TPTP Problem File: ITP036^2.p
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%------------------------------------------------------------------------------
% File : ITP036^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Coincidence problem prob_1096__7244526_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Coincidence/prob_1096__7244526_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 381 ( 102 unt; 64 typ; 0 def)
% Number of atoms : 987 ( 225 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 6103 ( 105 ~; 4 |; 224 &;5351 @)
% ( 0 <=>; 419 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 11 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 151 ( 151 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 59 usr; 12 con; 0-13 aty)
% Number of variables : 1642 ( 72 ^;1488 !; 12 ?;1642 :)
% ( 70 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:32:27.253
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext,type,
denota1663640101rp_ext: $tType > $tType > $tType > $tType > $tType ).
thf(ty_t_Finite__Cartesian__Product_Ovec,type,
finite_Cartesian_vec: $tType > $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Syntax_Oformula,type,
formula: $tType > $tType > $tType > $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Syntax_Otrm,type,
trm: $tType > $tType > $tType ).
thf(ty_t_Syntax_Ohp,type,
hp: $tType > $tType > $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_c,type,
c: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (50)
thf(sy_cl_Cardinality_OCARD__1,type,
cARD_1:
!>[A: $tType] : $o ).
thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
ordere890947078_space:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osup,type,
sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd,type,
euclid1678468529ng_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring,type,
euclid1155270486miring:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca1785829860lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__hp,type,
coinci32832645ide_hp:
!>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__hp_H,type,
coinci554967470ide_hp:
!>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > $o ) ).
thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Oode__sem__equiv,type,
coinci1495174342_equiv:
!>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
thf(sy_c_Denotational__Semantics_OIagree,type,
denotational_Iagree:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) > $o ) ).
thf(sy_c_Denotational__Semantics_OVagree,type,
denotational_Vagree:
!>[C: $tType] : ( ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( set @ ( sum_sum @ C @ C ) ) > $o ) ).
thf(sy_c_Denotational__Semantics_Oprog__sem,type,
denota1661140910og_sem:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( hp @ A @ B @ C ) > ( set @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) ) ).
thf(sy_c_Ids_Oids,type,
ids:
!>[Sz: $tType,Sf: $tType,Sc: $tType] : ( Sz > Sz > Sz > Sf > Sf > Sf > Sc > Sc > Sc > Sc > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Static__Semantics_OFVP,type,
static_FVP:
!>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
thf(sy_c_Static__Semantics_OMBV,type,
static_MBV:
!>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
thf(sy_c_Static__Semantics_OSIGP,type,
static_SIGP:
!>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
thf(sy_c_Syntax_Ohp_OChoice,type,
choice:
!>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( hp @ A @ B @ C ) > ( hp @ A @ B @ C ) ) ).
thf(sy_c_Syntax_Ohpsafe,type,
hpsafe:
!>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > $o ) ).
thf(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_I____,type,
i: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
thf(sy_v_J____,type,
j: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
thf(sy_v_V____,type,
v: set @ ( sum_sum @ c @ c ) ).
thf(sy_v__092_060mu_062_H_H____,type,
mu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ).
thf(sy_v__092_060mu_062_H____,type,
mu2: finite_Cartesian_vec @ real @ c ).
thf(sy_v__092_060mu_062____,type,
mu3: finite_Cartesian_vec @ real @ c ).
thf(sy_v__092_060nu_0621_H____,type,
nu_1: finite_Cartesian_vec @ real @ c ).
thf(sy_v__092_060nu_0621____,type,
nu_12: finite_Cartesian_vec @ real @ c ).
thf(sy_v__092_060nu_0622_H____,type,
nu_2: finite_Cartesian_vec @ real @ c ).
thf(sy_v__092_060nu_0622____,type,
nu_22: finite_Cartesian_vec @ real @ c ).
thf(sy_v_a____,type,
a2: hp @ a @ b @ c ).
thf(sy_v_b____,type,
b2: hp @ a @ b @ c ).
% Relevant facts (256)
thf(fact_0_hpsafe__Choice_OIH_I2_J,axiom,
coinci554967470ide_hp @ a @ b @ c @ b2 ).
% hpsafe_Choice.IH(2)
thf(fact_1_hpsafe__Choice_OIH_I1_J,axiom,
coinci554967470ide_hp @ a @ b @ c @ a2 ).
% hpsafe_Choice.IH(1)
thf(fact_2_prog__sem_H,axiom,
member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_22 @ nu_2 ) @ mu ) @ ( denota1661140910og_sem @ a @ b @ c @ j @ ( choice @ a @ b @ c @ a2 @ b2 ) ) ).
% prog_sem'
thf(fact_3_agree_H,axiom,
denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) @ mu @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) @ v ) ).
% agree'
thf(fact_4_prog__sem,axiom,
member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_22 @ nu_2 ) @ mu ) @ ( denota1661140910og_sem @ a @ b @ c @ j @ b2 ) ).
% prog_sem
thf(fact_5_safe_I2_J,axiom,
hpsafe @ a @ b @ c @ b2 ).
% safe(2)
thf(fact_6_safe_I1_J,axiom,
hpsafe @ a @ b @ c @ a2 ).
% safe(1)
thf(fact_7_agree,axiom,
denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) @ mu @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ b2 ) @ v ) ).
% agree
thf(fact_8__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060mu_062_H_H_O_A_092_060lbrakk_062_I_I_092_060nu_0622_M_A_092_060nu_0622_H_J_M_A_092_060mu_062_H_H_J_A_092_060in_062_Aprog__sem_AJ_Ab_059_AVagree_A_I_092_060mu_062_M_A_092_060mu_062_H_J_A_092_060mu_062_H_H_A_IMBV_Ab_A_092_060union_062_AV_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Mu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )] :
( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_22 @ nu_2 ) @ Mu ) @ ( denota1661140910og_sem @ a @ b @ c @ j @ b2 ) )
=> ~ ( denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) @ Mu @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ b2 ) @ v ) ) ) ).
% \<open>\<And>thesis. (\<And>\<mu>''. \<lbrakk>((\<nu>2, \<nu>2'), \<mu>'') \<in> prog_sem J b; Vagree (\<mu>, \<mu>') \<mu>'' (MBV b \<union> V)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_9_VA,axiom,
denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_22 @ nu_2 ) @ v ).
% VA
thf(fact_10_sup__Pair__Pair,axiom,
! [A: $tType,B: $tType] :
( ( ( sup @ B )
& ( sup @ A ) )
=> ! [A2: A,B2: B,C2: A,D: B] :
( ( sup_sup @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
= ( product_Pair @ A @ B @ ( sup_sup @ A @ A2 @ C2 ) @ ( sup_sup @ B @ B2 @ D ) ) ) ) ).
% sup_Pair_Pair
thf(fact_11_sem,axiom,
member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) ) @ ( denota1661140910og_sem @ a @ b @ c @ i @ ( choice @ a @ b @ c @ a2 @ b2 ) ) ).
% sem
thf(fact_12_sub,axiom,
ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVP @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) @ v ).
% sub
thf(fact_13_False,axiom,
~ ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) ) @ ( denota1661140910og_sem @ a @ b @ c @ i @ a2 ) ) ).
% False
thf(fact_14_hp_Oinject_I6_J,axiom,
! [C: $tType,B: $tType,A: $tType,X61: hp @ A @ B @ C,X62: hp @ A @ B @ C,Y61: hp @ A @ B @ C,Y62: hp @ A @ B @ C] :
( ( ( choice @ A @ B @ C @ X61 @ X62 )
= ( choice @ A @ B @ C @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% hp.inject(6)
thf(fact_15_UnCI,axiom,
! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
( ( ~ ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ A3 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_16_Un__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( ( member @ A @ C2 @ A3 )
| ( member @ A @ C2 @ B3 ) ) ) ).
% Un_iff
thf(fact_17_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F: A > B,G: A > B,X: A] : ( sup_sup @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% sup_apply
thf(fact_18_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_19_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A] :
( ( sup_sup @ A @ X2 @ X2 )
= X2 ) ) ).
% sup_idem
thf(fact_20_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.left_idem
thf(fact_21_subsetI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% subsetI
thf(fact_22_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_23_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.right_idem
thf(fact_24_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A] :
( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) )
= ( sup_sup @ A @ X2 @ Y ) ) ) ).
% sup_left_idem
thf(fact_25_sub2,axiom,
ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVP @ a @ b @ c @ b2 ) @ v ).
% sub2
thf(fact_26_sub1,axiom,
ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVP @ a @ b @ c @ a2 ) @ v ).
% sub1
thf(fact_27_Pair__le,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: B,C2: A,D: B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
= ( ( ord_less_eq @ A @ A2 @ C2 )
& ( ord_less_eq @ B @ B2 @ D ) ) ) ) ).
% Pair_le
thf(fact_28_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_29_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z )
= ( ( ord_less_eq @ A @ X2 @ Z )
& ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% le_sup_iff
thf(fact_30_Un__subset__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C3 )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
& ( ord_less_eq @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_31_hpsafe__Choice__simps,axiom,
! [C: $tType,B: $tType,A: $tType,A2: hp @ A @ B @ C,B2: hp @ A @ B @ C] :
( ( hpsafe @ A @ B @ C @ ( choice @ A @ B @ C @ A2 @ B2 ) )
= ( ( hpsafe @ A @ B @ C @ A2 )
& ( hpsafe @ A @ B @ C @ B2 ) ) ) ).
% hpsafe_Choice_simps
thf(fact_32_sem2,axiom,
member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) ) @ ( denota1661140910og_sem @ a @ b @ c @ i @ b2 ) ).
% sem2
thf(fact_33_eitherSem,axiom,
( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) ) @ ( denota1661140910og_sem @ a @ b @ c @ i @ a2 ) )
| ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) ) @ ( denota1661140910og_sem @ a @ b @ c @ i @ b2 ) ) ) ).
% eitherSem
thf(fact_34_in__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_35_subsetD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_36_equalityE,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_37_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
! [X: A] :
( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_38_equalityD1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_39_equalityD2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_40_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A4 )
=> ( member @ A @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_41_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_42_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_43_subset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_44_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z2: set @ A] : Y2 = Z2 )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
( ! [X3: A] :
( ( F2 @ X3 )
= ( G2 @ X3 ) )
=> ( F2 = G2 ) ) ).
% ext
thf(fact_49_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_50_Pair__mono,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [X2: A,X4: A,Y: B,Y3: B] :
( ( ord_less_eq @ A @ X2 @ X4 )
=> ( ( ord_less_eq @ B @ Y @ Y3 )
=> ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) ) ) ) ) ).
% Pair_mono
thf(fact_51_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_52_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_53_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( sup_sup @ A @ A5 @ B5 )
= B5 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_54_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( sup_sup @ A @ A5 @ B5 )
= A5 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_55_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded2
thf(fact_56_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded1
thf(fact_57_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( A5
= ( sup_sup @ A @ A5 @ B5 ) ) ) ) ) ).
% sup.order_iff
thf(fact_58_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% sup.boundedI
thf(fact_59_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.boundedE
thf(fact_60_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( sup_sup @ A @ X2 @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_61_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( sup_sup @ A @ X2 @ Y )
= X2 ) ) ) ).
% sup_absorb1
thf(fact_62_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb2
thf(fact_63_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb1
thf(fact_64_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F2: A > A > A,X2: A,Y: A] :
( ! [X3: A,Y4: A] : ( ord_less_eq @ A @ X3 @ ( F2 @ X3 @ Y4 ) )
=> ( ! [X3: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ ( F2 @ X3 @ Y4 ) )
=> ( ! [X3: A,Y4: A,Z3: A] :
( ( ord_less_eq @ A @ Y4 @ X3 )
=> ( ( ord_less_eq @ A @ Z3 @ X3 )
=> ( ord_less_eq @ A @ ( F2 @ Y4 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup @ A @ X2 @ Y )
= ( F2 @ X2 @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_65_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( sup_sup @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% sup.orderI
thf(fact_66_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2
= ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.orderE
thf(fact_67_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y5: A] :
( ( sup_sup @ A @ X @ Y5 )
= Y5 ) ) ) ) ).
% le_iff_sup
thf(fact_68_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X2: A,Z: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ Z @ X2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z ) @ X2 ) ) ) ) ).
% sup_least
thf(fact_69_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,C2: A,B2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C2 @ D ) ) ) ) ) ).
% sup_mono
thf(fact_70_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,D: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_71_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ X2 @ B2 )
=> ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_72_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X2 @ A2 )
=> ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_73_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X2: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% sup_ge2
thf(fact_74_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% sup_ge1
thf(fact_75_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,X2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X2 )
=> ( ( ord_less_eq @ A @ B2 @ X2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X2 ) ) ) ) ).
% le_supI
thf(fact_76_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A,X2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq @ A @ A2 @ X2 )
=> ~ ( ord_less_eq @ A @ B2 @ X2 ) ) ) ) ).
% le_supE
thf(fact_77_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_78_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X2: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_79_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_80_subset__UnE,axiom,
! [A: $tType,C3: set @ A,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ! [A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ A3 )
=> ! [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ B3 )
=> ( C3
!= ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_81_Un__absorb2,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
= A3 ) ) ).
% Un_absorb2
thf(fact_82_Un__absorb1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
= B3 ) ) ).
% Un_absorb1
thf(fact_83_Un__upper2,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_upper2
thf(fact_84_Un__upper1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_upper1
thf(fact_85_Un__least,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C3 ) ) ) ).
% Un_least
thf(fact_86_Un__mono,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,B3: set @ A,D2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ D2 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_87_hpsafe__fsafe_Ohpsafe__Choice,axiom,
! [C: $tType,B: $tType,A: $tType,A2: hp @ A @ B @ C,B2: hp @ A @ B @ C] :
( ( hpsafe @ A @ B @ C @ A2 )
=> ( ( hpsafe @ A @ B @ C @ B2 )
=> ( hpsafe @ A @ B @ C @ ( choice @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% hpsafe_fsafe.hpsafe_Choice
thf(fact_88_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X2 @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_89_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C2 ) )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_90_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X: A,Y5: A] : ( sup_sup @ A @ Y5 @ X ) ) ) ) ).
% sup_commute
thf(fact_91_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A5: A,B5: A] : ( sup_sup @ A @ B5 @ A5 ) ) ) ) ).
% sup.commute
thf(fact_92_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z )
= ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% sup_assoc
thf(fact_93_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C2 )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_94_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,K: A,B2: A,A2: A] :
( ( B3
= ( sup_sup @ A @ K @ B2 ) )
=> ( ( sup_sup @ A @ A2 @ B3 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_95_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( sup_sup @ A @ K @ A2 ) )
=> ( ( sup_sup @ A @ A3 @ B2 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_96_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F: A > B,G: A > B,X: A] : ( sup_sup @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% sup_fun_def
thf(fact_97_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X: A,Y5: A] : ( sup_sup @ A @ Y5 @ X ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_98_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z )
= ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_99_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X2 @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_100_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A] :
( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) )
= ( sup_sup @ A @ X2 @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_101_Un__left__commute,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_102_Un__left__absorb,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_left_absorb
thf(fact_103_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] : ( sup_sup @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_104_Un__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_105_Un__assoc,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Un_assoc
thf(fact_106_ball__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ( P @ X ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( P @ X ) )
& ! [X: A] :
( ( member @ A @ X @ B3 )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_107_bex__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
( ( ? [X: A] :
( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
& ( P @ X ) ) )
= ( ? [X: A] :
( ( member @ A @ X @ A3 )
& ( P @ X ) )
| ? [X: A] :
( ( member @ A @ X @ B3 )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_108_UnI2,axiom,
! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
( ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_109_UnI1,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_110_UnE,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ( ~ ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% UnE
thf(fact_111_IA,axiom,
denotational_Iagree @ a @ b @ c @ i @ j @ ( static_SIGP @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) ).
% IA
thf(fact_112_IH1,axiom,
! [I: denota1663640101rp_ext @ a @ b @ c @ product_unit,J: denota1663640101rp_ext @ a @ b @ c @ product_unit] :
( ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Mu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),V: set @ ( sum_sum @ c @ c )] :
( ( denotational_Iagree @ a @ b @ c @ I @ J @ ( static_SIGP @ a @ b @ c @ a2 ) )
=> ( ( denotational_Vagree @ c @ Nu @ Nu2 @ V )
=> ( ( ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVP @ a @ b @ c @ a2 ) @ V )
=> ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu @ Mu2 ) @ ( denota1661140910og_sem @ a @ b @ c @ I @ a2 ) )
=> ? [Mu3: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )] :
( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu2 @ Mu3 ) @ ( denota1661140910og_sem @ a @ b @ c @ J @ a2 ) )
& ( denotational_Vagree @ c @ Mu2 @ Mu3 @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ a2 ) @ V ) ) ) ) ) ) )
& ( coinci1495174342_equiv @ a @ b @ c @ a2 @ I ) ) ).
% IH1
thf(fact_113_IH2,axiom,
! [I: denota1663640101rp_ext @ a @ b @ c @ product_unit,J: denota1663640101rp_ext @ a @ b @ c @ product_unit] :
( ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Mu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),V: set @ ( sum_sum @ c @ c )] :
( ( denotational_Iagree @ a @ b @ c @ I @ J @ ( static_SIGP @ a @ b @ c @ b2 ) )
=> ( ( denotational_Vagree @ c @ Nu @ Nu2 @ V )
=> ( ( ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVP @ a @ b @ c @ b2 ) @ V )
=> ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu @ Mu2 ) @ ( denota1661140910og_sem @ a @ b @ c @ I @ b2 ) )
=> ? [Mu3: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )] :
( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu2 @ Mu3 ) @ ( denota1661140910og_sem @ a @ b @ c @ J @ b2 ) )
& ( denotational_Vagree @ c @ Mu2 @ Mu3 @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ b2 ) @ V ) ) ) ) ) ) )
& ( coinci1495174342_equiv @ a @ b @ c @ b2 @ I ) ) ).
% IH2
thf(fact_114_coincide__hp_H__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( ( coinci554967470ide_hp @ A @ B @ C )
= ( ^ [Alpha: hp @ A @ B @ C] :
! [I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( coinci32832645ide_hp @ A @ B @ C @ Alpha @ I2 @ J2 )
& ( coinci1495174342_equiv @ A @ B @ C @ Alpha @ I2 ) ) ) ) ) ).
% coincide_hp'_def
thf(fact_115_FVP_Osimps_I6_J,axiom,
! [C: $tType,B: $tType,A: $tType,Alpha2: hp @ A @ B @ C,Beta: hp @ A @ B @ C] :
( ( static_FVP @ A @ B @ C @ ( choice @ A @ B @ C @ Alpha2 @ Beta ) )
= ( sup_sup @ ( set @ ( sum_sum @ C @ C ) ) @ ( static_FVP @ A @ B @ C @ Alpha2 ) @ ( static_FVP @ A @ B @ C @ Beta ) ) ) ).
% FVP.simps(6)
thf(fact_116_agree__supset,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [B3: set @ ( sum_sum @ A @ A ),A3: set @ ( sum_sum @ A @ A ),Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu4: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ B3 @ A3 )
=> ( ( denotational_Vagree @ A @ Nu3 @ Nu4 @ A3 )
=> ( denotational_Vagree @ A @ Nu3 @ Nu4 @ B3 ) ) ) ) ).
% agree_supset
thf(fact_117_agree__sub,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A3: set @ ( sum_sum @ A @ A ),B3: set @ ( sum_sum @ A @ A ),Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Omega: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ A3 @ B3 )
=> ( ( denotational_Vagree @ A @ Nu3 @ Omega @ B3 )
=> ( denotational_Vagree @ A @ Nu3 @ Omega @ A3 ) ) ) ) ).
% agree_sub
thf(fact_118_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B7 ) )
= ( ( A2 = A7 )
& ( B2 = B7 ) ) ) ).
% old.prod.inject
thf(fact_119_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_120_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_121_Ssub_I2_J,axiom,
ord_less_eq @ ( set @ ( sum_sum @ a @ ( sum_sum @ b @ c ) ) ) @ ( static_SIGP @ a @ b @ c @ b2 ) @ ( static_SIGP @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) ).
% Ssub(2)
thf(fact_122_Ssub_I1_J,axiom,
ord_less_eq @ ( set @ ( sum_sum @ a @ ( sum_sum @ b @ c ) ) ) @ ( static_SIGP @ a @ b @ c @ a2 ) @ ( static_SIGP @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) ).
% Ssub(1)
thf(fact_123_IA2,axiom,
denotational_Iagree @ a @ b @ c @ i @ j @ ( static_SIGP @ a @ b @ c @ b2 ) ).
% IA2
thf(fact_124_IA1,axiom,
denotational_Iagree @ a @ b @ c @ i @ j @ ( static_SIGP @ a @ b @ c @ a2 ) ).
% IA1
thf(fact_125_coincide__hp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( ( coinci32832645ide_hp @ A @ B @ C )
= ( ^ [Alpha: hp @ A @ B @ C,I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu6: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Mu4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),V2: set @ ( sum_sum @ C @ C )] :
( ( denotational_Iagree @ A @ B @ C @ I2 @ J2 @ ( static_SIGP @ A @ B @ C @ Alpha ) )
=> ( ( denotational_Vagree @ C @ Nu5 @ Nu6 @ V2 )
=> ( ( ord_less_eq @ ( set @ ( sum_sum @ C @ C ) ) @ ( static_FVP @ A @ B @ C @ Alpha ) @ V2 )
=> ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu5 @ Mu4 ) @ ( denota1661140910og_sem @ A @ B @ C @ I2 @ Alpha ) )
=> ? [Mu5: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu6 @ Mu5 ) @ ( denota1661140910og_sem @ A @ B @ C @ J2 @ Alpha ) )
& ( denotational_Vagree @ C @ Mu4 @ Mu5 @ ( sup_sup @ ( set @ ( sum_sum @ C @ C ) ) @ ( static_MBV @ A @ B @ C @ Alpha ) @ V2 ) ) ) ) ) ) ) ) ) ) ).
% coincide_hp_def
thf(fact_126_Iagree__sub,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [A3: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),B3: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( ord_less_eq @ ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) @ A3 @ B3 )
=> ( ( denotational_Iagree @ A @ B @ C @ I3 @ J3 @ B3 )
=> ( denotational_Iagree @ A @ B @ C @ I3 @ J3 @ A3 ) ) ) ) ).
% Iagree_sub
thf(fact_127_Iagree__refl,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,A3: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] : ( denotational_Iagree @ A @ B @ C @ I3 @ I3 @ A3 ) ) ).
% Iagree_refl
thf(fact_128_Iagree__comm,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [A3: denota1663640101rp_ext @ A @ B @ C @ product_unit,B3: denota1663640101rp_ext @ A @ B @ C @ product_unit,V3: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] :
( ( denotational_Iagree @ A @ B @ C @ A3 @ B3 @ V3 )
=> ( denotational_Iagree @ A @ B @ C @ B3 @ A3 @ V3 ) ) ) ).
% Iagree_comm
thf(fact_129_SIGP_Osimps_I6_J,axiom,
! [C: $tType,B: $tType,A: $tType,A2: hp @ A @ B @ C,B2: hp @ A @ B @ C] :
( ( static_SIGP @ A @ B @ C @ ( choice @ A @ B @ C @ A2 @ B2 ) )
= ( sup_sup @ ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) @ ( static_SIGP @ A @ B @ C @ A2 ) @ ( static_SIGP @ A @ B @ C @ B2 ) ) ) ).
% SIGP.simps(6)
thf(fact_130_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_131_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_132_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_133_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A8: A,B8: A] :
( ( ord_less_eq @ A @ A8 @ B8 )
=> ( P @ A8 @ B8 ) )
=> ( ! [A8: A,B8: A] :
( ( P @ B8 @ A8 )
=> ( P @ A8 @ B8 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_134_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_135_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% order_trans
thf(fact_136_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_137_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_138_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_139_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_140_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv
thf(fact_141_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_142_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_143_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% le_cases
thf(fact_144_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 = Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% eq_refl
thf(fact_145_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linear
thf(fact_146_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( X2 = Y ) ) ) ) ).
% antisym
thf(fact_147_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [X: A,Y5: A] :
( ( ord_less_eq @ A @ X @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% eq_iff
thf(fact_148_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F2: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_149_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_150_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_151_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_152_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F: A > B,G: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_153_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G2 ) ) ) ).
% le_funI
thf(fact_154_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_funE
thf(fact_155_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_funD
thf(fact_156_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X3: A,Y4: B] :
( P2
= ( product_Pair @ A @ B @ X3 @ Y4 ) ) ).
% surj_pair
thf(fact_157_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A8: A,B8: B] : ( P @ ( product_Pair @ A @ B @ A8 @ B8 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_158_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B7 ) )
=> ~ ( ( A2 = A7 )
=> ( B2 != B7 ) ) ) ).
% Pair_inject
thf(fact_159_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A8: A,B8: B] :
( Y
!= ( product_Pair @ A @ B @ A8 @ B8 ) ) ).
% old.prod.exhaust
thf(fact_160_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A8: A,B8: B] : ( P @ ( product_Pair @ A @ B @ A8 @ B8 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_161_agree__comm,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),B3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),V3: set @ ( sum_sum @ A @ A )] :
( ( denotational_Vagree @ A @ A3 @ B3 @ V3 )
=> ( denotational_Vagree @ A @ B3 @ A3 @ V3 ) ) ) ).
% agree_comm
thf(fact_162_agree__refl,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A3: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu3 @ Nu3 @ A3 ) ) ).
% agree_refl
thf(fact_163_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A8: A,B8: B,C4: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A8 @ ( product_Pair @ B @ C @ B8 @ C4 ) ) ) ).
% prod_cases3
thf(fact_164_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) )] :
~ ! [A8: A,B8: B,C4: C,D4: D3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ D3 ) @ B8 @ ( product_Pair @ C @ D3 @ C4 @ D4 ) ) ) ) ).
% prod_cases4
thf(fact_165_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D3: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) )] :
~ ! [A8: A,B8: B,C4: C,D4: D3,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ E ) @ C4 @ ( product_Pair @ D3 @ E @ D4 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_166_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D3: $tType,E: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) )] :
~ ! [A8: A,B8: B,C4: C,D4: D3,E2: E,F4: F3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D3 @ ( product_prod @ E @ F3 ) @ D4 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_167_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D3: $tType,E: $tType,F3: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
~ ! [A8: A,B8: B,C4: C,D4: D3,E2: E,F4: F3,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C4 @ ( product_Pair @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_168_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A8: A,B8: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A8 @ ( product_Pair @ B @ C @ B8 @ C4 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_169_prod__induct4,axiom,
! [D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) )] :
( ! [A8: A,B8: B,C4: C,D4: D3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ D3 ) @ B8 @ ( product_Pair @ C @ D3 @ C4 @ D4 ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct4
thf(fact_170_prod__induct5,axiom,
! [E: $tType,D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) )] :
( ! [A8: A,B8: B,C4: C,D4: D3,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ E ) @ C4 @ ( product_Pair @ D3 @ E @ D4 @ E2 ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct5
thf(fact_171_prod__induct6,axiom,
! [F3: $tType,E: $tType,D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) )] :
( ! [A8: A,B8: B,C4: C,D4: D3,E2: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D3 @ ( product_prod @ E @ F3 ) @ D4 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct6
thf(fact_172_prod__induct7,axiom,
! [G3: $tType,F3: $tType,E: $tType,D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
( ! [A8: A,B8: B,C4: C,D4: D3,E2: E,F4: F3,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C4 @ ( product_Pair @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct7
thf(fact_173_agree__union,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Omega: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A3: set @ ( sum_sum @ A @ A ),B3: set @ ( sum_sum @ A @ A )] :
( ( denotational_Vagree @ A @ Nu3 @ Omega @ A3 )
=> ( ( denotational_Vagree @ A @ Nu3 @ Omega @ B3 )
=> ( denotational_Vagree @ A @ Nu3 @ Omega @ ( sup_sup @ ( set @ ( sum_sum @ A @ A ) ) @ A3 @ B3 ) ) ) ) ) ).
% agree_union
thf(fact_174_prog__sem_Osimps_I5_J,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,Alpha2: hp @ A @ B @ C,Beta: hp @ A @ B @ C] :
( ( denota1661140910og_sem @ A @ B @ C @ I3 @ ( choice @ A @ B @ C @ Alpha2 @ Beta ) )
= ( sup_sup @ ( set @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) @ ( denota1661140910og_sem @ A @ B @ C @ I3 @ Alpha2 ) @ ( denota1661140910og_sem @ A @ B @ C @ I3 @ Beta ) ) ) ) ).
% prog_sem.simps(5)
thf(fact_175_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_176_ids_Ocoincide__hp__def,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,B: $tType,A: $tType,C: $tType] :
( ( ( finite_finite @ C )
& ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Alpha2: hp @ A @ B @ C,I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( coinci32832645ide_hp @ A @ B @ C @ Alpha2 @ I3 @ J3 )
= ( ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu6: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Mu4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),V2: set @ ( sum_sum @ C @ C )] :
( ( denotational_Iagree @ A @ B @ C @ I3 @ J3 @ ( static_SIGP @ A @ B @ C @ Alpha2 ) )
=> ( ( denotational_Vagree @ C @ Nu5 @ Nu6 @ V2 )
=> ( ( ord_less_eq @ ( set @ ( sum_sum @ C @ C ) ) @ ( static_FVP @ A @ B @ C @ Alpha2 ) @ V2 )
=> ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu5 @ Mu4 ) @ ( denota1661140910og_sem @ A @ B @ C @ I3 @ Alpha2 ) )
=> ? [Mu5: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu6 @ Mu5 ) @ ( denota1661140910og_sem @ A @ B @ C @ J3 @ Alpha2 ) )
& ( denotational_Vagree @ C @ Mu4 @ Mu5 @ ( sup_sup @ ( set @ ( sum_sum @ C @ C ) ) @ ( static_MBV @ A @ B @ C @ Alpha2 ) @ V2 ) ) ) ) ) ) ) ) ) ) ) ).
% ids.coincide_hp_def
thf(fact_177_euclid__ext__aux_Ocases,axiom,
! [A: $tType] :
( ( euclid1678468529ng_gcd @ A )
=> ! [X2: product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) )] :
~ ! [S: A,S2: A,T3: A,T4: A,R: A,R2: A] :
( X2
!= ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) ) @ S @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) @ S2 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) @ T3 @ ( product_Pair @ A @ ( product_prod @ A @ A ) @ T4 @ ( product_Pair @ A @ A @ R @ R2 ) ) ) ) ) ) ) ).
% euclid_ext_aux.cases
thf(fact_178_ids_Ocoincide__hp_H__def,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Alpha2: hp @ A @ B @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( coinci554967470ide_hp @ A @ B @ C @ Alpha2 )
= ( ! [I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( coinci32832645ide_hp @ A @ B @ C @ Alpha2 @ I2 @ J2 )
& ( coinci1495174342_equiv @ A @ B @ C @ Alpha2 @ I2 ) ) ) ) ) ) ).
% ids.coincide_hp'_def
thf(fact_179_ids_Oseq__sem_Ocases,axiom,
! [Sz: $tType,Sc: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X2: product_prod @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) )] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ~ ! [I4: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,S3: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
( X2
!= ( product_Pair @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) @ I4 @ S3 ) ) ) ) ).
% ids.seq_sem.cases
thf(fact_180_ids_Oseq__sem_Oinduct,axiom,
! [Sz: $tType,Sc: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) > ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) > $o,A0: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,A1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ! [I4: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,X_1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] : ( P @ I4 @ X_1 )
=> ( P @ A0 @ A1 ) ) ) ) ).
% ids.seq_sem.induct
thf(fact_181_ids_Osingleton_Oinduct,axiom,
! [Sf: $tType,Sc: $tType,A: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( trm @ A @ Sz ) > Sz > $o,A0: trm @ A @ Sz,A1: Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ! [T4: trm @ A @ Sz,X_1: Sz] : ( P @ T4 @ X_1 )
=> ( P @ A0 @ A1 ) ) ) ) ).
% ids.singleton.induct
thf(fact_182_ids_Osingleton_Ocases,axiom,
! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X2: product_prod @ ( trm @ A @ Sz ) @ Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ~ ! [T4: trm @ A @ Sz,I5: Sz] :
( X2
!= ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ T4 @ I5 ) ) ) ) ).
% ids.singleton.cases
thf(fact_183_gcd_Ocases,axiom,
! [A: $tType] :
( ( euclid1155270486miring @ A )
=> ! [X2: product_prod @ A @ A] :
~ ! [A8: A,B8: A] :
( X2
!= ( product_Pair @ A @ A @ A8 @ B8 ) ) ) ).
% gcd.cases
thf(fact_184_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C2 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_185_same__fstI,axiom,
! [B: $tType,A: $tType,P: A > $o,X2: A,Y3: B,Y: B,R3: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P @ X2 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y3 @ Y ) @ ( R3 @ X2 ) )
=> ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( product_Pair @ A @ B @ X2 @ Y ) ) @ ( same_fst @ A @ B @ P @ R3 ) ) ) ) ).
% same_fstI
thf(fact_186_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ A ),As: A > B] :
! [I6: A,J4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I6 @ J4 ) @ R4 )
=> ( ord_less_eq @ B @ ( As @ I6 ) @ ( As @ J4 ) ) ) ) ) ) ).
% relChain_def
thf(fact_187_FunctionFrechet_Ocases,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ C )
& ( finite_finite @ B ) )
=> ! [X2: product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A] :
~ ! [I4: denota1663640101rp_ext @ A @ B @ C @ product_unit,I5: A] :
( X2
!= ( product_Pair @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A @ I4 @ I5 ) ) ) ).
% FunctionFrechet.cases
thf(fact_188_ids_Ovne23,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid2 != Vid3 ) ) ) ).
% ids.vne23
thf(fact_189_ids_Ovne13,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid1 != Vid3 ) ) ) ).
% ids.vne13
thf(fact_190_ids_Ovne12,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid1 != Vid2 ) ) ) ).
% ids.vne12
thf(fact_191_ids_Oid__simps_I24_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid4 != Pid3 ) ) ) ).
% ids.id_simps(24)
thf(fact_192_ids_Oid__simps_I23_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid4 != Pid2 ) ) ) ).
% ids.id_simps(23)
thf(fact_193_ids_Oid__simps_I22_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid4 != Pid1 ) ) ) ).
% ids.id_simps(22)
thf(fact_194_ids_Oid__simps_I21_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid3 != Pid1 ) ) ) ).
% ids.id_simps(21)
thf(fact_195_ids_Oid__simps_I20_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid3 != Pid2 ) ) ) ).
% ids.id_simps(20)
thf(fact_196_ids_Oid__simps_I19_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid2 != Pid1 ) ) ) ).
% ids.id_simps(19)
thf(fact_197_ids_Oid__simps_I18_J,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid3 != Fid1 ) ) ) ).
% ids.id_simps(18)
thf(fact_198_ids_Oid__simps_I17_J,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid3 != Fid2 ) ) ) ).
% ids.id_simps(17)
thf(fact_199_ids_Oid__simps_I16_J,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid2 != Fid1 ) ) ) ).
% ids.id_simps(16)
thf(fact_200_ids_Oid__simps_I15_J,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid3 != Vid1 ) ) ) ).
% ids.id_simps(15)
thf(fact_201_ids_Oid__simps_I14_J,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid3 != Vid2 ) ) ) ).
% ids.id_simps(14)
thf(fact_202_ids_Oid__simps_I13_J,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid2 != Vid1 ) ) ) ).
% ids.id_simps(13)
thf(fact_203_ids_Oid__simps_I12_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid3 != Pid4 ) ) ) ).
% ids.id_simps(12)
thf(fact_204_ids_Oid__simps_I11_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid2 != Pid4 ) ) ) ).
% ids.id_simps(11)
thf(fact_205_ids_Oid__simps_I10_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid4 ) ) ) ).
% ids.id_simps(10)
thf(fact_206_ids_Oid__simps_I9_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid3 ) ) ) ).
% ids.id_simps(9)
thf(fact_207_ids_Oid__simps_I8_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid2 != Pid3 ) ) ) ).
% ids.id_simps(8)
thf(fact_208_ids_Oid__simps_I7_J,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid2 ) ) ) ).
% ids.id_simps(7)
thf(fact_209_ids_Oid__simps_I6_J,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid1 != Fid3 ) ) ) ).
% ids.id_simps(6)
thf(fact_210_ids_Oid__simps_I5_J,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid2 != Fid3 ) ) ) ).
% ids.id_simps(5)
thf(fact_211_ids_Oid__simps_I4_J,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid1 != Fid2 ) ) ) ).
% ids.id_simps(4)
thf(fact_212_ids_Oid__simps_I3_J,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid1 != Vid3 ) ) ) ).
% ids.id_simps(3)
thf(fact_213_ids_Oid__simps_I2_J,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid2 != Vid3 ) ) ) ).
% ids.id_simps(2)
thf(fact_214_ids_Oid__simps_I1_J,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid1 != Vid2 ) ) ) ).
% ids.id_simps(1)
thf(fact_215_ids__def,axiom,
! [Sf: $tType,Sc: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc )
& ( finite_finite @ Sf ) )
=> ( ( ids @ Sz @ Sf @ Sc )
= ( ^ [Vid12: Sz,Vid22: Sz,Vid32: Sz,Fid12: Sf,Fid22: Sf,Fid32: Sf,Pid12: Sc,Pid22: Sc,Pid32: Sc,Pid42: Sc] :
( ( Vid12 != Vid22 )
& ( Vid22 != Vid32 )
& ( Vid12 != Vid32 )
& ( Fid12 != Fid22 )
& ( Fid22 != Fid32 )
& ( Fid12 != Fid32 )
& ( Pid12 != Pid22 )
& ( Pid22 != Pid32 )
& ( Pid12 != Pid32 )
& ( Pid12 != Pid42 )
& ( Pid22 != Pid42 )
& ( Pid32 != Pid42 ) ) ) ) ) ).
% ids_def
thf(fact_216_ids_Ofne12,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid1 != Fid2 ) ) ) ).
% ids.fne12
thf(fact_217_ids_Ofne13,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid1 != Fid3 ) ) ) ).
% ids.fne13
thf(fact_218_ids_Ofne23,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid2 != Fid3 ) ) ) ).
% ids.fne23
thf(fact_219_ids_Ointro,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( Vid1 != Vid2 )
=> ( ( Vid2 != Vid3 )
=> ( ( Vid1 != Vid3 )
=> ( ( Fid1 != Fid2 )
=> ( ( Fid2 != Fid3 )
=> ( ( Fid1 != Fid3 )
=> ( ( Pid1 != Pid2 )
=> ( ( Pid2 != Pid3 )
=> ( ( Pid1 != Pid3 )
=> ( ( Pid1 != Pid4 )
=> ( ( Pid2 != Pid4 )
=> ( ( Pid3 != Pid4 )
=> ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ids.intro
thf(fact_220_ids_Opne12,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid2 ) ) ) ).
% ids.pne12
thf(fact_221_ids_Opne13,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid3 ) ) ) ).
% ids.pne13
thf(fact_222_ids_Opne14,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid4 ) ) ) ).
% ids.pne14
thf(fact_223_ids_Opne23,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid2 != Pid3 ) ) ) ).
% ids.pne23
thf(fact_224_ids_Opne24,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid2 != Pid4 ) ) ) ).
% ids.pne24
thf(fact_225_ids_Opne34,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid3 != Pid4 ) ) ) ).
% ids.pne34
thf(fact_226_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A,Q: A > $o] :
( ( P @ X2 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X2 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ A @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_227_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A] :
( ( P @ X2 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X2 ) )
=> ( ( order_Greatest @ A @ P )
= X2 ) ) ) ) ).
% Greatest_equality
thf(fact_228_fst__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( sup @ A )
& ( sup @ B ) )
=> ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
( ( product_fst @ A @ B @ ( sup_sup @ ( product_prod @ A @ B ) @ X2 @ Y ) )
= ( sup_sup @ A @ ( product_fst @ A @ B @ X2 ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% fst_sup
thf(fact_229_snd__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( sup @ A )
& ( sup @ B ) )
=> ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
( ( product_snd @ A @ B @ ( sup_sup @ ( product_prod @ A @ B ) @ X2 @ Y ) )
= ( sup_sup @ B @ ( product_snd @ A @ B @ X2 ) @ ( product_snd @ A @ B @ Y ) ) ) ) ).
% snd_sup
thf(fact_230_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_231_snd__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ X2 @ Y )
=> ( ord_less_eq @ B @ ( product_snd @ A @ B @ X2 ) @ ( product_snd @ A @ B @ Y ) ) ) ) ).
% snd_mono
thf(fact_232_fst__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ X2 @ Y )
=> ( ord_less_eq @ A @ ( product_fst @ A @ B @ X2 ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% fst_mono
thf(fact_233_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y2: product_prod @ A @ B,Z2: product_prod @ A @ B] : Y2 = Z2 )
= ( ^ [S4: product_prod @ A @ B,T2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S4 )
= ( product_fst @ A @ B @ T2 ) )
& ( ( product_snd @ A @ B @ S4 )
= ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_234_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_235_prod__eqI,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P2 )
= ( product_fst @ A @ B @ Q2 ) )
=> ( ( ( product_snd @ A @ B @ P2 )
= ( product_snd @ A @ B @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_236_less__eq__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( ord_less_eq @ ( product_prod @ A @ B ) )
= ( ^ [X: product_prod @ A @ B,Y5: product_prod @ A @ B] :
( ( ord_less_eq @ A @ ( product_fst @ A @ B @ X ) @ ( product_fst @ A @ B @ Y5 ) )
& ( ord_less_eq @ B @ ( product_snd @ A @ B @ X ) @ ( product_snd @ A @ B @ Y5 ) ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_237_sup__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( ( sup @ A )
& ( sup @ B ) )
=> ( ( sup_sup @ ( product_prod @ A @ B ) )
= ( ^ [X: product_prod @ A @ B,Y5: product_prod @ A @ B] : ( product_Pair @ A @ B @ ( sup_sup @ A @ ( product_fst @ A @ B @ X ) @ ( product_fst @ A @ B @ Y5 ) ) @ ( sup_sup @ B @ ( product_snd @ A @ B @ X ) @ ( product_snd @ A @ B @ Y5 ) ) ) ) ) ) ).
% sup_prod_def
thf(fact_238_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X22: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_239_fst__eqD,axiom,
! [B: $tType,A: $tType,X2: A,Y: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y ) )
= A2 )
=> ( X2 = A2 ) ) ).
% fst_eqD
thf(fact_240_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_241_snd__eqD,axiom,
! [B: $tType,A: $tType,X2: B,Y: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_242_surjective__pairing,axiom,
! [B: $tType,A: $tType,T5: product_prod @ A @ B] :
( T5
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T5 ) @ ( product_snd @ A @ B @ T5 ) ) ) ).
% surjective_pairing
thf(fact_243_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_244_exI__realizer,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Y: A,X2: B] :
( ( P @ Y @ X2 )
=> ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_245_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_246_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X2: A,Y: B,A2: product_prod @ A @ B] :
( ( P @ X2 @ Y )
=> ( ( A2
= ( product_Pair @ A @ B @ X2 @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_247_sndI,axiom,
! [A: $tType,B: $tType,X2: product_prod @ A @ B,Y: A,Z: B] :
( ( X2
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_snd @ A @ B @ X2 )
= Z ) ) ).
% sndI
thf(fact_248_fstI,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,Y: A,Z: B] :
( ( X2
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_fst @ A @ B @ X2 )
= Y ) ) ).
% fstI
thf(fact_249_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P2 ) )
= ( ? [A5: B] :
( P2
= ( product_Pair @ B @ A @ A5 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_250_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
( ( A2
= ( product_fst @ A @ B @ P2 ) )
= ( ? [B5: B] :
( P2
= ( product_Pair @ A @ B @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_251_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_252_fst__swap,axiom,
! [A: $tType,B: $tType,X2: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X2 ) )
= ( product_snd @ B @ A @ X2 ) ) ).
% fst_swap
thf(fact_253_swap__swap,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
= P2 ) ).
% swap_swap
thf(fact_254_swap__simp,axiom,
! [A: $tType,B: $tType,X2: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
= ( product_Pair @ A @ B @ Y @ X2 ) ) ).
% swap_simp
thf(fact_255_snd__swap,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X2 ) )
= ( product_fst @ A @ B @ X2 ) ) ).
% snd_swap
% Subclasses (1)
thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( type @ A ) ) ).
% Type constructors (56)
thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( ordere890947078_space @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( ordere890947078_space @ A10 ) )
=> ( ordere890947078_space @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_2,axiom,
ordere890947078_space @ real ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A9: $tType,A10: $tType] :
( ( semilattice_sup @ A10 )
=> ( semilattice_sup @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A10: $tType] :
( ( preorder @ A10 )
=> ( preorder @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A9: $tType,A10: $tType] :
( ( lattice @ A10 )
=> ( lattice @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A10: $tType] :
( ( order @ A10 )
=> ( order @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A10: $tType] :
( ( ord @ A10 )
=> ( ord @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Osup,axiom,
! [A9: $tType,A10: $tType] :
( ( semilattice_sup @ A10 )
=> ( sup @ ( A9 > A10 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_3,axiom,
! [A9: $tType] : ( semilattice_sup @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 )
=> ( finite_finite @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_6,axiom,
! [A9: $tType] : ( lattice @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Osup_9,axiom,
! [A9: $tType] : ( sup @ ( set @ A9 ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_10,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_11,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_12,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_13,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Lattices_Osup_16,axiom,
sup @ $o ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_17,axiom,
semilattice_sup @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_18,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_19,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_20,axiom,
lattice @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_21,axiom,
order @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_22,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Lattices_Osup_23,axiom,
sup @ real ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_24,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( sum_sum @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__sup_25,axiom,
! [A9: $tType,A10: $tType] :
( ( ( semilattice_sup @ A9 )
& ( semilattice_sup @ A10 ) )
=> ( semilattice_sup @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Opreorder_26,axiom,
! [A9: $tType,A10: $tType] :
( ( ( preorder @ A9 )
& ( preorder @ A10 ) )
=> ( preorder @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_27,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Lattices_Olattice_28,axiom,
! [A9: $tType,A10: $tType] :
( ( ( lattice @ A9 )
& ( lattice @ A10 ) )
=> ( lattice @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder_29,axiom,
! [A9: $tType,A10: $tType] :
( ( ( order @ A9 )
& ( order @ A10 ) )
=> ( order @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oord_30,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ord @ A9 )
& ( ord @ A10 ) )
=> ( ord @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Lattices_Osup_31,axiom,
! [A9: $tType,A10: $tType] :
( ( ( sup @ A9 )
& ( sup @ A10 ) )
=> ( sup @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_32,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_33,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_34,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_35,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_36,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_37,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_38,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osup_39,axiom,
sup @ product_unit ).
thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Osemilattice__sup_40,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( semilattice_sup @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Opreorder_41,axiom,
! [A9: $tType,A10: $tType] :
( ( ( order @ A9 )
& ( finite_finite @ A10 ) )
=> ( preorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_42,axiom,
! [A9: $tType,A10: $tType] :
( ( ( linorder @ A9 )
& ( cARD_1 @ A10 ) )
=> ( linorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_43,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Olattice_44,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( lattice @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oorder_45,axiom,
! [A9: $tType,A10: $tType] :
( ( ( order @ A9 )
& ( finite_finite @ A10 ) )
=> ( order @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oord_46,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ord @ A9 )
& ( finite_finite @ A10 ) )
=> ( ord @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Osup_47,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( sup @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% Free types (3)
thf(tfree_0,hypothesis,
finite_finite @ c ).
thf(tfree_1,hypothesis,
finite_finite @ b ).
thf(tfree_2,hypothesis,
finite_finite @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
? [Mu6: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )] :
( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_22 @ nu_2 ) @ Mu6 ) @ ( denota1661140910og_sem @ a @ b @ c @ j @ ( choice @ a @ b @ c @ a2 @ b2 ) ) )
& ( denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) @ Mu6 @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) @ v ) ) ) ).
%------------------------------------------------------------------------------