53 thf(fact_115_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A4:A,B4:A,C3:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A4) @ B4)) @ C3) = (((plus_plus @ A) @ A4) @ (((plus_plus @ A) @ B4) @ C3))))))))).
% 2.89/1.53 thf(fact_116_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (K = L)) => ((((plus_plus @ A) @ I) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 2.89/1.53 thf(fact_117_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:A,K:A,A4:A,B4:A]: (((A2 = (((plus_plus @ A) @ K) @ A4)) => ((((plus_plus @ A) @ A2) @ B4) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A4) @ B4)))))))))).
% 2.89/1.53 thf(fact_118_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B2:A,K:A,B4:A,A4:A]: (((B2 = (((plus_plus @ A) @ K) @ B4)) => ((((plus_plus @ A) @ A4) @ B2) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A4) @ B4)))))))))).
% 2.89/1.53 thf(fact_119_set__plus__elim, axiom, (! [A:$tType]: (((plus @ A) => (! [X:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ X) @ (((plus_plus @ (set @ A)) @ A2) @ B2)) => (~ ((! [A3:A,B3:A]: (((X = (((plus_plus @ A) @ A3) @ B3)) => ((((member @ A) @ A3) @ A2) => (~ ((((member @ A) @ B3) @ B2)))))))))))))))).
% 2.89/1.53 thf(fact_120_add_Oassoc, axiom, (! [A:$tType]: (((semigroup_add @ A) => (! [A4:A,B4:A,C3:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A4) @ B4)) @ C3) = (((plus_plus @ A) @ A4) @ (((plus_plus @ A) @ B4) @ C3))))))))).
% 2.89/1.53 thf(fact_121_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,B4:A,C3:A]: ((((((plus_plus @ A) @ A4) @ B4) = (((plus_plus @ A) @ A4) @ C3)) = (B4 = C3)))))))).
% 2.89/1.53 thf(fact_122_add_Oright__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [B4:A,A4:A,C3:A]: ((((((plus_plus @ A) @ B4) @ A4) = (((plus_plus @ A) @ C3) @ A4)) = (B4 = C3)))))))).
% 2.89/1.53 thf(fact_123_add_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((plus_plus @ A) = (^ [A5:A,B5:A]: ((((plus_plus @ A) @ B5) @ A5)))))))).
% 2.89/1.53 thf(fact_124_add_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [B4:A,A4:A,C3:A]: (((((plus_plus @ A) @ B4) @ (((plus_plus @ A) @ A4) @ C3)) = (((plus_plus @ A) @ A4) @ (((plus_plus @ A) @ B4) @ C3))))))))).
% 2.89/1.53 thf(fact_125_add__left__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A4:A,B4:A,C3:A]: ((((((plus_plus @ A) @ A4) @ B4) = (((plus_plus @ A) @ A4) @ C3)) => (B4 = C3)))))))).
% 2.89/1.53 thf(fact_126_add__right__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B4:A,A4:A,C3:A]: ((((((plus_plus @ A) @ B4) @ A4) = (((plus_plus @ A) @ C3) @ A4)) => (B4 = C3)))))))).
% 2.89/1.53 thf(fact_127_transfer__forall__transfer_I4_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2))))) @ rev_implies) @ (transfer_forall @ A)) @ (transfer_forall @ B)))))).
% 2.89/1.53 thf(fact_128_transfer__forall__transfer_I5_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ rev_implies)) @ rev_implies) @ (transfer_forall @ A)) @ (transfer_forall @ B)))))).
% 2.89/1.53 thf(fact_129_Domainp__transfer, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,B2:(A > (B > $o)),A2:(C > (D > $o))]: (((((bi_total @ A) @ B) @ B2) => ((((((((bNF_rel_fun @ (C > (A > $o))) @ (D > (B > $o))) @ (C > $o)) @ (D > $o)) @ ((((((bNF_rel_fun @ C) @ D) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ B2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ A2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2))))) @ ((domainp @ C) @ A)) @ ((domainp @ D) @ B)))))).
% 2.89/1.53 thf(fact_130_folding__idem_Oaxioms_I2_J, axiom, (! [B:$tType,A:$tType,F:(A > (B > B))]: (((((finite_folding_idem @ A) @ B) @ F) => (((finite1921348288axioms @ A) @ B) @ F))))).
% 2.89/1.53 thf(fact_131_OO__eq, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((((((relcompp @ A) @ B) @ B) @ R) @ (^ [Y4:B,Z2:B]: ((Y4 = Z2)))) = R)))).
% 2.89/1.53 thf(fact_132_Domainp__iff, axiom, (! [B:$tType,A:$tType]: ((((domainp @ A) @ B) = (^ [T2:(A > (B > $o)),X3:A]: ((? [X6:B]: (((T2 @ X3) @ X6))))))))).
% 2.89/1.53 thf(fact_133_Domainp__refl, axiom, (! [B:$tType,A:$tType]: ((((domainp @ A) @ B) = ((domainp @ A) @ B))))).
% 2.89/1.53 thf(fact_134_rev__implies__def, axiom, (rev_implies = (^ [X3:$o,Y3:$o]: ((Y3 => X3))))).
% 2.89/1.53 thf(fact_135_folding__idem_Ocomp__fun__idem, axiom, (! [B:$tType,A:$tType,F:(A > (B > B)),X:A]: (((((finite_folding_idem @ A) @ B) @ F) => ((((((comp @ B) @ B) @ B) @ (F @ X)) @ (F @ X)) = (F @ X)))))).
% 2.89/1.53 thf(fact_136_relcompp_OrelcompI, axiom, (! [A:$tType,B:$tType,C:$tType,R4:(A > (B > $o)),A4:A,B4:B,S:(B > (C > $o)),C3:C]: ((((R4 @ A4) @ B4) => (((S @ B4) @ C3) => (((((((relcompp @ A) @ B) @ C) @ R4) @ S) @ A4) @ C3)))))).
% 2.89/1.53 thf(fact_137_relcompp_Oinducts, axiom, (! [B:$tType,A:$tType,C:$tType,R4:(A > (B > $o)),S:(B > (C > $o)),X1:A,X22:C,P:(A > (C > $o))]: (((((((((relcompp @ A) @ B) @ C) @ R4) @ S) @ X1) @ X22) => ((! [A3:A,B3:B,C4:C]: ((((R4 @ A3) @ B3) => (((S @ B3) @ C4) => ((P @ A3) @ C4))))) => ((P @ X1) @ X22)))))).
% 2.89/1.53 thf(fact_138_relcompp__assoc, axiom, (! [A:$tType,D:$tType,B:$tType,C:$tType,R4:(A > (D > $o)),S:(D > (C > $o)),T3:(C > (B > $o))]: (((((((relcompp @ A) @ C) @ B) @ (((((relcompp @ A) @ D) @ C) @ R4) @ S)) @ T3) = (((((relcompp @ A) @ D) @ B) @ R4) @ (((((relcompp @ D) @ C) @ B) @ S) @ T3)))))).
% 2.89/1.53 thf(fact_139_relcompp__apply, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((relcompp @ A) @ B) @ C) = (^ [R5:(A > (B > $o)),S5:(B > (C > $o)),A5:A,C5:C]: ((? [B5:B]: ((((R5 @ A5) @ B5) & ((S5 @ B5) @ C5)))))))))).
% 2.89/1.53 thf(fact_140_relcompp_Osimps, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((relcompp @ A) @ B) @ C) = (^ [R6:(A > (B > $o)),S3:(B > (C > $o)),A1:A,A22:C]: ((? [A5:A,B5:B,C5:C]: (((A1 = A5) & ((A22 = C5) & (((R6 @ A5) @ B5) & ((S3 @ B5) @ C5)))))))))))).
% 2.89/1.53 thf(fact_141_relcompp_Ocases, axiom, (! [A:$tType,B:$tType,C:$tType,R4:(A > (B > $o)),S:(B > (C > $o)),A12:A,A23:C]: (((((((((relcompp @ A) @ B) @ C) @ R4) @ S) @ A12) @ A23) => (~ ((! [B3:B]: ((((R4 @ A12) @ B3) => (~ (((S @ B3) @ A23)))))))))))).
% 2.89/1.53 thf(fact_142_relcomppE, axiom, (! [A:$tType,B:$tType,C:$tType,R4:(A > (B > $o)),S:(B > (C > $o)),A4:A,C3:C]: (((((((((relcompp @ A) @ B) @ C) @ R4) @ S) @ A4) @ C3) => (~ ((! [B3:B]: ((((R4 @ A4) @ B3) => (~ (((S @ B3) @ C3)))))))))))).
% 2.89/1.53 thf(fact_143_eq__OO, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((((((relcompp @ A) @ A) @ B) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2)))) @ R) = R)))).
% 2.89/1.53 thf(fact_144_folding__idem__def, axiom, (! [B:$tType,A:$tType]: ((((finite_folding_idem @ A) @ B) = (^ [F2:(A > (B > B))]: (((((finite_folding @ A) @ B) @ F2) & (((finite1921348288axioms @ A) @ B) @ F2)))))))).
% 2.89/1.53 thf(fact_145_folding__idem_Ointro, axiom, (! [B:$tType,A:$tType,F:(A > (B > B))]: (((((finite_folding @ A) @ B) @ F) => ((((finite1921348288axioms @ A) @ B) @ F) => (((finite_folding_idem @ A) @ B) @ F)))))).
% 2.89/1.53 thf(fact_146_reflp__transfer_I5_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > (A > $o))) @ (B > (B > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ rev_implies) @ (reflp @ A)) @ (reflp @ B)))))).
% 2.89/1.53 thf(fact_147_reflp__transfer_I4_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > (A > $o))) @ (B > (B > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ rev_implies))) @ rev_implies) @ (reflp @ A)) @ (reflp @ B)))))).
% 2.89/1.53 thf(fact_148_fun_Orel__reflp, axiom, (! [D:$tType,A:$tType,R:(A > (A > $o))]: ((((reflp @ A) @ R) => ((reflp @ (D > A)) @ ((((((bNF_rel_fun @ D) @ D) @ A) @ A) @ (^ [Y4:D,Z2:D]: ((Y4 = Z2)))) @ R)))))).
% 2.89/1.53 thf(fact_149_DEADID_Orel__reflp, axiom, (! [A:$tType]: (((reflp @ A) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2))))))).
% 2.89/1.53 thf(fact_150_folding__def, axiom, (! [B:$tType,A:$tType]: ((((finite_folding @ A) @ B) = (^ [F2:(A > (B > B))]: ((! [Y3:A,X3:A]: (((((((comp @ B) @ B) @ B) @ (F2 @ Y3)) @ (F2 @ X3)) = (((((comp @ B) @ B) @ B) @ (F2 @ X3)) @ (F2 @ Y3))))))))))).
% 2.89/1.53 thf(fact_151_folding_Ointro, axiom, (! [B:$tType,A:$tType,F:(A > (B > B))]: (((! [Y2:A,X2:A]: (((((((comp @ B) @ B) @ B) @ (F @ Y2)) @ (F @ X2)) = (((((comp @ B) @ B) @ B) @ (F @ X2)) @ (F @ Y2))))) => (((finite_folding @ A) @ B) @ F))))).
% 2.89/1.53 thf(fact_152_folding_Ocomp__fun__commute, axiom, (! [B:$tType,A:$tType,F:(A > (B > B)),Y:A,X:A]: (((((finite_folding @ A) @ B) @ F) => ((((((comp @ B) @ B) @ B) @ (F @ Y)) @ (F @ X)) = (((((comp @ B) @ B) @ B) @ (F @ X)) @ (F @ Y))))))).
% 2.89/1.53 thf(fact_153_folding__idem_Oaxioms_I1_J, axiom, (! [B:$tType,A:$tType,F:(A > (B > B))]: (((((finite_folding_idem @ A) @ B) @ F) => (((finite_folding @ A) @ B) @ F))))).
% 2.89/1.53 thf(fact_154_reflp__transfer_I1_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > (A > $o))) @ (B > (B > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))) @ (reflp @ A)) @ (reflp @ B)))))).
% 2.89/1.53 thf(fact_155_left__total__parametric, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((bi_total @ C) @ D) @ B2) => ((((((((bNF_rel_fun @ (A > (C > $o))) @ (B > (D > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (C > $o)) @ (D > $o)) @ A2) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ B2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))) @ ((left_total @ A) @ C)) @ ((left_total @ B) @ D))))))).
% 2.89/1.53 thf(fact_156_reflp__transfer_I2_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((right_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > (A > $o))) @ (B > (B > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (=>)))) @ (=>)) @ (reflp @ A)) @ (reflp @ B)))))).
% 2.89/1.53 thf(fact_157_reflp__transfer_I3_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((right_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > (A > $o))) @ (B > (B > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ (=>)) @ (reflp @ A)) @ (reflp @ B)))))).
% 2.89/1.53 thf(fact_158_left__totalE, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o))]: (((((left_total @ A) @ B) @ R) => (! [X7:A]: ((? [X_1:B]: (((R @ X7) @ X_1))))))))).
% 2.89/1.53 thf(fact_159_left__totalI, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((! [X2:A]: ((? [X_12:B]: (((R @ X2) @ X_12))))) => (((left_total @ A) @ B) @ R))))).
% 2.89/1.53 thf(fact_160_right__totalE, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o)),Y:B]: (((((right_total @ A) @ B) @ A2) => (~ ((! [X2:A]: ((~ (((A2 @ X2) @ Y))))))))))).
% 2.89/1.53 thf(fact_161_right__totalI, axiom, (! [A:$tType,B:$tType,A2:(B > (A > $o))]: (((! [Y2:A]: ((? [X7:B]: (((A2 @ X7) @ Y2))))) => (((right_total @ B) @ A) @ A2))))).
% 2.89/1.53 thf(fact_162_left__total__eq, axiom, (! [A:$tType]: ((((left_total @ A) @ A) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2))))))).
% 2.89/1.53 thf(fact_163_left__total__def, axiom, (! [B:$tType,A:$tType]: ((((left_total @ A) @ B) = (^ [R5:(A > (B > $o))]: ((! [X3:A]: ((? [X6:B]: (((R5 @ X3) @ X6))))))))))).
% 2.89/1.53 thf(fact_164_right__total__eq, axiom, (! [A:$tType]: ((((right_total @ A) @ A) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2))))))).
% 2.89/1.53 thf(fact_165_right__total__def, axiom, (! [B:$tType,A:$tType]: ((((right_total @ A) @ B) = (^ [R5:(A > (B > $o))]: ((! [Y3:B]: ((? [X3:A]: (((R5 @ X3) @ Y3))))))))))).
% 2.89/1.53 thf(fact_166_bi__total__alt__def, axiom, (! [B:$tType,A:$tType]: ((((bi_total @ A) @ B) = (^ [A6:(A > (B > $o))]: (((((left_total @ A) @ B) @ A6) & (((right_total @ A) @ B) @ A6)))))))).
% 2.89/1.53 thf(fact_167_bi__totalI, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((((left_total @ A) @ B) @ R) => ((((right_total @ A) @ B) @ R) => (((bi_total @ A) @ B) @ R)))))).
% 2.89/1.53 thf(fact_168_left__total__OO, axiom, (! [A:$tType,C:$tType,B:$tType,R:(A > (B > $o)),S2:(B > (C > $o))]: (((((left_total @ A) @ B) @ R) => ((((left_total @ B) @ C) @ S2) => (((left_total @ A) @ C) @ (((((relcompp @ A) @ B) @ C) @ R) @ S2))))))).
% 2.89/1.53 thf(fact_169_right__total__OO, axiom, (! [A:$tType,C:$tType,B:$tType,A2:(A > (B > $o)),B2:(B > (C > $o))]: (((((right_total @ A) @ B) @ A2) => ((((right_total @ B) @ C) @ B2) => (((right_total @ A) @ C) @ (((((relcompp @ A) @ B) @ C) @ A2) @ B2))))))).
% 2.89/1.53 thf(fact_170_pcr__Domainp__total, axiom, (! [A:$tType,B:$tType,C:$tType,B2:(A > (B > $o)),A2:(C > (A > $o)),P:(C > $o)]: (((((left_total @ A) @ B) @ B2) => (((((domainp @ C) @ A) @ A2) = P) => ((((domainp @ C) @ B) @ (((((relcompp @ C) @ A) @ B) @ A2) @ B2)) = P)))))).
% 2.89/1.53 thf(fact_171_pcr__Domainp__par__left__total, axiom, (! [A:$tType,B:$tType,C:$tType,B2:(A > (B > $o)),P:(A > $o),A2:(C > (A > $o)),P2:(C > $o)]: ((((((domainp @ A) @ B) @ B2) = P) => ((((left_total @ C) @ A) @ A2) => (((((((((bNF_rel_fun @ C) @ A) @ $o) @ $o) @ A2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))) @ P2) @ P) => ((((domainp @ C) @ B) @ (((((relcompp @ C) @ A) @ B) @ A2) @ B2)) = P2))))))).
% 2.89/1.53 thf(fact_172_transfer__forall__transfer_I3_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((right_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (=>))) @ (=>)) @ (transfer_forall @ A)) @ (transfer_forall @ B)))))).
% 2.89/1.53 thf(fact_173_transfer__forall__transfer_I2_J, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((right_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2))))) @ (=>)) @ (transfer_forall @ A)) @ (transfer_forall @ B)))))).
% 2.89/1.53 thf(fact_174_right__total__parametric, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((bi_total @ C) @ D) @ B2) => ((((((((bNF_rel_fun @ (A > (C > $o))) @ (B > (D > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (C > $o)) @ (D > $o)) @ A2) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ B2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))) @ ((right_total @ A) @ C)) @ ((right_total @ B) @ D))))))).
% 2.89/1.53 thf(fact_175_Domainp__forall__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((right_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2))))) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))) @ ((transfer_bforall @ A) @ (((domainp @ A) @ B) @ A2))) @ (transfer_forall @ B)))))).
% 2.89/1.53 thf(fact_176_nchotomy__relcomppE, axiom, (! [C:$tType,B:$tType,A:$tType,D:$tType,F:(B > A),R4:(C > (A > $o)),S:(A > (D > $o)),A4:C,C3:D]: (((! [Y2:A]: ((? [X7:B]: ((Y2 = (F @ X7)))))) => ((((((((relcompp @ C) @ A) @ D) @ R4) @ S) @ A4) @ C3) => (~ ((! [B3:B]: ((((R4 @ A4) @ (F @ B3)) => (~ (((S @ (F @ B3)) @ C3))))))))))))).
% 2.89/1.53 thf(fact_177_pos__fun__distr, axiom, (! [E:$tType,C:$tType,A:$tType,B:$tType,D:$tType,F3:$tType,R:(A > (E > $o)),S2:(B > (F3 > $o)),R8:(E > (C > $o)),S6:(F3 > (D > $o))]: ((((ord_less_eq @ ((A > B) > ((C > D) > $o))) @ (((((relcompp @ (A > B)) @ (E > F3)) @ (C > D)) @ ((((((bNF_rel_fun @ A) @ E) @ B) @ F3) @ R) @ S2)) @ ((((((bNF_rel_fun @ E) @ C) @ F3) @ D) @ R8) @ S6))) @ ((((((bNF_rel_fun @ A) @ C) @ B) @ D) @ (((((relcompp @ A) @ E) @ C) @ R) @ R8)) @ (((((relcompp @ B) @ F3) @ D) @ S2) @ S6)))))).
% 2.89/1.53 thf(fact_178_add__le__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C3:A,A4:A,B4:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C3) @ A4)) @ (((plus_plus @ A) @ C3) @ B4)) = (((ord_less_eq @ A) @ A4) @ B4)))))))).
% 2.89/1.53 thf(fact_179_add__le__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A4:A,C3:A,B4:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A4) @ C3)) @ (((plus_plus @ A) @ B4) @ C3)) = (((ord_less_eq @ A) @ A4) @ B4)))))))).
% 2.89/1.53 thf(fact_180_reflp__eq, axiom, (! [A:$tType]: (((reflp @ A) = ((ord_less_eq @ (A > (A > $o))) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2)))))))).
% 2.89/1.53 thf(fact_181_reflp__ge__eq, axiom, (! [A:$tType,R:(A > (A > $o))]: ((((reflp @ A) @ R) => (((ord_less_eq @ (A > (A > $o))) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2)))) @ R))))).
% 2.89/1.53 thf(fact_182_leq__OOI, axiom, (! [A:$tType,R:(A > (A > $o))]: (((R = (^ [Y4:A,Z2:A]: ((Y4 = Z2)))) => (((ord_less_eq @ (A > (A > $o))) @ R) @ (((((relcompp @ A) @ A) @ A) @ R) @ R)))))).
% 2.89/1.53 thf(fact_183_relcompp__mono, axiom, (! [A:$tType,C:$tType,B:$tType,R9:(A > (B > $o)),R4:(A > (B > $o)),S7:(B > (C > $o)),S:(B > (C > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ R9) @ R4) => ((((ord_less_eq @ (B > (C > $o))) @ S7) @ S) => (((ord_less_eq @ (A > (C > $o))) @ (((((relcompp @ A) @ B) @ C) @ R9) @ S7)) @ (((((relcompp @ A) @ B) @ C) @ R4) @ S))))))).
% 2.89/1.53 thf(fact_184_predicate2D__conj, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Q:(A > (B > $o)),R:$o,X:A,Y:B]: ((((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) & R) => (R & (((P @ X) @ Y) => ((Q @ X) @ Y))))))).
% 2.89/1.53 thf(fact_185_refl__ge__eq, axiom, (! [A:$tType,R:(A > (A > $o))]: (((! [X2:A]: (((R @ X2) @ X2))) => (((ord_less_eq @ (A > (A > $o))) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2)))) @ R))))).
% 2.89/1.53 thf(fact_186_ge__eq__refl, axiom, (! [A:$tType,R:(A > (A > $o)),X:A]: (((((ord_less_eq @ (A > (A > $o))) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2)))) @ R) => ((R @ X) @ X))))).
% 2.89/1.53 thf(fact_187_add__mono__thms__linordered__semiring_I3_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (K = L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.89/1.53 thf(fact_188_add__mono__thms__linordered__semiring_I2_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.89/1.53 thf(fact_189_add__mono__thms__linordered__semiring_I1_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.89/1.53 thf(fact_190_add__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A4:A,B4:A,C3:A,D2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ C3) @ D2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A4) @ C3)) @ (((plus_plus @ A) @ B4) @ D2)))))))))).
% 2.89/1.53 thf(fact_191_add__left__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A4:A,B4:A,C3:A]: (((((ord_less_eq @ A) @ A4) @ B4) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ C3) @ A4)) @ (((plus_plus @ A) @ C3) @ B4))))))))).
% 2.89/1.53 thf(fact_192_less__eqE, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => (~ ((! [C4:A]: ((B4 != (((plus_plus @ A) @ A4) @ C4))))))))))))).
% 2.89/1.53 thf(fact_193_add__right__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A4:A,B4:A,C3:A]: (((((ord_less_eq @ A) @ A4) @ B4) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A4) @ C3)) @ (((plus_plus @ A) @ B4) @ C3))))))))).
% 2.89/1.53 thf(fact_194_le__iff__add, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => ((ord_less_eq @ A) = (^ [A5:A,B5:A]: ((? [C5:A]: ((B5 = (((plus_plus @ A) @ A5) @ C5))))))))))).
% 2.89/1.53 thf(fact_195_add__le__imp__le__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C3:A,A4:A,B4:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C3) @ A4)) @ (((plus_plus @ A) @ C3) @ B4)) => (((ord_less_eq @ A) @ A4) @ B4)))))))).
% 2.89/1.53 thf(fact_196_add__le__imp__le__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A4:A,C3:A,B4:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A4) @ C3)) @ (((plus_plus @ A) @ B4) @ C3)) => (((ord_less_eq @ A) @ A4) @ B4)))))))).
% 2.89/1.53 thf(fact_197_fun_Orel__mono, axiom, (! [D:$tType,B:$tType,A:$tType,R:(A > (B > $o)),Ra:(A > (B > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ R) @ Ra) => (((ord_less_eq @ ((D > A) > ((D > B) > $o))) @ ((((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y4:D,Z2:D]: ((Y4 = Z2)))) @ R)) @ ((((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y4:D,Z2:D]: ((Y4 = Z2)))) @ Ra)))))).
% 2.89/1.53 thf(fact_198_fun__mono, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,C2:(A > (B > $o)),A2:(A > (B > $o)),B2:(C > (D > $o)),D3:(C > (D > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ C2) @ A2) => ((((ord_less_eq @ (C > (D > $o))) @ B2) @ D3) => (((ord_less_eq @ ((A > C) > ((B > D) > $o))) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2)) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ C2) @ D3))))))).
% 2.89/1.53 thf(fact_199_lfp_Omonotone__if__bot, axiom, (! [B:$tType,A:$tType]: (((comple187826305attice @ A) => (! [Bound:A,G:(A > B),Bot:B,F:(A > B),Ord:(B > (B > $o))]: (((! [X2:A]: ((((((ord_less_eq @ A) @ X2) @ Bound) => ((G @ X2) = Bot)) & ((~ ((((ord_less_eq @ A) @ X2) @ Bound))) => ((G @ X2) = (F @ X2)))))) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => ((~ ((((ord_less_eq @ A) @ X2) @ Bound))) => ((Ord @ (F @ X2)) @ (F @ Y2)))))) => ((! [X2:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Bound))) => ((Ord @ Bot) @ (F @ X2))))) => (((Ord @ Bot) @ Bot) => (((((comple1396247847notone @ A) @ B) @ (ord_less_eq @ A)) @ Ord) @ G))))))))))).
% 2.89/1.53 thf(fact_200_mono__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType]: ((((order @ B) & ((order @ D) & ((order @ C) & (order @ A)))) => (! [A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((bi_total @ A) @ B) @ A2) => (((((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2))))) @ (ord_less_eq @ A)) @ (ord_less_eq @ B)) => (((((((((bNF_rel_fun @ C) @ D) @ (C > $o)) @ (D > $o)) @ B2) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ B2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2))))) @ (ord_less_eq @ C)) @ (ord_less_eq @ D)) => ((((((((bNF_rel_fun @ (A > C)) @ (B > D)) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2)) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))) @ ((order_mono @ A) @ C)) @ ((order_mono @ B) @ D))))))))))).
% 2.89/1.53 thf(fact_201_neg__fun__distr1, axiom, (! [D:$tType,A:$tType,B:$tType,C:$tType,E:$tType,F3:$tType,R:(A > (B > $o)),R8:(B > (C > $o)),S2:(D > (F3 > $o)),S6:(F3 > (E > $o))]: (((((left_unique @ A) @ B) @ R) => ((((right_total @ A) @ B) @ R) => ((((right_unique @ B) @ C) @ R8) => ((((left_total @ B) @ C) @ R8) => (((ord_less_eq @ ((A > D) > ((C > E) > $o))) @ ((((((bNF_rel_fun @ A) @ C) @ D) @ E) @ (((((relcompp @ A) @ B) @ C) @ R) @ R8)) @ (((((relcompp @ D) @ F3) @ E) @ S2) @ S6))) @ (((((relcompp @ (A > D)) @ (B > F3)) @ (C > E)) @ ((((((bNF_rel_fun @ A) @ B) @ D) @ F3) @ R) @ S2)) @ ((((((bNF_rel_fun @ B) @ C) @ F3) @ E) @ R8) @ S6)))))))))).
% 2.89/1.53 thf(fact_202_set__plus__mono2, axiom, (! [A:$tType]: (((plus @ A) => (! [C2:(set @ A),D3:(set @ A),E2:(set @ A),F5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C2) @ D3) => ((((ord_less_eq @ (set @ A)) @ E2) @ F5) => (((ord_less_eq @ (set @ A)) @ (((plus_plus @ (set @ A)) @ C2) @ E2)) @ (((plus_plus @ (set @ A)) @ D3) @ F5)))))))))).
% 2.89/1.53 thf(fact_203_set__plus__mono2__b, axiom, (! [A:$tType]: (((plus @ A) => (! [C2:(set @ A),D3:(set @ A),E2:(set @ A),F5:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ C2) @ D3) => ((((ord_less_eq @ (set @ A)) @ E2) @ F5) => ((((member @ A) @ X) @ (((plus_plus @ (set @ A)) @ C2) @ E2)) => (((member @ A) @ X) @ (((plus_plus @ (set @ A)) @ D3) @ F5))))))))))).
% 2.89/1.53 thf(fact_204_left__uniqueD, axiom, (! [B:$tType,A:$tType,A2:(A > (B > $o)),X:A,Z:B,Y:A]: (((((left_unique @ A) @ B) @ A2) => (((A2 @ X) @ Z) => (((A2 @ Y) @ Z) => (X = Y))))))).
% 2.89/1.53 thf(fact_205_left__uniqueI, axiom, (! [B:$tType,A:$tType,A2:(A > (B > $o))]: (((! [X2:A,Y2:A,Z3:B]: ((((A2 @ X2) @ Z3) => (((A2 @ Y2) @ Z3) => (X2 = Y2))))) => (((left_unique @ A) @ B) @ A2))))).
% 2.89/1.53 thf(fact_206_right__uniqueD, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o)),X:A,Y:B,Z:B]: (((((right_unique @ A) @ B) @ A2) => (((A2 @ X) @ Y) => (((A2 @ X) @ Z) => (Y = Z))))))).
% 2.89/1.53 thf(fact_207_right__uniqueI, axiom, (! [B:$tType,A:$tType,A2:(A > (B > $o))]: (((! [X2:A,Y2:B,Z3:B]: ((((A2 @ X2) @ Y2) => (((A2 @ X2) @ Z3) => (Y2 = Z3))))) => (((right_unique @ A) @ B) @ A2))))).
% 2.89/1.53 thf(fact_208_left__unique__eq, axiom, (! [A:$tType]: ((((left_unique @ A) @ A) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2))))))).
% 2.89/1.53 thf(fact_209_left__unique__def, axiom, (! [B:$tType,A:$tType]: ((((left_unique @ A) @ B) = (^ [R5:(A > (B > $o))]: ((! [X3:A,Y3:A,Z4:B]: ((((R5 @ X3) @ Z4) => (((R5 @ Y3) @ Z4) => (X3 = Y3))))))))))).
% 2.89/1.53 thf(fact_210_right__unique__eq, axiom, (! [A:$tType]: ((((right_unique @ A) @ A) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2))))))).
% 2.89/1.53 thf(fact_211_right__unique__def, axiom, (! [B:$tType,A:$tType]: ((((right_unique @ A) @ B) = (^ [R5:(A > (B > $o))]: ((! [X3:A,Y3:B,Z4:B]: ((((R5 @ X3) @ Y3) => (((R5 @ X3) @ Z4) => (Y3 = Z4))))))))))).
% 2.89/1.53 thf(fact_212_left__unique__OO, axiom, (! [A:$tType,C:$tType,B:$tType,R:(A > (B > $o)),S2:(B > (C > $o))]: (((((left_unique @ A) @ B) @ R) => ((((left_unique @ B) @ C) @ S2) => (((left_unique @ A) @ C) @ (((((relcompp @ A) @ B) @ C) @ R) @ S2))))))).
% 2.89/1.53 thf(fact_213_right__unique__OO, axiom, (! [A:$tType,C:$tType,B:$tType,A2:(A > (B > $o)),B2:(B > (C > $o))]: (((((right_unique @ A) @ B) @ A2) => ((((right_unique @ B) @ C) @ B2) => (((right_unique @ A) @ C) @ (((((relcompp @ A) @ B) @ C) @ A2) @ B2))))))).
% 2.89/1.53 thf(fact_214_functional__converse__relation, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((((left_unique @ A) @ B) @ R) => ((((right_total @ A) @ B) @ R) => (! [Y6:B]: ((? [X2:A]: ((((R @ X2) @ Y6) & (! [Ya3:A]: ((((R @ Ya3) @ Y6) => (Ya3 = X2)))))))))))))).
% 2.89/1.53 thf(fact_215_functional__relation, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o))]: (((((right_unique @ A) @ B) @ R) => ((((left_total @ A) @ B) @ R) => (! [X7:A]: ((? [Xa2:B]: ((((R @ X7) @ Xa2) & (! [Y6:B]: ((((R @ X7) @ Y6) => (Y6 = Xa2)))))))))))))).
% 2.89/1.53 thf(fact_216_right__unique__fun, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((right_total @ A) @ B) @ A2) => ((((right_unique @ C) @ D) @ B2) => (((right_unique @ (A > C)) @ (B > D)) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2))))))).
% 2.89/1.53 thf(fact_217_left__unique__fun, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((left_total @ A) @ B) @ A2) => ((((left_unique @ C) @ D) @ B2) => (((left_unique @ (A > C)) @ (B > D)) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2))))))).
% 2.89/1.53 thf(fact_218_right__total__fun, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((right_unique @ A) @ B) @ A2) => ((((right_total @ C) @ D) @ B2) => (((right_total @ (A > C)) @ (B > D)) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2))))))).
% 2.89/1.53 thf(fact_219_left__total__fun, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((left_unique @ A) @ B) @ A2) => ((((left_total @ C) @ D) @ B2) => (((left_total @ (A > C)) @ (B > D)) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2))))))).
% 2.89/1.53 thf(fact_220_right__unique__alt__def2, axiom, (! [B:$tType,A:$tType]: ((((right_unique @ A) @ B) = (^ [R5:(A > (B > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ R5) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ R5) @ (=>))) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2)))) @ (^ [Y4:B,Z2:B]: ((Y4 = Z2)))))))))).
% 2.89/1.53 thf(fact_221_eq__imp__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((right_unique @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (=>))) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2)))) @ (^ [Y4:B,Z2:B]: ((Y4 = Z2)))))))).
% 2.89/1.53 thf(fact_222_neg__fun__distr2, axiom, (! [F3:$tType,E:$tType,A:$tType,B:$tType,D:$tType,C:$tType,R8:(A > (B > $o)),S6:(C > (D > $o)),R:(E > (A > $o)),S2:(F3 > (C > $o))]: (((((right_unique @ A) @ B) @ R8) => ((((left_total @ A) @ B) @ R8) => ((((left_unique @ C) @ D) @ S6) => ((((right_total @ C) @ D) @ S6) => (((ord_less_eq @ ((E > F3) > ((B > D) > $o))) @ ((((((bNF_rel_fun @ E) @ B) @ F3) @ D) @ (((((relcompp @ E) @ A) @ B) @ R) @ R8)) @ (((((relcompp @ F3) @ C) @ D) @ S2) @ S6))) @ (((((relcompp @ (E > F3)) @ (A > C)) @ (B > D)) @ ((((((bNF_rel_fun @ E) @ A) @ F3) @ C) @ R) @ S2)) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ R8) @ S6)))))))))).
% 2.89/1.53 thf(fact_223_fixp__mono, axiom, (! [A:$tType]: (((comple1141879883l_ccpo @ A) => (! [F:(A > A),G:(A > A)]: ((((((((partial_fun_ord @ A) @ A) @ A) @ (ord_less_eq @ A)) @ F) @ G) => ((((((comple1396247847notone @ A) @ A) @ (ord_less_eq @ A)) @ (ord_less_eq @ A)) @ F) => ((((((comple1396247847notone @ A) @ A) @ (ord_less_eq @ A)) @ (ord_less_eq @ A)) @ G) => (((ord_less_eq @ A) @ ((comple939513234o_fixp @ A) @ F)) @ ((comple939513234o_fixp @ A) @ G))))))))))).
% 2.89/1.53 thf(fact_224_left__unique__parametric, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((bi_unique @ A) @ B) @ A2) => ((((bi_total @ A) @ B) @ A2) => ((((bi_total @ C) @ D) @ B2) => ((((((((bNF_rel_fun @ (A > (C > $o))) @ (B > (D > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (C > $o)) @ (D > $o)) @ A2) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ B2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))) @ ((left_unique @ A) @ C)) @ ((left_unique @ B) @ D)))))))).
% 2.89/1.53 thf(fact_225_bi__uniqueI, axiom, (! [B:$tType,A:$tType,R:(A > (B > $o))]: (((((left_unique @ A) @ B) @ R) => ((((right_unique @ A) @ B) @ R) => (((bi_unique @ A) @ B) @ R)))))).
% 2.89/1.53 thf(fact_226_bi__unique__alt__def, axiom, (! [B:$tType,A:$tType]: ((((bi_unique @ A) @ B) = (^ [A6:(A > (B > $o))]: (((((left_unique @ A) @ B) @ A6) & (((right_unique @ A) @ B) @ A6)))))))).
% 2.89/1.53 thf(fact_227_bi__unique__OO, axiom, (! [A:$tType,C:$tType,B:$tType,A2:(A > (B > $o)),B2:(B > (C > $o))]: (((((bi_unique @ A) @ B) @ A2) => ((((bi_unique @ B) @ C) @ B2) => (((bi_unique @ A) @ C) @ (((((relcompp @ A) @ B) @ C) @ A2) @ B2))))))).
% 2.89/1.53 thf(fact_228_bi__unique__def, axiom, (! [B:$tType,A:$tType]: ((((bi_unique @ A) @ B) = (^ [R5:(A > (B > $o))]: (((! [X3:A,Y3:B,Z4:B]: ((((R5 @ X3) @ Y3) => (((R5 @ X3) @ Z4) => (Y3 = Z4))))) & (! [X3:A,Y3:A,Z4:B]: ((((R5 @ X3) @ Z4) => (((R5 @ Y3) @ Z4) => (X3 = Y3)))))))))))).
% 2.89/1.53 thf(fact_229_bi__unique__eq, axiom, (! [A:$tType]: ((((bi_unique @ A) @ A) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2))))))).
% 2.89/1.53 thf(fact_230_bi__uniqueDr, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o)),X:A,Y:B,Z:B]: (((((bi_unique @ A) @ B) @ A2) => (((A2 @ X) @ Y) => (((A2 @ X) @ Z) => (Y = Z))))))).
% 2.89/1.53 thf(fact_231_bi__uniqueDl, axiom, (! [B:$tType,A:$tType,A2:(A > (B > $o)),X:A,Y:B,Z:A]: (((((bi_unique @ A) @ B) @ A2) => (((A2 @ X) @ Y) => (((A2 @ Z) @ Y) => (X = Z))))))).
% 2.89/1.53 thf(fact_232_bi__unique__fun, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((bi_unique @ C) @ D) @ B2) => (((bi_unique @ (A > C)) @ (B > D)) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2))))))).
% 2.89/1.53 thf(fact_233_bi__unique__alt__def2, axiom, (! [B:$tType,A:$tType]: ((((bi_unique @ A) @ B) = (^ [R5:(A > (B > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ R5) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ R5) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2))))) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2)))) @ (^ [Y4:B,Z2:B]: ((Y4 = Z2)))))))))).
% 2.89/1.53 thf(fact_234_eq__transfer, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((bi_unique @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2))))) @ (^ [Y4:A,Z2:A]: ((Y4 = Z2)))) @ (^ [Y4:B,Z2:B]: ((Y4 = Z2)))))))).
% 2.89/1.53 thf(fact_235_bi__total__fun, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((bi_unique @ A) @ B) @ A2) => ((((bi_total @ C) @ D) @ B2) => (((bi_total @ (A > C)) @ (B > D)) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2))))))).
% 2.89/1.53 thf(fact_236_right__unique__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((right_total @ A) @ B) @ A2) => ((((right_total @ C) @ D) @ B2) => ((((bi_unique @ C) @ D) @ B2) => ((((((((bNF_rel_fun @ (A > (C > $o))) @ (B > (D > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (C > $o)) @ (D > $o)) @ A2) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ B2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ (=>)) @ ((right_unique @ A) @ C)) @ ((right_unique @ B) @ D)))))))).
% 2.89/1.53 thf(fact_237_left__unique__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((right_total @ A) @ B) @ A2) => ((((right_total @ C) @ D) @ B2) => ((((bi_unique @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > (C > $o))) @ (B > (D > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (C > $o)) @ (D > $o)) @ A2) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ B2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ (=>)) @ ((left_unique @ A) @ C)) @ ((left_unique @ B) @ D)))))))).
% 2.89/1.53 thf(fact_238_right__unique__parametric, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((bi_total @ A) @ B) @ A2) => ((((bi_unique @ C) @ D) @ B2) => ((((bi_total @ C) @ D) @ B2) => ((((((((bNF_rel_fun @ (A > (C > $o))) @ (B > (D > $o))) @ $o) @ $o) @ ((((((bNF_rel_fun @ A) @ B) @ (C > $o)) @ (D > $o)) @ A2) @ ((((((bNF_rel_fun @ C) @ D) @ $o) @ $o) @ B2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))) @ ((right_unique @ A) @ C)) @ ((right_unique @ B) @ D)))))))).
% 2.89/1.53 thf(fact_239_fixp__lowerbound, axiom, (! [A:$tType]: (((comple1141879883l_ccpo @ A) => (! [F:(A > A),Z:A]: (((((((comple1396247847notone @ A) @ A) @ (ord_less_eq @ A)) @ (ord_less_eq @ A)) @ F) => ((((ord_less_eq @ A) @ (F @ Z)) @ Z) => (((ord_less_eq @ A) @ ((comple939513234o_fixp @ A) @ F)) @ Z))))))))).
% 2.89/1.53 thf(fact_240_fixp__unfold, axiom, (! [A:$tType]: (((comple1141879883l_ccpo @ A) => (! [F:(A > A)]: (((((((comple1396247847notone @ A) @ A) @ (ord_less_eq @ A)) @ (ord_less_eq @ A)) @ F) => (((comple939513234o_fixp @ A) @ F) = (F @ ((comple939513234o_fixp @ A) @ F)))))))))).
% 2.89/1.53 thf(fact_241_monotone__def, axiom, (! [B:$tType,A:$tType]: ((((comple1396247847notone @ A) @ B) = (^ [Orda:(A > (A > $o)),Ordb:(B > (B > $o)),F2:(A > B)]: ((! [X3:A,Y3:A]: ((((Orda @ X3) @ Y3) => ((Ordb @ (F2 @ X3)) @ (F2 @ Y3))))))))))).
% 2.89/1.53 thf(fact_242_monotoneI, axiom, (! [B:$tType,A:$tType,Orda2:(A > (A > $o)),Ordb2:(B > (B > $o)),F:(A > B)]: (((! [X2:A,Y2:A]: ((((Orda2 @ X2) @ Y2) => ((Ordb2 @ (F @ X2)) @ (F @ Y2))))) => (((((comple1396247847notone @ A) @ B) @ Orda2) @ Ordb2) @ F))))).
% 2.89/1.53 thf(fact_243_monotoneD, axiom, (! [B:$tType,A:$tType,Orda2:(A > (A > $o)),Ordb2:(B > (B > $o)),F:(A > B),X:A,Y:A]: (((((((comple1396247847notone @ A) @ B) @ Orda2) @ Ordb2) @ F) => (((Orda2 @ X) @ Y) => ((Ordb2 @ (F @ X)) @ (F @ Y))))))).
% 2.89/1.53 thf(fact_244_fun__upd__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A2:(A > (B > $o)),B2:(C > (D > $o))]: (((((bi_unique @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > C)) @ (B > D)) @ (A > (C > (A > C)))) @ (B > (D > (B > D)))) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2)) @ ((((((bNF_rel_fun @ A) @ B) @ (C > (A > C))) @ (D > (B > D))) @ A2) @ ((((((bNF_rel_fun @ C) @ D) @ (A > C)) @ (B > D)) @ B2) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ A2) @ B2)))) @ ((fun_upd @ A) @ C)) @ ((fun_upd @ B) @ D)))))).
% 2.89/1.53 thf(fact_245_rtranclp__parametric, axiom, (! [A:$tType,B:$tType,A2:(A > (B > $o))]: (((((bi_unique @ A) @ B) @ A2) => ((((bi_total @ A) @ B) @ A2) => ((((((((bNF_rel_fun @ (A > (A > $o))) @ (B > (B > $o))) @ (A > (A > $o))) @ (B > (B > $o))) @ ((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ ((((((bNF_rel_fun @ A) @ B) @ (A > $o)) @ (B > $o)) @ A2) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A2) @ (^ [Y4:$o,Z2:$o]: ((Y4 = Z2)))))) @ (transitive_rtranclp @ A)) @ (transitive_rtranclp @ B))))))).
% 2.89/1.53 thf(fact_246_fun__upd__apply, axiom, (! [A:$tType,B:$tType]: ((((fun_upd @ B) @ A) = (^ [F2:(B > A),X3:B,Y3:A,Z4:B]: (((((if @ A) @ (Z4 = X3)) @ Y3) @ (F2 @ Z4)))))))).
% 2.89/1.53 thf(fact_247_fun__upd__triv, axiom, (! [B:$tType,A:$tType,F:(A > B),X:A]: (((((((fun_upd @ A) @ B) @ F) @ X) @ (F @ X)) = F)))).
% 2.89/1.53 thf(fact_248_fun__upd__upd, axiom, (! [A:$tType,B:$tType,F:(A > B),X:A,Y:B,Z:B]: (((((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ F) @ X) @ Y)) @ X) @ Z) = (((((fun_upd @ A) @ B) @ F) @ X) @ Z))))).
% 2.89/1.53 thf(fact_249_fun__upd__idem__iff, axiom, (! [A:$tType,B:$tType,F:(A > B),X:A,Y:B]: ((((((((fun_upd @ A) @ B) @ F) @ X) @ Y) = F) = ((F @ X) = Y))))).
% 2.89/1.53 thf(fact_250_fun__upd__twist, axiom, (! [A:$tType,B:$tType,A4:A,C3:A,M3:(A > B),B4:B,D2:B]: (((A4 != C3) => ((((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ M3) @ A4) @ B4)) @ C3) @ D2) = (((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ M3) @ C3) @ D2)) @ A4) @ B4)))))).
% 2.89/1.53 thf(fact_251_fun__upd__other, axiom, (! [B:$tType,A:$tType,Z:A,X:A,F:(A > B),Y:B]: (((Z != X) => (((((((fun_upd @ A) @ B) @ F) @ X) @ Y) @ Z) = (F @ Z)))))).
% 2.89/1.53 thf(fact_252_fun__upd__same, axiom, (! [B:$tType,A:$tType,F:(B > A),X:B,Y:A]: ((((((((fun_upd @ B) @ A) @ F) @ X) @ Y) @ X) = Y)))).
% 2.89/1.53 thf(fact_253_fun__upd__idem, axiom, (! [A:$tType,B:$tType,F:(B > A),X:B,Y:A]: ((((F @ X) = Y) => ((((((fun_upd @ B) @ A) @ F) @ X) @ Y) = F))))).
% 2.89/1.53 thf(fact_254_fun__upd__eqD, axiom, (! [A:$tType,B:$tType,F:(A > B),X:A,Y:B,G:(A > B),Z:B]: ((((((((fun_upd @ A) @ B) @ F) @ X) @ Y) = (((((fun_upd @ A) @ B) @ G) @ X) @ Z)) => (Y = Z))))).
% 2.89/1.53 thf(fact_255_fun__upd__def, axiom, (! [B:$tType,A:$tType]: ((((fun_upd @ A) @ B) = (^ [F2:(A > B),A5:A,B5:B,X3:A]: (((((if @ B) @ (X3 = A5)) @ B5) @ (F2 @ X3)))))))).
% 2.89/1.53 thf(tcon_Multiset_Omultiset___Orderings_Opreorder, axiom, (! [A7:$tType]: (((preorder @ A7) => (preorder @ (multiset @ A7)))))).
% 2.89/1.53 thf(tcon_HOL_Obool___Orderings_Opreorder_1, axiom, (preorder @ $o)).
% 2.89/1.53 thf(tcon_Set_Oset___Orderings_Opreorder_2, axiom, (! [A7:$tType]: ((preorder @ (set @ A7))))).
% 2.89/1.53 thf(tcon_fun___Orderings_Opreorder_3, axiom, (! [A7:$tType,A8:$tType]: (((preorder @ A8) => (preorder @ (A7 > A8)))))).
% 2.89/1.53 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A7:$tType,A8:$tType]: (((comple187826305attice @ A8) => (comple187826305attice @ (A7 > A8)))))).
% 2.89/1.53 thf(tcon_fun___Complete__Partial__Order_Occpo, axiom, (! [A7:$tType,A8:$tType]: (((comple187826305attice @ A8) => (comple1141879883l_ccpo @ (A7 > A8)))))).
% 2.89/1.53 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: (((order @ A8) => (order @ (A7 > A8)))))).
% 2.89/1.53 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_4, axiom, (! [A7:$tType]: ((comple187826305attice @ (set @ A7))))).
% 2.89/1.53 thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_5, axiom, (! [A7:$tType]: ((comple1141879883l_ccpo @ (set @ A7))))).
% 2.89/1.53 thf(tcon_Set_Oset___Groups_Oab__semigroup__add, axiom, (! [A7:$tType]: (((ab_semigroup_add @ A7) => (ab_semigroup_add @ (set @ A7)))))).
% 2.89/1.53 thf(tcon_Set_Oset___Groups_Ocomm__monoid__add, axiom, (! [A7:$tType]: (((comm_monoid_add @ A7) => (comm_monoid_add @ (set @ A7)))))).
% 2.89/1.53 thf(tcon_Set_Oset___Groups_Osemigroup__add, axiom, (! [A7:$tType]: (((semigroup_add @ A7) => (semigroup_add @ (set @ A7)))))).
% 2.89/1.53 thf(tcon_Set_Oset___Orderings_Oorder_6, axiom, (! [A7:$tType]: ((order @ (set @ A7))))).
% 2.89/1.53 thf(tcon_Set_Oset___Groups_Oplus, axiom, (! [A7:$tType]: (((plus @ A7) => (plus @ (set @ A7)))))).
% 2.89/1.53 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_7, axiom, (comple187826305attice @ $o)).
% 2.89/1.53 thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_8, axiom, (comple1141879883l_ccpo @ $o)).
% 2.89/1.53 thf(tcon_HOL_Obool___Orderings_Oorder_9, axiom, (order @ $o)).
% 2.89/1.53 thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add, axiom, (! [A7:$tType]: (((preorder @ A7) => (ordere779506340up_add @ (multiset @ A7)))))).
% 2.89/1.53 thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add, axiom, (! [A7:$tType]: ((cancel_semigroup_add @ (multiset @ A7))))).
% 2.89/1.53 thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_10, axiom, (! [A7:$tType]: ((ab_semigroup_add @ (multiset @ A7))))).
% 2.89/1.53 thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_11, axiom, (! [A7:$tType]: ((comm_monoid_add @ (multiset @ A7))))).
% 2.89/1.53 thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_12, axiom, (! [A7:$tType]: ((semigroup_add @ (multiset @ A7))))).
% 2.89/1.53 thf(tcon_Multiset_Omultiset___Orderings_Oorder_13, axiom, (! [A7:$tType]: (((preorder @ A7) => (order @ (multiset @ A7)))))).
% 2.89/1.53 thf(tcon_Multiset_Omultiset___Groups_Oplus_14, axiom, (! [A7:$tType]: ((plus @ (multiset @ A7))))).
% 2.89/1.53 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.89/1.53 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.89/1.53 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.89/1.53 thf(conj_0, conjecture, (((finite100568337ommute @ a) @ c) @ f1)).
% 2.89/1.53 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.XftS7e7A06/DTF2DTF9803.p
% 2.89/1.53 ------------------------
% 2.89/1.56 ---- Cleaned THF ---
% 2.89/1.56 thf(ty_t_Multiset_Omultiset,type,
% 2.89/1.56 multiset: $tType > $tType ).
% 2.89/1.56
% 2.89/1.56 thf(ty_t_FSet_Ofset,type,
% 2.89/1.56 fset: $tType > $tType ).
% 2.89/1.56
% 2.89/1.56 thf(ty_t_Set_Oset,type,
% 2.89/1.56 set: $tType > $tType ).
% 2.89/1.56
% 2.89/1.56 thf(ty_tf_d,type,
% 2.89/1.56 d: $tType ).
% 2.89/1.56
% 2.89/1.56 thf(ty_tf_c,type,
% 2.89/1.56 c: $tType ).
% 2.89/1.56
% 2.89/1.56 thf(ty_tf_b,type,
% 2.89/1.56 b: $tType ).
% 2.89/1.56
% 2.89/1.56 thf(ty_tf_a,type,
% 2.89/1.56 a: $tType ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Orderings_Opreorder,type,
% 2.89/1.56 preorder:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_HOL_Otype,type,
% 2.89/1.56 type:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Oplus,type,
% 2.89/1.56 plus:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Orderings_Oorder,type,
% 2.89/1.56 order:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Ogroup__add,type,
% 2.89/1.56 group_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Osemigroup__add,type,
% 2.89/1.56 semigroup_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 2.89/1.56 comm_monoid_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 2.89/1.56 ab_semigroup_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Complete__Partial__Order_Occpo,type,
% 2.89/1.56 comple1141879883l_ccpo:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.89/1.56 cancel_semigroup_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 2.89/1.56 ordere779506340up_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 2.89/1.56 comple187826305attice:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.89/1.56 canoni770627133id_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 2.89/1.56 ordere236663937imp_le:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_BNF__Def_Orel__fun,type,
% 2.89/1.56 bNF_rel_fun:
% 2.89/1.56 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp,type,
% 2.89/1.56 comple939513234o_fixp:
% 2.89/1.56 !>[A: $tType] : ( ( A > A ) > A ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Complete__Partial__Order_Omonotone,type,
% 2.89/1.56 comple1396247847notone:
% 2.89/1.56 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_FSet_Offold,type,
% 2.89/1.56 ffold:
% 2.89/1.56 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( fset @ A ) > B ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
% 2.89/1.56 finite100568337ommute:
% 2.89/1.56 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Finite__Set_Ocomp__fun__idem,type,
% 2.89/1.56 finite_comp_fun_idem:
% 2.89/1.56 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Finite__Set_Ocomp__fun__idem__axioms,type,
% 2.89/1.56 finite852775215axioms:
% 2.89/1.56 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Finite__Set_Ofold__graph,type,
% 2.89/1.56 finite_fold_graph:
% 2.89/1.56 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B > $o ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Finite__Set_Ofolding,type,
% 2.89/1.56 finite_folding:
% 2.89/1.56 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Finite__Set_Ofolding__idem,type,
% 2.89/1.56 finite_folding_idem:
% 2.89/1.56 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Finite__Set_Ofolding__idem__axioms,type,
% 2.89/1.56 finite1921348288axioms:
% 2.89/1.56 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Fun_Ocomp,type,
% 2.89/1.56 comp:
% 2.89/1.56 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Fun_Ofun__upd,type,
% 2.89/1.56 fun_upd:
% 2.89/1.56 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Fun_Omap__fun,type,
% 2.89/1.56 map_fun:
% 2.89/1.56 !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.89/1.56 plus_plus:
% 2.89/1.56 !>[A: $tType] : ( A > A > A ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_If,type,
% 2.89/1.56 if:
% 2.89/1.56 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Multiset_Ofold__mset,type,
% 2.89/1.56 fold_mset:
% 2.89/1.56 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( multiset @ A ) > B ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.89/1.56 ord_less_eq:
% 2.89/1.56 !>[A: $tType] : ( A > A > $o ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Orderings_Oorder__class_Omono,type,
% 2.89/1.56 order_mono:
% 2.89/1.56 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Partial__Function_Ofun__ord,type,
% 2.89/1.56 partial_fun_ord:
% 2.89/1.57 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_Quotient_OBex1__rel,type,
% 2.89/1.57 bex1_rel:
% 2.89/1.57 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > $o ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_Quotient_OQuotient3,type,
% 2.89/1.57 quotient3:
% 2.89/1.57 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_Relation_ODomainp,type,
% 2.89/1.57 domainp:
% 2.89/1.57 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_Relation_Oreflp,type,
% 2.89/1.57 reflp:
% 2.89/1.57 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_Relation_Orelcompp,type,
% 2.89/1.57 relcompp:
% 2.89/1.57 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_Set_OCollect,type,
% 2.89/1.57 collect:
% 2.89/1.57 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_Transfer_Obi__total,type,
% 2.89/1.57 bi_total:
% 2.89/1.57 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_Transfer_Obi__unique,type,
% 2.89/1.57 bi_unique:
% 2.89/1.57 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_Transfer_Oleft__total,type,
% 2.89/1.57 left_total:
% 2.89/1.57 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_Transfer_Oleft__unique,type,
% 2.89/1.57 left_unique:
% 2.89/1.57 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_Transfer_Orev__implies,type,
% 2.89/1.57 rev_implies: $o > $o > $o ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_Transfer_Oright__total,type,
% 2.89/1.57 right_total:
% 2.89/1.57 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_Transfer_Oright__unique,type,
% 2.89/1.57 right_unique:
% 2.89/1.57 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_Transfer_Otransfer__bforall,type,
% 2.89/1.57 transfer_bforall:
% 2.89/1.57 !>[A: $tType] : ( ( A > $o ) > ( A > $o ) > $o ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_Transfer_Otransfer__forall,type,
% 2.89/1.57 transfer_forall:
% 2.89/1.57 !>[A: $tType] : ( ( A > $o ) > $o ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_Transitive__Closure_Ortranclp,type,
% 2.89/1.57 transitive_rtranclp:
% 2.89/1.57 !>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_c_member,type,
% 2.89/1.57 member:
% 2.89/1.57 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.89/1.57
% 2.89/1.57 thf(sy_v_A,type,
% 2.89/1.57 a2: a > b > $o ).
% 2.89/1.57
% 2.89/1.57 thf(sy_v_B,type,
% 2.89/1.57 b2: c > d > $o ).
% 2.89/1.57
% 2.89/1.57 thf(sy_v_f1,type,
% 2.89/1.57 f1: a > c > c ).
% 2.89/1.57
% 2.89/1.57 thf(sy_v_f2,type,
% 2.89/1.57 f2: b > d > d ).
% 2.89/1.57
% 2.89/1.57 thf(fact_0_assms_I2_J,axiom,
% 2.89/1.57 finite100568337ommute @ a @ c @ f1 ).
% 2.89/1.57
% 2.89/1.57 thf(fact_1_comp__fun__commute_Ofun__left__comm,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,F: A > B > B,Y: A,X: A,Z: B] :
% 2.89/1.57 ( ( finite100568337ommute @ A @ B @ F )
% 2.89/1.57 => ( ( F @ Y @ ( F @ X @ Z ) )
% 2.89/1.57 = ( F @ X @ ( F @ Y @ Z ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_2_assms_I3_J,axiom,
% 2.89/1.57 finite100568337ommute @ b @ d @ f2 ).
% 2.89/1.57
% 2.89/1.57 thf(fact_3__C12_C,axiom,
% 2.89/1.57 bNF_rel_fun @ a @ b @ ( c > c ) @ ( d > d ) @ a2 @ ( bNF_rel_fun @ c @ d @ c @ d @ b2 @ b2 ) @ f1 @ f2 ).
% 2.89/1.57
% 2.89/1.57 thf(fact_4_comp__fun__commute_Ofold__mset__fusion,axiom,
% 2.89/1.57 ! [C: $tType,B: $tType,A: $tType,F: A > B > B,G: A > C > C,H: C > B,W: C,A2: multiset @ A] :
% 2.89/1.57 ( ( finite100568337ommute @ A @ B @ F )
% 2.89/1.57 => ( ( finite100568337ommute @ A @ C @ G )
% 2.89/1.57 => ( ! [X2: A,Y2: C] :
% 2.89/1.57 ( ( H @ ( G @ X2 @ Y2 ) )
% 2.89/1.57 = ( F @ X2 @ ( H @ Y2 ) ) )
% 2.89/1.57 => ( ( H @ ( fold_mset @ A @ C @ G @ W @ A2 ) )
% 2.89/1.57 = ( fold_mset @ A @ B @ F @ ( H @ W ) @ A2 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_5_comp__fun__commute_Ofold__mset__fun__left__comm,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B,X: A,S: B,M: multiset @ A] :
% 2.89/1.57 ( ( finite100568337ommute @ A @ B @ F )
% 2.89/1.57 => ( ( F @ X @ ( fold_mset @ A @ B @ F @ S @ M ) )
% 2.89/1.57 = ( fold_mset @ A @ B @ F @ ( F @ X @ S ) @ M ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_6_comp__fun__idem_Oaxioms_I1_J,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B] :
% 2.89/1.57 ( ( finite_comp_fun_idem @ A @ B @ F )
% 2.89/1.57 => ( finite100568337ommute @ A @ B @ F ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_7_comp__fun__commute_Ofold__graph__determ,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,F: A > B > B,Z: B,A2: set @ A,X: B,Y: B] :
% 2.89/1.57 ( ( finite100568337ommute @ A @ B @ F )
% 2.89/1.57 => ( ( finite_fold_graph @ A @ B @ F @ Z @ A2 @ X )
% 2.89/1.57 => ( ( finite_fold_graph @ A @ B @ F @ Z @ A2 @ Y )
% 2.89/1.57 => ( Y = X ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_8_comp__fun__commute_Offold__fun__left__comm,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B,X: A,Z: B,A2: fset @ A] :
% 2.89/1.57 ( ( finite100568337ommute @ A @ B @ F )
% 2.89/1.57 => ( ( F @ X @ ( ffold @ A @ B @ F @ Z @ A2 ) )
% 2.89/1.57 = ( ffold @ A @ B @ F @ ( F @ X @ Z ) @ A2 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_9_comp__fun__commute_Ocomp__comp__fun__commute,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,C: $tType,F: A > B > B,G: C > A] :
% 2.89/1.57 ( ( finite100568337ommute @ A @ B @ F )
% 2.89/1.57 => ( finite100568337ommute @ C @ B @ ( comp @ A @ ( B > B ) @ C @ F @ G ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_10_comp__fun__commute__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( finite100568337ommute @ A @ B )
% 2.89/1.57 = ( ^ [F2: A > B > B] :
% 2.89/1.57 ! [Y3: A,X3: A] :
% 2.89/1.57 ( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) )
% 2.89/1.57 = ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_11_comp__fun__commute_Ointro,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B] :
% 2.89/1.57 ( ! [Y2: A,X2: A] :
% 2.89/1.57 ( ( comp @ B @ B @ B @ ( F @ Y2 ) @ ( F @ X2 ) )
% 2.89/1.57 = ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.89/1.57 => ( finite100568337ommute @ A @ B @ F ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_12_comp__fun__commute_Ocomp__fun__commute,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B,Y: A,X: A] :
% 2.89/1.57 ( ( finite100568337ommute @ A @ B @ F )
% 2.89/1.57 => ( ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) )
% 2.89/1.57 = ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_13_fold__graph__closed__eq,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,A2: set @ A,B2: set @ B,F: A > B > B,G: A > B > B,Z: B] :
% 2.89/1.57 ( ! [A3: A,B3: B] :
% 2.89/1.57 ( ( member @ A @ A3 @ A2 )
% 2.89/1.57 => ( ( member @ B @ B3 @ B2 )
% 2.89/1.57 => ( ( F @ A3 @ B3 )
% 2.89/1.57 = ( G @ A3 @ B3 ) ) ) )
% 2.89/1.57 => ( ! [A3: A,B3: B] :
% 2.89/1.57 ( ( member @ A @ A3 @ A2 )
% 2.89/1.57 => ( ( member @ B @ B3 @ B2 )
% 2.89/1.57 => ( member @ B @ ( G @ A3 @ B3 ) @ B2 ) ) )
% 2.89/1.57 => ( ( member @ B @ Z @ B2 )
% 2.89/1.57 => ( ( finite_fold_graph @ A @ B @ F @ Z @ A2 )
% 2.89/1.57 = ( finite_fold_graph @ A @ B @ G @ Z @ A2 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_14_fold__graph__closed__lemma,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,G: A > B > B,Z: B,A2: set @ A,X: B,B2: set @ B,F: A > B > B] :
% 2.89/1.57 ( ( finite_fold_graph @ A @ B @ G @ Z @ A2 @ X )
% 2.89/1.57 => ( ! [A3: A,B3: B] :
% 2.89/1.57 ( ( member @ A @ A3 @ A2 )
% 2.89/1.57 => ( ( member @ B @ B3 @ B2 )
% 2.89/1.57 => ( ( F @ A3 @ B3 )
% 2.89/1.57 = ( G @ A3 @ B3 ) ) ) )
% 2.89/1.57 => ( ! [A3: A,B3: B] :
% 2.89/1.57 ( ( member @ A @ A3 @ A2 )
% 2.89/1.57 => ( ( member @ B @ B3 @ B2 )
% 2.89/1.57 => ( member @ B @ ( G @ A3 @ B3 ) @ B2 ) ) )
% 2.89/1.57 => ( ( member @ B @ Z @ B2 )
% 2.89/1.57 => ( ( finite_fold_graph @ A @ B @ F @ Z @ A2 @ X )
% 2.89/1.57 & ( member @ B @ X @ B2 ) ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_15_comp__fun__idem_Ocomp__fun__idem,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B,X: A] :
% 2.89/1.57 ( ( finite_comp_fun_idem @ A @ B @ F )
% 2.89/1.57 => ( ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ X ) )
% 2.89/1.57 = ( F @ X ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_16_comp__fun__idem_Ofun__left__idem,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,F: A > B > B,X: A,Z: B] :
% 2.89/1.57 ( ( finite_comp_fun_idem @ A @ B @ F )
% 2.89/1.57 => ( ( F @ X @ ( F @ X @ Z ) )
% 2.89/1.57 = ( F @ X @ Z ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_17_comp__fun__idem_Ocomp__comp__fun__idem,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,C: $tType,F: A > B > B,G: C > A] :
% 2.89/1.57 ( ( finite_comp_fun_idem @ A @ B @ F )
% 2.89/1.57 => ( finite_comp_fun_idem @ C @ B @ ( comp @ A @ ( B > B ) @ C @ F @ G ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_18_comp__fun__commute_Ocommute__left__comp,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,C: $tType,F: A > B > B,Y: A,X: A,G: C > B] :
% 2.89/1.57 ( ( finite100568337ommute @ A @ B @ F )
% 2.89/1.57 => ( ( comp @ B @ B @ C @ ( F @ Y ) @ ( comp @ B @ B @ C @ ( F @ X ) @ G ) )
% 2.89/1.57 = ( comp @ B @ B @ C @ ( F @ X ) @ ( comp @ B @ B @ C @ ( F @ Y ) @ G ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_19_comp__apply,axiom,
% 2.89/1.57 ! [C: $tType,A: $tType,B: $tType] :
% 2.89/1.57 ( ( comp @ B @ A @ C )
% 2.89/1.57 = ( ^ [F2: B > A,G2: C > B,X3: C] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_20_rel__funI,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F: A > C,G: B > D] :
% 2.89/1.57 ( ! [X2: A,Y2: B] :
% 2.89/1.57 ( ( A2 @ X2 @ Y2 )
% 2.89/1.57 => ( B2 @ ( F @ X2 ) @ ( G @ Y2 ) ) )
% 2.89/1.57 => ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_21_If__transfer,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 2.89/1.57 ( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( B > B > B )
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A2 @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) )
% 2.89/1.57 @ ( if @ A )
% 2.89/1.57 @ ( if @ B ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_22_rel__fun__def__butlast,axiom,
% 2.89/1.57 ! [B: $tType,D: $tType,C: $tType,E: $tType,F3: $tType,A: $tType,R: A > B > $o,S2: C > E > $o,T: D > F3 > $o,F: A > C > D,G: B > E > F3] :
% 2.89/1.57 ( ( bNF_rel_fun @ A @ B @ ( C > D ) @ ( E > F3 ) @ R @ ( bNF_rel_fun @ C @ E @ D @ F3 @ S2 @ T ) @ F @ G )
% 2.89/1.57 = ( ! [X3: A,Y3: B] :
% 2.89/1.57 ( ( R @ X3 @ Y3 )
% 2.89/1.57 => ( bNF_rel_fun @ C @ E @ D @ F3 @ S2 @ T @ ( F @ X3 ) @ ( G @ Y3 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_23_o__rsp_I2_J,axiom,
% 2.89/1.57 ! [E: $tType,F3: $tType,H2: $tType,G3: $tType,R1: E > F3 > $o] :
% 2.89/1.57 ( bNF_rel_fun @ ( G3 > H2 ) @ ( G3 > H2 ) @ ( ( E > G3 ) > E > H2 ) @ ( ( F3 > G3 ) > F3 > H2 )
% 2.89/1.57 @ ^ [Y4: G3 > H2,Z2: G3 > H2] : ( Y4 = Z2 )
% 2.89/1.57 @ ( bNF_rel_fun @ ( E > G3 ) @ ( F3 > G3 ) @ ( E > H2 ) @ ( F3 > H2 )
% 2.89/1.57 @ ( bNF_rel_fun @ E @ F3 @ G3 @ G3 @ R1
% 2.89/1.57 @ ^ [Y4: G3,Z2: G3] : ( Y4 = Z2 ) )
% 2.89/1.57 @ ( bNF_rel_fun @ E @ F3 @ H2 @ H2 @ R1
% 2.89/1.57 @ ^ [Y4: H2,Z2: H2] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ( comp @ G3 @ H2 @ E )
% 2.89/1.57 @ ( comp @ G3 @ H2 @ F3 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_24_o__rsp_I1_J,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,R2: A > C > $o,R3: B > D > $o,R1: E > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F3 > C ) > F3 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R2 @ R3 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F3 > C ) @ ( E > B ) @ ( F3 > D ) @ ( bNF_rel_fun @ E @ F3 @ A @ C @ R1 @ R2 ) @ ( bNF_rel_fun @ E @ F3 @ B @ D @ R1 @ R3 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F3 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_25_fun_Omap__transfer,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,D: $tType,G3: $tType,F3: $tType,Rb: A > F3 > $o,Sd: B > G3 > $o] :
% 2.89/1.57 ( bNF_rel_fun @ ( A > B ) @ ( F3 > G3 ) @ ( ( D > A ) > D > B ) @ ( ( D > F3 ) > D > G3 ) @ ( bNF_rel_fun @ A @ F3 @ B @ G3 @ Rb @ Sd )
% 2.89/1.57 @ ( bNF_rel_fun @ ( D > A ) @ ( D > F3 ) @ ( D > B ) @ ( D > G3 )
% 2.89/1.57 @ ( bNF_rel_fun @ D @ D @ A @ F3
% 2.89/1.57 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 2.89/1.57 @ Rb )
% 2.89/1.57 @ ( bNF_rel_fun @ D @ D @ B @ G3
% 2.89/1.57 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 2.89/1.57 @ Sd ) )
% 2.89/1.57 @ ( comp @ A @ B @ D )
% 2.89/1.57 @ ( comp @ F3 @ G3 @ D ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_26_comp__transfer,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,B2: A > C > $o,C2: B > D > $o,A2: E > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F3 > C ) > F3 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ B2 @ C2 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F3 > C ) @ ( E > B ) @ ( F3 > D ) @ ( bNF_rel_fun @ E @ F3 @ A @ C @ A2 @ B2 ) @ ( bNF_rel_fun @ E @ F3 @ B @ D @ A2 @ C2 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F3 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_27_comp__fun__idem_Ointro,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B] :
% 2.89/1.57 ( ( finite100568337ommute @ A @ B @ F )
% 2.89/1.57 => ( ( finite852775215axioms @ A @ B @ F )
% 2.89/1.57 => ( finite_comp_fun_idem @ A @ B @ F ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_28_comp__fun__idem__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( finite_comp_fun_idem @ A @ B )
% 2.89/1.57 = ( ^ [F2: A > B > B] :
% 2.89/1.57 ( ( finite100568337ommute @ A @ B @ F2 )
% 2.89/1.57 & ( finite852775215axioms @ A @ B @ F2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_29_comp__fun__idem_Oaxioms_I2_J,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B] :
% 2.89/1.57 ( ( finite_comp_fun_idem @ A @ B @ F )
% 2.89/1.57 => ( finite852775215axioms @ A @ B @ F ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_30_comp__fun__idem__axioms_Ointro,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B] :
% 2.89/1.57 ( ! [X2: A] :
% 2.89/1.57 ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
% 2.89/1.57 = ( F @ X2 ) )
% 2.89/1.57 => ( finite852775215axioms @ A @ B @ F ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_31_comp__fun__idem__axioms__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( finite852775215axioms @ A @ B )
% 2.89/1.57 = ( ^ [F2: A > B > B] :
% 2.89/1.57 ! [X3: A] :
% 2.89/1.57 ( ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
% 2.89/1.57 = ( F2 @ X3 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_32_fun_Orel__transfer,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,C: $tType,E: $tType,D: $tType,Sa: A > C > $o,Sc: B > E > $o] :
% 2.89/1.57 ( bNF_rel_fun @ ( A > B > $o ) @ ( C > E > $o ) @ ( ( D > A ) > ( D > B ) > $o ) @ ( ( D > C ) > ( D > E ) > $o )
% 2.89/1.57 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( E > $o ) @ Sa
% 2.89/1.57 @ ( bNF_rel_fun @ B @ E @ $o @ $o @ Sc
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ( bNF_rel_fun @ ( D > A ) @ ( D > C ) @ ( ( D > B ) > $o ) @ ( ( D > E ) > $o )
% 2.89/1.57 @ ( bNF_rel_fun @ D @ D @ A @ C
% 2.89/1.57 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 2.89/1.57 @ Sa )
% 2.89/1.57 @ ( bNF_rel_fun @ ( D > B ) @ ( D > E ) @ $o @ $o
% 2.89/1.57 @ ( bNF_rel_fun @ D @ D @ B @ E
% 2.89/1.57 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 2.89/1.57 @ Sc )
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ( bNF_rel_fun @ D @ D @ A @ B
% 2.89/1.57 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 ) )
% 2.89/1.57 @ ( bNF_rel_fun @ D @ D @ C @ E
% 2.89/1.57 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_33_fun_Orel__refl,axiom,
% 2.89/1.57 ! [B: $tType,D: $tType,Ra: B > B > $o,X: D > B] :
% 2.89/1.57 ( ! [X2: B] : ( Ra @ X2 @ X2 )
% 2.89/1.57 => ( bNF_rel_fun @ D @ D @ B @ B
% 2.89/1.57 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 2.89/1.57 @ Ra
% 2.89/1.57 @ X
% 2.89/1.57 @ X ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_34_fun_Orel__eq,axiom,
% 2.89/1.57 ! [A: $tType,D: $tType] :
% 2.89/1.57 ( ( bNF_rel_fun @ D @ D @ A @ A
% 2.89/1.57 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.89/1.57 = ( ^ [Y4: D > A,Z2: D > A] : ( Y4 = Z2 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_35_rel__fun__mono_H,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,Y5: A > B > $o,X4: A > B > $o,A2: C > D > $o,B2: C > D > $o,F: A > C,G: B > D] :
% 2.89/1.57 ( ! [X2: A,Y2: B] :
% 2.89/1.57 ( ( Y5 @ X2 @ Y2 )
% 2.89/1.57 => ( X4 @ X2 @ Y2 ) )
% 2.89/1.57 => ( ! [X2: C,Y2: D] :
% 2.89/1.57 ( ( A2 @ X2 @ Y2 )
% 2.89/1.57 => ( B2 @ X2 @ Y2 ) )
% 2.89/1.57 => ( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A2 @ F @ G )
% 2.89/1.57 => ( bNF_rel_fun @ A @ B @ C @ D @ Y5 @ B2 @ F @ G ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_36_rel__fun__mono,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,X4: A > B > $o,A2: C > D > $o,F: A > C,G: B > D,Y5: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A2 @ F @ G )
% 2.89/1.57 => ( ! [X2: A,Y2: B] :
% 2.89/1.57 ( ( Y5 @ X2 @ Y2 )
% 2.89/1.57 => ( X4 @ X2 @ Y2 ) )
% 2.89/1.57 => ( ! [X2: C,Y2: D] :
% 2.89/1.57 ( ( A2 @ X2 @ Y2 )
% 2.89/1.57 => ( B2 @ X2 @ Y2 ) )
% 2.89/1.57 => ( bNF_rel_fun @ A @ B @ C @ D @ Y5 @ B2 @ F @ G ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_37_let__rsp,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o] :
% 2.89/1.57 ( bNF_rel_fun @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ R1 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ C @ D @ ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 ) @ R2 )
% 2.89/1.57 @ ^ [S3: A,F2: A > C] : ( F2 @ S3 )
% 2.89/1.57 @ ^ [S3: B,F2: B > D] : ( F2 @ S3 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_38_rel__funD,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F: A > C,G: B > D,X: A,Y: B] :
% 2.89/1.57 ( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G )
% 2.89/1.57 => ( ( A2 @ X @ Y )
% 2.89/1.57 => ( B2 @ ( F @ X ) @ ( G @ Y ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_39_rewriteR__comp__comp2,axiom,
% 2.89/1.57 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H: A > C,R12: D > B,R22: A > D,F: B > E,L: D > E] :
% 2.89/1.57 ( ( ( comp @ C @ B @ A @ G @ H )
% 2.89/1.57 = ( comp @ D @ B @ A @ R12 @ R22 ) )
% 2.89/1.57 => ( ( ( comp @ B @ E @ D @ F @ R12 )
% 2.89/1.57 = L )
% 2.89/1.57 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H )
% 2.89/1.57 = ( comp @ D @ E @ A @ L @ R22 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_40_rewriteL__comp__comp2,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L2: A > D,H: E > A,R4: E > D] :
% 2.89/1.57 ( ( ( comp @ C @ B @ A @ F @ G )
% 2.89/1.57 = ( comp @ D @ B @ A @ L1 @ L2 ) )
% 2.89/1.57 => ( ( ( comp @ A @ D @ E @ L2 @ H )
% 2.89/1.57 = R4 )
% 2.89/1.57 => ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H ) )
% 2.89/1.57 = ( comp @ D @ B @ E @ L1 @ R4 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_41_rewriteR__comp__comp,axiom,
% 2.89/1.57 ! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R4: A > B,F: B > D] :
% 2.89/1.57 ( ( ( comp @ C @ B @ A @ G @ H )
% 2.89/1.57 = R4 )
% 2.89/1.57 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H )
% 2.89/1.57 = ( comp @ B @ D @ A @ F @ R4 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_42_rewriteL__comp__comp,axiom,
% 2.89/1.57 ! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L: A > B,H: D > A] :
% 2.89/1.57 ( ( ( comp @ C @ B @ A @ F @ G )
% 2.89/1.57 = L )
% 2.89/1.57 => ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H ) )
% 2.89/1.57 = ( comp @ A @ B @ D @ L @ H ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_43_fun_Omap__comp,axiom,
% 2.89/1.57 ! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
% 2.89/1.57 ( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
% 2.89/1.57 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_44_comp__apply__eq,axiom,
% 2.89/1.57 ! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H: D > A,K: C > D] :
% 2.89/1.57 ( ( ( F @ ( G @ X ) )
% 2.89/1.57 = ( H @ ( K @ X ) ) )
% 2.89/1.57 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 2.89/1.57 = ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_45_mem__Collect__eq,axiom,
% 2.89/1.57 ! [A: $tType,A4: A,P: A > $o] :
% 2.89/1.57 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 2.89/1.57 = ( P @ A4 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_46_Collect__mem__eq,axiom,
% 2.89/1.57 ! [A: $tType,A2: set @ A] :
% 2.89/1.57 ( ( collect @ A
% 2.89/1.57 @ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
% 2.89/1.57 = A2 ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_47_Collect__cong,axiom,
% 2.89/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.89/1.57 ( ! [X2: A] :
% 2.89/1.57 ( ( P @ X2 )
% 2.89/1.57 = ( Q @ X2 ) )
% 2.89/1.57 => ( ( collect @ A @ P )
% 2.89/1.57 = ( collect @ A @ Q ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_48_ext,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.89/1.57 ( ! [X2: A] :
% 2.89/1.57 ( ( F @ X2 )
% 2.89/1.57 = ( G @ X2 ) )
% 2.89/1.57 => ( F = G ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_49_comp__eq__dest__lhs,axiom,
% 2.89/1.57 ! [C: $tType,B: $tType,A: $tType,A4: C > B,B4: A > C,C3: A > B,V: A] :
% 2.89/1.57 ( ( ( comp @ C @ B @ A @ A4 @ B4 )
% 2.89/1.57 = C3 )
% 2.89/1.57 => ( ( A4 @ ( B4 @ V ) )
% 2.89/1.57 = ( C3 @ V ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_50_comp__eq__elim,axiom,
% 2.89/1.57 ! [C: $tType,B: $tType,D: $tType,A: $tType,A4: C > B,B4: A > C,C3: D > B,D2: A > D] :
% 2.89/1.57 ( ( ( comp @ C @ B @ A @ A4 @ B4 )
% 2.89/1.57 = ( comp @ D @ B @ A @ C3 @ D2 ) )
% 2.89/1.57 => ! [V2: A] :
% 2.89/1.57 ( ( A4 @ ( B4 @ V2 ) )
% 2.89/1.57 = ( C3 @ ( D2 @ V2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_51_comp__eq__dest,axiom,
% 2.89/1.57 ! [C: $tType,B: $tType,D: $tType,A: $tType,A4: C > B,B4: A > C,C3: D > B,D2: A > D,V: A] :
% 2.89/1.57 ( ( ( comp @ C @ B @ A @ A4 @ B4 )
% 2.89/1.57 = ( comp @ D @ B @ A @ C3 @ D2 ) )
% 2.89/1.57 => ( ( A4 @ ( B4 @ V ) )
% 2.89/1.57 = ( C3 @ ( D2 @ V ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_52_comp__assoc,axiom,
% 2.89/1.57 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
% 2.89/1.57 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
% 2.89/1.57 = ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_53_comp__def,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,B: $tType] :
% 2.89/1.57 ( ( comp @ B @ C @ A )
% 2.89/1.57 = ( ^ [F2: B > C,G2: A > B,X3: A] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_54_Let__transfer,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( bNF_rel_fun @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ A2 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ C @ D @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) @ B2 )
% 2.89/1.57 @ ^ [S3: A,F2: A > C] : ( F2 @ S3 )
% 2.89/1.57 @ ^ [S3: B,F2: B > D] : ( F2 @ S3 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_55_type__copy__map__cong0,axiom,
% 2.89/1.57 ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X: C,N: D > A,H: C > D,F: A > E] :
% 2.89/1.57 ( ( ( M @ ( G @ X ) )
% 2.89/1.57 = ( N @ ( H @ X ) ) )
% 2.89/1.57 => ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M ) @ G @ X )
% 2.89/1.57 = ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N ) @ H @ X ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_56_function__factors__right,axiom,
% 2.89/1.57 ! [C: $tType,B: $tType,A: $tType,G: B > C,F: A > C] :
% 2.89/1.57 ( ( ! [X3: A] :
% 2.89/1.57 ? [Y3: B] :
% 2.89/1.57 ( ( G @ Y3 )
% 2.89/1.57 = ( F @ X3 ) ) )
% 2.89/1.57 = ( ? [H3: A > B] :
% 2.89/1.57 ( F
% 2.89/1.57 = ( comp @ B @ C @ A @ G @ H3 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_57_function__factors__left,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,B: $tType,G: A > B,F: A > C] :
% 2.89/1.57 ( ( ! [X3: A,Y3: A] :
% 2.89/1.57 ( ( ( G @ X3 )
% 2.89/1.57 = ( G @ Y3 ) )
% 2.89/1.57 => ( ( F @ X3 )
% 2.89/1.57 = ( F @ Y3 ) ) ) )
% 2.89/1.57 = ( ? [H3: B > C] :
% 2.89/1.57 ( F
% 2.89/1.57 = ( comp @ B @ C @ A @ H3 @ G ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_58_comp__cong,axiom,
% 2.89/1.57 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X: C,F4: D > A,G4: E > D,X5: E] :
% 2.89/1.57 ( ( ( F @ ( G @ X ) )
% 2.89/1.57 = ( F4 @ ( G4 @ X5 ) ) )
% 2.89/1.57 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 2.89/1.57 = ( comp @ D @ A @ E @ F4 @ G4 @ X5 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_59_if__rsp,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 2.89/1.57 => ( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( A > A > A )
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 2.89/1.57 @ ( bNF_rel_fun @ A @ A @ ( A > A ) @ ( A > A ) @ R @ ( bNF_rel_fun @ A @ A @ A @ A @ R @ R ) )
% 2.89/1.57 @ ( if @ A )
% 2.89/1.57 @ ( if @ A ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_60_rel__funD2,axiom,
% 2.89/1.57 ! [B: $tType,C: $tType,A: $tType,A2: A > A > $o,B2: B > C > $o,F: A > B,G: A > C,X: A] :
% 2.89/1.57 ( ( bNF_rel_fun @ A @ A @ B @ C @ A2 @ B2 @ F @ G )
% 2.89/1.57 => ( ( A2 @ X @ X )
% 2.89/1.57 => ( B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_61_apply__rsp_H,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o,F: A > C,G: B > D,X: A,Y: B] :
% 2.89/1.57 ( ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 @ F @ G )
% 2.89/1.57 => ( ( R1 @ X @ Y )
% 2.89/1.57 => ( R2 @ ( F @ X ) @ ( G @ Y ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_62_rel__funE,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F: A > C,G: B > D,X: A,Y: B] :
% 2.89/1.57 ( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G )
% 2.89/1.57 => ( ( A2 @ X @ Y )
% 2.89/1.57 => ( B2 @ ( F @ X ) @ ( G @ Y ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_63_quot__rel__rsp,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 2.89/1.57 => ( bNF_rel_fun @ A @ A @ ( A > $o ) @ ( A > $o ) @ R
% 2.89/1.57 @ ( bNF_rel_fun @ A @ A @ $o @ $o @ R
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 2.89/1.57 @ R
% 2.89/1.57 @ R ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_64_Quotient3__rep__reflp,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A4: B] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 2.89/1.57 => ( R @ ( Rep @ A4 ) @ ( Rep @ A4 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_65_Quotient3__rep__abs,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 2.89/1.57 => ( ( R @ R4 @ R4 )
% 2.89/1.57 => ( R @ ( Rep @ ( Abs @ R4 ) ) @ R4 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_66_Quotient3__rel__rep,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A4: B,B4: B] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 2.89/1.57 => ( ( R @ ( Rep @ A4 ) @ ( Rep @ B4 ) )
% 2.89/1.57 = ( A4 = B4 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_67_Quotient3__rel__abs,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S: A] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 2.89/1.57 => ( ( R @ R4 @ S )
% 2.89/1.57 => ( ( Abs @ R4 )
% 2.89/1.57 = ( Abs @ S ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_68_Quotient3__abs__rep,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A4: B] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 2.89/1.57 => ( ( Abs @ ( Rep @ A4 ) )
% 2.89/1.57 = A4 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_69_rep__abs__rsp__left,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 2.89/1.57 => ( ( R @ X1 @ X22 )
% 2.89/1.57 => ( R @ ( Rep @ ( Abs @ X1 ) ) @ X22 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_70_Quotient3__refl2,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S: A] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 2.89/1.57 => ( ( R @ R4 @ S )
% 2.89/1.57 => ( R @ S @ S ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_71_Quotient3__refl1,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S: A] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 2.89/1.57 => ( ( R @ R4 @ S )
% 2.89/1.57 => ( R @ R4 @ R4 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_72_Quotient3__rel,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S: A] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 2.89/1.57 => ( ( ( R @ R4 @ R4 )
% 2.89/1.57 & ( R @ S @ S )
% 2.89/1.57 & ( ( Abs @ R4 )
% 2.89/1.57 = ( Abs @ S ) ) )
% 2.89/1.57 = ( R @ R4 @ S ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_73_Quotient3__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( quotient3 @ A @ B )
% 2.89/1.57 = ( ^ [R5: A > A > $o,Abs2: A > B,Rep2: B > A] :
% 2.89/1.57 ( ! [A5: B] :
% 2.89/1.57 ( ( Abs2 @ ( Rep2 @ A5 ) )
% 2.89/1.57 = A5 )
% 2.89/1.57 & ! [A5: B] : ( R5 @ ( Rep2 @ A5 ) @ ( Rep2 @ A5 ) )
% 2.89/1.57 & ! [R6: A,S3: A] :
% 2.89/1.57 ( ( R5 @ R6 @ S3 )
% 2.89/1.57 = ( ( R5 @ R6 @ R6 )
% 2.89/1.57 & ( R5 @ S3 @ S3 )
% 2.89/1.57 & ( ( Abs2 @ R6 )
% 2.89/1.57 = ( Abs2 @ S3 ) ) ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_74_rep__abs__rsp,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 2.89/1.57 => ( ( R @ X1 @ X22 )
% 2.89/1.57 => ( R @ X1 @ ( Rep @ ( Abs @ X22 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_75_equals__rsp,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,Xa: A,Xb: A,Ya: A,Yb: A] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 2.89/1.57 => ( ( R @ Xa @ Xb )
% 2.89/1.57 => ( ( R @ Ya @ Yb )
% 2.89/1.57 => ( ( R @ Xa @ Ya )
% 2.89/1.57 = ( R @ Xb @ Yb ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_76_Quotient3I,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,Abs: B > A,Rep: A > B,R: B > B > $o] :
% 2.89/1.57 ( ! [A3: A] :
% 2.89/1.57 ( ( Abs @ ( Rep @ A3 ) )
% 2.89/1.57 = A3 )
% 2.89/1.57 => ( ! [A3: A] : ( R @ ( Rep @ A3 ) @ ( Rep @ A3 ) )
% 2.89/1.57 => ( ! [R7: B,S4: B] :
% 2.89/1.57 ( ( R @ R7 @ S4 )
% 2.89/1.57 = ( ( R @ R7 @ R7 )
% 2.89/1.57 & ( R @ S4 @ S4 )
% 2.89/1.57 & ( ( Abs @ R7 )
% 2.89/1.57 = ( Abs @ S4 ) ) ) )
% 2.89/1.57 => ( quotient3 @ B @ A @ R @ Abs @ Rep ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_77_cond__prs,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,R: A > A > $o,Absf: A > B,Repf: B > A,A4: $o,B4: B,C3: B] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R @ Absf @ Repf )
% 2.89/1.57 => ( ( A4
% 2.89/1.57 => ( ( Absf @ ( if @ A @ A4 @ ( Repf @ B4 ) @ ( Repf @ C3 ) ) )
% 2.89/1.57 = B4 ) )
% 2.89/1.57 & ( ~ A4
% 2.89/1.57 => ( ( Absf @ ( if @ A @ A4 @ ( Repf @ B4 ) @ ( Repf @ C3 ) ) )
% 2.89/1.57 = C3 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_78_apply__rspQ3,axiom,
% 2.89/1.57 ! [B: $tType,C: $tType,A: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R2: C > C > $o,F: A > C,G: A > C,X: A,Y: A] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
% 2.89/1.57 => ( ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 @ F @ G )
% 2.89/1.57 => ( ( R1 @ X @ Y )
% 2.89/1.57 => ( R2 @ ( F @ X ) @ ( G @ Y ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_79_apply__rspQ3_H_H,axiom,
% 2.89/1.57 ! [C: $tType,A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,S2: C > C > $o,F: A > C,X: B] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R @ Abs @ Rep )
% 2.89/1.57 => ( ( bNF_rel_fun @ A @ A @ C @ C @ R @ S2 @ F @ F )
% 2.89/1.57 => ( S2 @ ( F @ ( Rep @ X ) ) @ ( F @ ( Rep @ X ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_80_bex1__rel__rsp,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > A > $o,Absf: A > B,Repf: B > A] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R @ Absf @ Repf )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > $o ) @ ( A > $o ) @ $o @ $o
% 2.89/1.57 @ ( bNF_rel_fun @ A @ A @ $o @ $o @ R
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 2.89/1.57 @ ( bex1_rel @ A @ R )
% 2.89/1.57 @ ( bex1_rel @ A @ R ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_81_map__fun__parametric,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,E: $tType,F3: $tType,H2: $tType,G3: $tType,D: $tType,C: $tType,A2: A > C > $o,B2: B > D > $o,C2: E > G3 > $o,D3: F3 > H2 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > F3 ) > ( B > E ) > A > F3 ) @ ( ( G3 > H2 ) > ( D > G3 ) > C > H2 ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ ( E > F3 ) @ ( G3 > H2 ) @ ( ( B > E ) > A > F3 ) @ ( ( D > G3 ) > C > H2 ) @ ( bNF_rel_fun @ E @ G3 @ F3 @ H2 @ C2 @ D3 ) @ ( bNF_rel_fun @ ( B > E ) @ ( D > G3 ) @ ( A > F3 ) @ ( C > H2 ) @ ( bNF_rel_fun @ B @ D @ E @ G3 @ B2 @ C2 ) @ ( bNF_rel_fun @ A @ C @ F3 @ H2 @ A2 @ D3 ) ) ) @ ( map_fun @ A @ B @ E @ F3 ) @ ( map_fun @ C @ D @ G3 @ H2 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_82_folding__idem__axioms__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( finite1921348288axioms @ A @ B )
% 2.89/1.57 = ( ^ [F2: A > B > B] :
% 2.89/1.57 ! [X3: A] :
% 2.89/1.57 ( ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
% 2.89/1.57 = ( F2 @ X3 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_83_folding__idem__axioms_Ointro,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B] :
% 2.89/1.57 ( ! [X2: A] :
% 2.89/1.57 ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
% 2.89/1.57 = ( F @ X2 ) )
% 2.89/1.57 => ( finite1921348288axioms @ A @ B @ F ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_84_fun__quotient3,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs22: C > D,Rep22: D > C] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
% 2.89/1.57 => ( ( quotient3 @ C @ D @ R2 @ Abs22 @ Rep22 )
% 2.89/1.57 => ( quotient3 @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 ) @ ( map_fun @ B @ A @ C @ D @ Rep12 @ Abs22 ) @ ( map_fun @ A @ B @ D @ C @ Abs12 @ Rep22 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_85_fun__ord__parametric,axiom,
% 2.89/1.57 ! [C: $tType,D: $tType,A: $tType,B: $tType,F3: $tType,E: $tType,C2: A > B > $o,A2: C > E > $o,B2: D > F3 > $o] :
% 2.89/1.57 ( ( bi_total @ A @ B @ C2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( C > D > $o ) @ ( E > F3 > $o ) @ ( ( A > C ) > ( A > D ) > $o ) @ ( ( B > E ) > ( B > F3 ) > $o )
% 2.89/1.57 @ ( bNF_rel_fun @ C @ E @ ( D > $o ) @ ( F3 > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ D @ F3 @ $o @ $o @ B2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ( bNF_rel_fun @ ( A > C ) @ ( B > E ) @ ( ( A > D ) > $o ) @ ( ( B > F3 ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ E @ C2 @ A2 )
% 2.89/1.57 @ ( bNF_rel_fun @ ( A > D ) @ ( B > F3 ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ D @ F3 @ C2 @ B2 )
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ( partial_fun_ord @ C @ D @ A )
% 2.89/1.57 @ ( partial_fun_ord @ E @ F3 @ B ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_86_comp__fun__commute_Ofold__mset__union,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B,S: B,M: multiset @ A,N: multiset @ A] :
% 2.89/1.57 ( ( finite100568337ommute @ A @ B @ F )
% 2.89/1.57 => ( ( fold_mset @ A @ B @ F @ S @ ( plus_plus @ ( multiset @ A ) @ M @ N ) )
% 2.89/1.57 = ( fold_mset @ A @ B @ F @ ( fold_mset @ A @ B @ F @ S @ M ) @ N ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_87_map__fun__apply,axiom,
% 2.89/1.57 ! [D: $tType,A: $tType,C: $tType,B: $tType] :
% 2.89/1.57 ( ( map_fun @ B @ C @ D @ A )
% 2.89/1.57 = ( ^ [F2: B > C,G2: D > A,H3: C > D,X3: B] : ( G2 @ ( H3 @ ( F2 @ X3 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_88_o__prs_I1_J,axiom,
% 2.89/1.57 ! [C: $tType,E: $tType,A: $tType,B: $tType,F3: $tType,D: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R2: C > C > $o,Abs23: C > D,Rep23: D > C,R3: E > E > $o,Abs3: E > F3,Rep3: F3 > E] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
% 2.89/1.57 => ( ( quotient3 @ C @ D @ R2 @ Abs23 @ Rep23 )
% 2.89/1.57 => ( ( quotient3 @ E @ F3 @ R3 @ Abs3 @ Rep3 )
% 2.89/1.57 => ( ( map_fun @ ( D > F3 ) @ ( C > E ) @ ( ( A > C ) > A > E ) @ ( ( B > D ) > B > F3 ) @ ( map_fun @ C @ D @ F3 @ E @ Abs23 @ Rep3 ) @ ( map_fun @ ( B > D ) @ ( A > C ) @ ( A > E ) @ ( B > F3 ) @ ( map_fun @ A @ B @ D @ C @ Abs1 @ Rep23 ) @ ( map_fun @ B @ A @ E @ F3 @ Rep1 @ Abs3 ) ) @ ( comp @ C @ E @ A ) )
% 2.89/1.57 = ( comp @ D @ F3 @ B ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_89_let__prs,axiom,
% 2.89/1.57 ! [C: $tType,A: $tType,B: $tType,D: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R2: C > C > $o,Abs23: C > D,Rep23: D > C] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
% 2.89/1.57 => ( ( quotient3 @ C @ D @ R2 @ Abs23 @ Rep23 )
% 2.89/1.57 => ( ( map_fun @ D @ C @ ( ( C > A ) > A ) @ ( ( D > B ) > B ) @ Rep23 @ ( map_fun @ ( D > B ) @ ( C > A ) @ A @ B @ ( map_fun @ C @ D @ B @ A @ Abs23 @ Rep1 ) @ Abs1 )
% 2.89/1.57 @ ^ [S3: C,F2: C > A] : ( F2 @ S3 ) )
% 2.89/1.57 = ( ^ [S3: D,F2: D > B] : ( F2 @ S3 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_90_bi__total__eq,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( bi_total @ A @ A
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_91_bi__total__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( bi_total @ A @ B )
% 2.89/1.57 = ( ^ [R5: A > B > $o] :
% 2.89/1.57 ( ! [X3: A] :
% 2.89/1.57 ? [X6: B] : ( R5 @ X3 @ X6 )
% 2.89/1.57 & ! [Y3: B] :
% 2.89/1.57 ? [X3: A] : ( R5 @ X3 @ Y3 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_92_union__assoc,axiom,
% 2.89/1.57 ! [A: $tType,M: multiset @ A,N: multiset @ A,K2: multiset @ A] :
% 2.89/1.57 ( ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M @ N ) @ K2 )
% 2.89/1.57 = ( plus_plus @ ( multiset @ A ) @ M @ ( plus_plus @ ( multiset @ A ) @ N @ K2 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_93_union__lcomm,axiom,
% 2.89/1.57 ! [A: $tType,M: multiset @ A,N: multiset @ A,K2: multiset @ A] :
% 2.89/1.57 ( ( plus_plus @ ( multiset @ A ) @ M @ ( plus_plus @ ( multiset @ A ) @ N @ K2 ) )
% 2.89/1.57 = ( plus_plus @ ( multiset @ A ) @ N @ ( plus_plus @ ( multiset @ A ) @ M @ K2 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_94_union__commute,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( plus_plus @ ( multiset @ A ) )
% 2.89/1.57 = ( ^ [M2: multiset @ A,N2: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ N2 @ M2 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_95_union__left__cancel,axiom,
% 2.89/1.57 ! [A: $tType,K2: multiset @ A,M: multiset @ A,N: multiset @ A] :
% 2.89/1.57 ( ( ( plus_plus @ ( multiset @ A ) @ K2 @ M )
% 2.89/1.57 = ( plus_plus @ ( multiset @ A ) @ K2 @ N ) )
% 2.89/1.57 = ( M = N ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_96_union__right__cancel,axiom,
% 2.89/1.57 ! [A: $tType,M: multiset @ A,K2: multiset @ A,N: multiset @ A] :
% 2.89/1.57 ( ( ( plus_plus @ ( multiset @ A ) @ M @ K2 )
% 2.89/1.57 = ( plus_plus @ ( multiset @ A ) @ N @ K2 ) )
% 2.89/1.57 = ( M = N ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_97_multi__union__self__other__eq,axiom,
% 2.89/1.57 ! [A: $tType,A2: multiset @ A,X4: multiset @ A,Y5: multiset @ A] :
% 2.89/1.57 ( ( ( plus_plus @ ( multiset @ A ) @ A2 @ X4 )
% 2.89/1.57 = ( plus_plus @ ( multiset @ A ) @ A2 @ Y5 ) )
% 2.89/1.57 => ( X4 = Y5 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_98_comp__fun__commute__plus__mset,axiom,
% 2.89/1.57 ! [A: $tType] : ( finite100568337ommute @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_99_bex1__rel__aux,axiom,
% 2.89/1.57 ! [A: $tType,R: A > A > $o,X: A > $o,Y: A > $o] :
% 2.89/1.57 ( ! [Xa2: A,Ya2: A] :
% 2.89/1.57 ( ( R @ Xa2 @ Ya2 )
% 2.89/1.57 => ( ( X @ Xa2 )
% 2.89/1.57 = ( Y @ Ya2 ) ) )
% 2.89/1.57 => ( ( bex1_rel @ A @ R @ X )
% 2.89/1.57 => ( bex1_rel @ A @ R @ Y ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_100_bex1__rel__aux2,axiom,
% 2.89/1.57 ! [A: $tType,R: A > A > $o,X: A > $o,Y: A > $o] :
% 2.89/1.57 ( ! [Xa2: A,Ya2: A] :
% 2.89/1.57 ( ( R @ Xa2 @ Ya2 )
% 2.89/1.57 => ( ( X @ Xa2 )
% 2.89/1.57 = ( Y @ Ya2 ) ) )
% 2.89/1.57 => ( ( bex1_rel @ A @ R @ Y )
% 2.89/1.57 => ( bex1_rel @ A @ R @ X ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_101_map__fun_Ocompositionality,axiom,
% 2.89/1.57 ! [D: $tType,F3: $tType,C: $tType,E: $tType,B: $tType,A: $tType,F: E > C,G: D > F3,H: C > A,I: B > D,Fun: A > B] :
% 2.89/1.57 ( ( map_fun @ E @ C @ D @ F3 @ F @ G @ ( map_fun @ C @ A @ B @ D @ H @ I @ Fun ) )
% 2.89/1.57 = ( map_fun @ E @ A @ B @ F3 @ ( comp @ C @ A @ E @ H @ F ) @ ( comp @ D @ F3 @ B @ G @ I ) @ Fun ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_102_map__fun_Ocomp,axiom,
% 2.89/1.57 ! [E: $tType,C: $tType,A: $tType,F3: $tType,D: $tType,B: $tType,F: E > C,G: D > F3,H: C > A,I: B > D] :
% 2.89/1.57 ( ( comp @ ( C > D ) @ ( E > F3 ) @ ( A > B ) @ ( map_fun @ E @ C @ D @ F3 @ F @ G ) @ ( map_fun @ C @ A @ B @ D @ H @ I ) )
% 2.89/1.57 = ( map_fun @ E @ A @ B @ F3 @ ( comp @ C @ A @ E @ H @ F ) @ ( comp @ D @ F3 @ B @ G @ I ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_103_map__fun__def,axiom,
% 2.89/1.57 ! [B: $tType,D: $tType,A: $tType,C: $tType] :
% 2.89/1.57 ( ( map_fun @ C @ A @ B @ D )
% 2.89/1.57 = ( ^ [F2: C > A,G2: B > D,H3: A > B] : ( comp @ A @ D @ C @ ( comp @ B @ D @ A @ G2 @ H3 ) @ F2 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_104_add__right__cancel,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( cancel_semigroup_add @ A )
% 2.89/1.57 => ! [B4: A,A4: A,C3: A] :
% 2.89/1.57 ( ( ( plus_plus @ A @ B4 @ A4 )
% 2.89/1.57 = ( plus_plus @ A @ C3 @ A4 ) )
% 2.89/1.57 = ( B4 = C3 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_105_add__left__cancel,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( cancel_semigroup_add @ A )
% 2.89/1.57 => ! [A4: A,B4: A,C3: A] :
% 2.89/1.57 ( ( ( plus_plus @ A @ A4 @ B4 )
% 2.89/1.57 = ( plus_plus @ A @ A4 @ C3 ) )
% 2.89/1.57 = ( B4 = C3 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_106_set__plus__intro,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( plus @ A )
% 2.89/1.57 => ! [A4: A,C2: set @ A,B4: A,D3: set @ A] :
% 2.89/1.57 ( ( member @ A @ A4 @ C2 )
% 2.89/1.57 => ( ( member @ A @ B4 @ D3 )
% 2.89/1.57 => ( member @ A @ ( plus_plus @ A @ A4 @ B4 ) @ ( plus_plus @ ( set @ A ) @ C2 @ D3 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_107_transfer__forall__transfer_I1_J,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 2.89/1.57 ( ( bi_total @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 2.89/1.57 @ ( transfer_forall @ A )
% 2.89/1.57 @ ( transfer_forall @ B ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_108_monotone__parametric,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( ( bi_total @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( ( C > C > $o ) > ( A > C ) > $o ) @ ( ( D > D > $o ) > ( B > D ) > $o )
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ( bNF_rel_fun @ ( C > C > $o ) @ ( D > D > $o ) @ ( ( A > C ) > $o ) @ ( ( B > D ) > $o )
% 2.89/1.57 @ ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B2
% 2.89/1.57 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 )
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ( comple1396247847notone @ A @ C )
% 2.89/1.57 @ ( comple1396247847notone @ B @ D ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_109_relcompp__transfer,axiom,
% 2.89/1.57 ! [C: $tType,A: $tType,E: $tType,F3: $tType,B: $tType,D: $tType,B2: A > B > $o,A2: C > D > $o,C2: E > F3 > $o] :
% 2.89/1.57 ( ( bi_total @ A @ B @ B2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( ( A > E > $o ) > C > E > $o ) @ ( ( B > F3 > $o ) > D > F3 > $o )
% 2.89/1.57 @ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ B2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ( bNF_rel_fun @ ( A > E > $o ) @ ( B > F3 > $o ) @ ( C > E > $o ) @ ( D > F3 > $o )
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ ( E > $o ) @ ( F3 > $o ) @ B2
% 2.89/1.57 @ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ( bNF_rel_fun @ C @ D @ ( E > $o ) @ ( F3 > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) ) )
% 2.89/1.57 @ ( relcompp @ C @ A @ E )
% 2.89/1.57 @ ( relcompp @ D @ B @ F3 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_110_fun_Orel__compp,axiom,
% 2.89/1.57 ! [A: $tType,D: $tType,C: $tType,B: $tType,R: A > B > $o,S2: B > C > $o] :
% 2.89/1.57 ( ( bNF_rel_fun @ D @ D @ A @ C
% 2.89/1.57 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 2.89/1.57 @ ( relcompp @ A @ B @ C @ R @ S2 ) )
% 2.89/1.57 = ( relcompp @ ( D > A ) @ ( D > B ) @ ( D > C )
% 2.89/1.57 @ ( bNF_rel_fun @ D @ D @ A @ B
% 2.89/1.57 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 2.89/1.57 @ R )
% 2.89/1.57 @ ( bNF_rel_fun @ D @ D @ B @ C
% 2.89/1.57 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 2.89/1.57 @ S2 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_111_eq__comp__r,axiom,
% 2.89/1.57 ! [A: $tType,R: A > A > $o] :
% 2.89/1.57 ( ( relcompp @ A @ A @ A
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 2.89/1.57 @ ( relcompp @ A @ A @ A @ R
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) )
% 2.89/1.57 = R ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_112_bi__total__OO,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,B: $tType,A2: A > B > $o,B2: B > C > $o] :
% 2.89/1.57 ( ( bi_total @ A @ B @ A2 )
% 2.89/1.57 => ( ( bi_total @ B @ C @ B2 )
% 2.89/1.57 => ( bi_total @ A @ C @ ( relcompp @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_113_OOO__quotient3,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,C: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R2: B > B > $o,Abs23: B > C,Rep23: C > B,R23: A > A > $o] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
% 2.89/1.57 => ( ( quotient3 @ B @ C @ R2 @ Abs23 @ Rep23 )
% 2.89/1.57 => ( ! [X2: A,Y2: A] :
% 2.89/1.57 ( ( R23 @ X2 @ Y2 )
% 2.89/1.57 => ( ( R1 @ X2 @ X2 )
% 2.89/1.57 => ( ( R1 @ Y2 @ Y2 )
% 2.89/1.57 => ( R2 @ ( Abs1 @ X2 ) @ ( Abs1 @ Y2 ) ) ) ) )
% 2.89/1.57 => ( ! [X2: B,Y2: B] :
% 2.89/1.57 ( ( R2 @ X2 @ Y2 )
% 2.89/1.57 => ( R23 @ ( Rep1 @ X2 ) @ ( Rep1 @ Y2 ) ) )
% 2.89/1.57 => ( quotient3 @ A @ C @ ( relcompp @ A @ A @ A @ R1 @ ( relcompp @ A @ A @ A @ R23 @ R1 ) ) @ ( comp @ B @ C @ A @ Abs23 @ Abs1 ) @ ( comp @ B @ A @ C @ Rep1 @ Rep23 ) ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_114_OOO__eq__quotient3,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,C: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,Abs23: B > C,Rep23: C > B] :
% 2.89/1.57 ( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
% 2.89/1.57 => ( ( quotient3 @ B @ C
% 2.89/1.57 @ ^ [Y4: B,Z2: B] : ( Y4 = Z2 )
% 2.89/1.57 @ Abs23
% 2.89/1.57 @ Rep23 )
% 2.89/1.57 => ( quotient3 @ A @ C
% 2.89/1.57 @ ( relcompp @ A @ A @ A @ R1
% 2.89/1.57 @ ( relcompp @ A @ A @ A
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 2.89/1.57 @ R1 ) )
% 2.89/1.57 @ ( comp @ B @ C @ A @ Abs23 @ Abs1 )
% 2.89/1.57 @ ( comp @ B @ A @ C @ Rep1 @ Rep23 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_115_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( ab_semigroup_add @ A )
% 2.89/1.57 => ! [A4: A,B4: A,C3: A] :
% 2.89/1.57 ( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B4 ) @ C3 )
% 2.89/1.57 = ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B4 @ C3 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_116_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( ordere779506340up_add @ A )
% 2.89/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.89/1.57 ( ( ( I = J )
% 2.89/1.57 & ( K = L ) )
% 2.89/1.57 => ( ( plus_plus @ A @ I @ K )
% 2.89/1.57 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_117_group__cancel_Oadd1,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( comm_monoid_add @ A )
% 2.89/1.57 => ! [A2: A,K: A,A4: A,B4: A] :
% 2.89/1.57 ( ( A2
% 2.89/1.57 = ( plus_plus @ A @ K @ A4 ) )
% 2.89/1.57 => ( ( plus_plus @ A @ A2 @ B4 )
% 2.89/1.57 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A4 @ B4 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_118_group__cancel_Oadd2,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( comm_monoid_add @ A )
% 2.89/1.57 => ! [B2: A,K: A,B4: A,A4: A] :
% 2.89/1.57 ( ( B2
% 2.89/1.57 = ( plus_plus @ A @ K @ B4 ) )
% 2.89/1.57 => ( ( plus_plus @ A @ A4 @ B2 )
% 2.89/1.57 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A4 @ B4 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_119_set__plus__elim,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( plus @ A )
% 2.89/1.57 => ! [X: A,A2: set @ A,B2: set @ A] :
% 2.89/1.57 ( ( member @ A @ X @ ( plus_plus @ ( set @ A ) @ A2 @ B2 ) )
% 2.89/1.57 => ~ ! [A3: A,B3: A] :
% 2.89/1.57 ( ( X
% 2.89/1.57 = ( plus_plus @ A @ A3 @ B3 ) )
% 2.89/1.57 => ( ( member @ A @ A3 @ A2 )
% 2.89/1.57 => ~ ( member @ A @ B3 @ B2 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_120_add_Oassoc,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( semigroup_add @ A )
% 2.89/1.57 => ! [A4: A,B4: A,C3: A] :
% 2.89/1.57 ( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B4 ) @ C3 )
% 2.89/1.57 = ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B4 @ C3 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_121_add_Oleft__cancel,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( group_add @ A )
% 2.89/1.57 => ! [A4: A,B4: A,C3: A] :
% 2.89/1.57 ( ( ( plus_plus @ A @ A4 @ B4 )
% 2.89/1.57 = ( plus_plus @ A @ A4 @ C3 ) )
% 2.89/1.57 = ( B4 = C3 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_122_add_Oright__cancel,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( group_add @ A )
% 2.89/1.57 => ! [B4: A,A4: A,C3: A] :
% 2.89/1.57 ( ( ( plus_plus @ A @ B4 @ A4 )
% 2.89/1.57 = ( plus_plus @ A @ C3 @ A4 ) )
% 2.89/1.57 = ( B4 = C3 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_123_add_Ocommute,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( ab_semigroup_add @ A )
% 2.89/1.57 => ( ( plus_plus @ A )
% 2.89/1.57 = ( ^ [A5: A,B5: A] : ( plus_plus @ A @ B5 @ A5 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_124_add_Oleft__commute,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( ab_semigroup_add @ A )
% 2.89/1.57 => ! [B4: A,A4: A,C3: A] :
% 2.89/1.57 ( ( plus_plus @ A @ B4 @ ( plus_plus @ A @ A4 @ C3 ) )
% 2.89/1.57 = ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B4 @ C3 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_125_add__left__imp__eq,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( cancel_semigroup_add @ A )
% 2.89/1.57 => ! [A4: A,B4: A,C3: A] :
% 2.89/1.57 ( ( ( plus_plus @ A @ A4 @ B4 )
% 2.89/1.57 = ( plus_plus @ A @ A4 @ C3 ) )
% 2.89/1.57 => ( B4 = C3 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_126_add__right__imp__eq,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( cancel_semigroup_add @ A )
% 2.89/1.57 => ! [B4: A,A4: A,C3: A] :
% 2.89/1.57 ( ( ( plus_plus @ A @ B4 @ A4 )
% 2.89/1.57 = ( plus_plus @ A @ C3 @ A4 ) )
% 2.89/1.57 => ( B4 = C3 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_127_transfer__forall__transfer_I4_J,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 2.89/1.57 ( ( bi_total @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 2.89/1.57 @ rev_implies
% 2.89/1.57 @ ( transfer_forall @ A )
% 2.89/1.57 @ ( transfer_forall @ B ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_128_transfer__forall__transfer_I5_J,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 2.89/1.57 ( ( bi_total @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ rev_implies ) @ rev_implies @ ( transfer_forall @ A ) @ ( transfer_forall @ B ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_129_Domainp__transfer,axiom,
% 2.89/1.57 ! [C: $tType,A: $tType,B: $tType,D: $tType,B2: A > B > $o,A2: C > D > $o] :
% 2.89/1.57 ( ( bi_total @ A @ B @ B2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( C > $o ) @ ( D > $o )
% 2.89/1.57 @ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ B2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ A2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 2.89/1.57 @ ( domainp @ C @ A )
% 2.89/1.57 @ ( domainp @ D @ B ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_130_folding__idem_Oaxioms_I2_J,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B] :
% 2.89/1.57 ( ( finite_folding_idem @ A @ B @ F )
% 2.89/1.57 => ( finite1921348288axioms @ A @ B @ F ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_131_OO__eq,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > B > $o] :
% 2.89/1.57 ( ( relcompp @ A @ B @ B @ R
% 2.89/1.57 @ ^ [Y4: B,Z2: B] : ( Y4 = Z2 ) )
% 2.89/1.57 = R ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_132_Domainp__iff,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( domainp @ A @ B )
% 2.89/1.57 = ( ^ [T2: A > B > $o,X3: A] :
% 2.89/1.57 ? [X6: B] : ( T2 @ X3 @ X6 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_133_Domainp__refl,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( domainp @ A @ B )
% 2.89/1.57 = ( domainp @ A @ B ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_134_rev__implies__def,axiom,
% 2.89/1.57 ( rev_implies
% 2.89/1.57 = ( ^ [X3: $o,Y3: $o] :
% 2.89/1.57 ( Y3
% 2.89/1.57 => X3 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_135_folding__idem_Ocomp__fun__idem,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B,X: A] :
% 2.89/1.57 ( ( finite_folding_idem @ A @ B @ F )
% 2.89/1.57 => ( ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ X ) )
% 2.89/1.57 = ( F @ X ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_136_relcompp_OrelcompI,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,C: $tType,R4: A > B > $o,A4: A,B4: B,S: B > C > $o,C3: C] :
% 2.89/1.57 ( ( R4 @ A4 @ B4 )
% 2.89/1.57 => ( ( S @ B4 @ C3 )
% 2.89/1.57 => ( relcompp @ A @ B @ C @ R4 @ S @ A4 @ C3 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_137_relcompp_Oinducts,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,C: $tType,R4: A > B > $o,S: B > C > $o,X1: A,X22: C,P: A > C > $o] :
% 2.89/1.57 ( ( relcompp @ A @ B @ C @ R4 @ S @ X1 @ X22 )
% 2.89/1.57 => ( ! [A3: A,B3: B,C4: C] :
% 2.89/1.57 ( ( R4 @ A3 @ B3 )
% 2.89/1.57 => ( ( S @ B3 @ C4 )
% 2.89/1.57 => ( P @ A3 @ C4 ) ) )
% 2.89/1.57 => ( P @ X1 @ X22 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_138_relcompp__assoc,axiom,
% 2.89/1.57 ! [A: $tType,D: $tType,B: $tType,C: $tType,R4: A > D > $o,S: D > C > $o,T3: C > B > $o] :
% 2.89/1.57 ( ( relcompp @ A @ C @ B @ ( relcompp @ A @ D @ C @ R4 @ S ) @ T3 )
% 2.89/1.57 = ( relcompp @ A @ D @ B @ R4 @ ( relcompp @ D @ C @ B @ S @ T3 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_139_relcompp__apply,axiom,
% 2.89/1.57 ! [C: $tType,B: $tType,A: $tType] :
% 2.89/1.57 ( ( relcompp @ A @ B @ C )
% 2.89/1.57 = ( ^ [R5: A > B > $o,S5: B > C > $o,A5: A,C5: C] :
% 2.89/1.57 ? [B5: B] :
% 2.89/1.57 ( ( R5 @ A5 @ B5 )
% 2.89/1.57 & ( S5 @ B5 @ C5 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_140_relcompp_Osimps,axiom,
% 2.89/1.57 ! [C: $tType,B: $tType,A: $tType] :
% 2.89/1.57 ( ( relcompp @ A @ B @ C )
% 2.89/1.57 = ( ^ [R6: A > B > $o,S3: B > C > $o,A1: A,A22: C] :
% 2.89/1.57 ? [A5: A,B5: B,C5: C] :
% 2.89/1.57 ( ( A1 = A5 )
% 2.89/1.57 & ( A22 = C5 )
% 2.89/1.57 & ( R6 @ A5 @ B5 )
% 2.89/1.57 & ( S3 @ B5 @ C5 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_141_relcompp_Ocases,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,C: $tType,R4: A > B > $o,S: B > C > $o,A12: A,A23: C] :
% 2.89/1.57 ( ( relcompp @ A @ B @ C @ R4 @ S @ A12 @ A23 )
% 2.89/1.57 => ~ ! [B3: B] :
% 2.89/1.57 ( ( R4 @ A12 @ B3 )
% 2.89/1.57 => ~ ( S @ B3 @ A23 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_142_relcomppE,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,C: $tType,R4: A > B > $o,S: B > C > $o,A4: A,C3: C] :
% 2.89/1.57 ( ( relcompp @ A @ B @ C @ R4 @ S @ A4 @ C3 )
% 2.89/1.57 => ~ ! [B3: B] :
% 2.89/1.57 ( ( R4 @ A4 @ B3 )
% 2.89/1.57 => ~ ( S @ B3 @ C3 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_143_eq__OO,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > B > $o] :
% 2.89/1.57 ( ( relcompp @ A @ A @ B
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 2.89/1.57 @ R )
% 2.89/1.57 = R ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_144_folding__idem__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( finite_folding_idem @ A @ B )
% 2.89/1.57 = ( ^ [F2: A > B > B] :
% 2.89/1.57 ( ( finite_folding @ A @ B @ F2 )
% 2.89/1.57 & ( finite1921348288axioms @ A @ B @ F2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_145_folding__idem_Ointro,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B] :
% 2.89/1.57 ( ( finite_folding @ A @ B @ F )
% 2.89/1.57 => ( ( finite1921348288axioms @ A @ B @ F )
% 2.89/1.57 => ( finite_folding_idem @ A @ B @ F ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_146_reflp__transfer_I5_J,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 2.89/1.57 ( ( bi_total @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ rev_implies
% 2.89/1.57 @ ( reflp @ A )
% 2.89/1.57 @ ( reflp @ B ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_147_reflp__transfer_I4_J,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 2.89/1.57 ( ( bi_total @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ rev_implies ) ) @ rev_implies @ ( reflp @ A ) @ ( reflp @ B ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_148_fun_Orel__reflp,axiom,
% 2.89/1.57 ! [D: $tType,A: $tType,R: A > A > $o] :
% 2.89/1.57 ( ( reflp @ A @ R )
% 2.89/1.57 => ( reflp @ ( D > A )
% 2.89/1.57 @ ( bNF_rel_fun @ D @ D @ A @ A
% 2.89/1.57 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 2.89/1.57 @ R ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_149_DEADID_Orel__reflp,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( reflp @ A
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_150_folding__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( finite_folding @ A @ B )
% 2.89/1.57 = ( ^ [F2: A > B > B] :
% 2.89/1.57 ! [Y3: A,X3: A] :
% 2.89/1.57 ( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) )
% 2.89/1.57 = ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_151_folding_Ointro,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B] :
% 2.89/1.57 ( ! [Y2: A,X2: A] :
% 2.89/1.57 ( ( comp @ B @ B @ B @ ( F @ Y2 ) @ ( F @ X2 ) )
% 2.89/1.57 = ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.89/1.57 => ( finite_folding @ A @ B @ F ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_152_folding_Ocomp__fun__commute,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B,Y: A,X: A] :
% 2.89/1.57 ( ( finite_folding @ A @ B @ F )
% 2.89/1.57 => ( ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) )
% 2.89/1.57 = ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_153_folding__idem_Oaxioms_I1_J,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B > B] :
% 2.89/1.57 ( ( finite_folding_idem @ A @ B @ F )
% 2.89/1.57 => ( finite_folding @ A @ B @ F ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_154_reflp__transfer_I1_J,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 2.89/1.57 ( ( bi_total @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 2.89/1.57 @ ( reflp @ A )
% 2.89/1.57 @ ( reflp @ B ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_155_left__total__parametric,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( ( bi_total @ A @ B @ A2 )
% 2.89/1.57 => ( ( bi_total @ C @ D @ B2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 2.89/1.57 @ ( left_total @ A @ C )
% 2.89/1.57 @ ( left_total @ B @ D ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_156_reflp__transfer_I2_J,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 2.89/1.57 ( ( right_total @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ (=>) ) ) @ (=>) @ ( reflp @ A ) @ ( reflp @ B ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_157_reflp__transfer_I3_J,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 2.89/1.57 ( ( right_total @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ (=>)
% 2.89/1.57 @ ( reflp @ A )
% 2.89/1.57 @ ( reflp @ B ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_158_left__totalE,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,R: A > B > $o] :
% 2.89/1.57 ( ( left_total @ A @ B @ R )
% 2.89/1.57 => ! [X7: A] :
% 2.89/1.57 ? [X_1: B] : ( R @ X7 @ X_1 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_159_left__totalI,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > B > $o] :
% 2.89/1.57 ( ! [X2: A] :
% 2.89/1.57 ? [X_12: B] : ( R @ X2 @ X_12 )
% 2.89/1.57 => ( left_total @ A @ B @ R ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_160_right__totalE,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o,Y: B] :
% 2.89/1.57 ( ( right_total @ A @ B @ A2 )
% 2.89/1.57 => ~ ! [X2: A] :
% 2.89/1.57 ~ ( A2 @ X2 @ Y ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_161_right__totalI,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: B > A > $o] :
% 2.89/1.57 ( ! [Y2: A] :
% 2.89/1.57 ? [X7: B] : ( A2 @ X7 @ Y2 )
% 2.89/1.57 => ( right_total @ B @ A @ A2 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_162_left__total__eq,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( left_total @ A @ A
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_163_left__total__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( left_total @ A @ B )
% 2.89/1.57 = ( ^ [R5: A > B > $o] :
% 2.89/1.57 ! [X3: A] :
% 2.89/1.57 ? [X6: B] : ( R5 @ X3 @ X6 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_164_right__total__eq,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( right_total @ A @ A
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_165_right__total__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( right_total @ A @ B )
% 2.89/1.57 = ( ^ [R5: A > B > $o] :
% 2.89/1.57 ! [Y3: B] :
% 2.89/1.57 ? [X3: A] : ( R5 @ X3 @ Y3 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_166_bi__total__alt__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( bi_total @ A @ B )
% 2.89/1.57 = ( ^ [A6: A > B > $o] :
% 2.89/1.57 ( ( left_total @ A @ B @ A6 )
% 2.89/1.57 & ( right_total @ A @ B @ A6 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_167_bi__totalI,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > B > $o] :
% 2.89/1.57 ( ( left_total @ A @ B @ R )
% 2.89/1.57 => ( ( right_total @ A @ B @ R )
% 2.89/1.57 => ( bi_total @ A @ B @ R ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_168_left__total__OO,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,B: $tType,R: A > B > $o,S2: B > C > $o] :
% 2.89/1.57 ( ( left_total @ A @ B @ R )
% 2.89/1.57 => ( ( left_total @ B @ C @ S2 )
% 2.89/1.57 => ( left_total @ A @ C @ ( relcompp @ A @ B @ C @ R @ S2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_169_right__total__OO,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,B: $tType,A2: A > B > $o,B2: B > C > $o] :
% 2.89/1.57 ( ( right_total @ A @ B @ A2 )
% 2.89/1.57 => ( ( right_total @ B @ C @ B2 )
% 2.89/1.57 => ( right_total @ A @ C @ ( relcompp @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_170_pcr__Domainp__total,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,C: $tType,B2: A > B > $o,A2: C > A > $o,P: C > $o] :
% 2.89/1.57 ( ( left_total @ A @ B @ B2 )
% 2.89/1.57 => ( ( ( domainp @ C @ A @ A2 )
% 2.89/1.57 = P )
% 2.89/1.57 => ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
% 2.89/1.57 = P ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_171_pcr__Domainp__par__left__total,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,C: $tType,B2: A > B > $o,P: A > $o,A2: C > A > $o,P2: C > $o] :
% 2.89/1.57 ( ( ( domainp @ A @ B @ B2 )
% 2.89/1.57 = P )
% 2.89/1.57 => ( ( left_total @ C @ A @ A2 )
% 2.89/1.57 => ( ( bNF_rel_fun @ C @ A @ $o @ $o @ A2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 2.89/1.57 @ P2
% 2.89/1.57 @ P )
% 2.89/1.57 => ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
% 2.89/1.57 = P2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_172_transfer__forall__transfer_I3_J,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 2.89/1.57 ( ( right_total @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ (=>) ) @ (=>) @ ( transfer_forall @ A ) @ ( transfer_forall @ B ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_173_transfer__forall__transfer_I2_J,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 2.89/1.57 ( ( right_total @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 2.89/1.57 @ (=>)
% 2.89/1.57 @ ( transfer_forall @ A )
% 2.89/1.57 @ ( transfer_forall @ B ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_174_right__total__parametric,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( ( bi_total @ A @ B @ A2 )
% 2.89/1.57 => ( ( bi_total @ C @ D @ B2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 2.89/1.57 @ ( right_total @ A @ C )
% 2.89/1.57 @ ( right_total @ B @ D ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_175_Domainp__forall__transfer,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 2.89/1.57 ( ( right_total @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 2.89/1.57 @ ( transfer_bforall @ A @ ( domainp @ A @ B @ A2 ) )
% 2.89/1.57 @ ( transfer_forall @ B ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_176_nchotomy__relcomppE,axiom,
% 2.89/1.57 ! [C: $tType,B: $tType,A: $tType,D: $tType,F: B > A,R4: C > A > $o,S: A > D > $o,A4: C,C3: D] :
% 2.89/1.57 ( ! [Y2: A] :
% 2.89/1.57 ? [X7: B] :
% 2.89/1.57 ( Y2
% 2.89/1.57 = ( F @ X7 ) )
% 2.89/1.57 => ( ( relcompp @ C @ A @ D @ R4 @ S @ A4 @ C3 )
% 2.89/1.57 => ~ ! [B3: B] :
% 2.89/1.57 ( ( R4 @ A4 @ ( F @ B3 ) )
% 2.89/1.57 => ~ ( S @ ( F @ B3 ) @ C3 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_177_pos__fun__distr,axiom,
% 2.89/1.57 ! [E: $tType,C: $tType,A: $tType,B: $tType,D: $tType,F3: $tType,R: A > E > $o,S2: B > F3 > $o,R8: E > C > $o,S6: F3 > D > $o] : ( ord_less_eq @ ( ( A > B ) > ( C > D ) > $o ) @ ( relcompp @ ( A > B ) @ ( E > F3 ) @ ( C > D ) @ ( bNF_rel_fun @ A @ E @ B @ F3 @ R @ S2 ) @ ( bNF_rel_fun @ E @ C @ F3 @ D @ R8 @ S6 ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ ( relcompp @ A @ E @ C @ R @ R8 ) @ ( relcompp @ B @ F3 @ D @ S2 @ S6 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_178_add__le__cancel__left,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( ordere236663937imp_le @ A )
% 2.89/1.57 => ! [C3: A,A4: A,B4: A] :
% 2.89/1.57 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A4 ) @ ( plus_plus @ A @ C3 @ B4 ) )
% 2.89/1.57 = ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_179_add__le__cancel__right,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( ordere236663937imp_le @ A )
% 2.89/1.57 => ! [A4: A,C3: A,B4: A] :
% 2.89/1.57 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B4 @ C3 ) )
% 2.89/1.57 = ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_180_reflp__eq,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( reflp @ A )
% 2.89/1.57 = ( ord_less_eq @ ( A > A > $o )
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_181_reflp__ge__eq,axiom,
% 2.89/1.57 ! [A: $tType,R: A > A > $o] :
% 2.89/1.57 ( ( reflp @ A @ R )
% 2.89/1.57 => ( ord_less_eq @ ( A > A > $o )
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 2.89/1.57 @ R ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_182_leq__OOI,axiom,
% 2.89/1.57 ! [A: $tType,R: A > A > $o] :
% 2.89/1.57 ( ( R
% 2.89/1.57 = ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) )
% 2.89/1.57 => ( ord_less_eq @ ( A > A > $o ) @ R @ ( relcompp @ A @ A @ A @ R @ R ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_183_relcompp__mono,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,B: $tType,R9: A > B > $o,R4: A > B > $o,S7: B > C > $o,S: B > C > $o] :
% 2.89/1.57 ( ( ord_less_eq @ ( A > B > $o ) @ R9 @ R4 )
% 2.89/1.57 => ( ( ord_less_eq @ ( B > C > $o ) @ S7 @ S )
% 2.89/1.57 => ( ord_less_eq @ ( A > C > $o ) @ ( relcompp @ A @ B @ C @ R9 @ S7 ) @ ( relcompp @ A @ B @ C @ R4 @ S ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_184_predicate2D__conj,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X: A,Y: B] :
% 2.89/1.57 ( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 2.89/1.57 & R )
% 2.89/1.57 => ( R
% 2.89/1.57 & ( ( P @ X @ Y )
% 2.89/1.57 => ( Q @ X @ Y ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_185_refl__ge__eq,axiom,
% 2.89/1.57 ! [A: $tType,R: A > A > $o] :
% 2.89/1.57 ( ! [X2: A] : ( R @ X2 @ X2 )
% 2.89/1.57 => ( ord_less_eq @ ( A > A > $o )
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 2.89/1.57 @ R ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_186_ge__eq__refl,axiom,
% 2.89/1.57 ! [A: $tType,R: A > A > $o,X: A] :
% 2.89/1.57 ( ( ord_less_eq @ ( A > A > $o )
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 2.89/1.57 @ R )
% 2.89/1.57 => ( R @ X @ X ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_187_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( ordere779506340up_add @ A )
% 2.89/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.89/1.57 ( ( ( ord_less_eq @ A @ I @ J )
% 2.89/1.57 & ( K = L ) )
% 2.89/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_188_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( ordere779506340up_add @ A )
% 2.89/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.89/1.57 ( ( ( I = J )
% 2.89/1.57 & ( ord_less_eq @ A @ K @ L ) )
% 2.89/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_189_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( ordere779506340up_add @ A )
% 2.89/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.89/1.57 ( ( ( ord_less_eq @ A @ I @ J )
% 2.89/1.57 & ( ord_less_eq @ A @ K @ L ) )
% 2.89/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_190_add__mono,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( ordere779506340up_add @ A )
% 2.89/1.57 => ! [A4: A,B4: A,C3: A,D2: A] :
% 2.89/1.57 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.89/1.57 => ( ( ord_less_eq @ A @ C3 @ D2 )
% 2.89/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B4 @ D2 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_191_add__left__mono,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( ordere779506340up_add @ A )
% 2.89/1.57 => ! [A4: A,B4: A,C3: A] :
% 2.89/1.57 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.89/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A4 ) @ ( plus_plus @ A @ C3 @ B4 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_192_less__eqE,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( canoni770627133id_add @ A )
% 2.89/1.57 => ! [A4: A,B4: A] :
% 2.89/1.57 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.89/1.57 => ~ ! [C4: A] :
% 2.89/1.57 ( B4
% 2.89/1.57 != ( plus_plus @ A @ A4 @ C4 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_193_add__right__mono,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( ordere779506340up_add @ A )
% 2.89/1.57 => ! [A4: A,B4: A,C3: A] :
% 2.89/1.57 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.89/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B4 @ C3 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_194_le__iff__add,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( canoni770627133id_add @ A )
% 2.89/1.57 => ( ( ord_less_eq @ A )
% 2.89/1.57 = ( ^ [A5: A,B5: A] :
% 2.89/1.57 ? [C5: A] :
% 2.89/1.57 ( B5
% 2.89/1.57 = ( plus_plus @ A @ A5 @ C5 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_195_add__le__imp__le__left,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( ordere236663937imp_le @ A )
% 2.89/1.57 => ! [C3: A,A4: A,B4: A] :
% 2.89/1.57 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A4 ) @ ( plus_plus @ A @ C3 @ B4 ) )
% 2.89/1.57 => ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_196_add__le__imp__le__right,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( ordere236663937imp_le @ A )
% 2.89/1.57 => ! [A4: A,C3: A,B4: A] :
% 2.89/1.57 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C3 ) @ ( plus_plus @ A @ B4 @ C3 ) )
% 2.89/1.57 => ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_197_fun_Orel__mono,axiom,
% 2.89/1.57 ! [D: $tType,B: $tType,A: $tType,R: A > B > $o,Ra: A > B > $o] :
% 2.89/1.57 ( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra )
% 2.89/1.57 => ( ord_less_eq @ ( ( D > A ) > ( D > B ) > $o )
% 2.89/1.57 @ ( bNF_rel_fun @ D @ D @ A @ B
% 2.89/1.57 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 2.89/1.57 @ R )
% 2.89/1.57 @ ( bNF_rel_fun @ D @ D @ A @ B
% 2.89/1.57 @ ^ [Y4: D,Z2: D] : ( Y4 = Z2 )
% 2.89/1.57 @ Ra ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_198_fun__mono,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,D: $tType,C: $tType,C2: A > B > $o,A2: A > B > $o,B2: C > D > $o,D3: C > D > $o] :
% 2.89/1.57 ( ( ord_less_eq @ ( A > B > $o ) @ C2 @ A2 )
% 2.89/1.57 => ( ( ord_less_eq @ ( C > D > $o ) @ B2 @ D3 )
% 2.89/1.57 => ( ord_less_eq @ ( ( A > C ) > ( B > D ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ A @ B @ C @ D @ C2 @ D3 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_199_lfp_Omonotone__if__bot,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( comple187826305attice @ A )
% 2.89/1.57 => ! [Bound: A,G: A > B,Bot: B,F: A > B,Ord: B > B > $o] :
% 2.89/1.57 ( ! [X2: A] :
% 2.89/1.57 ( ( ( ord_less_eq @ A @ X2 @ Bound )
% 2.89/1.57 => ( ( G @ X2 )
% 2.89/1.57 = Bot ) )
% 2.89/1.57 & ( ~ ( ord_less_eq @ A @ X2 @ Bound )
% 2.89/1.57 => ( ( G @ X2 )
% 2.89/1.57 = ( F @ X2 ) ) ) )
% 2.89/1.57 => ( ! [X2: A,Y2: A] :
% 2.89/1.57 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.89/1.57 => ( ~ ( ord_less_eq @ A @ X2 @ Bound )
% 2.89/1.57 => ( Ord @ ( F @ X2 ) @ ( F @ Y2 ) ) ) )
% 2.89/1.57 => ( ! [X2: A] :
% 2.89/1.57 ( ~ ( ord_less_eq @ A @ X2 @ Bound )
% 2.89/1.57 => ( Ord @ Bot @ ( F @ X2 ) ) )
% 2.89/1.57 => ( ( Ord @ Bot @ Bot )
% 2.89/1.57 => ( comple1396247847notone @ A @ B @ ( ord_less_eq @ A ) @ Ord @ G ) ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_200_mono__transfer,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType] :
% 2.89/1.57 ( ( ( order @ B )
% 2.89/1.57 & ( order @ D )
% 2.89/1.57 & ( order @ C )
% 2.89/1.57 & ( order @ A ) )
% 2.89/1.57 => ! [A2: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( ( bi_total @ A @ B @ A2 )
% 2.89/1.57 => ( ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 2.89/1.57 @ ( ord_less_eq @ A )
% 2.89/1.57 @ ( ord_less_eq @ B ) )
% 2.89/1.57 => ( ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B2
% 2.89/1.57 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 2.89/1.57 @ ( ord_less_eq @ C )
% 2.89/1.57 @ ( ord_less_eq @ D ) )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 )
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 2.89/1.57 @ ( order_mono @ A @ C )
% 2.89/1.57 @ ( order_mono @ B @ D ) ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_201_neg__fun__distr1,axiom,
% 2.89/1.57 ! [D: $tType,A: $tType,B: $tType,C: $tType,E: $tType,F3: $tType,R: A > B > $o,R8: B > C > $o,S2: D > F3 > $o,S6: F3 > E > $o] :
% 2.89/1.57 ( ( left_unique @ A @ B @ R )
% 2.89/1.57 => ( ( right_total @ A @ B @ R )
% 2.89/1.57 => ( ( right_unique @ B @ C @ R8 )
% 2.89/1.57 => ( ( left_total @ B @ C @ R8 )
% 2.89/1.57 => ( ord_less_eq @ ( ( A > D ) > ( C > E ) > $o ) @ ( bNF_rel_fun @ A @ C @ D @ E @ ( relcompp @ A @ B @ C @ R @ R8 ) @ ( relcompp @ D @ F3 @ E @ S2 @ S6 ) ) @ ( relcompp @ ( A > D ) @ ( B > F3 ) @ ( C > E ) @ ( bNF_rel_fun @ A @ B @ D @ F3 @ R @ S2 ) @ ( bNF_rel_fun @ B @ C @ F3 @ E @ R8 @ S6 ) ) ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_202_set__plus__mono2,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( plus @ A )
% 2.89/1.57 => ! [C2: set @ A,D3: set @ A,E2: set @ A,F5: set @ A] :
% 2.89/1.57 ( ( ord_less_eq @ ( set @ A ) @ C2 @ D3 )
% 2.89/1.57 => ( ( ord_less_eq @ ( set @ A ) @ E2 @ F5 )
% 2.89/1.57 => ( ord_less_eq @ ( set @ A ) @ ( plus_plus @ ( set @ A ) @ C2 @ E2 ) @ ( plus_plus @ ( set @ A ) @ D3 @ F5 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_203_set__plus__mono2__b,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( plus @ A )
% 2.89/1.57 => ! [C2: set @ A,D3: set @ A,E2: set @ A,F5: set @ A,X: A] :
% 2.89/1.57 ( ( ord_less_eq @ ( set @ A ) @ C2 @ D3 )
% 2.89/1.57 => ( ( ord_less_eq @ ( set @ A ) @ E2 @ F5 )
% 2.89/1.57 => ( ( member @ A @ X @ ( plus_plus @ ( set @ A ) @ C2 @ E2 ) )
% 2.89/1.57 => ( member @ A @ X @ ( plus_plus @ ( set @ A ) @ D3 @ F5 ) ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_204_left__uniqueD,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,A2: A > B > $o,X: A,Z: B,Y: A] :
% 2.89/1.57 ( ( left_unique @ A @ B @ A2 )
% 2.89/1.57 => ( ( A2 @ X @ Z )
% 2.89/1.57 => ( ( A2 @ Y @ Z )
% 2.89/1.57 => ( X = Y ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_205_left__uniqueI,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,A2: A > B > $o] :
% 2.89/1.57 ( ! [X2: A,Y2: A,Z3: B] :
% 2.89/1.57 ( ( A2 @ X2 @ Z3 )
% 2.89/1.57 => ( ( A2 @ Y2 @ Z3 )
% 2.89/1.57 => ( X2 = Y2 ) ) )
% 2.89/1.57 => ( left_unique @ A @ B @ A2 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_206_right__uniqueD,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o,X: A,Y: B,Z: B] :
% 2.89/1.57 ( ( right_unique @ A @ B @ A2 )
% 2.89/1.57 => ( ( A2 @ X @ Y )
% 2.89/1.57 => ( ( A2 @ X @ Z )
% 2.89/1.57 => ( Y = Z ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_207_right__uniqueI,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,A2: A > B > $o] :
% 2.89/1.57 ( ! [X2: A,Y2: B,Z3: B] :
% 2.89/1.57 ( ( A2 @ X2 @ Y2 )
% 2.89/1.57 => ( ( A2 @ X2 @ Z3 )
% 2.89/1.57 => ( Y2 = Z3 ) ) )
% 2.89/1.57 => ( right_unique @ A @ B @ A2 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_208_left__unique__eq,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( left_unique @ A @ A
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_209_left__unique__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( left_unique @ A @ B )
% 2.89/1.57 = ( ^ [R5: A > B > $o] :
% 2.89/1.57 ! [X3: A,Y3: A,Z4: B] :
% 2.89/1.57 ( ( R5 @ X3 @ Z4 )
% 2.89/1.57 => ( ( R5 @ Y3 @ Z4 )
% 2.89/1.57 => ( X3 = Y3 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_210_right__unique__eq,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( right_unique @ A @ A
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_211_right__unique__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( right_unique @ A @ B )
% 2.89/1.57 = ( ^ [R5: A > B > $o] :
% 2.89/1.57 ! [X3: A,Y3: B,Z4: B] :
% 2.89/1.57 ( ( R5 @ X3 @ Y3 )
% 2.89/1.57 => ( ( R5 @ X3 @ Z4 )
% 2.89/1.57 => ( Y3 = Z4 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_212_left__unique__OO,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,B: $tType,R: A > B > $o,S2: B > C > $o] :
% 2.89/1.57 ( ( left_unique @ A @ B @ R )
% 2.89/1.57 => ( ( left_unique @ B @ C @ S2 )
% 2.89/1.57 => ( left_unique @ A @ C @ ( relcompp @ A @ B @ C @ R @ S2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_213_right__unique__OO,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,B: $tType,A2: A > B > $o,B2: B > C > $o] :
% 2.89/1.57 ( ( right_unique @ A @ B @ A2 )
% 2.89/1.57 => ( ( right_unique @ B @ C @ B2 )
% 2.89/1.57 => ( right_unique @ A @ C @ ( relcompp @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_214_functional__converse__relation,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > B > $o] :
% 2.89/1.57 ( ( left_unique @ A @ B @ R )
% 2.89/1.57 => ( ( right_total @ A @ B @ R )
% 2.89/1.57 => ! [Y6: B] :
% 2.89/1.57 ? [X2: A] :
% 2.89/1.57 ( ( R @ X2 @ Y6 )
% 2.89/1.57 & ! [Ya3: A] :
% 2.89/1.57 ( ( R @ Ya3 @ Y6 )
% 2.89/1.57 => ( Ya3 = X2 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_215_functional__relation,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,R: A > B > $o] :
% 2.89/1.57 ( ( right_unique @ A @ B @ R )
% 2.89/1.57 => ( ( left_total @ A @ B @ R )
% 2.89/1.57 => ! [X7: A] :
% 2.89/1.57 ? [Xa2: B] :
% 2.89/1.57 ( ( R @ X7 @ Xa2 )
% 2.89/1.57 & ! [Y6: B] :
% 2.89/1.57 ( ( R @ X7 @ Y6 )
% 2.89/1.57 => ( Y6 = Xa2 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_216_right__unique__fun,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( ( right_total @ A @ B @ A2 )
% 2.89/1.57 => ( ( right_unique @ C @ D @ B2 )
% 2.89/1.57 => ( right_unique @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_217_left__unique__fun,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( ( left_total @ A @ B @ A2 )
% 2.89/1.57 => ( ( left_unique @ C @ D @ B2 )
% 2.89/1.57 => ( left_unique @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_218_right__total__fun,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( ( right_unique @ A @ B @ A2 )
% 2.89/1.57 => ( ( right_total @ C @ D @ B2 )
% 2.89/1.57 => ( right_total @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_219_left__total__fun,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( ( left_unique @ A @ B @ A2 )
% 2.89/1.57 => ( ( left_total @ C @ D @ B2 )
% 2.89/1.57 => ( left_total @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_220_right__unique__alt__def2,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( right_unique @ A @ B )
% 2.89/1.57 = ( ^ [R5: A > B > $o] :
% 2.89/1.57 ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ R5 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5 @ (=>) )
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 2.89/1.57 @ ^ [Y4: B,Z2: B] : ( Y4 = Z2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_221_eq__imp__transfer,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 2.89/1.57 ( ( right_unique @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ (=>) )
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 2.89/1.57 @ ^ [Y4: B,Z2: B] : ( Y4 = Z2 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_222_neg__fun__distr2,axiom,
% 2.89/1.57 ! [F3: $tType,E: $tType,A: $tType,B: $tType,D: $tType,C: $tType,R8: A > B > $o,S6: C > D > $o,R: E > A > $o,S2: F3 > C > $o] :
% 2.89/1.57 ( ( right_unique @ A @ B @ R8 )
% 2.89/1.57 => ( ( left_total @ A @ B @ R8 )
% 2.89/1.57 => ( ( left_unique @ C @ D @ S6 )
% 2.89/1.57 => ( ( right_total @ C @ D @ S6 )
% 2.89/1.57 => ( ord_less_eq @ ( ( E > F3 ) > ( B > D ) > $o ) @ ( bNF_rel_fun @ E @ B @ F3 @ D @ ( relcompp @ E @ A @ B @ R @ R8 ) @ ( relcompp @ F3 @ C @ D @ S2 @ S6 ) ) @ ( relcompp @ ( E > F3 ) @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ E @ A @ F3 @ C @ R @ S2 ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R8 @ S6 ) ) ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_223_fixp__mono,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( comple1141879883l_ccpo @ A )
% 2.89/1.57 => ! [F: A > A,G: A > A] :
% 2.89/1.57 ( ( partial_fun_ord @ A @ A @ A @ ( ord_less_eq @ A ) @ F @ G )
% 2.89/1.57 => ( ( comple1396247847notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 2.89/1.57 => ( ( comple1396247847notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ G )
% 2.89/1.57 => ( ord_less_eq @ A @ ( comple939513234o_fixp @ A @ F ) @ ( comple939513234o_fixp @ A @ G ) ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_224_left__unique__parametric,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( ( bi_unique @ A @ B @ A2 )
% 2.89/1.57 => ( ( bi_total @ A @ B @ A2 )
% 2.89/1.57 => ( ( bi_total @ C @ D @ B2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 2.89/1.57 @ ( left_unique @ A @ C )
% 2.89/1.57 @ ( left_unique @ B @ D ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_225_bi__uniqueI,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,R: A > B > $o] :
% 2.89/1.57 ( ( left_unique @ A @ B @ R )
% 2.89/1.57 => ( ( right_unique @ A @ B @ R )
% 2.89/1.57 => ( bi_unique @ A @ B @ R ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_226_bi__unique__alt__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( bi_unique @ A @ B )
% 2.89/1.57 = ( ^ [A6: A > B > $o] :
% 2.89/1.57 ( ( left_unique @ A @ B @ A6 )
% 2.89/1.57 & ( right_unique @ A @ B @ A6 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_227_bi__unique__OO,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,B: $tType,A2: A > B > $o,B2: B > C > $o] :
% 2.89/1.57 ( ( bi_unique @ A @ B @ A2 )
% 2.89/1.57 => ( ( bi_unique @ B @ C @ B2 )
% 2.89/1.57 => ( bi_unique @ A @ C @ ( relcompp @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_228_bi__unique__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( bi_unique @ A @ B )
% 2.89/1.57 = ( ^ [R5: A > B > $o] :
% 2.89/1.57 ( ! [X3: A,Y3: B,Z4: B] :
% 2.89/1.57 ( ( R5 @ X3 @ Y3 )
% 2.89/1.57 => ( ( R5 @ X3 @ Z4 )
% 2.89/1.57 => ( Y3 = Z4 ) ) )
% 2.89/1.57 & ! [X3: A,Y3: A,Z4: B] :
% 2.89/1.57 ( ( R5 @ X3 @ Z4 )
% 2.89/1.57 => ( ( R5 @ Y3 @ Z4 )
% 2.89/1.57 => ( X3 = Y3 ) ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_229_bi__unique__eq,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( bi_unique @ A @ A
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_230_bi__uniqueDr,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o,X: A,Y: B,Z: B] :
% 2.89/1.57 ( ( bi_unique @ A @ B @ A2 )
% 2.89/1.57 => ( ( A2 @ X @ Y )
% 2.89/1.57 => ( ( A2 @ X @ Z )
% 2.89/1.57 => ( Y = Z ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_231_bi__uniqueDl,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,A2: A > B > $o,X: A,Y: B,Z: A] :
% 2.89/1.57 ( ( bi_unique @ A @ B @ A2 )
% 2.89/1.57 => ( ( A2 @ X @ Y )
% 2.89/1.57 => ( ( A2 @ Z @ Y )
% 2.89/1.57 => ( X = Z ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_232_bi__unique__fun,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( ( bi_total @ A @ B @ A2 )
% 2.89/1.57 => ( ( bi_unique @ C @ D @ B2 )
% 2.89/1.57 => ( bi_unique @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_233_bi__unique__alt__def2,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( bi_unique @ A @ B )
% 2.89/1.57 = ( ^ [R5: A > B > $o] :
% 2.89/1.57 ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ R5
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 2.89/1.57 @ ^ [Y4: B,Z2: B] : ( Y4 = Z2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_234_eq__transfer,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 2.89/1.57 ( ( bi_unique @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
% 2.89/1.57 @ ^ [Y4: A,Z2: A] : ( Y4 = Z2 )
% 2.89/1.57 @ ^ [Y4: B,Z2: B] : ( Y4 = Z2 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_235_bi__total__fun,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( ( bi_unique @ A @ B @ A2 )
% 2.89/1.57 => ( ( bi_total @ C @ D @ B2 )
% 2.89/1.57 => ( bi_total @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_236_right__unique__transfer,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( ( right_total @ A @ B @ A2 )
% 2.89/1.57 => ( ( right_total @ C @ D @ B2 )
% 2.89/1.57 => ( ( bi_unique @ C @ D @ B2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ (=>)
% 2.89/1.57 @ ( right_unique @ A @ C )
% 2.89/1.57 @ ( right_unique @ B @ D ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_237_left__unique__transfer,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( ( right_total @ A @ B @ A2 )
% 2.89/1.57 => ( ( right_total @ C @ D @ B2 )
% 2.89/1.57 => ( ( bi_unique @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ (=>)
% 2.89/1.57 @ ( left_unique @ A @ C )
% 2.89/1.57 @ ( left_unique @ B @ D ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_238_right__unique__parametric,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( ( bi_total @ A @ B @ A2 )
% 2.89/1.57 => ( ( bi_unique @ C @ D @ B2 )
% 2.89/1.57 => ( ( bi_total @ C @ D @ B2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
% 2.89/1.57 @ ( right_unique @ A @ C )
% 2.89/1.57 @ ( right_unique @ B @ D ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_239_fixp__lowerbound,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( comple1141879883l_ccpo @ A )
% 2.89/1.57 => ! [F: A > A,Z: A] :
% 2.89/1.57 ( ( comple1396247847notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 2.89/1.57 => ( ( ord_less_eq @ A @ ( F @ Z ) @ Z )
% 2.89/1.57 => ( ord_less_eq @ A @ ( comple939513234o_fixp @ A @ F ) @ Z ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_240_fixp__unfold,axiom,
% 2.89/1.57 ! [A: $tType] :
% 2.89/1.57 ( ( comple1141879883l_ccpo @ A )
% 2.89/1.57 => ! [F: A > A] :
% 2.89/1.57 ( ( comple1396247847notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 2.89/1.57 => ( ( comple939513234o_fixp @ A @ F )
% 2.89/1.57 = ( F @ ( comple939513234o_fixp @ A @ F ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_241_monotone__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( comple1396247847notone @ A @ B )
% 2.89/1.57 = ( ^ [Orda: A > A > $o,Ordb: B > B > $o,F2: A > B] :
% 2.89/1.57 ! [X3: A,Y3: A] :
% 2.89/1.57 ( ( Orda @ X3 @ Y3 )
% 2.89/1.57 => ( Ordb @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_242_monotoneI,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,Orda2: A > A > $o,Ordb2: B > B > $o,F: A > B] :
% 2.89/1.57 ( ! [X2: A,Y2: A] :
% 2.89/1.57 ( ( Orda2 @ X2 @ Y2 )
% 2.89/1.57 => ( Ordb2 @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.89/1.57 => ( comple1396247847notone @ A @ B @ Orda2 @ Ordb2 @ F ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_243_monotoneD,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,Orda2: A > A > $o,Ordb2: B > B > $o,F: A > B,X: A,Y: A] :
% 2.89/1.57 ( ( comple1396247847notone @ A @ B @ Orda2 @ Ordb2 @ F )
% 2.89/1.57 => ( ( Orda2 @ X @ Y )
% 2.89/1.57 => ( Ordb2 @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_244_fun__upd__transfer,axiom,
% 2.89/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
% 2.89/1.57 ( ( bi_unique @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( A > C > A > C ) @ ( B > D > B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ A @ B @ ( C > A > C ) @ ( D > B > D ) @ A2 @ ( bNF_rel_fun @ C @ D @ ( A > C ) @ ( B > D ) @ B2 @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) @ ( fun_upd @ A @ C ) @ ( fun_upd @ B @ D ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_245_rtranclp__parametric,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A2: A > B > $o] :
% 2.89/1.57 ( ( bi_unique @ A @ B @ A2 )
% 2.89/1.57 => ( ( bi_total @ A @ B @ A2 )
% 2.89/1.57 => ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( A > A > $o ) @ ( B > B > $o )
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
% 2.89/1.57 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
% 2.89/1.57 @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) ) )
% 2.89/1.57 @ ( transitive_rtranclp @ A )
% 2.89/1.57 @ ( transitive_rtranclp @ B ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_246_fun__upd__apply,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType] :
% 2.89/1.57 ( ( fun_upd @ B @ A )
% 2.89/1.57 = ( ^ [F2: B > A,X3: B,Y3: A,Z4: B] : ( if @ A @ ( Z4 = X3 ) @ Y3 @ ( F2 @ Z4 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_247_fun__upd__triv,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 2.89/1.57 ( ( fun_upd @ A @ B @ F @ X @ ( F @ X ) )
% 2.89/1.57 = F ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_248_fun__upd__upd,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,Z: B] :
% 2.89/1.57 ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ X @ Z )
% 2.89/1.57 = ( fun_upd @ A @ B @ F @ X @ Z ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_249_fun__upd__idem__iff,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B] :
% 2.89/1.57 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 2.89/1.57 = F )
% 2.89/1.57 = ( ( F @ X )
% 2.89/1.57 = Y ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_250_fun__upd__twist,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,A4: A,C3: A,M3: A > B,B4: B,D2: B] :
% 2.89/1.57 ( ( A4 != C3 )
% 2.89/1.57 => ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M3 @ A4 @ B4 ) @ C3 @ D2 )
% 2.89/1.57 = ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M3 @ C3 @ D2 ) @ A4 @ B4 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_251_fun__upd__other,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,Z: A,X: A,F: A > B,Y: B] :
% 2.89/1.57 ( ( Z != X )
% 2.89/1.57 => ( ( fun_upd @ A @ B @ F @ X @ Y @ Z )
% 2.89/1.57 = ( F @ Z ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_252_fun__upd__same,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType,F: B > A,X: B,Y: A] :
% 2.89/1.57 ( ( fun_upd @ B @ A @ F @ X @ Y @ X )
% 2.89/1.57 = Y ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_253_fun__upd__idem,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,F: B > A,X: B,Y: A] :
% 2.89/1.57 ( ( ( F @ X )
% 2.89/1.57 = Y )
% 2.89/1.57 => ( ( fun_upd @ B @ A @ F @ X @ Y )
% 2.89/1.57 = F ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_254_fun__upd__eqD,axiom,
% 2.89/1.57 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,G: A > B,Z: B] :
% 2.89/1.57 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 2.89/1.57 = ( fun_upd @ A @ B @ G @ X @ Z ) )
% 2.89/1.57 => ( Y = Z ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(fact_255_fun__upd__def,axiom,
% 2.89/1.57 ! [B: $tType,A: $tType] :
% 2.89/1.57 ( ( fun_upd @ A @ B )
% 2.89/1.57 = ( ^ [F2: A > B,A5: A,B5: B,X3: A] : ( if @ B @ ( X3 = A5 ) @ B5 @ ( F2 @ X3 ) ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_Multiset_Omultiset___Orderings_Opreorder,axiom,
% 2.89/1.57 ! [A7: $tType] :
% 2.89/1.57 ( ( preorder @ A7 )
% 2.89/1.57 => ( preorder @ ( multiset @ A7 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_HOL_Obool___Orderings_Opreorder_1,axiom,
% 2.89/1.57 preorder @ $o ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
% 2.89/1.57 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_fun___Orderings_Opreorder_3,axiom,
% 2.89/1.57 ! [A7: $tType,A8: $tType] :
% 2.89/1.57 ( ( preorder @ A8 )
% 2.89/1.57 => ( preorder @ ( A7 > A8 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 2.89/1.57 ! [A7: $tType,A8: $tType] :
% 2.89/1.57 ( ( comple187826305attice @ A8 )
% 2.89/1.57 => ( comple187826305attice @ ( A7 > A8 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
% 2.89/1.57 ! [A7: $tType,A8: $tType] :
% 2.89/1.57 ( ( comple187826305attice @ A8 )
% 2.89/1.57 => ( comple1141879883l_ccpo @ ( A7 > A8 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.89/1.57 ! [A7: $tType,A8: $tType] :
% 2.89/1.57 ( ( order @ A8 )
% 2.89/1.57 => ( order @ ( A7 > A8 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_4,axiom,
% 2.89/1.57 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_5,axiom,
% 2.89/1.57 ! [A7: $tType] : ( comple1141879883l_ccpo @ ( set @ A7 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_Set_Oset___Groups_Oab__semigroup__add,axiom,
% 2.89/1.57 ! [A7: $tType] :
% 2.89/1.57 ( ( ab_semigroup_add @ A7 )
% 2.89/1.57 => ( ab_semigroup_add @ ( set @ A7 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_Set_Oset___Groups_Ocomm__monoid__add,axiom,
% 2.89/1.57 ! [A7: $tType] :
% 2.89/1.57 ( ( comm_monoid_add @ A7 )
% 2.89/1.57 => ( comm_monoid_add @ ( set @ A7 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_Set_Oset___Groups_Osemigroup__add,axiom,
% 2.89/1.57 ! [A7: $tType] :
% 2.89/1.57 ( ( semigroup_add @ A7 )
% 2.89/1.57 => ( semigroup_add @ ( set @ A7 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 2.89/1.57 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_Set_Oset___Groups_Oplus,axiom,
% 2.89/1.57 ! [A7: $tType] :
% 2.89/1.57 ( ( plus @ A7 )
% 2.89/1.57 => ( plus @ ( set @ A7 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_7,axiom,
% 2.89/1.57 comple187826305attice @ $o ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_8,axiom,
% 2.89/1.57 comple1141879883l_ccpo @ $o ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 2.89/1.57 order @ $o ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add,axiom,
% 2.89/1.57 ! [A7: $tType] :
% 2.89/1.57 ( ( preorder @ A7 )
% 2.89/1.57 => ( ordere779506340up_add @ ( multiset @ A7 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add,axiom,
% 2.89/1.57 ! [A7: $tType] : ( cancel_semigroup_add @ ( multiset @ A7 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_10,axiom,
% 2.89/1.57 ! [A7: $tType] : ( ab_semigroup_add @ ( multiset @ A7 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_11,axiom,
% 2.89/1.57 ! [A7: $tType] : ( comm_monoid_add @ ( multiset @ A7 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_12,axiom,
% 2.89/1.57 ! [A7: $tType] : ( semigroup_add @ ( multiset @ A7 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_Multiset_Omultiset___Orderings_Oorder_13,axiom,
% 2.89/1.57 ! [A7: $tType] :
% 2.89/1.57 ( ( preorder @ A7 )
% 2.89/1.57 => ( order @ ( multiset @ A7 ) ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(tcon_Multiset_Omultiset___Groups_Oplus_14,axiom,
% 2.89/1.57 ! [A7: $tType] : ( plus @ ( multiset @ A7 ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(help_If_3_1_T,axiom,
% 2.89/1.57 ! [P: $o] :
% 2.89/1.57 ( ( P = $true )
% 2.89/1.57 | ( P = $false ) ) ).
% 2.89/1.57
% 2.89/1.57 thf(help_If_2_1_T,axiom,
% 2.89/1.57 ! [A: $tType,X: A,Y: A] :
% 2.89/1.57 ( ( if @ A @ $false @ X @ Y )
% 2.89/1.57 = Y ) ).
% 2.89/1.57
% 2.89/1.57 thf(help_If_1_1_T,axiom,
% 2.89/1.57 ! [A: $tType,X: A,Y: A] :
% 2.89/1.57 ( ( if @ A @ $true @ X @ Y )
% 2.89/1.57 = X ) ).
% 2.89/1.57
% 2.89/1.57 thf(conj_0,conjecture,
% 2.89/1.57 finite100568337ommute @ a @ c @ f1 ).
% 2.89/1.57 ------------------------
% 2.98/1.61 % (10081)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_9803 for (2999ds/183Mi)
% 2.98/1.61 % (10082)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_9803 for (2999ds/4Mi)
% 2.98/1.61 % (10086)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_9803 for (2999ds/275Mi)
% 2.98/1.61 % (10082)Instruction limit reached!
% 2.98/1.61 % (10082)------------------------------
% 2.98/1.61 % (10082)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.61 % (10082)Termination reason: Unknown
% 2.98/1.61 % (10082)Termination phase: shuffling
% 2.98/1.61
% 2.98/1.61 % (10082)Memory used [KB]: 1663
% 2.98/1.61 % (10082)Time elapsed: 0.003 s
% 2.98/1.61 % (10082)Instructions burned: 4 (million)
% 2.98/1.61 % (10082)------------------------------
% 2.98/1.61 % (10082)------------------------------
% 2.98/1.61 % (10083)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_9803 for (2999ds/27Mi)
% 2.98/1.61 % (10084)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_9803 for (2999ds/2Mi)
% 2.98/1.61 % (10085)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_9803 for (2999ds/2Mi)
% 2.98/1.61 % (10087)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_9803 for (2999ds/18Mi)
% 2.98/1.62 % (10084)Instruction limit reached!
% 2.98/1.62 % (10084)------------------------------
% 2.98/1.62 % (10084)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.62 % (10084)Termination reason: Unknown
% 2.98/1.62 % (10084)Termination phase: shuffling
% 2.98/1.62
% 2.98/1.62 % (10084)Memory used [KB]: 1535
% 2.98/1.62 % (10084)Time elapsed: 0.004 s
% 2.98/1.62 % (10084)Instructions burned: 3 (million)
% 2.98/1.62 % (10084)------------------------------
% 2.98/1.62 % (10084)------------------------------
% 2.98/1.62 % (10085)Instruction limit reached!
% 2.98/1.62 % (10085)------------------------------
% 2.98/1.62 % (10085)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.62 % (10085)Termination reason: Unknown
% 2.98/1.62 % (10085)Termination phase: shuffling
% 2.98/1.62
% 2.98/1.62 % (10085)Memory used [KB]: 1535
% 2.98/1.62 % (10085)Time elapsed: 0.004 s
% 2.98/1.62 % (10085)Instructions burned: 3 (million)
% 2.98/1.62 % (10085)------------------------------
% 2.98/1.62 % (10085)------------------------------
% 2.98/1.62 % (10088)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9803 for (2999ds/3Mi)
% 2.98/1.62 % (10088)Instruction limit reached!
% 2.98/1.62 % (10088)------------------------------
% 2.98/1.62 % (10088)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.62 % (10088)Termination reason: Unknown
% 2.98/1.62 % (10088)Termination phase: shuffling
% 2.98/1.62
% 2.98/1.62 % (10088)Memory used [KB]: 1535
% 2.98/1.62 % (10088)Time elapsed: 0.002 s
% 2.98/1.62 % (10088)Instructions burned: 3 (million)
% 2.98/1.62 % (10088)------------------------------
% 2.98/1.62 % (10088)------------------------------
% 2.98/1.63 % (10087)Instruction limit reached!
% 2.98/1.63 % (10087)------------------------------
% 2.98/1.63 % (10087)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.63 % (10087)Termination reason: Unknown
% 2.98/1.63 % (10087)Termination phase: shuffling
% 2.98/1.63
% 2.98/1.63 % (10087)Memory used [KB]: 1918
% 2.98/1.63 % (10087)Time elapsed: 0.014 s
% 2.98/1.63 % (10087)Instructions burned: 19 (million)
% 2.98/1.63 % (10087)------------------------------
% 2.98/1.63 % (10087)------------------------------
% 2.98/1.63 % (10091)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_9803 for (2999ds/3Mi)
% 2.98/1.63 % (10091)Instruction limit reached!
% 2.98/1.63 % (10091)------------------------------
% 2.98/1.63 % (10091)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.63 % (10091)Termination reason: Unknown
% 2.98/1.63 % (10091)Termination phase: shuffling
% 2.98/1.63
% 2.98/1.63 % (10091)Memory used [KB]: 1535
% 2.98/1.63 % (10091)Time elapsed: 0.002 s
% 2.98/1.63 % (10091)Instructions burned: 3 (million)
% 2.98/1.63 % (10091)------------------------------
% 2.98/1.63 % (10091)------------------------------
% 2.98/1.63 % (10083)Instruction limit reached!
% 2.98/1.63 % (10083)------------------------------
% 2.98/1.63 % (10083)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.63 % (10083)Termination reason: Unknown
% 2.98/1.63 % (10083)Termination phase: shuffling
% 2.98/1.63
% 2.98/1.63 % (10083)Memory used [KB]: 2046
% 2.98/1.63 % (10083)Time elapsed: 0.019 s
% 2.98/1.63 % (10083)Instructions burned: 27 (million)
% 2.98/1.63 % (10083)------------------------------
% 2.98/1.63 % (10083)------------------------------
% 2.98/1.63 % (10089)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_9803 for (2999ds/37Mi)
% 2.98/1.63 % (10090)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_9803 for (2999ds/15Mi)
% 2.98/1.64 % (10086)First to succeed.
% 2.98/1.64 % (10086)Refutation found. Thanks to Tanya!
% 2.98/1.64 % SZS status Theorem for DTF2THF_9803
% 2.98/1.64 % SZS output start Proof for DTF2THF_9803
% 2.98/1.64 thf(type_def_6, type, fset: ($tType) > $tType).
% 2.98/1.64 thf(type_def_7, type, set: ($tType) > $tType).
% 2.98/1.64 thf(type_def_8, type, multiset: ($tType) > $tType).
% 2.98/1.64 thf(type_def_9, type, a: $tType).
% 2.98/1.64 thf(type_def_10, type, b: $tType).
% 2.98/1.64 thf(type_def_11, type, c: $tType).
% 2.98/1.64 thf(type_def_12, type, d: $tType).
% 2.98/1.64 thf(func_def_0, type, multiset: ($tType) > $tType).
% 2.98/1.64 thf(func_def_1, type, fset: ($tType) > $tType).
% 2.98/1.64 thf(func_def_2, type, set: ($tType) > $tType).
% 2.98/1.64 thf(func_def_3, type, d: $tType).
% 2.98/1.64 thf(func_def_4, type, c: $tType).
% 2.98/1.64 thf(func_def_5, type, b: $tType).
% 2.98/1.64 thf(func_def_6, type, a: $tType).
% 2.98/1.64 thf(func_def_7, type, preorder: !>[X0: $tType]:($o)).
% 2.98/1.64 thf(func_def_8, type, type: !>[X0: $tType]:($o)).
% 2.98/1.64 thf(func_def_9, type, plus: !>[X0: $tType]:($o)).
% 2.98/1.64 thf(func_def_10, type, order: !>[X0: $tType]:($o)).
% 2.98/1.64 thf(func_def_11, type, group_add: !>[X0: $tType]:($o)).
% 2.98/1.64 thf(func_def_12, type, semigroup_add: !>[X0: $tType]:($o)).
% 2.98/1.64 thf(func_def_13, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 2.98/1.64 thf(func_def_14, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 2.98/1.64 thf(func_def_15, type, comple1141879883l_ccpo: !>[X0: $tType]:($o)).
% 2.98/1.64 thf(func_def_16, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 2.98/1.64 thf(func_def_17, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 2.98/1.64 thf(func_def_18, type, comple187826305attice: !>[X0: $tType]:($o)).
% 2.98/1.64 thf(func_def_19, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.98/1.64 thf(func_def_20, type, ordere236663937imp_le: !>[X0: $tType]:($o)).
% 2.98/1.64 thf(func_def_21, type, bNF_rel_fun: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1 > $o) > (X2 > X3 > $o) > (X0 > X2) > (X1 > X3) > $o)).
% 2.98/1.64 thf(func_def_22, type, comple939513234o_fixp: !>[X0: $tType]:((X0 > X0) > X0)).
% 2.98/1.64 thf(func_def_23, type, comple1396247847notone: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X1 > $o) > (X0 > X1) > $o)).
% 2.98/1.64 thf(func_def_24, type, ffold: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > X1 > fset @ X0 > X1)).
% 2.98/1.64 thf(func_def_25, type, finite100568337ommute: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > $o)).
% 2.98/1.64 thf(func_def_26, type, finite_comp_fun_idem: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > $o)).
% 2.98/1.64 thf(func_def_27, type, finite852775215axioms: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > $o)).
% 2.98/1.64 thf(func_def_28, type, finite_fold_graph: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > X1 > set @ X0 > X1 > $o)).
% 2.98/1.64 thf(func_def_29, type, finite_folding: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > $o)).
% 2.98/1.64 thf(func_def_30, type, finite_folding_idem: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > $o)).
% 2.98/1.64 thf(func_def_31, type, finite1921348288axioms: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > $o)).
% 2.98/1.64 thf(func_def_32, type, comp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X0) > X2 > X1)).
% 2.98/1.64 thf(func_def_33, type, fun_upd: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0 > X1 > X0 > X1)).
% 2.98/1.64 thf(func_def_34, type, map_fun: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > (X1 > X2) > X0 > X3)).
% 2.98/1.64 thf(func_def_35, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.98/1.64 thf(func_def_36, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.98/1.64 thf(func_def_37, type, fold_mset: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > X1 > multiset @ X0 > X1)).
% 2.98/1.64 thf(func_def_38, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.98/1.64 thf(func_def_39, type, order_mono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 2.98/1.64 thf(func_def_40, type, partial_fun_ord: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X2 > X0) > (X2 > X1) > $o)).
% 2.98/1.64 thf(func_def_41, type, bex1_rel: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > $o) > $o)).
% 2.98/1.64 thf(func_def_42, type, quotient3: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > (X1 > X0) > $o)).
% 2.98/1.64 thf(func_def_43, type, domainp: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X0 > $o)).
% 2.98/1.64 thf(func_def_44, type, reflp: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 2.98/1.64 thf(func_def_45, type, relcompp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > X0 > X2 > $o)).
% 2.98/1.64 thf(func_def_46, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.98/1.64 thf(func_def_47, type, bi_total: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > $o)).
% 2.98/1.64 thf(func_def_48, type, bi_unique: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > $o)).
% 2.98/1.64 thf(func_def_49, type, left_total: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > $o)).
% 2.98/1.64 thf(func_def_50, type, left_unique: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > $o)).
% 2.98/1.64 thf(func_def_51, type, rev_implies: $o > $o > $o).
% 2.98/1.64 thf(func_def_52, type, right_total: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > $o)).
% 2.98/1.64 thf(func_def_53, type, right_unique: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > $o)).
% 2.98/1.64 thf(func_def_54, type, transfer_bforall: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > $o)).
% 2.98/1.64 thf(func_def_55, type, transfer_forall: !>[X0: $tType]:((X0 > $o) > $o)).
% 2.98/1.64 thf(func_def_56, type, transitive_rtranclp: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > $o)).
% 2.98/1.64 thf(func_def_57, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.98/1.64 thf(func_def_58, type, a2: a > b > $o).
% 2.98/1.64 thf(func_def_59, type, b2: c > d > $o).
% 2.98/1.64 thf(func_def_60, type, f1: a > c > c).
% 2.98/1.64 thf(func_def_61, type, f2: b > d > d).
% 2.98/1.64 thf(func_def_78, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0 > X0) > X1)).
% 2.98/1.64 thf(func_def_79, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > X0 > X0) > X1)).
% 2.98/1.64 thf(func_def_80, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType]:(X2 > (X2 > X0) > (X1 > X0) > X1)).
% 2.98/1.64 thf(func_def_81, type, sK3: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X0) > (X1 > X0) > X2 > X1)).
% 2.98/1.64 thf(func_def_82, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X0) > (X1 > X0) > X2)).
% 2.98/1.64 thf(func_def_83, type, sK5: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.98/1.64 thf(func_def_84, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X0 > X2) > X2 > X1)).
% 2.98/1.64 thf(func_def_85, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X0 > X2) > X0)).
% 2.98/1.64 thf(func_def_86, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X0 > X2) > X0)).
% 2.98/1.64 thf(func_def_88, type, ph10: !>[X0: $tType]:(X0)).
% 2.98/1.64 thf(f904,plain,(
% 2.98/1.64 $false),
% 2.98/1.64 inference(subsumption_resolution,[status(thm)],[f862,f877])).
% 2.98/1.64 thf(f877,plain,(
% 2.98/1.64 ((finite100568337ommute @ a @ c @ f1) = $true)),
% 2.98/1.64 inference(cnf_transformation,[status(thm)],[f442])).
% 2.98/1.64 thf(f442,plain,(
% 2.98/1.64 ((finite100568337ommute @ a @ c @ f1) = $true)),
% 2.98/1.64 inference(fool_elimination,[status(thm)],[f441])).
% 2.98/1.64 thf(f441,plain,(
% 2.98/1.64 (finite100568337ommute @ a @ c @ f1)),
% 2.98/1.64 inference(rectify,[status(thm)],[f1])).
% 2.98/1.64 thf(f1,axiom,(
% 2.98/1.64 (finite100568337ommute @ a @ c @ f1)),
% 2.98/1.64 file('/export/starexec/sandbox2/tmp/tmp.XftS7e7A06/DTF2THF_9803.p',fact_0_assms_I2_J)).
% 2.98/1.64 thf(f862,plain,(
% 2.98/1.64 ((finite100568337ommute @ a @ c @ f1) != $true)),
% 2.98/1.64 inference(cnf_transformation,[status(thm)],[f797])).
% 2.98/1.64 thf(f797,plain,(
% 2.98/1.64 ((finite100568337ommute @ a @ c @ f1) != $true)),
% 2.98/1.64 inference(flattening,[status(thm)],[f532])).
% 2.98/1.64 thf(f532,plain,(
% 2.98/1.64 ~((finite100568337ommute @ a @ c @ f1) = $true)),
% 2.98/1.64 inference(fool_elimination,[status(thm)],[f531])).
% 2.98/1.64 thf(f531,plain,(
% 2.98/1.64 ~(finite100568337ommute @ a @ c @ f1)),
% 2.98/1.64 inference(rectify,[status(thm)],[f285])).
% 2.98/1.64 thf(f285,negated_conjecture,(
% 2.98/1.64 ~(finite100568337ommute @ a @ c @ f1)),
% 2.98/1.64 inference(negated_conjecture,[status(cth)],[f284])).
% 2.98/1.64 thf(f284,conjecture,(
% 2.98/1.64 (finite100568337ommute @ a @ c @ f1)),
% 2.98/1.64 file('/export/starexec/sandbox2/tmp/tmp.XftS7e7A06/DTF2THF_9803.p',conj_0)).
% 2.98/1.64 % SZS output end Proof for DTF2THF_9803
% 2.98/1.64 % (10086)------------------------------
% 2.98/1.64 % (10086)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.98/1.64 % (10086)Termination reason: Refutation
% 2.98/1.64
% 2.98/1.64 % (10086)Memory used [KB]: 7164
% 2.98/1.64 % (10086)Time elapsed: 0.030 s
% 2.98/1.64 % (10086)Instructions burned: 86 (million)
% 2.98/1.64 % (10086)------------------------------
% 2.98/1.64 % (10086)------------------------------
% 2.98/1.64 % (10080)Success in time 0.062 s
%------------------------------------------------------------------------------