TPTP Problem File: ITP026^1.p
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%------------------------------------------------------------------------------
% File : ITP026^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Algebra8 problem prob_1404__6432666_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_1404__6432666_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.31 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 493 ( 128 unt; 122 typ; 0 def)
% Number of atoms : 1156 ( 221 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 4133 ( 74 ~; 13 |; 56 &;3386 @)
% ( 0 <=>; 604 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 375 ( 375 >; 0 *; 0 +; 0 <<)
% Number of symbols : 105 ( 103 usr; 16 con; 0-5 aty)
% Number of variables : 1109 ( 166 ^; 904 !; 39 ?;1109 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:46:47.396
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
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% Explicit typings (103)
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bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__e_J,type,
bot_bot_set_e: set_e ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
ord_le157368549_nat_a: set_nat_a > set_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__b_J_J,type,
ord_le1296932838_nat_b: set_nat_b > set_nat_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__e_J,type,
ord_less_eq_set_e: set_e > set_e > $o ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__b_J,type,
collect_nat_b: ( ( nat > b ) > $o ) > set_nat_b ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mtf__a_J,type,
insert_nat_a: ( nat > a ) > set_nat_a > set_nat_a ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mtf__b_J,type,
insert_nat_b: ( nat > b ) > set_nat_b > set_nat_b ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__b_J,type,
member_nat_b: ( nat > b ) > set_nat_b > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__e_J,type,
member_nat_e: ( nat > e ) > set_nat_e > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__e,type,
member_e: e > set_e > $o ).
thf(sy_v_H,type,
h: set_a ).
thf(sy_v_H1,type,
h1: set_a ).
thf(sy_v_M,type,
m: carrie722926983_a_b_c ).
thf(sy_v_N,type,
n: carrie1821755406_e_b_f ).
thf(sy_v_R,type,
r: carrie1950868226xt_b_d ).
thf(sy_v_g,type,
g: nat > a ).
thf(sy_v_h,type,
h2: a ).
thf(sy_v_m,type,
m2: nat ).
thf(sy_v_t,type,
t: nat > b ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_xa,type,
xa: nat ).
% Relevant facts (349)
thf(fact_0_sc__Ring,axiom,
ring_b_d @ r ).
% sc_Ring
thf(fact_1_submodule__subset1,axiom,
! [H: set_a,H2: a] :
( ( submodule_b_d_a_c @ r @ m @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ H2 @ ( carrie2021454486_a_b_c @ m ) ) ) ) ).
% submodule_subset1
thf(fact_2_submodule__whole,axiom,
submodule_b_d_a_c @ r @ m @ ( carrie2021454486_a_b_c @ m ) ).
% submodule_whole
thf(fact_3_submodule__subset,axiom,
! [H: set_a] :
( ( submodule_b_d_a_c @ r @ m @ H )
=> ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) ) ) ).
% submodule_subset
thf(fact_4_free__generator__sub,axiom,
! [H: set_a] :
( ( free_g1087686480_d_a_c @ r @ m @ H )
=> ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) ) ) ).
% free_generator_sub
thf(fact_5_elem__fgs,axiom,
! [A: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ ( algebr1152250919_d_a_c @ r @ m @ A ) ) ) ) ).
% elem_fgs
thf(fact_6_fgs__sub__carrier,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ r @ m @ A ) @ ( carrie2021454486_a_b_c @ m ) ) ) ).
% fgs_sub_carrier
thf(fact_7_fgs__mono,axiom,
! [H: set_a,J: set_a,K: set_a] :
( ( free_g1087686480_d_a_c @ r @ m @ H )
=> ( ( ord_less_eq_set_a @ J @ K )
=> ( ( ord_less_eq_set_a @ K @ H )
=> ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ r @ m @ J ) @ ( algebr1152250919_d_a_c @ r @ m @ K ) ) ) ) ) ).
% fgs_mono
thf(fact_8_fgs__submodule,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ m ) )
=> ( submodule_b_d_a_c @ r @ m @ ( algebr1152250919_d_a_c @ r @ m @ A ) ) ) ).
% fgs_submodule
thf(fact_9_linear__comb__eq,axiom,
! [H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( l_comb_b_d_a_c @ r @ m @ N @ S @ F )
= ( l_comb_b_d_a_c @ r @ m @ N @ S @ G ) ) ) ) ) ) ) ).
% linear_comb_eq
thf(fact_10_linear__comb__eqTr,axiom,
! [H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) )
& ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
& ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
& ! [X2: nat] :
( ( member_nat @ X2
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) )
=> ( ( l_comb_b_d_a_c @ r @ m @ N @ S @ F )
= ( l_comb_b_d_a_c @ r @ m @ N @ S @ G ) ) ) ) ).
% linear_comb_eqTr
thf(fact_11_l__comb__mem,axiom,
! [A2: set_b,H: set_a,S: nat > b,N: nat,M: nat > a] :
( ( ideal_b_d @ r @ A2 )
=> ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : A2 ) )
=> ( ( member_nat_a @ M
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S @ M ) @ ( carrie2021454486_a_b_c @ m ) ) ) ) ) ) ).
% l_comb_mem
thf(fact_12_liear__comb__memTr,axiom,
! [A2: set_b,H: set_a,N: nat] :
( ( ideal_b_d @ r @ A2 )
=> ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ! [S2: nat > b,M2: nat > a] :
( ( ( member_nat_b @ S2
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : A2 ) )
& ( member_nat_a @ M2
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S2 @ M2 ) @ ( carrie2021454486_a_b_c @ m ) ) ) ) ) ).
% liear_comb_memTr
thf(fact_13_Ann__is__ideal,axiom,
ideal_b_d @ r @ ( annihilator_b_d_a_c @ r @ m ) ).
% Ann_is_ideal
thf(fact_14_mem__fgs__l__comb,axiom,
! [K: set_a,X: a] :
( ( K != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ K @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_a @ X @ ( algebr1152250919_d_a_c @ r @ m @ K ) )
=> ? [N2: nat,X2: nat > a] :
( ( member_nat_a @ X2
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : K ) )
& ? [Xa: nat > b] :
( ( member_nat_b @ Xa
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) )
& ( X
= ( l_comb_b_d_a_c @ r @ m @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ).
% mem_fgs_l_comb
thf(fact_15_Module__axioms,axiom,
module_a_b_c_d @ m @ r ).
% Module_axioms
thf(fact_16_quotient__of__submodules__is__ideal,axiom,
! [P: set_a,Q: set_a] :
( ( submodule_b_d_a_c @ r @ m @ P )
=> ( ( submodule_b_d_a_c @ r @ m @ Q )
=> ( ideal_b_d @ r @ ( quotie1153752712_d_a_c @ r @ m @ P @ Q ) ) ) ) ).
% quotient_of_submodules_is_ideal
thf(fact_17_smodule__ideal__coeff__is__Submodule,axiom,
! [A2: set_b] :
( ( ideal_b_d @ r @ A2 )
=> ( submodule_b_d_a_c @ r @ m @ ( smodul818989740_d_a_c @ r @ m @ A2 ) ) ) ).
% smodule_ideal_coeff_is_Submodule
thf(fact_18_mem__smodule__ideal__coeff,axiom,
! [A2: set_b,X: a] :
( ( ideal_b_d @ r @ A2 )
=> ( ( member_a @ X @ ( smodul818989740_d_a_c @ r @ m @ A2 ) )
=> ? [N2: nat,X2: nat > b] :
( ( member_nat_b @ X2
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : A2 ) )
& ? [Xa: nat > a] :
( ( member_nat_a @ Xa
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : ( carrie2021454486_a_b_c @ m ) ) )
& ( X
= ( l_comb_b_d_a_c @ r @ m @ N2 @ X2 @ Xa ) ) ) ) ) ) ).
% mem_smodule_ideal_coeff
thf(fact_19_Module_Oelem__fgs,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A: set_e,X: e] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( ord_less_eq_set_e @ A @ ( carrie730238621_e_b_f @ M3 ) )
=> ( ( member_e @ X @ A )
=> ( member_e @ X @ ( algebr168361254_d_e_f @ R @ M3 @ A ) ) ) ) ) ).
% Module.elem_fgs
thf(fact_20_Module_Oelem__fgs,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A: set_a,X: a] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( ord_less_eq_set_a @ A @ ( carrie1074654371t_unit @ M3 ) )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ ( algebr1039611956t_unit @ R @ M3 @ A ) ) ) ) ) ).
% Module.elem_fgs
thf(fact_21_Module_Oelem__fgs,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A: set_a,X: a] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ M3 ) )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ ( algebr1152250919_d_a_c @ R @ M3 @ A ) ) ) ) ) ).
% Module.elem_fgs
thf(fact_22_Module_Ofgs__mono,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e,J: set_e,K: set_e] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( free_g103796815_d_e_f @ R @ M3 @ H )
=> ( ( ord_less_eq_set_e @ J @ K )
=> ( ( ord_less_eq_set_e @ K @ H )
=> ( ord_less_eq_set_e @ ( algebr168361254_d_e_f @ R @ M3 @ J ) @ ( algebr168361254_d_e_f @ R @ M3 @ K ) ) ) ) ) ) ).
% Module.fgs_mono
thf(fact_23_Module_Ofgs__mono,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a,J: set_a,K: set_a] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( free_g637607517t_unit @ R @ M3 @ H )
=> ( ( ord_less_eq_set_a @ J @ K )
=> ( ( ord_less_eq_set_a @ K @ H )
=> ( ord_less_eq_set_a @ ( algebr1039611956t_unit @ R @ M3 @ J ) @ ( algebr1039611956t_unit @ R @ M3 @ K ) ) ) ) ) ) ).
% Module.fgs_mono
thf(fact_24_Module_Ofgs__mono,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a,J: set_a,K: set_a] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( free_g1087686480_d_a_c @ R @ M3 @ H )
=> ( ( ord_less_eq_set_a @ J @ K )
=> ( ( ord_less_eq_set_a @ K @ H )
=> ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ R @ M3 @ J ) @ ( algebr1152250919_d_a_c @ R @ M3 @ K ) ) ) ) ) ) ).
% Module.fgs_mono
thf(fact_25_Module_Ofgs__submodule,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A: set_e] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( ord_less_eq_set_e @ A @ ( carrie730238621_e_b_f @ M3 ) )
=> ( submodule_b_d_e_f @ R @ M3 @ ( algebr168361254_d_e_f @ R @ M3 @ A ) ) ) ) ).
% Module.fgs_submodule
thf(fact_26_Module_Ofgs__submodule,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A: set_a] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( ord_less_eq_set_a @ A @ ( carrie1074654371t_unit @ M3 ) )
=> ( submod903911234t_unit @ R @ M3 @ ( algebr1039611956t_unit @ R @ M3 @ A ) ) ) ) ).
% Module.fgs_submodule
thf(fact_27_Module_Ofgs__submodule,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A: set_a] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ M3 ) )
=> ( submodule_b_d_a_c @ R @ M3 @ ( algebr1152250919_d_a_c @ R @ M3 @ A ) ) ) ) ).
% Module.fgs_submodule
thf(fact_28_Module_Omem__fgs__l__comb,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,K: set_e,X: e] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( K != bot_bot_set_e )
=> ( ( ord_less_eq_set_e @ K @ ( carrie730238621_e_b_f @ M3 ) )
=> ( ( member_e @ X @ ( algebr168361254_d_e_f @ R @ M3 @ K ) )
=> ? [N2: nat,X2: nat > e] :
( ( member_nat_e @ X2
@ ( pi_nat_e
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : K ) )
& ? [Xa: nat > b] :
( ( member_nat_b @ Xa
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
& ( X
= ( l_comb_b_d_e_f @ R @ M3 @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ) ).
% Module.mem_fgs_l_comb
thf(fact_29_Module_Omem__fgs__l__comb,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,K: set_a,X: a] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( K != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ K @ ( carrie1074654371t_unit @ M3 ) )
=> ( ( member_a @ X @ ( algebr1039611956t_unit @ R @ M3 @ K ) )
=> ? [N2: nat,X2: nat > a] :
( ( member_nat_a @ X2
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : K ) )
& ? [Xa: nat > b] :
( ( member_nat_b @ Xa
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
& ( X
= ( l_comb1138968323t_unit @ R @ M3 @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ) ).
% Module.mem_fgs_l_comb
thf(fact_30_Module_Omem__fgs__l__comb,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,K: set_a,X: a] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( K != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ K @ ( carrie2021454486_a_b_c @ M3 ) )
=> ( ( member_a @ X @ ( algebr1152250919_d_a_c @ R @ M3 @ K ) )
=> ? [N2: nat,X2: nat > a] :
( ( member_nat_a @ X2
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : K ) )
& ? [Xa: nat > b] :
( ( member_nat_b @ Xa
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
& ( X
= ( l_comb_b_d_a_c @ R @ M3 @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ) ).
% Module.mem_fgs_l_comb
thf(fact_31_Module_Ofgs__sub__carrier,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A: set_e] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( ord_less_eq_set_e @ A @ ( carrie730238621_e_b_f @ M3 ) )
=> ( ord_less_eq_set_e @ ( algebr168361254_d_e_f @ R @ M3 @ A ) @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ).
% Module.fgs_sub_carrier
thf(fact_32_Module_Ofgs__sub__carrier,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A: set_a] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( ord_less_eq_set_a @ A @ ( carrie1074654371t_unit @ M3 ) )
=> ( ord_less_eq_set_a @ ( algebr1039611956t_unit @ R @ M3 @ A ) @ ( carrie1074654371t_unit @ M3 ) ) ) ) ).
% Module.fgs_sub_carrier
thf(fact_33_Module_Ofgs__sub__carrier,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A: set_a] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( ord_less_eq_set_a @ A @ ( carrie2021454486_a_b_c @ M3 ) )
=> ( ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ R @ M3 @ A ) @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ).
% Module.fgs_sub_carrier
thf(fact_34_misom__self,axiom,
misomo1282343797_c_a_c @ r @ m @ m ).
% misom_self
thf(fact_35_HOM__is__module,axiom,
! [N3: carrie1821755406_e_b_f] :
( ( module_e_b_f_d @ N3 @ r )
=> ( module1648870817unit_d @ ( hOM_b_d_a_c_e_f @ r @ m @ N3 ) @ r ) ) ).
% HOM_is_module
thf(fact_36_HOM__is__module,axiom,
! [N3: carrie722926983_a_b_c] :
( ( module_a_b_c_d @ N3 @ r )
=> ( module589927589unit_d @ ( hOM_b_d_a_c_a_c @ r @ m @ N3 ) @ r ) ) ).
% HOM_is_module
thf(fact_37_HOM__is__module,axiom,
! [N3: carrie1963041556t_unit] :
( ( module1821517916unit_d @ N3 @ r )
=> ( module589927589unit_d @ ( hOM_b_717364638t_unit @ r @ m @ N3 ) @ r ) ) ).
% HOM_is_module
thf(fact_38_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_39_generator__sub__carrier,axiom,
! [H: set_a] :
( ( generator_b_d_a_c @ r @ m @ H )
=> ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) ) ) ).
% generator_sub_carrier
thf(fact_40_free__generator__generator,axiom,
! [H: set_a] :
( ( free_g1087686480_d_a_c @ r @ m @ H )
=> ( generator_b_d_a_c @ r @ m @ H ) ) ).
% free_generator_generator
thf(fact_41_Module_Ol__comb__mem,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A2: set_b,H: set_e,S: nat > b,N: nat,M: nat > e] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( ideal_b_d @ R @ A2 )
=> ( ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) )
=> ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : A2 ) )
=> ( ( member_nat_e @ M
@ ( pi_nat_e
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( member_e @ ( l_comb_b_d_e_f @ R @ M3 @ N @ S @ M ) @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ) ) ) ).
% Module.l_comb_mem
thf(fact_42_Module_Ol__comb__mem,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A2: set_b,H: set_a,S: nat > b,N: nat,M: nat > a] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( ideal_b_d @ R @ A2 )
=> ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) )
=> ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : A2 ) )
=> ( ( member_nat_a @ M
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( member_a @ ( l_comb_b_d_a_c @ R @ M3 @ N @ S @ M ) @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ) ) ) ).
% Module.l_comb_mem
thf(fact_43_Module_Ol__comb__mem,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A2: set_b,H: set_a,S: nat > b,N: nat,M: nat > a] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( ideal_b_d @ R @ A2 )
=> ( ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) )
=> ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : A2 ) )
=> ( ( member_nat_a @ M
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( member_a @ ( l_comb1138968323t_unit @ R @ M3 @ N @ S @ M ) @ ( carrie1074654371t_unit @ M3 ) ) ) ) ) ) ) ).
% Module.l_comb_mem
thf(fact_44_Module_Oliear__comb__memTr,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A2: set_b,H: set_e,N: nat] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( ideal_b_d @ R @ A2 )
=> ( ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) )
=> ! [S2: nat > b,M2: nat > e] :
( ( ( member_nat_b @ S2
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : A2 ) )
& ( member_nat_e @ M2
@ ( pi_nat_e
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) ) )
=> ( member_e @ ( l_comb_b_d_e_f @ R @ M3 @ N @ S2 @ M2 ) @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ) ) ).
% Module.liear_comb_memTr
thf(fact_45_Module_Oliear__comb__memTr,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A2: set_b,H: set_a,N: nat] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( ideal_b_d @ R @ A2 )
=> ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) )
=> ! [S2: nat > b,M2: nat > a] :
( ( ( member_nat_b @ S2
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : A2 ) )
& ( member_nat_a @ M2
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) ) )
=> ( member_a @ ( l_comb_b_d_a_c @ R @ M3 @ N @ S2 @ M2 ) @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ) ) ).
% Module.liear_comb_memTr
thf(fact_46_Module_Oliear__comb__memTr,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A2: set_b,H: set_a,N: nat] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( ideal_b_d @ R @ A2 )
=> ( ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) )
=> ! [S2: nat > b,M2: nat > a] :
( ( ( member_nat_b @ S2
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : A2 ) )
& ( member_nat_a @ M2
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) ) )
=> ( member_a @ ( l_comb1138968323t_unit @ R @ M3 @ N @ S2 @ M2 ) @ ( carrie1074654371t_unit @ M3 ) ) ) ) ) ) ).
% Module.liear_comb_memTr
thf(fact_47_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_48_Module_Olinear__comb__eq,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e,S: nat > b,N: nat,F: nat > e,G: nat > e] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) )
=> ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
=> ( ( member_nat_e @ F
@ ( pi_nat_e
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( ( member_nat_e @ G
@ ( pi_nat_e
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( l_comb_b_d_e_f @ R @ M3 @ N @ S @ F )
= ( l_comb_b_d_e_f @ R @ M3 @ N @ S @ G ) ) ) ) ) ) ) ) ).
% Module.linear_comb_eq
thf(fact_49_Module_Olinear__comb__eq,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) )
=> ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( l_comb_b_d_a_c @ R @ M3 @ N @ S @ F )
= ( l_comb_b_d_a_c @ R @ M3 @ N @ S @ G ) ) ) ) ) ) ) ) ).
% Module.linear_comb_eq
thf(fact_50_Module_Olinear__comb__eq,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) )
=> ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( l_comb1138968323t_unit @ R @ M3 @ N @ S @ F )
= ( l_comb1138968323t_unit @ R @ M3 @ N @ S @ G ) ) ) ) ) ) ) ) ).
% Module.linear_comb_eq
thf(fact_51_Module_Olinear__comb__eqTr,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e,S: nat > b,N: nat,F: nat > e,G: nat > e] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) )
=> ( ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
& ( member_nat_e @ F
@ ( pi_nat_e
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
& ( member_nat_e @ G
@ ( pi_nat_e
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
& ! [X2: nat] :
( ( member_nat @ X2
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) )
=> ( ( l_comb_b_d_e_f @ R @ M3 @ N @ S @ F )
= ( l_comb_b_d_e_f @ R @ M3 @ N @ S @ G ) ) ) ) ) ).
% Module.linear_comb_eqTr
thf(fact_52_Module_Olinear__comb__eqTr,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) )
=> ( ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
& ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
& ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
& ! [X2: nat] :
( ( member_nat @ X2
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) )
=> ( ( l_comb_b_d_a_c @ R @ M3 @ N @ S @ F )
= ( l_comb_b_d_a_c @ R @ M3 @ N @ S @ G ) ) ) ) ) ).
% Module.linear_comb_eqTr
thf(fact_53_Module_Olinear__comb__eqTr,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) )
=> ( ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ R ) ) )
& ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
& ( member_nat_a @ G
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
& ! [X2: nat] :
( ( member_nat @ X2
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) )
=> ( ( l_comb1138968323t_unit @ R @ M3 @ N @ S @ F )
= ( l_comb1138968323t_unit @ R @ M3 @ N @ S @ G ) ) ) ) ) ).
% Module.linear_comb_eqTr
thf(fact_54_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_55_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_56_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_57_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_58_misom__sym,axiom,
! [N3: carrie1821755406_e_b_f] :
( ( module_e_b_f_d @ N3 @ r )
=> ( ( misomo298454132_c_e_f @ r @ m @ N3 )
=> ( misomo1752826100_f_a_c @ r @ N3 @ m ) ) ) ).
% misom_sym
thf(fact_59_misom__sym,axiom,
! [N3: carrie1963041556t_unit] :
( ( module1821517916unit_d @ N3 @ r )
=> ( ( misomo1857159554t_unit @ r @ m @ N3 )
=> ( misomo1935876354it_a_c @ r @ N3 @ m ) ) ) ).
% misom_sym
thf(fact_60_misom__sym,axiom,
! [N3: carrie722926983_a_b_c] :
( ( module_a_b_c_d @ N3 @ r )
=> ( ( misomo1282343797_c_a_c @ r @ m @ N3 )
=> ( misomo1282343797_c_a_c @ r @ N3 @ m ) ) ) ).
% misom_sym
thf(fact_61_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_62_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_63_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_64_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_65_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_66_mem__Collect__eq,axiom,
! [A: nat > a,P: ( nat > a ) > $o] :
( ( member_nat_a @ A @ ( collect_nat_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
! [A: nat > b,P: ( nat > b ) > $o] :
( ( member_nat_b @ A @ ( collect_nat_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A2: set_nat_a] :
( ( collect_nat_a
@ ^ [X3: nat > a] : ( member_nat_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A2: set_nat_b] :
( ( collect_nat_b
@ ^ [X3: nat > b] : ( member_nat_b @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_74_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_75_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_76_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_77_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_78_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_79_Suc__diff__diff,axiom,
! [M: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_80_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_81_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_82_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_83_misom__trans,axiom,
! [L: carrie1821755406_e_b_f,N3: carrie1821755406_e_b_f] :
( ( module_e_b_f_d @ L @ r )
=> ( ( module_e_b_f_d @ N3 @ r )
=> ( ( misomo1752826100_f_a_c @ r @ L @ m )
=> ( ( misomo298454132_c_e_f @ r @ m @ N3 )
=> ( misomo768936435_f_e_f @ r @ L @ N3 ) ) ) ) ) ).
% misom_trans
thf(fact_84_misom__trans,axiom,
! [L: carrie1821755406_e_b_f,N3: carrie1963041556t_unit] :
( ( module_e_b_f_d @ L @ r )
=> ( ( module1821517916unit_d @ N3 @ r )
=> ( ( misomo1752826100_f_a_c @ r @ L @ m )
=> ( ( misomo1857159554t_unit @ r @ m @ N3 )
=> ( misomo368157697t_unit @ r @ L @ N3 ) ) ) ) ) ).
% misom_trans
thf(fact_85_misom__trans,axiom,
! [L: carrie1963041556t_unit,N3: carrie1821755406_e_b_f] :
( ( module1821517916unit_d @ L @ r )
=> ( ( module_e_b_f_d @ N3 @ r )
=> ( ( misomo1935876354it_a_c @ r @ L @ m )
=> ( ( misomo298454132_c_e_f @ r @ m @ N3 )
=> ( misomo951986689it_e_f @ r @ L @ N3 ) ) ) ) ) ).
% misom_trans
thf(fact_86_misom__trans,axiom,
! [L: carrie1963041556t_unit,N3: carrie1963041556t_unit] :
( ( module1821517916unit_d @ L @ r )
=> ( ( module1821517916unit_d @ N3 @ r )
=> ( ( misomo1935876354it_a_c @ r @ L @ m )
=> ( ( misomo1857159554t_unit @ r @ m @ N3 )
=> ( misomo493403663t_unit @ r @ L @ N3 ) ) ) ) ) ).
% misom_trans
thf(fact_87_misom__trans,axiom,
! [L: carrie1821755406_e_b_f,N3: carrie722926983_a_b_c] :
( ( module_e_b_f_d @ L @ r )
=> ( ( module_a_b_c_d @ N3 @ r )
=> ( ( misomo1752826100_f_a_c @ r @ L @ m )
=> ( ( misomo1282343797_c_a_c @ r @ m @ N3 )
=> ( misomo1752826100_f_a_c @ r @ L @ N3 ) ) ) ) ) ).
% misom_trans
thf(fact_88_misom__trans,axiom,
! [L: carrie1963041556t_unit,N3: carrie722926983_a_b_c] :
( ( module1821517916unit_d @ L @ r )
=> ( ( module_a_b_c_d @ N3 @ r )
=> ( ( misomo1935876354it_a_c @ r @ L @ m )
=> ( ( misomo1282343797_c_a_c @ r @ m @ N3 )
=> ( misomo1935876354it_a_c @ r @ L @ N3 ) ) ) ) ) ).
% misom_trans
thf(fact_89_misom__trans,axiom,
! [L: carrie722926983_a_b_c,N3: carrie1821755406_e_b_f] :
( ( module_a_b_c_d @ L @ r )
=> ( ( module_e_b_f_d @ N3 @ r )
=> ( ( misomo1282343797_c_a_c @ r @ L @ m )
=> ( ( misomo298454132_c_e_f @ r @ m @ N3 )
=> ( misomo298454132_c_e_f @ r @ L @ N3 ) ) ) ) ) ).
% misom_trans
thf(fact_90_misom__trans,axiom,
! [L: carrie722926983_a_b_c,N3: carrie1963041556t_unit] :
( ( module_a_b_c_d @ L @ r )
=> ( ( module1821517916unit_d @ N3 @ r )
=> ( ( misomo1282343797_c_a_c @ r @ L @ m )
=> ( ( misomo1857159554t_unit @ r @ m @ N3 )
=> ( misomo1857159554t_unit @ r @ L @ N3 ) ) ) ) ) ).
% misom_trans
thf(fact_91_misom__trans,axiom,
! [L: carrie722926983_a_b_c,N3: carrie722926983_a_b_c] :
( ( module_a_b_c_d @ L @ r )
=> ( ( module_a_b_c_d @ N3 @ r )
=> ( ( misomo1282343797_c_a_c @ r @ L @ m )
=> ( ( misomo1282343797_c_a_c @ r @ m @ N3 )
=> ( misomo1282343797_c_a_c @ r @ L @ N3 ) ) ) ) ) ).
% misom_trans
thf(fact_92_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_93_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_94_Module_OHOM__is__module,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( module_e_b_f_d @ N3 @ R )
=> ( module241733029unit_d @ ( hOM_b_d_e_f_e_f @ R @ M3 @ N3 ) @ R ) ) ) ).
% Module.HOM_is_module
thf(fact_95_Module_OHOM__is__module,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( module_a_b_c_d @ N3 @ R )
=> ( module1330273449unit_d @ ( hOM_b_d_e_f_a_c @ R @ M3 @ N3 ) @ R ) ) ) ).
% Module.HOM_is_module
thf(fact_96_Module_OHOM__is__module,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( module1821517916unit_d @ N3 @ R )
=> ( module1330273449unit_d @ ( hOM_b_1375846429t_unit @ R @ M3 @ N3 ) @ R ) ) ) ).
% Module.HOM_is_module
thf(fact_97_Module_OHOM__is__module,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( module_e_b_f_d @ N3 @ R )
=> ( module1648870817unit_d @ ( hOM_b_d_a_c_e_f @ R @ M3 @ N3 ) @ R ) ) ) ).
% Module.HOM_is_module
thf(fact_98_Module_OHOM__is__module,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( module_a_b_c_d @ N3 @ R )
=> ( module589927589unit_d @ ( hOM_b_d_a_c_a_c @ R @ M3 @ N3 ) @ R ) ) ) ).
% Module.HOM_is_module
thf(fact_99_Module_OHOM__is__module,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( module1821517916unit_d @ N3 @ R )
=> ( module589927589unit_d @ ( hOM_b_717364638t_unit @ R @ M3 @ N3 ) @ R ) ) ) ).
% Module.HOM_is_module
thf(fact_100_Module_OHOM__is__module,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( module_e_b_f_d @ N3 @ R )
=> ( module1648870817unit_d @ ( hOM_b_1959675421it_e_f @ R @ M3 @ N3 ) @ R ) ) ) ).
% Module.HOM_is_module
thf(fact_101_Module_OHOM__is__module,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( module_a_b_c_d @ N3 @ R )
=> ( module589927589unit_d @ ( hOM_b_796081438it_a_c @ R @ M3 @ N3 ) @ R ) ) ) ).
% Module.HOM_is_module
thf(fact_102_Module_OHOM__is__module,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( module1821517916unit_d @ N3 @ R )
=> ( module589927589unit_d @ ( hOM_b_1103679019t_unit @ R @ M3 @ N3 ) @ R ) ) ) ).
% Module.HOM_is_module
thf(fact_103_Module_Omisom__trans,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1821755406_e_b_f,N3: carrie1821755406_e_b_f] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( module_e_b_f_d @ L @ R )
=> ( ( module_e_b_f_d @ N3 @ R )
=> ( ( misomo768936435_f_e_f @ R @ L @ M3 )
=> ( ( misomo768936435_f_e_f @ R @ M3 @ N3 )
=> ( misomo768936435_f_e_f @ R @ L @ N3 ) ) ) ) ) ) ).
% Module.misom_trans
thf(fact_104_Module_Omisom__trans,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1821755406_e_b_f,N3: carrie722926983_a_b_c] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( module_e_b_f_d @ L @ R )
=> ( ( module_a_b_c_d @ N3 @ R )
=> ( ( misomo768936435_f_e_f @ R @ L @ M3 )
=> ( ( misomo1752826100_f_a_c @ R @ M3 @ N3 )
=> ( misomo1752826100_f_a_c @ R @ L @ N3 ) ) ) ) ) ) ).
% Module.misom_trans
thf(fact_105_Module_Omisom__trans,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1821755406_e_b_f,N3: carrie1963041556t_unit] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( module_e_b_f_d @ L @ R )
=> ( ( module1821517916unit_d @ N3 @ R )
=> ( ( misomo768936435_f_e_f @ R @ L @ M3 )
=> ( ( misomo368157697t_unit @ R @ M3 @ N3 )
=> ( misomo368157697t_unit @ R @ L @ N3 ) ) ) ) ) ) ).
% Module.misom_trans
thf(fact_106_Module_Omisom__trans,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie722926983_a_b_c,N3: carrie1821755406_e_b_f] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( module_a_b_c_d @ L @ R )
=> ( ( module_e_b_f_d @ N3 @ R )
=> ( ( misomo298454132_c_e_f @ R @ L @ M3 )
=> ( ( misomo768936435_f_e_f @ R @ M3 @ N3 )
=> ( misomo298454132_c_e_f @ R @ L @ N3 ) ) ) ) ) ) ).
% Module.misom_trans
thf(fact_107_Module_Omisom__trans,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie722926983_a_b_c,N3: carrie1963041556t_unit] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( module_a_b_c_d @ L @ R )
=> ( ( module1821517916unit_d @ N3 @ R )
=> ( ( misomo298454132_c_e_f @ R @ L @ M3 )
=> ( ( misomo368157697t_unit @ R @ M3 @ N3 )
=> ( misomo1857159554t_unit @ R @ L @ N3 ) ) ) ) ) ) ).
% Module.misom_trans
thf(fact_108_Module_Omisom__trans,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1963041556t_unit,N3: carrie1821755406_e_b_f] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( module1821517916unit_d @ L @ R )
=> ( ( module_e_b_f_d @ N3 @ R )
=> ( ( misomo951986689it_e_f @ R @ L @ M3 )
=> ( ( misomo768936435_f_e_f @ R @ M3 @ N3 )
=> ( misomo951986689it_e_f @ R @ L @ N3 ) ) ) ) ) ) ).
% Module.misom_trans
thf(fact_109_Module_Omisom__trans,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1963041556t_unit,N3: carrie722926983_a_b_c] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( module1821517916unit_d @ L @ R )
=> ( ( module_a_b_c_d @ N3 @ R )
=> ( ( misomo951986689it_e_f @ R @ L @ M3 )
=> ( ( misomo1752826100_f_a_c @ R @ M3 @ N3 )
=> ( misomo1935876354it_a_c @ R @ L @ N3 ) ) ) ) ) ) ).
% Module.misom_trans
thf(fact_110_Module_Omisom__trans,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,L: carrie1963041556t_unit,N3: carrie1963041556t_unit] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( module1821517916unit_d @ L @ R )
=> ( ( module1821517916unit_d @ N3 @ R )
=> ( ( misomo951986689it_e_f @ R @ L @ M3 )
=> ( ( misomo368157697t_unit @ R @ M3 @ N3 )
=> ( misomo493403663t_unit @ R @ L @ N3 ) ) ) ) ) ) ).
% Module.misom_trans
thf(fact_111_Module_Omisom__trans,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,L: carrie1821755406_e_b_f,N3: carrie1821755406_e_b_f] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( module_e_b_f_d @ L @ R )
=> ( ( module_e_b_f_d @ N3 @ R )
=> ( ( misomo1752826100_f_a_c @ R @ L @ M3 )
=> ( ( misomo298454132_c_e_f @ R @ M3 @ N3 )
=> ( misomo768936435_f_e_f @ R @ L @ N3 ) ) ) ) ) ) ).
% Module.misom_trans
thf(fact_112_Module_Omisom__trans,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,L: carrie1821755406_e_b_f,N3: carrie1963041556t_unit] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( module_e_b_f_d @ L @ R )
=> ( ( module1821517916unit_d @ N3 @ R )
=> ( ( misomo1752826100_f_a_c @ R @ L @ M3 )
=> ( ( misomo1857159554t_unit @ R @ M3 @ N3 )
=> ( misomo368157697t_unit @ R @ L @ N3 ) ) ) ) ) ) ).
% Module.misom_trans
thf(fact_113_Module_Omisom__self,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d] :
( ( module_e_b_f_d @ M3 @ R )
=> ( misomo768936435_f_e_f @ R @ M3 @ M3 ) ) ).
% Module.misom_self
thf(fact_114_Module_Omisom__self,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d] :
( ( module1821517916unit_d @ M3 @ R )
=> ( misomo493403663t_unit @ R @ M3 @ M3 ) ) ).
% Module.misom_self
thf(fact_115_Module_Omisom__self,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d] :
( ( module_a_b_c_d @ M3 @ R )
=> ( misomo1282343797_c_a_c @ R @ M3 @ M3 ) ) ).
% Module.misom_self
thf(fact_116_Module_Omisom__sym,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( module_e_b_f_d @ N3 @ R )
=> ( ( misomo768936435_f_e_f @ R @ M3 @ N3 )
=> ( misomo768936435_f_e_f @ R @ N3 @ M3 ) ) ) ) ).
% Module.misom_sym
thf(fact_117_Module_Omisom__sym,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( module_a_b_c_d @ N3 @ R )
=> ( ( misomo1752826100_f_a_c @ R @ M3 @ N3 )
=> ( misomo298454132_c_e_f @ R @ N3 @ M3 ) ) ) ) ).
% Module.misom_sym
thf(fact_118_Module_Omisom__sym,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( module1821517916unit_d @ N3 @ R )
=> ( ( misomo368157697t_unit @ R @ M3 @ N3 )
=> ( misomo951986689it_e_f @ R @ N3 @ M3 ) ) ) ) ).
% Module.misom_sym
thf(fact_119_Module_Omisom__sym,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( module_e_b_f_d @ N3 @ R )
=> ( ( misomo298454132_c_e_f @ R @ M3 @ N3 )
=> ( misomo1752826100_f_a_c @ R @ N3 @ M3 ) ) ) ) ).
% Module.misom_sym
thf(fact_120_Module_Omisom__sym,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( module1821517916unit_d @ N3 @ R )
=> ( ( misomo1857159554t_unit @ R @ M3 @ N3 )
=> ( misomo1935876354it_a_c @ R @ N3 @ M3 ) ) ) ) ).
% Module.misom_sym
thf(fact_121_Module_Omisom__sym,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie1821755406_e_b_f] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( module_e_b_f_d @ N3 @ R )
=> ( ( misomo951986689it_e_f @ R @ M3 @ N3 )
=> ( misomo368157697t_unit @ R @ N3 @ M3 ) ) ) ) ).
% Module.misom_sym
thf(fact_122_Module_Omisom__sym,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( module_a_b_c_d @ N3 @ R )
=> ( ( misomo1935876354it_a_c @ R @ M3 @ N3 )
=> ( misomo1857159554t_unit @ R @ N3 @ M3 ) ) ) ) ).
% Module.misom_sym
thf(fact_123_Module_Omisom__sym,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,N3: carrie1963041556t_unit] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( module1821517916unit_d @ N3 @ R )
=> ( ( misomo493403663t_unit @ R @ M3 @ N3 )
=> ( misomo493403663t_unit @ R @ N3 @ M3 ) ) ) ) ).
% Module.misom_sym
thf(fact_124_Module_Omisom__sym,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,N3: carrie722926983_a_b_c] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( module_a_b_c_d @ N3 @ R )
=> ( ( misomo1282343797_c_a_c @ R @ M3 @ N3 )
=> ( misomo1282343797_c_a_c @ R @ N3 @ M3 ) ) ) ) ).
% Module.misom_sym
thf(fact_125_Module_Ofree__generator__generator,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( free_g103796815_d_e_f @ R @ M3 @ H )
=> ( generator_b_d_e_f @ R @ M3 @ H ) ) ) ).
% Module.free_generator_generator
thf(fact_126_Module_Ofree__generator__generator,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( free_g637607517t_unit @ R @ M3 @ H )
=> ( genera1692266857t_unit @ R @ M3 @ H ) ) ) ).
% Module.free_generator_generator
thf(fact_127_Module_Ofree__generator__generator,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( free_g1087686480_d_a_c @ R @ M3 @ H )
=> ( generator_b_d_a_c @ R @ M3 @ H ) ) ) ).
% Module.free_generator_generator
thf(fact_128_Module_Ogenerator__sub__carrier,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( generator_b_d_e_f @ R @ M3 @ H )
=> ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ).
% Module.generator_sub_carrier
thf(fact_129_Module_Ogenerator__sub__carrier,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( generator_b_d_a_c @ R @ M3 @ H )
=> ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ).
% Module.generator_sub_carrier
thf(fact_130_Module_Ogenerator__sub__carrier,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( genera1692266857t_unit @ R @ M3 @ H )
=> ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) ) ) ) ).
% Module.generator_sub_carrier
thf(fact_131_Module_OAnn__is__ideal,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ideal_b_d @ R @ ( annihilator_b_d_e_f @ R @ M3 ) ) ) ).
% Module.Ann_is_ideal
thf(fact_132_Module_OAnn__is__ideal,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ideal_b_d @ R @ ( annihi259882159t_unit @ R @ M3 ) ) ) ).
% Module.Ann_is_ideal
thf(fact_133_Module_OAnn__is__ideal,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ideal_b_d @ R @ ( annihilator_b_d_a_c @ R @ M3 ) ) ) ).
% Module.Ann_is_ideal
thf(fact_134_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_135_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_136_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_137_le__trans,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_138_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_139_Suc__inject,axiom,
! [X: nat,Y4: nat] :
( ( ( suc @ X )
= ( suc @ Y4 ) )
=> ( X = Y4 ) ) ).
% Suc_inject
thf(fact_140_linorder__neqE__nat,axiom,
! [X: nat,Y4: nat] :
( ( X != Y4 )
=> ( ~ ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ Y4 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_141_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_142_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_143_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_144_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_145_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_146_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_147_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_148_diff__commute,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J3 ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J3 ) ) ).
% diff_commute
thf(fact_149_Module_Osmodule__ideal__coeff__is__Submodule,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A2: set_b] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( ideal_b_d @ R @ A2 )
=> ( submodule_b_d_e_f @ R @ M3 @ ( smodul1982583723_d_e_f @ R @ M3 @ A2 ) ) ) ) ).
% Module.smodule_ideal_coeff_is_Submodule
thf(fact_150_Module_Osmodule__ideal__coeff__is__Submodule,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A2: set_b] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( ideal_b_d @ R @ A2 )
=> ( submod903911234t_unit @ R @ M3 @ ( smodul132175289t_unit @ R @ M3 @ A2 ) ) ) ) ).
% Module.smodule_ideal_coeff_is_Submodule
thf(fact_151_Module_Osmodule__ideal__coeff__is__Submodule,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A2: set_b] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( ideal_b_d @ R @ A2 )
=> ( submodule_b_d_a_c @ R @ M3 @ ( smodul818989740_d_a_c @ R @ M3 @ A2 ) ) ) ) ).
% Module.smodule_ideal_coeff_is_Submodule
thf(fact_152_Module_Oquotient__of__submodules__is__ideal,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,P: set_e,Q: set_e] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( submodule_b_d_e_f @ R @ M3 @ P )
=> ( ( submodule_b_d_e_f @ R @ M3 @ Q )
=> ( ideal_b_d @ R @ ( quotie169863047_d_e_f @ R @ M3 @ P @ Q ) ) ) ) ) ).
% Module.quotient_of_submodules_is_ideal
thf(fact_153_Module_Oquotient__of__submodules__is__ideal,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,P: set_a,Q: set_a] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( submod903911234t_unit @ R @ M3 @ P )
=> ( ( submod903911234t_unit @ R @ M3 @ Q )
=> ( ideal_b_d @ R @ ( quotie1196826005t_unit @ R @ M3 @ P @ Q ) ) ) ) ) ).
% Module.quotient_of_submodules_is_ideal
thf(fact_154_Module_Oquotient__of__submodules__is__ideal,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,P: set_a,Q: set_a] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( submodule_b_d_a_c @ R @ M3 @ P )
=> ( ( submodule_b_d_a_c @ R @ M3 @ Q )
=> ( ideal_b_d @ R @ ( quotie1153752712_d_a_c @ R @ M3 @ P @ Q ) ) ) ) ) ).
% Module.quotient_of_submodules_is_ideal
thf(fact_155_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y: nat,Z: nat] :
( ( R @ X2 @ Y )
=> ( ( R @ Y @ Z )
=> ( R @ X2 @ Z ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_156_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_157_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_158_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_159_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_160_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_161_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_162_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_163_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_164_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_165_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_166_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_167_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_168_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_169_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_170_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ zero_zero_nat )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_171_old_Onat_Oexhaust,axiom,
! [Y4: nat] :
( ( Y4 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y4
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_172_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_173_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_174_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_175_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_176_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X2: nat,Y: nat] :
( ( P @ X2 @ Y )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_177_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_178_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_179_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_180_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_181_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_182_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J3: nat] :
( ! [I2: nat,J4: nat] :
( ( ord_less_nat @ I2 @ J4 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J4 ) ) )
=> ( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_183_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_184_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_185_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N4: nat] :
( ( ord_less_nat @ M6 @ N4 )
| ( M6 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_186_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_187_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N4: nat] :
( ( ord_less_eq_nat @ M6 @ N4 )
& ( M6 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_188_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_189_strict__inc__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J3 )
=> ( ! [I2: nat] :
( ( J3
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_190_less__Suc__induct,axiom,
! [I: nat,J3: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J3 )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J4: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J4 )
=> ( ( ord_less_nat @ J4 @ K3 )
=> ( ( P @ I2 @ J4 )
=> ( ( P @ J4 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J3 ) ) ) ) ).
% less_Suc_induct
thf(fact_191_less__trans__Suc,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_192_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_193_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_194_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_195_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_196_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_197_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_198_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_199_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_200_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_201_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_202_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K2 )
=> ~ ! [J4: nat] :
( ( ord_less_nat @ I @ J4 )
=> ( K2
!= ( suc @ J4 ) ) ) ) ).
% Suc_lessE
thf(fact_203_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_204_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J4: nat] :
( ( ord_less_nat @ I @ J4 )
=> ( K2
!= ( suc @ J4 ) ) ) ) ) ).
% Nat.lessE
thf(fact_205_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_206_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_207_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_208_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_209_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_210_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_211_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_212_diff__le__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% diff_le_mono2
thf(fact_213_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_214_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_215_diff__le__mono,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_216_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_217_le__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_218_eq__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_219_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_220_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_221_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_222_less__imp__diff__less,axiom,
! [J3: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J3 @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J3 @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_223_diff__less__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L2 )
=> ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_224_Module_Osc__Ring,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ring_b_d @ R ) ) ).
% Module.sc_Ring
thf(fact_225_Module_Osc__Ring,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ring_b_d @ R ) ) ).
% Module.sc_Ring
thf(fact_226_Module_Osc__Ring,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ring_b_d @ R ) ) ).
% Module.sc_Ring
thf(fact_227_Module_Omem__smodule__ideal__coeff,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,A2: set_b,X: e] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( ideal_b_d @ R @ A2 )
=> ( ( member_e @ X @ ( smodul1982583723_d_e_f @ R @ M3 @ A2 ) )
=> ? [N2: nat,X2: nat > b] :
( ( member_nat_b @ X2
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : A2 ) )
& ? [Xa: nat > e] :
( ( member_nat_e @ Xa
@ ( pi_nat_e
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : ( carrie730238621_e_b_f @ M3 ) ) )
& ( X
= ( l_comb_b_d_e_f @ R @ M3 @ N2 @ X2 @ Xa ) ) ) ) ) ) ) ).
% Module.mem_smodule_ideal_coeff
thf(fact_228_Module_Omem__smodule__ideal__coeff,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,A2: set_b,X: a] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( ideal_b_d @ R @ A2 )
=> ( ( member_a @ X @ ( smodul132175289t_unit @ R @ M3 @ A2 ) )
=> ? [N2: nat,X2: nat > b] :
( ( member_nat_b @ X2
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : A2 ) )
& ? [Xa: nat > a] :
( ( member_nat_a @ Xa
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : ( carrie1074654371t_unit @ M3 ) ) )
& ( X
= ( l_comb1138968323t_unit @ R @ M3 @ N2 @ X2 @ Xa ) ) ) ) ) ) ) ).
% Module.mem_smodule_ideal_coeff
thf(fact_229_Module_Omem__smodule__ideal__coeff,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,A2: set_b,X: a] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( ideal_b_d @ R @ A2 )
=> ( ( member_a @ X @ ( smodul818989740_d_a_c @ R @ M3 @ A2 ) )
=> ? [N2: nat,X2: nat > b] :
( ( member_nat_b @ X2
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : A2 ) )
& ? [Xa: nat > a] :
( ( member_nat_a @ Xa
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : ( carrie2021454486_a_b_c @ M3 ) ) )
& ( X
= ( l_comb_b_d_a_c @ R @ M3 @ N2 @ X2 @ Xa ) ) ) ) ) ) ) ).
% Module.mem_smodule_ideal_coeff
thf(fact_230_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_231_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_set_a @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_232_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_233_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_234_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_235_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_236_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_237_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_238_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_239_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_240_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_241_inc__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( P @ J3 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J3 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_242_dec__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J3 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J3 ) ) ) ) ).
% dec_induct
thf(fact_243_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_244_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_245_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_246_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_247_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_248_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_249_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_250_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_251_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_252_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_253_less__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_254_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_255_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_256_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_257_Module_Osubmodule__subset1,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e,H2: e] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( submodule_b_d_e_f @ R @ M3 @ H )
=> ( ( member_e @ H2 @ H )
=> ( member_e @ H2 @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ) ).
% Module.submodule_subset1
thf(fact_258_Module_Osubmodule__subset1,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a,H2: a] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( submodule_b_d_a_c @ R @ M3 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ H2 @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ) ).
% Module.submodule_subset1
thf(fact_259_Module_Osubmodule__subset1,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a,H2: a] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( submod903911234t_unit @ R @ M3 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ H2 @ ( carrie1074654371t_unit @ M3 ) ) ) ) ) ).
% Module.submodule_subset1
thf(fact_260_Module_Osubmodule__whole,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d] :
( ( module_e_b_f_d @ M3 @ R )
=> ( submodule_b_d_e_f @ R @ M3 @ ( carrie730238621_e_b_f @ M3 ) ) ) ).
% Module.submodule_whole
thf(fact_261_Module_Osubmodule__whole,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d] :
( ( module_a_b_c_d @ M3 @ R )
=> ( submodule_b_d_a_c @ R @ M3 @ ( carrie2021454486_a_b_c @ M3 ) ) ) ).
% Module.submodule_whole
thf(fact_262_Module_Osubmodule__whole,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d] :
( ( module1821517916unit_d @ M3 @ R )
=> ( submod903911234t_unit @ R @ M3 @ ( carrie1074654371t_unit @ M3 ) ) ) ).
% Module.submodule_whole
thf(fact_263_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_264_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_265_Module_Osubmodule__subset,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( submodule_b_d_e_f @ R @ M3 @ H )
=> ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ).
% Module.submodule_subset
thf(fact_266_Module_Osubmodule__subset,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( submodule_b_d_a_c @ R @ M3 @ H )
=> ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ).
% Module.submodule_subset
thf(fact_267_Module_Osubmodule__subset,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( submod903911234t_unit @ R @ M3 @ H )
=> ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) ) ) ) ).
% Module.submodule_subset
thf(fact_268_Module_Ofree__generator__sub,axiom,
! [M3: carrie1821755406_e_b_f,R: carrie1950868226xt_b_d,H: set_e] :
( ( module_e_b_f_d @ M3 @ R )
=> ( ( free_g103796815_d_e_f @ R @ M3 @ H )
=> ( ord_less_eq_set_e @ H @ ( carrie730238621_e_b_f @ M3 ) ) ) ) ).
% Module.free_generator_sub
thf(fact_269_Module_Ofree__generator__sub,axiom,
! [M3: carrie722926983_a_b_c,R: carrie1950868226xt_b_d,H: set_a] :
( ( module_a_b_c_d @ M3 @ R )
=> ( ( free_g1087686480_d_a_c @ R @ M3 @ H )
=> ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ M3 ) ) ) ) ).
% Module.free_generator_sub
thf(fact_270_Module_Ofree__generator__sub,axiom,
! [M3: carrie1963041556t_unit,R: carrie1950868226xt_b_d,H: set_a] :
( ( module1821517916unit_d @ M3 @ R )
=> ( ( free_g637607517t_unit @ R @ M3 @ H )
=> ( ord_less_eq_set_a @ H @ ( carrie1074654371t_unit @ M3 ) ) ) ) ).
% Module.free_generator_sub
thf(fact_271_eSum__in__SubmoduleTr,axiom,
! [H: set_a,K: set_a,F: nat > a,N: nat,S: nat > b] :
( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ord_less_eq_set_a @ K @ H )
=> ( ( ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : K ) )
& ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) ) )
=> ( ( l_comb1138968323t_unit @ r @ ( mdl_a_b_c @ m @ ( algebr1152250919_d_a_c @ r @ m @ K ) ) @ N @ S @ F )
= ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) ) ) ) ) ).
% eSum_in_SubmoduleTr
thf(fact_272_eSum__in__Submodule,axiom,
! [H: set_a,K: set_a,F: nat > a,N: nat,S: nat > b] :
( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ord_less_eq_set_a @ K @ H )
=> ( ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : K ) )
=> ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) )
=> ( ( l_comb1138968323t_unit @ r @ ( mdl_a_b_c @ m @ ( algebr1152250919_d_a_c @ r @ m @ K ) ) @ N @ S @ F )
= ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) ) ) ) ) ) ).
% eSum_in_Submodule
thf(fact_273_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_274_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_275_l__span__closed2Tr,axiom,
! [A2: set_b,H: set_a,S: nat > b,N: nat,F: nat > a] :
( ( ideal_b_d @ r @ A2 )
=> ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : A2 ) )
& ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ) ).
% l_span_closed2Tr
thf(fact_276_l__span__closed2,axiom,
! [A2: set_b,H: set_a,S: nat > b,N: nat,F: nat > a] :
( ( ideal_b_d @ r @ A2 )
=> ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : A2 ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ) ) ).
% l_span_closed2
thf(fact_277_linear__comb0__2Tr,axiom,
! [A2: set_b,S: nat > b,N: nat,M: nat > a] :
( ( ideal_b_d @ r @ A2 )
=> ( ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : A2 ) )
& ( member_nat_a @ M
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) ) )
=> ( ( l_comb_b_d_a_c @ r @ m @ N @ S @ M )
= ( zero_a1261444626_a_b_c @ m ) ) ) ) ).
% linear_comb0_2Tr
thf(fact_278_module__inc__zero,axiom,
member_a @ ( zero_a1261444626_a_b_c @ m ) @ ( carrie2021454486_a_b_c @ m ) ).
% module_inc_zero
thf(fact_279_psubsetI,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_280_subsetI,axiom,
! [A2: set_nat_a,B2: set_nat_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_nat_a @ X2 @ B2 ) )
=> ( ord_le157368549_nat_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_281_subsetI,axiom,
! [A2: set_nat_b,B2: set_nat_b] :
( ! [X2: nat > b] :
( ( member_nat_b @ X2 @ A2 )
=> ( member_nat_b @ X2 @ B2 ) )
=> ( ord_le1296932838_nat_b @ A2 @ B2 ) ) ).
% subsetI
thf(fact_282_subsetI,axiom,
! [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_283_subset__antisym,axiom,
! [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_284_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_285_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_286_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_287_empty__iff,axiom,
! [C: nat > a] :
~ ( member_nat_a @ C @ bot_bot_set_nat_a ) ).
% empty_iff
thf(fact_288_empty__iff,axiom,
! [C: nat > b] :
~ ( member_nat_b @ C @ bot_bot_set_nat_b ) ).
% empty_iff
thf(fact_289_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_290_all__not__in__conv,axiom,
! [A2: set_nat_a] :
( ( ! [X3: nat > a] :
~ ( member_nat_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat_a ) ) ).
% all_not_in_conv
thf(fact_291_all__not__in__conv,axiom,
! [A2: set_nat_b] :
( ( ! [X3: nat > b] :
~ ( member_nat_b @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat_b ) ) ).
% all_not_in_conv
thf(fact_292_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_293_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_294_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_295_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_296_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_297_Diff__insert0,axiom,
! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ( minus_1788767276_nat_a @ A2 @ ( insert_nat_a @ X @ B2 ) )
= ( minus_1788767276_nat_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_298_Diff__insert0,axiom,
! [X: nat > b,A2: set_nat_b,B2: set_nat_b] :
( ~ ( member_nat_b @ X @ A2 )
=> ( ( minus_780847917_nat_b @ A2 @ ( insert_nat_b @ X @ B2 ) )
= ( minus_780847917_nat_b @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_299_Diff__insert0,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_300_insert__Diff1,axiom,
! [X: nat > a,B2: set_nat_a,A2: set_nat_a] :
( ( member_nat_a @ X @ B2 )
=> ( ( minus_1788767276_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
= ( minus_1788767276_nat_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_301_insert__Diff1,axiom,
! [X: nat > b,B2: set_nat_b,A2: set_nat_b] :
( ( member_nat_b @ X @ B2 )
=> ( ( minus_780847917_nat_b @ ( insert_nat_b @ X @ A2 ) @ B2 )
= ( minus_780847917_nat_b @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_302_insert__Diff1,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_303_insertCI,axiom,
! [A: nat > a,B2: set_nat_a,B: nat > a] :
( ( ~ ( member_nat_a @ A @ B2 )
=> ( A = B ) )
=> ( member_nat_a @ A @ ( insert_nat_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_304_insertCI,axiom,
! [A: nat > b,B2: set_nat_b,B: nat > b] :
( ( ~ ( member_nat_b @ A @ B2 )
=> ( A = B ) )
=> ( member_nat_b @ A @ ( insert_nat_b @ B @ B2 ) ) ) ).
% insertCI
thf(fact_305_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_306_insert__iff,axiom,
! [A: nat > a,B: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ ( insert_nat_a @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_307_insert__iff,axiom,
! [A: nat > b,B: nat > b,A2: set_nat_b] :
( ( member_nat_b @ A @ ( insert_nat_b @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_308_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_309_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_310_submodule__inc__0,axiom,
! [H: set_a] :
( ( submodule_b_d_a_c @ r @ m @ H )
=> ( member_a @ ( zero_a1261444626_a_b_c @ m ) @ H ) ) ).
% submodule_inc_0
thf(fact_311_mdl__carrier,axiom,
! [H: set_a] :
( ( submodule_b_d_a_c @ r @ m @ H )
=> ( ( carrie1074654371t_unit @ ( mdl_a_b_c @ m @ H ) )
= H ) ) ).
% mdl_carrier
thf(fact_312_mdl__is__module,axiom,
! [H: set_a] :
( ( submodule_b_d_a_c @ r @ m @ H )
=> ( module1821517916unit_d @ ( mdl_a_b_c @ m @ H ) @ r ) ) ).
% mdl_is_module
thf(fact_313_submodule__of__mdl,axiom,
! [H: set_a,N3: set_a] :
( ( submodule_b_d_a_c @ r @ m @ H )
=> ( ( submodule_b_d_a_c @ r @ m @ N3 )
=> ( ( ord_less_eq_set_a @ H @ N3 )
=> ( submod903911234t_unit @ r @ ( mdl_a_b_c @ m @ N3 ) @ H ) ) ) ) ).
% submodule_of_mdl
thf(fact_314_h__in__linear__span,axiom,
! [H: set_a,H2: a] :
( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ H2 @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) ) ) ).
% h_in_linear_span
thf(fact_315_l__span__cont__H,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( ord_less_eq_set_a @ H @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) ) ).
% l_span_cont_H
thf(fact_316_l__span__l__span,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) )
= ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) ) ).
% l_span_l_span
thf(fact_317_l__span__gen__mono,axiom,
! [K: set_a,H: set_a,A2: set_b] :
( ( ord_less_eq_set_a @ K @ H )
=> ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ideal_b_d @ r @ A2 )
=> ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A2 @ K ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ) ).
% l_span_gen_mono
thf(fact_318_linear__span__sub,axiom,
! [A2: set_b,H: set_a] :
( ( ideal_b_d @ r @ A2 )
=> ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) @ ( carrie2021454486_a_b_c @ m ) ) ) ) ).
% linear_span_sub
thf(fact_319_l__span__sub__submodule,axiom,
! [A2: set_b,N3: set_a,H: set_a] :
( ( ideal_b_d @ r @ A2 )
=> ( ( submodule_b_d_a_c @ r @ m @ N3 )
=> ( ( ord_less_eq_set_a @ H @ N3 )
=> ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) @ N3 ) ) ) ) ).
% l_span_sub_submodule
thf(fact_320_fgs__generator,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( genera1692266857t_unit @ r @ ( mdl_a_b_c @ m @ ( algebr1152250919_d_a_c @ r @ m @ H ) ) @ H ) ) ).
% fgs_generator
thf(fact_321_submodule__0,axiom,
submodule_b_d_a_c @ r @ m @ ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ).
% submodule_0
thf(fact_322_empty__fgs,axiom,
( ( algebr1152250919_d_a_c @ r @ m @ bot_bot_set_a )
= ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) ).
% empty_fgs
thf(fact_323_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_324_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_325_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_326_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_327_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_328_Diff__eq__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_329_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_330_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_331_insert__subset,axiom,
! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( ord_le157368549_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
= ( ( member_nat_a @ X @ B2 )
& ( ord_le157368549_nat_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_332_insert__subset,axiom,
! [X: nat > b,A2: set_nat_b,B2: set_nat_b] :
( ( ord_le1296932838_nat_b @ ( insert_nat_b @ X @ A2 ) @ B2 )
= ( ( member_nat_b @ X @ B2 )
& ( ord_le1296932838_nat_b @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_333_insert__subset,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( ( member_a @ X @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_334_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_335_singletonI,axiom,
! [A: nat > a] : ( member_nat_a @ A @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) ) ).
% singletonI
thf(fact_336_singletonI,axiom,
! [A: nat > b] : ( member_nat_b @ A @ ( insert_nat_b @ A @ bot_bot_set_nat_b ) ) ).
% singletonI
thf(fact_337_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_338_linear__span__subModule,axiom,
! [A2: set_b,H: set_a] :
( ( ideal_b_d @ r @ A2 )
=> ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( submodule_b_d_a_c @ r @ m @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ).
% linear_span_subModule
thf(fact_339_l__spanA__l__span,axiom,
! [A2: set_b,H: set_a] :
( ( ideal_b_d @ r @ A2 )
=> ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( linear_span_b_d_a_c @ r @ m @ A2 @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) )
= ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ).
% l_spanA_l_span
thf(fact_340_lin__span__coeff__mono,axiom,
! [A2: set_b,H: set_a] :
( ( ideal_b_d @ r @ A2 )
=> ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( ord_less_eq_set_a @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H ) ) ) ) ).
% lin_span_coeff_mono
thf(fact_341_linear__span__inc__0,axiom,
! [A2: set_b,H: set_a] :
( ( ideal_b_d @ r @ A2 )
=> ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( member_a @ ( zero_a1261444626_a_b_c @ m ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ).
% linear_span_inc_0
thf(fact_342_generator__generator,axiom,
! [H: set_a,H1: set_a] :
( ( generator_b_d_a_c @ r @ m @ H )
=> ( ( ord_less_eq_set_a @ H1 @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( ord_less_eq_set_a @ H @ ( linear_span_b_d_a_c @ r @ m @ ( carrie2079586589xt_b_d @ r ) @ H1 ) )
=> ( generator_b_d_a_c @ r @ m @ H1 ) ) ) ) ).
% generator_generator
thf(fact_343_l__span__zero,axiom,
! [A2: set_b] :
( ( ideal_b_d @ r @ A2 )
=> ( ( linear_span_b_d_a_c @ r @ m @ A2 @ ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) ) ).
% l_span_zero
thf(fact_344_l__span__closed3,axiom,
! [A2: set_b,H: set_a] :
( ( ideal_b_d @ r @ A2 )
=> ( ( generator_b_d_a_c @ r @ m @ H )
=> ( ( ( smodul818989740_d_a_c @ r @ m @ A2 )
= ( carrie2021454486_a_b_c @ m ) )
=> ( ( linear_span_b_d_a_c @ r @ m @ A2 @ H )
= ( carrie2021454486_a_b_c @ m ) ) ) ) ) ).
% l_span_closed3
thf(fact_345_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X3: a] : X3 = A )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_346_l__comb__mem__linear__span,axiom,
! [A2: set_b,H: set_a,S: nat > b,N: nat,F: nat > a] :
( ( ideal_b_d @ r @ A2 )
=> ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : A2 ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ) ) ).
% l_comb_mem_linear_span
thf(fact_347_l__span__closed,axiom,
! [A2: set_b,H: set_a,S: nat > b,N: nat,F: nat > a] :
( ( ideal_b_d @ r @ A2 )
=> ( ( ord_less_eq_set_a @ H @ ( carrie2021454486_a_b_c @ m ) )
=> ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : A2 ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) )
=> ( member_a @ ( l_comb_b_d_a_c @ r @ m @ N @ S @ F ) @ ( linear_span_b_d_a_c @ r @ m @ A2 @ H ) ) ) ) ) ) ).
% l_span_closed
thf(fact_348_linear__comb0__2,axiom,
! [A2: set_b,S: nat > b,N: nat,M: nat > a] :
( ( ideal_b_d @ r @ A2 )
=> ( ( member_nat_b @ S
@ ( pi_nat_b
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : A2 ) )
=> ( ( member_nat_a @ M
@ ( pi_nat_a
@ ( collect_nat
@ ^ [J2: nat] : ( ord_less_eq_nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( insert_a @ ( zero_a1261444626_a_b_c @ m ) @ bot_bot_set_a ) ) )
=> ( ( l_comb_b_d_a_c @ r @ m @ N @ S @ M )
= ( zero_a1261444626_a_b_c @ m ) ) ) ) ) ).
% linear_comb0_2
% Conjectures (22)
thf(conj_0,hypothesis,
( member_nat_a @ g
@ ( pi_nat_a
@ ( collect_nat
@ ^ [K4: nat] : ( ord_less_eq_nat @ K4 @ m2 ) )
@ ^ [Uu: nat] : ( insert_a @ h2 @ h1 ) ) ) ).
thf(conj_1,hypothesis,
! [X4: nat] :
( ( ord_less_eq_nat @ X4 @ m2 )
=> ! [Y3: nat] :
( ( ord_less_eq_nat @ Y3 @ m2 )
=> ( ( ( g @ X4 )
= ( g @ Y3 ) )
=> ( X4 = Y3 ) ) ) ) ).
thf(conj_2,hypothesis,
( member_nat_b @ t
@ ( pi_nat_b
@ ( collect_nat
@ ^ [K4: nat] : ( ord_less_eq_nat @ K4 @ m2 ) )
@ ^ [Uu: nat] : ( carrie2079586589xt_b_d @ r ) ) ) ).
thf(conj_3,hypothesis,
( ( g @ m2 )
= h2 ) ).
thf(conj_4,hypothesis,
module_e_b_f_d @ n @ r ).
thf(conj_5,hypothesis,
free_g1087686480_d_a_c @ r @ m @ h ).
thf(conj_6,hypothesis,
ord_less_eq_set_a @ h1 @ h ).
thf(conj_7,hypothesis,
$true ).
thf(conj_8,hypothesis,
ideal_b_d @ r @ ( carrie2079586589xt_b_d @ r ) ).
thf(conj_9,hypothesis,
ord_less_eq_set_a @ h @ ( carrie2021454486_a_b_c @ m ) ).
thf(conj_10,hypothesis,
ord_less_eq_set_a @ h1 @ ( carrie2021454486_a_b_c @ m ) ).
thf(conj_11,hypothesis,
ord_less_eq_set_a @ ( algebr1152250919_d_a_c @ r @ m @ h1 ) @ ( carrie2021454486_a_b_c @ m ) ).
thf(conj_12,hypothesis,
member_a @ ( l_comb_b_d_a_c @ r @ m @ m2 @ t @ g ) @ ( algebr1152250919_d_a_c @ r @ m @ ( insert_a @ h2 @ h1 ) ) ).
thf(conj_13,hypothesis,
( x
= ( l_comb_b_d_a_c @ r @ m @ m2 @ t @ g ) ) ).
thf(conj_14,hypothesis,
member_a @ h2 @ h ).
thf(conj_15,hypothesis,
~ ( member_a @ h2 @ h1 ) ).
thf(conj_16,hypothesis,
ord_less_nat @ zero_zero_nat @ m2 ).
thf(conj_17,hypothesis,
member_a @ h2 @ ( carrie2021454486_a_b_c @ m ) ).
thf(conj_18,hypothesis,
submodule_b_d_a_c @ r @ m @ ( algebr1152250919_d_a_c @ r @ m @ ( insert_a @ h2 @ bot_bot_set_a ) ) ).
thf(conj_19,hypothesis,
ord_less_eq_nat @ xa @ ( minus_minus_nat @ m2 @ ( suc @ zero_zero_nat ) ) ).
thf(conj_20,hypothesis,
( ( ( g @ xa )
= h2 )
| ( member_a @ ( g @ xa ) @ h1 ) ) ).
thf(conj_21,conjecture,
member_a @ ( g @ xa ) @ h1 ).
%------------------------------------------------------------------------------