TPTP Problem File: ITP028^2.p
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%------------------------------------------------------------------------------
% File : ITP028^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Algebra8 problem prob_962__6421504_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 : Algebra8/prob_962__6421504_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 322 ( 122 unt; 50 typ; 0 def)
% Number of atoms : 645 ( 220 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4431 ( 65 ~; 9 |; 40 &;4008 @)
% ( 0 <=>; 309 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 421 ( 421 >; 0 *; 0 +; 0 <<)
% Number of symbols : 51 ( 48 usr; 7 con; 0-9 aty)
% Number of variables : 1350 ( 77 ^;1168 !; 30 ?;1350 :)
% ( 75 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:30:50.699
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_e,type,
e: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (46)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[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_Orderings_Oorder__top,type,
order_top:
!>[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_BNF__Def_Ocsquare,type,
bNF_csquare:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( set @ A ) > ( B > C ) > ( D > C ) > ( A > B ) > ( A > D ) > $o ) ).
thf(sy_c_BNF__Def_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
thf(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Fun__Def_Oin__rel,type,
fun_in_rel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > A > B > $o ) ).
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_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : 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_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
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_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
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_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
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_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_ORange,type,
range:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Relation_Oconverse,type,
converse:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ A ) ) ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
thf(sy_c_Relation_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_C,type,
c: set @ ( product_prod @ ( set @ a ) @ ( a > e ) ) ).
thf(sy_v_a1,type,
a1: set @ a ).
thf(sy_v_a2,type,
a2: set @ a ).
thf(sy_v_a3,type,
a3: set @ a ).
thf(sy_v_b1,type,
b1: a > e ).
thf(sy_v_b2,type,
b2: a > e ).
thf(sy_v_b3,type,
b3: a > e ).
thf(sy_v_thesis____,type,
thesis: $o ).
% Relevant facts (254)
thf(fact_0__092_060open_062_092_060forall_062a_092_060in_062C_O_A_092_060forall_062b_092_060in_062C_O_Afst_Aa_A_092_060subseteq_062_Afst_Ab_A_092_060or_062_Afst_Ab_A_092_060subseteq_062_Afst_Aa_092_060close_062,axiom,
! [X: product_prod @ ( set @ a ) @ ( a > e )] :
( ( member @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ X @ c )
=> ! [Xa: product_prod @ ( set @ a ) @ ( a > e )] :
( ( member @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ Xa @ c )
=> ( ( ord_less_eq @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) @ X ) @ ( product_fst @ ( set @ a ) @ ( a > e ) @ Xa ) )
| ( ord_less_eq @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) @ Xa ) @ ( product_fst @ ( set @ a ) @ ( a > e ) @ X ) ) ) ) ) ).
% \<open>\<forall>a\<in>C. \<forall>b\<in>C. fst a \<subseteq> fst b \<or> fst b \<subseteq> fst a\<close>
thf(fact_1__092_060open_062_092_060exists_062a_092_060in_062fst_A_096_AC_O_Aa1_A_092_060subseteq_062_Aa_A_092_060and_062_Aa2_A_092_060subseteq_062_Aa_A_092_060and_062_Aa3_A_092_060subseteq_062_Aa_092_060close_062,axiom,
? [X2: set @ a] :
( ( member @ ( set @ a ) @ X2 @ ( image @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) ) @ c ) )
& ( ord_less_eq @ ( set @ a ) @ a1 @ X2 )
& ( ord_less_eq @ ( set @ a ) @ a2 @ X2 )
& ( ord_less_eq @ ( set @ a ) @ a3 @ X2 ) ) ).
% \<open>\<exists>a\<in>fst ` C. a1 \<subseteq> a \<and> a2 \<subseteq> a \<and> a3 \<subseteq> a\<close>
thf(fact_2__092_060open_062a1_A_092_060subseteq_062_Aa2_A_092_060or_062_Aa2_A_092_060subseteq_062_Aa1_092_060close_062,axiom,
( ( ord_less_eq @ ( set @ a ) @ a1 @ a2 )
| ( ord_less_eq @ ( set @ a ) @ a2 @ a1 ) ) ).
% \<open>a1 \<subseteq> a2 \<or> a2 \<subseteq> a1\<close>
thf(fact_3__092_060open_062a1_A_092_060subseteq_062_Aa3_A_092_060or_062_Aa3_A_092_060subseteq_062_Aa1_092_060close_062,axiom,
( ( ord_less_eq @ ( set @ a ) @ a1 @ a3 )
| ( ord_less_eq @ ( set @ a ) @ a3 @ a1 ) ) ).
% \<open>a1 \<subseteq> a3 \<or> a3 \<subseteq> a1\<close>
thf(fact_4__092_060open_062a2_A_092_060subseteq_062_Aa3_A_092_060or_062_Aa3_A_092_060subseteq_062_Aa2_092_060close_062,axiom,
( ( ord_less_eq @ ( set @ a ) @ a2 @ a3 )
| ( ord_less_eq @ ( set @ a ) @ a3 @ a2 ) ) ).
% \<open>a2 \<subseteq> a3 \<or> a3 \<subseteq> a2\<close>
thf(fact_5_A_I3_J,axiom,
member @ ( set @ a ) @ a3 @ ( image @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) ) @ c ) ).
% A(3)
thf(fact_6_A_I2_J,axiom,
member @ ( set @ a ) @ a2 @ ( image @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) ) @ c ) ).
% A(2)
thf(fact_7_A_I1_J,axiom,
member @ ( set @ a ) @ a1 @ ( image @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) ) @ c ) ).
% A(1)
thf(fact_8__092_060open_062_Ia1_M_Ab1_J_A_092_060in_062_AC_092_060close_062,axiom,
member @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( product_Pair @ ( set @ a ) @ ( a > e ) @ a1 @ b1 ) @ c ).
% \<open>(a1, b1) \<in> C\<close>
thf(fact_9__092_060open_062_Ia2_M_Ab2_J_A_092_060in_062_AC_092_060close_062,axiom,
member @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( product_Pair @ ( set @ a ) @ ( a > e ) @ a2 @ b2 ) @ c ).
% \<open>(a2, b2) \<in> C\<close>
thf(fact_10__092_060open_062_Ia3_M_Ab3_J_A_092_060in_062_AC_092_060close_062,axiom,
member @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( product_Pair @ ( set @ a ) @ ( a > e ) @ a3 @ b3 ) @ c ).
% \<open>(a3, b3) \<in> C\<close>
thf(fact_11_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( member @ A @ X2 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_12_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_13_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X3: B,A2: set @ B] :
( ( B3
= ( F @ X3 ) )
=> ( ( member @ B @ X3 @ A2 )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_14_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_15_image__mono,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% image_mono
thf(fact_16_image__subsetI,axiom,
! [A: $tType,B: $tType,A2: set @ A,F: A > B,B2: set @ B] :
( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( member @ B @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_17_subset__imageE,axiom,
! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
=> ~ ! [C2: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C2 @ A2 )
=> ( B2
!= ( image @ B @ A @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_18_image__subset__iff,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( member @ A @ ( F @ X4 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_19_subset__image__iff,axiom,
! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
& ( B2
= ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_20_PairE__lemma,axiom,
! [A: $tType,B: $tType,P: product_prod @ A @ B] :
? [X2: A,Y: B] :
( P
= ( product_Pair @ A @ B @ X2 @ Y ) ) ).
% PairE_lemma
thf(fact_21_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_22_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
& ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_23_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_24_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > A > $o,A3: A,B3: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P2 @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P2 @ B5 @ A5 )
=> ( P2 @ A5 @ B5 ) )
=> ( P2 @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_25_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_26_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_27_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_28_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_29_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_30_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
& ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_31_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv
thf(fact_32_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_33_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% order.trans
thf(fact_34_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% le_cases
thf(fact_35_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( X3 = Y3 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% eq_refl
thf(fact_36_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% linear
thf(fact_37_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ) ).
% antisym
thf(fact_38_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_39_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F: A > B,C3: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_40_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X2: B,Y: B] :
( ( ord_less_eq @ B @ X2 @ Y )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_41_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
=> ( ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_42_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X2: B,Y: B] :
( ( ord_less_eq @ B @ X2 @ Y )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_43_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_44_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G2: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% le_funI
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P2: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G2: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G2 @ X2 ) )
=> ( F = G2 ) ) ).
% ext
thf(fact_49_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G2: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G2 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_funE
thf(fact_50_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G2: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G2 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_funD
thf(fact_51_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X3: A,A2: set @ A,B3: B,F: A > B] :
( ( member @ A @ X3 @ A2 )
=> ( ( B3
= ( F @ X3 ) )
=> ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_52_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( image @ B @ A @ F @ A2 ) )
=> ( P2 @ X2 ) )
=> ! [X: B] :
( ( member @ B @ X @ A2 )
=> ( P2 @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_53_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G2: A > B] :
( ( M = N )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ N )
=> ( ( F @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_54_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
( ? [X: A] :
( ( member @ A @ X @ ( image @ B @ A @ F @ A2 ) )
& ( P2 @ X ) )
=> ? [X2: B] :
( ( member @ B @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_55_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F: B > A,A2: set @ B] :
( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A2 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A2 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_56_imageI,axiom,
! [B: $tType,A: $tType,X3: A,A2: set @ A,F: A > B] :
( ( member @ A @ X3 @ A2 )
=> ( member @ B @ ( F @ X3 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% imageI
thf(fact_57_Collect__mono__iff,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
= ( ! [X4: A] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_58_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z: set @ A] : Y2 = Z )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_59_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_60_Collect__mono,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_61_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A6 )
=> ( member @ A @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_62_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_63_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( member @ A @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_64_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_65_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C3 @ A2 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_66_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_67_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_68_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A7: A,B7: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A7 @ B7 ) )
= ( ( A3 = A7 )
& ( B3 = B7 ) ) ) ).
% old.prod.inject
thf(fact_69_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A3: A,P: product_prod @ A @ B] :
( ( A3
= ( product_fst @ A @ B @ P ) )
= ( ? [B4: B] :
( P
= ( product_Pair @ A @ B @ A3 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_70_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X22: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_71_fst__eqD,axiom,
! [B: $tType,A: $tType,X3: A,Y3: B,A3: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y3 ) )
= A3 )
=> ( X3 = A3 ) ) ).
% fst_eqD
thf(fact_72_fstI,axiom,
! [B: $tType,A: $tType,X3: product_prod @ A @ B,Y3: A,Z2: B] :
( ( X3
= ( product_Pair @ A @ B @ Y3 @ Z2 ) )
=> ( ( product_fst @ A @ B @ X3 )
= Y3 ) ) ).
% fstI
thf(fact_73_all__subset__image,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: ( set @ A ) > $o] :
( ( ! [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F @ A2 ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A2 )
=> ( P2 @ ( image @ B @ A @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_74_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R: set @ ( product_prod @ A @ A ),As: A > B] :
! [I: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
=> ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_75_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o,X3: A] :
( ( P2 @ X3 )
=> ( ! [Y: A] :
( ( P2 @ Y )
=> ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ( order_Greatest @ A @ P2 )
= X3 ) ) ) ) ).
% Greatest_equality
thf(fact_76_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o,X3: A,Q: A > $o] :
( ( P2 @ X3 )
=> ( ! [Y: A] :
( ( P2 @ Y )
=> ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ! [X2: A] :
( ( P2 @ X2 )
=> ( ! [Y5: A] :
( ( P2 @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest @ A @ P2 ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_77_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_78_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
~ ! [A5: A,B5: B] :
( Y3
!= ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_79_prod__induct7,axiom,
! [G3: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
( ! [A5: A,B5: B,C5: C,D2: D,E2: E,F4: F3,G4: G3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct7
thf(fact_80_prod__induct6,axiom,
! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
( ! [A5: A,B5: B,C5: C,D2: D,E2: E,F4: F3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct6
thf(fact_81_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A5: A,B5: B,C5: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C5 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct5
thf(fact_82_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B5: B,C5: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C5 @ D2 ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct4
thf(fact_83_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B5: B,C5: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C5 ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct3
thf(fact_84_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
~ ! [A5: A,B5: B,C5: C,D2: D,E2: E,F4: F3,G4: G3] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_85_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
~ ! [A5: A,B5: B,C5: C,D2: D,E2: E,F4: F3] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_86_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A5: A,B5: B,C5: C,D2: D,E2: E] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C5 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_87_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B5: B,C5: C,D2: D] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C5 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_88_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B5: B,C5: C] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C5 ) ) ) ).
% prod_cases3
thf(fact_89_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A7: A,B7: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A7 @ B7 ) )
=> ~ ( ( A3 = A7 )
=> ( B3 != B7 ) ) ) ).
% Pair_inject
thf(fact_90_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_91_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( F1 @ A3 @ B3 ) ) ).
% old.prod.rec
thf(fact_92_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A3: B,B3: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( C3 @ A3 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_93_image__Fpow__mono,axiom,
! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( finite_Fpow @ B @ A2 ) ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_94_not__sub,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ? [A5: A] :
( ( member @ A @ A5 @ A2 )
& ~ ( member @ A @ A5 @ B2 ) ) ) ).
% not_sub
thf(fact_95_eqsets__sub,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% eqsets_sub
thf(fact_96_sub__which1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
( ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
| ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) )
=> ( ( member @ A @ X3 @ A2 )
=> ( ~ ( member @ A @ X3 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ) ).
% sub_which1
thf(fact_97_Fpow__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A2 ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% Fpow_mono
thf(fact_98_le__convert,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ B3 @ C3 ) ) ) ) ).
% le_convert
thf(fact_99_ge__convert,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ C3 @ B3 ) ) ) ) ).
% ge_convert
thf(fact_100_mem__in__image3,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,A2: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) )
=> ? [X2: B] :
( ( member @ B @ X2 @ A2 )
& ( B3
= ( F @ X2 ) ) ) ) ).
% mem_in_image3
thf(fact_101_mem__in__image2,axiom,
! [B: $tType,A: $tType,A3: A,A2: set @ A,F: A > B] :
( ( member @ A @ A3 @ A2 )
=> ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% mem_in_image2
thf(fact_102_mem__in__image1,axiom,
! [B: $tType,A: $tType,A2: set @ A,F: A > B,B2: set @ B,A3: A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( member @ B @ ( F @ X2 ) @ B2 ) )
=> ( ( member @ A @ A3 @ A2 )
=> ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% mem_in_image1
thf(fact_103_proper__subset,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ~ ( member @ A @ X3 @ A2 )
=> ( ( member @ A @ X3 @ B2 )
=> ( A2 != B2 ) ) ) ) ).
% proper_subset
thf(fact_104_not__subseteq,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A2 )
& ~ ( member @ A @ X2 @ B2 ) ) ) ).
% not_subseteq
thf(fact_105_subset__self,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_self
thf(fact_106_sets__not__eq,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 != B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A2 )
& ~ ( member @ A @ X2 @ B2 ) ) ) ) ).
% sets_not_eq
thf(fact_107_sub__which2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
( ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
| ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) )
=> ( ~ ( member @ A @ X3 @ A2 )
=> ( ( member @ A @ X3 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% sub_which2
thf(fact_108_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) ) @ ( pow @ A @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_109_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C4: B > A,D3: B > A,Sup: ( set @ A ) > A] :
( ( A2 = B2 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ B2 )
=> ( ( C4 @ X2 )
= ( D3 @ X2 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C4 @ A2 ) )
= ( Sup @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_110_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C4: B > A,D3: B > A,Inf: ( set @ A ) > A] :
( ( A2 = B2 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ B2 )
=> ( ( C4 @ X2 )
= ( D3 @ X2 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C4 @ A2 ) )
= ( Inf @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_111_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X2: A,Y: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ).
% subrelI
thf(fact_112_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S: B,R3: set @ ( product_prod @ A @ B ),S2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_113_PowI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) ) ) ).
% PowI
thf(fact_114_Pow__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% Pow_iff
thf(fact_115_Pow__top,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( set @ A ) @ A2 @ ( pow @ A @ A2 ) ) ).
% Pow_top
thf(fact_116_PowD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% PowD
thf(fact_117_Pow__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A2 ) @ ( pow @ A @ B2 ) ) ) ).
% Pow_mono
thf(fact_118_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
( ( ( image @ B @ A @ F @ A2 )
= B2 )
=> ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) )
= ( pow @ A @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_119_Fpow__subset__Pow,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A2 ) @ ( pow @ A @ A2 ) ) ).
% Fpow_subset_Pow
thf(fact_120_Cantors__paradox,axiom,
! [A: $tType,A2: set @ A] :
~ ? [F5: A > ( set @ A )] :
( ( image @ A @ ( set @ A ) @ F5 @ A2 )
= ( pow @ A @ A2 ) ) ).
% Cantors_paradox
thf(fact_121_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F2: ( product_prod @ B @ C ) > A,A4: B,B4: C] : ( F2 @ ( product_Pair @ B @ C @ A4 @ B4 ) ) ) ) ).
% curry_conv
thf(fact_122_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R: A > A > $o,F2: B > A,X4: B,Y4: B] : ( R @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) ).
% in_inv_imagep
thf(fact_123_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( A3 = B3 )
| ~ ( ord_less_eq @ A @ A3 @ B3 )
| ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% verit_la_disequality
thf(fact_124_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X5: $o > A,Y6: $o > A] :
( ( ord_less_eq @ A @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq @ A @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_125_curryI,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
( ( F @ ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ( product_curry @ A @ B @ $o @ F @ A3 @ B3 ) ) ).
% curryI
thf(fact_126_curryD,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
( ( product_curry @ A @ B @ $o @ F @ A3 @ B3 )
=> ( F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% curryD
thf(fact_127_curryE,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
( ( product_curry @ A @ B @ $o @ F @ A3 @ B3 )
=> ( F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% curryE
thf(fact_128_in__inv__image,axiom,
! [A: $tType,B: $tType,X3: A,Y3: A,R2: set @ ( product_prod @ B @ B ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( inv_image @ B @ A @ R2 @ F ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_129_fst__eq__Domain,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ R3 )
= ( domain @ A @ B @ R3 ) ) ).
% fst_eq_Domain
thf(fact_130_Domain__fst,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) ).
% Domain_fst
thf(fact_131_Range__mono,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S )
=> ( ord_less_eq @ ( set @ B ) @ ( range @ A @ B @ R2 ) @ ( range @ A @ B @ S ) ) ) ).
% Range_mono
thf(fact_132_RangeE,axiom,
! [A: $tType,B: $tType,B3: A,R2: set @ ( product_prod @ B @ A )] :
( ( member @ A @ B3 @ ( range @ B @ A @ R2 ) )
=> ~ ! [A5: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A5 @ B3 ) @ R2 ) ) ).
% RangeE
thf(fact_133_Range__iff,axiom,
! [A: $tType,B: $tType,A3: A,R2: set @ ( product_prod @ B @ A )] :
( ( member @ A @ A3 @ ( range @ B @ A @ R2 ) )
= ( ? [Y4: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y4 @ A3 ) @ R2 ) ) ) ).
% Range_iff
thf(fact_134_Range_Ocases,axiom,
! [B: $tType,A: $tType,A3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A3 @ ( range @ A @ B @ R2 ) )
=> ~ ! [A5: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ A3 ) @ R2 ) ) ).
% Range.cases
thf(fact_135_Range_Osimps,axiom,
! [B: $tType,A: $tType,A3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A3 @ ( range @ A @ B @ R2 ) )
= ( ? [A4: A,B4: B] :
( ( A3 = B4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R2 ) ) ) ) ).
% Range.simps
thf(fact_136_Range_Ointros,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R2 )
=> ( member @ B @ B3 @ ( range @ A @ B @ R2 ) ) ) ).
% Range.intros
thf(fact_137_Range_Oinducts,axiom,
! [A: $tType,B: $tType,X3: B,R2: set @ ( product_prod @ A @ B ),P2: B > $o] :
( ( member @ B @ X3 @ ( range @ A @ B @ R2 ) )
=> ( ! [A5: A,B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R2 )
=> ( P2 @ B5 ) )
=> ( P2 @ X3 ) ) ) ).
% Range.inducts
thf(fact_138_DomainE,axiom,
! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
=> ~ ! [B5: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R2 ) ) ).
% DomainE
thf(fact_139_Domain__iff,axiom,
! [A: $tType,B: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
= ( ? [Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y4 ) @ R2 ) ) ) ).
% Domain_iff
thf(fact_140_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
=> ~ ! [B5: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R2 ) ) ).
% Domain.cases
thf(fact_141_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
= ( ? [A4: A,B4: B] :
( ( A3 = A4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R2 ) ) ) ) ).
% Domain.simps
thf(fact_142_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R2 )
=> ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) ) ) ).
% Domain.DomainI
thf(fact_143_Domain_Oinducts,axiom,
! [B: $tType,A: $tType,X3: A,R2: set @ ( product_prod @ A @ B ),P2: A > $o] :
( ( member @ A @ X3 @ ( domain @ A @ B @ R2 ) )
=> ( ! [A5: A,B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R2 )
=> ( P2 @ A5 ) )
=> ( P2 @ X3 ) ) ) ).
% Domain.inducts
thf(fact_144_Domain__mono,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S )
=> ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R2 ) @ ( domain @ A @ B @ S ) ) ) ).
% Domain_mono
thf(fact_145_Range__converse,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( range @ B @ A @ ( converse @ A @ B @ R2 ) )
= ( domain @ A @ B @ R2 ) ) ).
% Range_converse
thf(fact_146_Domain__converse,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( domain @ A @ B @ ( converse @ B @ A @ R2 ) )
= ( range @ B @ A @ R2 ) ) ).
% Domain_converse
thf(fact_147_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y3: A,X3: B,A2: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ X3 ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A2 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ A2 ) ) ).
% pair_in_swap_image
thf(fact_148_swap__simp,axiom,
! [A: $tType,B: $tType,X3: B,Y3: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y3 ) )
= ( product_Pair @ A @ B @ Y3 @ X3 ) ) ).
% swap_simp
thf(fact_149_converse__converse,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( converse @ B @ A @ ( converse @ A @ B @ R2 ) )
= R2 ) ).
% converse_converse
thf(fact_150_converse__inject,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A )] :
( ( ( converse @ B @ A @ R2 )
= ( converse @ B @ A @ S ) )
= ( R2 = S ) ) ).
% converse_inject
thf(fact_151_swap__swap,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
= P ) ).
% swap_swap
thf(fact_152_converse__iff,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,R2: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( converse @ B @ A @ R2 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A3 ) @ R2 ) ) ).
% converse_iff
thf(fact_153_converse__mono,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S ) ) ).
% converse_mono
thf(fact_154_converse__inv__image,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ B @ B ),F: A > B] :
( ( converse @ A @ A @ ( inv_image @ B @ A @ R3 @ F ) )
= ( inv_image @ B @ A @ ( converse @ B @ B @ R3 ) @ F ) ) ).
% converse_inv_image
thf(fact_155_converse_Oinducts,axiom,
! [B: $tType,A: $tType,X1: B,X22: A,R2: set @ ( product_prod @ A @ B ),P2: B > A > $o] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X1 @ X22 ) @ ( converse @ A @ B @ R2 ) )
=> ( ! [A5: A,B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R2 )
=> ( P2 @ B5 @ A5 ) )
=> ( P2 @ X1 @ X22 ) ) ) ).
% converse.inducts
thf(fact_156_converse_Ointros,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R2 )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A3 ) @ ( converse @ A @ B @ R2 ) ) ) ).
% converse.intros
thf(fact_157_converse_Osimps,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R2 ) )
= ( ? [A4: A,B4: B] :
( ( A1 = B4 )
& ( A22 = A4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_158_converse_Ocases,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R2 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A22 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_159_converseE,axiom,
! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R2: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R2 ) )
=> ~ ! [X2: B,Y: A] :
( ( Yx
= ( product_Pair @ A @ B @ Y @ X2 ) )
=> ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y ) @ R2 ) ) ) ).
% converseE
thf(fact_160_converseD,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,R2: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( converse @ B @ A @ R2 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A3 ) @ R2 ) ) ).
% converseD
thf(fact_161_converse__subset__swap,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ ( converse @ B @ A @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( converse @ A @ B @ R2 ) @ S ) ) ).
% converse_subset_swap
thf(fact_162_surj__swap,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% surj_swap
thf(fact_163_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_164_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B ),F: A > C,G2: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A3 ) @ ( G2 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G2 ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_165_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_166_UNIV__I,axiom,
! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_167_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G2: D > B,A3: C,B3: D] :
( ( product_map_prod @ C @ A @ D @ B @ F @ G2 @ ( product_Pair @ C @ D @ A3 @ B3 ) )
= ( product_Pair @ A @ B @ ( F @ A3 ) @ ( G2 @ B3 ) ) ) ).
% map_prod_simp
thf(fact_168_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_169_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G2: D > B,X3: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G2 @ X3 ) )
= ( F @ ( product_fst @ C @ D @ X3 ) ) ) ).
% fst_map_prod
thf(fact_170_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G2: D > A,X3: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G2 @ X3 ) )
= ( G2 @ ( product_snd @ C @ D @ X3 ) ) ) ).
% snd_map_prod
thf(fact_171_converse__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( converse @ B @ A @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% converse_UNIV
thf(fact_172_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_173_fst__swap,axiom,
! [A: $tType,B: $tType,X3: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X3 ) )
= ( product_snd @ B @ A @ X3 ) ) ).
% fst_swap
thf(fact_174_snd__swap,axiom,
! [B: $tType,A: $tType,X3: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X3 ) )
= ( product_fst @ A @ B @ X3 ) ) ).
% snd_swap
thf(fact_175_range__fst,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_fst
thf(fact_176_range__snd,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_snd
thf(fact_177_rangeI,axiom,
! [A: $tType,B: $tType,F: B > A,X3: B] : ( member @ A @ ( F @ X3 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_178_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X3: B] :
( ( B3
= ( F @ X3 ) )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_179_surj__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y4: A] :
? [X4: B] :
( Y4
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_180_surjI,axiom,
! [B: $tType,A: $tType,G2: B > A,F: A > B] :
( ! [X2: A] :
( ( G2 @ ( F @ X2 ) )
= X2 )
=> ( ( image @ B @ A @ G2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_181_surjE,axiom,
! [A: $tType,B: $tType,F: B > A,Y3: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X2: B] :
( Y3
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_182_surjD,axiom,
! [A: $tType,B: $tType,F: B > A,Y3: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X2: B] :
( Y3
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_183_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G2: C > D] :
( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image @ C @ D @ G2 @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G2 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_184_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_185_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_186_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_187_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_188_sndI,axiom,
! [A: $tType,B: $tType,X3: product_prod @ A @ B,Y3: A,Z2: B] :
( ( X3
= ( product_Pair @ A @ B @ Y3 @ Z2 ) )
=> ( ( product_snd @ A @ B @ X3 )
= Z2 ) ) ).
% sndI
thf(fact_189_snd__eqD,axiom,
! [B: $tType,A: $tType,X3: B,Y3: A,A3: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y3 ) )
= A3 )
=> ( Y3 = A3 ) ) ).
% snd_eqD
thf(fact_190_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_191_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B3: A,P: product_prod @ B @ A] :
( ( B3
= ( product_snd @ B @ A @ P ) )
= ( ? [A4: B] :
( P
= ( product_Pair @ B @ A @ A4 @ B3 ) ) ) ) ).
% eq_snd_iff
thf(fact_192_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y2: product_prod @ A @ B,Z: product_prod @ A @ B] : Y2 = Z )
= ( ^ [S3: product_prod @ A @ B,T2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S3 )
= ( product_fst @ A @ B @ T2 ) )
& ( ( product_snd @ A @ B @ S3 )
= ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_193_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_194_prod__eqI,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,Q2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P )
= ( product_fst @ A @ B @ Q2 ) )
=> ( ( ( product_snd @ A @ B @ P )
= ( product_snd @ A @ B @ Q2 ) )
=> ( P = Q2 ) ) ) ).
% prod_eqI
thf(fact_195_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X2: A] : ( member @ A @ X2 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_196_UNIV__witness,axiom,
! [A: $tType] :
? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_197_surjective__pairing,axiom,
! [B: $tType,A: $tType,T3: product_prod @ A @ B] :
( T3
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T3 ) @ ( product_snd @ A @ B @ T3 ) ) ) ).
% surjective_pairing
thf(fact_198_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_199_snd__eq__Range,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A )] :
( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ R3 )
= ( range @ B @ A @ R3 ) ) ).
% snd_eq_Range
thf(fact_200_Range__snd,axiom,
! [A: $tType,B: $tType] :
( ( range @ B @ A )
= ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) ) ) ).
% Range_snd
thf(fact_201_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F: C > A,G2: D > B,R3: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G2 ) @ R3 ) )
=> ~ ! [X2: C,Y: D] :
( ( C3
= ( product_Pair @ A @ B @ ( F @ X2 ) @ ( G2 @ Y ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X2 @ Y ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_202_exI__realizer,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Y3: A,X3: B] :
( ( P2 @ Y3 @ X3 )
=> ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y3 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_203_conjI__realizer,axiom,
! [A: $tType,B: $tType,P2: A > $o,P: A,Q: B > $o,Q2: B] :
( ( P2 @ P )
=> ( ( Q @ Q2 )
=> ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_204_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,X3: A,Y3: B,A3: product_prod @ A @ B] :
( ( P2 @ X3 @ Y3 )
=> ( ( A3
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ( P2 @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_205_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_206_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_207_exE__realizer_H,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A] :
( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
=> ~ ! [X2: B,Y: A] :
~ ( P2 @ Y @ X2 ) ) ).
% exE_realizer'
thf(fact_208_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X6: set @ A,A2: set @ ( product_prod @ A @ B ),Y7: set @ B,P2: A > B > $o,Q: A > B > $o] :
( ( X6
= ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A2 ) )
=> ( ( Y7
= ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A2 ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ X6 )
=> ! [Xa2: B] :
( ( member @ B @ Xa2 @ Y7 )
=> ( ( P2 @ X2 @ Xa2 )
=> ( Q @ X2 @ Xa2 ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_209_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P4: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_210_top1I,axiom,
! [A: $tType,X3: A] : ( top_top @ ( A > $o ) @ X3 ) ).
% top1I
thf(fact_211_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
= F ) ).
% curry_case_prod
thf(fact_212_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
= F ) ).
% case_prod_curry
thf(fact_213_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X3: product_prod @ A @ B,A2: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A2 ) ) )
=> ( A2 @ ( product_fst @ A @ B @ X3 ) @ ( product_snd @ A @ B @ X3 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_214_prod_Ocase__eq__if,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F2: A > B > C,Prod3: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% prod.case_eq_if
thf(fact_215_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F2: B > C > A,P3: product_prod @ B @ C] : ( F2 @ ( product_fst @ B @ C @ P3 ) @ ( product_snd @ B @ C @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_216_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( produc2004651681e_prod @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% internal_case_prod_def
thf(fact_217_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A2: A > B > $o,B2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A2 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B2 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_218_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_219_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_220_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_221_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P4: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_222_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F: A > B > C,G2: A > B > C,P: product_prod @ A @ B] :
( ! [X2: A,Y: B] :
( ( ( product_Pair @ A @ B @ X2 @ Y )
= Q2 )
=> ( ( F @ X2 @ Y )
= ( G2 @ X2 @ Y ) ) )
=> ( ( P = Q2 )
=> ( ( product_case_prod @ A @ B @ C @ F @ P )
= ( product_case_prod @ A @ B @ C @ G2 @ Q2 ) ) ) ) ).
% split_cong
thf(fact_223_predicate2I,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Q: A > B > $o] :
( ! [X2: A,Y: B] :
( ( P2 @ X2 @ Y )
=> ( Q @ X2 @ Y ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q ) ) ).
% predicate2I
thf(fact_224_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C3: B > C > ( set @ A ),P: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P ) )
=> ~ ! [X2: B,Y: C] :
( ( P
= ( product_Pair @ B @ C @ X2 @ Y ) )
=> ~ ( member @ A @ Z2 @ ( C3 @ X2 @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_225_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,X3: A,Y3: B,Q: A > B > $o] :
( ( P2 @ X3 @ Y3 )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
=> ( Q @ X3 @ Y3 ) ) ) ).
% rev_predicate2D
thf(fact_226_predicate2D,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Q: A > B > $o,X3: A,Y3: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
=> ( ( P2 @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) ).
% predicate2D
thf(fact_227_refl__ge__eq,axiom,
! [A: $tType,R3: A > A > $o] :
( ! [X2: A] : ( R3 @ X2 @ X2 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y2: A,Z: A] : Y2 = Z
@ R3 ) ) ).
% refl_ge_eq
thf(fact_228_ge__eq__refl,axiom,
! [A: $tType,R3: A > A > $o,X3: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y2: A,Z: A] : Y2 = Z
@ R3 )
=> ( R3 @ X3 @ X3 ) ) ).
% ge_eq_refl
thf(fact_229_Collect__case__prod__in__rel__leE,axiom,
! [B: $tType,A: $tType,X6: set @ ( product_prod @ A @ B ),Y7: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( fun_in_rel @ A @ B @ Y7 ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ Y7 ) ) ).
% Collect_case_prod_in_rel_leE
thf(fact_230_Collect__case__prod__in__rel__leI,axiom,
! [B: $tType,A: $tType,X6: set @ ( product_prod @ A @ B ),Y7: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ Y7 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( fun_in_rel @ A @ B @ Y7 ) ) ) ) ) ).
% Collect_case_prod_in_rel_leI
thf(fact_231_in__rel__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_in_rel @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B ),X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R4 ) ) ) ).
% in_rel_def
thf(fact_232_csquare__fstOp__sndOp,axiom,
! [A: $tType,B: $tType,C: $tType,F: ( A > B > $o ) > ( product_prod @ A @ B ) > $o,P2: A > C > $o,Q: C > B > $o] : ( bNF_csquare @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ C @ ( product_prod @ C @ B ) @ ( collect @ ( product_prod @ A @ B ) @ ( F @ ( relcompp @ A @ C @ B @ P2 @ Q ) ) ) @ ( product_snd @ A @ C ) @ ( product_fst @ C @ B ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q ) ) ).
% csquare_fstOp_sndOp
thf(fact_233_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Q: A > B > $o,R3: $o,X3: A,Y3: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
& R3 )
=> ( R3
& ( ( P2 @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) ) ).
% predicate2D_conj
thf(fact_234_fstOp__in,axiom,
! [B: $tType,C: $tType,A: $tType,Ac2: product_prod @ A @ B,P2: A > C > $o,Q: C > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P2 @ Q ) ) ) )
=> ( member @ ( product_prod @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q @ Ac2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ P2 ) ) ) ) ).
% fstOp_in
thf(fact_235_sndOp__in,axiom,
! [A: $tType,B: $tType,C: $tType,Ac2: product_prod @ A @ B,P2: A > C > $o,Q: C > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P2 @ Q ) ) ) )
=> ( member @ ( product_prod @ C @ B ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q @ Ac2 ) @ ( collect @ ( product_prod @ C @ B ) @ ( product_case_prod @ C @ B @ $o @ Q ) ) ) ) ).
% sndOp_in
thf(fact_236_pick__middlep,axiom,
! [B: $tType,A: $tType,C: $tType,P2: A > B > $o,Q: B > C > $o,A3: A,C3: C] :
( ( relcompp @ A @ B @ C @ P2 @ Q @ A3 @ C3 )
=> ( ( P2 @ A3 @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q @ A3 @ C3 ) )
& ( Q @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q @ A3 @ C3 ) @ C3 ) ) ) ).
% pick_middlep
thf(fact_237_nchotomy__relcomppE,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: B > A,R2: C > A > $o,S: A > D > $o,A3: C,C3: D] :
( ! [Y: A] :
? [X: B] :
( Y
= ( F @ X ) )
=> ( ( relcompp @ C @ A @ D @ R2 @ S @ A3 @ C3 )
=> ~ ! [B5: B] :
( ( R2 @ A3 @ ( F @ B5 ) )
=> ~ ( S @ ( F @ B5 ) @ C3 ) ) ) ) ).
% nchotomy_relcomppE
thf(fact_238_relcompp_OrelcompI,axiom,
! [A: $tType,B: $tType,C: $tType,R2: A > B > $o,A3: A,B3: B,S: B > C > $o,C3: C] :
( ( R2 @ A3 @ B3 )
=> ( ( S @ B3 @ C3 )
=> ( relcompp @ A @ B @ C @ R2 @ S @ A3 @ C3 ) ) ) ).
% relcompp.relcompI
thf(fact_239_relcompp_Oinducts,axiom,
! [B: $tType,A: $tType,C: $tType,R2: A > B > $o,S: B > C > $o,X1: A,X22: C,P2: A > C > $o] :
( ( relcompp @ A @ B @ C @ R2 @ S @ X1 @ X22 )
=> ( ! [A5: A,B5: B,C5: C] :
( ( R2 @ A5 @ B5 )
=> ( ( S @ B5 @ C5 )
=> ( P2 @ A5 @ C5 ) ) )
=> ( P2 @ X1 @ X22 ) ) ) ).
% relcompp.inducts
thf(fact_240_relcompp__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R2: A > D > $o,S: D > C > $o,T3: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( relcompp @ A @ D @ C @ R2 @ S ) @ T3 )
= ( relcompp @ A @ D @ B @ R2 @ ( relcompp @ D @ C @ B @ S @ T3 ) ) ) ).
% relcompp_assoc
thf(fact_241_relcompp__apply,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcompp @ A @ B @ C )
= ( ^ [R4: A > B > $o,S4: B > C > $o,A4: A,C6: C] :
? [B4: B] :
( ( R4 @ A4 @ B4 )
& ( S4 @ B4 @ C6 ) ) ) ) ).
% relcompp_apply
thf(fact_242_relcompp_Osimps,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcompp @ A @ B @ C )
= ( ^ [R: A > B > $o,S3: B > C > $o,A12: A,A23: C] :
? [A4: A,B4: B,C6: C] :
( ( A12 = A4 )
& ( A23 = C6 )
& ( R @ A4 @ B4 )
& ( S3 @ B4 @ C6 ) ) ) ) ).
% relcompp.simps
thf(fact_243_relcompp_Ocases,axiom,
! [A: $tType,B: $tType,C: $tType,R2: A > B > $o,S: B > C > $o,A1: A,A22: C] :
( ( relcompp @ A @ B @ C @ R2 @ S @ A1 @ A22 )
=> ~ ! [B5: B] :
( ( R2 @ A1 @ B5 )
=> ~ ( S @ B5 @ A22 ) ) ) ).
% relcompp.cases
thf(fact_244_relcomppE,axiom,
! [A: $tType,B: $tType,C: $tType,R2: A > B > $o,S: B > C > $o,A3: A,C3: C] :
( ( relcompp @ A @ B @ C @ R2 @ S @ A3 @ C3 )
=> ~ ! [B5: B] :
( ( R2 @ A3 @ B5 )
=> ~ ( S @ B5 @ C3 ) ) ) ).
% relcomppE
thf(fact_245_eq__OO,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( relcompp @ A @ A @ B
@ ^ [Y2: A,Z: A] : Y2 = Z
@ R3 )
= R3 ) ).
% eq_OO
thf(fact_246_OO__eq,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( relcompp @ A @ B @ B @ R3
@ ^ [Y2: B,Z: B] : Y2 = Z )
= R3 ) ).
% OO_eq
thf(fact_247_csquare__def,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType] :
( ( bNF_csquare @ A @ B @ C @ D )
= ( ^ [A6: set @ A,F12: B > C,F22: D > C,P1: A > B,P22: A > D] :
! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ( F12 @ ( P1 @ X4 ) )
= ( F22 @ ( P22 @ X4 ) ) ) ) ) ) ).
% csquare_def
thf(fact_248_leq__OOI,axiom,
! [A: $tType,R3: A > A > $o] :
( ( R3
= ( ^ [Y2: A,Z: A] : Y2 = Z ) )
=> ( ord_less_eq @ ( A > A > $o ) @ R3 @ ( relcompp @ A @ A @ A @ R3 @ R3 ) ) ) ).
% leq_OOI
thf(fact_249_relcompp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R5: A > B > $o,R2: A > B > $o,S2: B > C > $o,S: B > C > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R5 @ R2 )
=> ( ( ord_less_eq @ ( B > C > $o ) @ S2 @ S )
=> ( ord_less_eq @ ( A > C > $o ) @ ( relcompp @ A @ B @ C @ R5 @ S2 ) @ ( relcompp @ A @ B @ C @ R2 @ S ) ) ) ) ).
% relcompp_mono
thf(fact_250_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B3: A,F: B > A,X3: B,C3: C,G2: B > C,A2: set @ B] :
( ( B3
= ( F @ X3 ) )
=> ( ( C3
= ( G2 @ X3 ) )
=> ( ( member @ B @ X3 @ A2 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B3 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A2 @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_251_strict__mono__inv,axiom,
! [A: $tType,B: $tType] :
( ( ( linorder @ B )
& ( linorder @ A ) )
=> ! [F: A > B,G2: B > A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ! [X2: A] :
( ( G2 @ ( F @ X2 ) )
= X2 )
=> ( order_strict_mono @ B @ A @ G2 ) ) ) ) ) ).
% strict_mono_inv
thf(fact_252_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X3: A,Y3: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y3 ) )
= ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ).
% strict_mono_less_eq
thf(fact_253_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X3: A,Y3: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
= ( X3 = Y3 ) ) ) ) ).
% strict_mono_eq
% Type constructors (16)
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A8: $tType,A9: $tType] :
( ( order_top @ A9 )
=> ( order_top @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A8: $tType,A9: $tType] :
( ( top @ A9 )
=> ( top @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_1,axiom,
! [A8: $tType] : ( order_top @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_4,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_6,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_9,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
ord @ $o ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [A10: set @ a] :
( ( member @ ( set @ a ) @ A10 @ ( image @ ( product_prod @ ( set @ a ) @ ( a > e ) ) @ ( set @ a ) @ ( product_fst @ ( set @ a ) @ ( a > e ) ) @ c ) )
=> ( ( ord_less_eq @ ( set @ a ) @ a1 @ A10 )
=> ( ( ord_less_eq @ ( set @ a ) @ a2 @ A10 )
=> ( ( ord_less_eq @ ( set @ a ) @ a3 @ A10 )
=> thesis ) ) ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------