TPTP Problem File: ITP118^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP118^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Minkowskis_Theorem problem prob_62__6246984_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 : Minkowskis_Theorem/prob_62__6246984_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 466 ( 241 unt; 102 typ; 0 def)
% Number of atoms : 767 ( 316 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 2505 ( 39 ~; 4 |; 25 &;2191 @)
% ( 0 <=>; 246 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Number of types : 17 ( 16 usr)
% Number of type conns : 153 ( 153 >; 0 *; 0 +; 0 <<)
% Number of symbols : 89 ( 86 usr; 18 con; 0-3 aty)
% Number of variables : 842 ( 83 ^; 740 !; 19 ?; 842 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:43:54.513
%------------------------------------------------------------------------------
% Could-be-implicit typings (16)
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thf(ty_n_t__Real__Oreal,type,
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thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__n,type,
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% Explicit typings (86)
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thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
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thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__nth_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_001t__Numeral____Type__Onum1,type,
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thf(sy_c_Finite__Cartesian__Product_Ovector__scalar__mult_001t__Real__Oreal_001tf__n,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mt__Numeral____Type__Onum1_J,type,
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minus_1196255695_int_n: finite964658038_int_n > finite964658038_int_n > finite964658038_int_n ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
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one_one_Numeral_num1: numeral_num1 ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_If_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mt__Numeral____Type__Onum1_J,type,
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thf(sy_c_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
if_Fin1668193290l_num1: $o > finite1183840848l_num1 > finite1183840848l_num1 > finite1183840848l_num1 ).
thf(sy_c_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
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thf(sy_c_Int_Oring__1__class_OInts_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
ring_11529266783l_num1: set_Fi1257234438l_num1 ).
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ring_1723899781real_n: set_Fi1058188332real_n ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
ring_1_Ints_int: set_int ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
ring_1_Ints_real: set_real ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
ring_1154201247l_num1: int > finite1183840848l_num1 ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
ring_11737179077real_n: int > finite1489363574real_n ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Minkowskis__Theorem__Mirabelle__uzuvqgwfeb_Oof__int__vec_001t__Numeral____Type__Onum1_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
minkow781993583real_n: finite2063899472l_num1 > finite121157254l_num1 ).
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minkow2063319845m1_int: finite2063899472l_num1 > finite2063899472l_num1 ).
thf(sy_c_Minkowskis__Theorem__Mirabelle__uzuvqgwfeb_Oof__int__vec_001t__Numeral____Type__Onum1_001t__Real__Oreal,type,
minkow19073400691_real: finite2063899472l_num1 > finite1183840848l_num1 ).
thf(sy_c_Minkowskis__Theorem__Mirabelle__uzuvqgwfeb_Oof__int__vec_001tf__n_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
minkow371489429real_n: finite964658038_int_n > finite318270636al_n_n ).
thf(sy_c_Minkowskis__Theorem__Mirabelle__uzuvqgwfeb_Oof__int__vec_001tf__n_001t__Int__Oint,type,
minkow1468704075_n_int: finite964658038_int_n > finite964658038_int_n ).
thf(sy_c_Minkowskis__Theorem__Mirabelle__uzuvqgwfeb_Oof__int__vec_001tf__n_001t__Real__Oreal,type,
minkow1134813771n_real: finite964658038_int_n > finite1489363574real_n ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
ord_le2035778724l_num1: finite1183840848l_num1 > finite1183840848l_num1 > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
ord_le1017090250real_n: finite1489363574real_n > finite1489363574real_n > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mt__Numeral____Type__Onum1_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_Mt__Numeral____Type__Onum1_J,type,
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thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Numeral____Type__Onum1_J,type,
real_V845814485l_num1: finite121157254l_num1 > finite121157254l_num1 > real ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
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thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
real_V783965509real_n: finite1489363574real_n > finite1489363574real_n > real ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
real_V1934908667t_real: real > real > real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_Mt__Numeral____Type__Onum1_J,type,
real_V316032763l_num1: finite463740460l_num1 > real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_Mtf__n_J,type,
real_V1121707553num1_n: finite337957458num1_n > real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
real_V400292255l_num1: finite1183840848l_num1 > real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
real_V739724485real_n: finite1489363574real_n > real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
real_V646646907m_real: real > real ).
thf(sy_c_Set_OCollect_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
collec321817931real_n: ( finite1489363574real_n > $o ) > set_Fi1058188332real_n ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_member_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
member1413569767l_num1: finite1183840848l_num1 > set_Fi1257234438l_num1 > $o ).
thf(sy_c_member_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
member1352538125real_n: finite1489363574real_n > set_Fi1058188332real_n > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_A,type,
a: set_Fi1058188332real_n ).
thf(sy_v_i____,type,
i: n ).
thf(sy_v_m____,type,
m: int ).
thf(sy_v_x____,type,
x: finite1489363574real_n ).
thf(sy_v_y____,type,
y: finite1489363574real_n ).
% Relevant facts (354)
thf(fact_0__C1_C_I2_J,axiom,
member1352538125real_n @ y @ a ).
% "1"(2)
thf(fact_1__C1_C_I1_J,axiom,
member1352538125real_n @ x @ a ).
% "1"(1)
thf(fact_2_m,axiom,
( ( ring_1_of_int_real @ m )
= ( minus_minus_real @ ( finite772340589real_n @ x @ i ) @ ( finite772340589real_n @ y @ i ) ) ) ).
% m
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_Areal__of__int_Am_A_061_Ax_A_E_Ai_A_N_Ay_A_E_Ai_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [M: int] :
( ( ring_1_of_int_real @ M )
!= ( minus_minus_real @ ( finite772340589real_n @ x @ i ) @ ( finite772340589real_n @ y @ i ) ) ) ).
% \<open>\<And>thesis. (\<And>m. real_of_int m = x $ i - y $ i \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_vector__minus__component,axiom,
! [X: finite2063899472l_num1,Y: finite2063899472l_num1,I: numeral_num1] :
( ( finite1800583751l_num1 @ ( minus_779068585l_num1 @ X @ Y ) @ I )
= ( minus_minus_int @ ( finite1800583751l_num1 @ X @ I ) @ ( finite1800583751l_num1 @ Y @ I ) ) ) ).
% vector_minus_component
thf(fact_5_vector__minus__component,axiom,
! [X: finite964658038_int_n,Y: finite964658038_int_n,I: n] :
( ( finite566214509_int_n @ ( minus_1196255695_int_n @ X @ Y ) @ I )
= ( minus_minus_int @ ( finite566214509_int_n @ X @ I ) @ ( finite566214509_int_n @ Y @ I ) ) ) ).
% vector_minus_component
thf(fact_6_vector__minus__component,axiom,
! [X: finite1489363574real_n,Y: finite1489363574real_n,I: n] :
( ( finite772340589real_n @ ( minus_1037315151real_n @ X @ Y ) @ I )
= ( minus_minus_real @ ( finite772340589real_n @ X @ I ) @ ( finite772340589real_n @ Y @ I ) ) ) ).
% vector_minus_component
thf(fact_7_vector__minus__component,axiom,
! [X: finite1183840848l_num1,Y: finite1183840848l_num1,I: numeral_num1] :
( ( finite1106245191l_num1 @ ( minus_249161513l_num1 @ X @ Y ) @ I )
= ( minus_minus_real @ ( finite1106245191l_num1 @ X @ I ) @ ( finite1106245191l_num1 @ Y @ I ) ) ) ).
% vector_minus_component
thf(fact_8_of__int__abs,axiom,
! [X: int] :
( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
= ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).
% of_int_abs
thf(fact_9_of__int__abs,axiom,
! [X: int] :
( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
= ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).
% of_int_abs
thf(fact_10_of__int__diff,axiom,
! [W: int,Z: int] :
( ( ring_1154201247l_num1 @ ( minus_minus_int @ W @ Z ) )
= ( minus_249161513l_num1 @ ( ring_1154201247l_num1 @ W ) @ ( ring_1154201247l_num1 @ Z ) ) ) ).
% of_int_diff
thf(fact_11_of__int__diff,axiom,
! [W: int,Z: int] :
( ( ring_11737179077real_n @ ( minus_minus_int @ W @ Z ) )
= ( minus_1037315151real_n @ ( ring_11737179077real_n @ W ) @ ( ring_11737179077real_n @ Z ) ) ) ).
% of_int_diff
thf(fact_12_of__int__diff,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
= ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% of_int_diff
thf(fact_13_of__int__diff,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
= ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_diff
thf(fact_14__092_060open_062x_A_E_Ai_A_N_Ay_A_E_Ai_A_092_060in_062_A_092_060int_062_092_060close_062,axiom,
member_real @ ( minus_minus_real @ ( finite772340589real_n @ x @ i ) @ ( finite772340589real_n @ y @ i ) ) @ ring_1_Ints_real ).
% \<open>x $ i - y $ i \<in> \<int>\<close>
thf(fact_15_of__int__vec__nth,axiom,
! [V: finite2063899472l_num1,N: numeral_num1] :
( ( finite1503577981l_num1 @ ( minkow781993583real_n @ V ) @ N )
= ( ring_11737179077real_n @ ( finite1800583751l_num1 @ V @ N ) ) ) ).
% of_int_vec_nth
thf(fact_16_of__int__vec__nth,axiom,
! [V: finite964658038_int_n,N: n] :
( ( finite625646243al_n_n @ ( minkow371489429real_n @ V ) @ N )
= ( ring_11737179077real_n @ ( finite566214509_int_n @ V @ N ) ) ) ).
% of_int_vec_nth
thf(fact_17_of__int__vec__nth,axiom,
! [V: finite2063899472l_num1,N: numeral_num1] :
( ( finite1800583751l_num1 @ ( minkow2063319845m1_int @ V ) @ N )
= ( ring_1_of_int_int @ ( finite1800583751l_num1 @ V @ N ) ) ) ).
% of_int_vec_nth
thf(fact_18_of__int__vec__nth,axiom,
! [V: finite964658038_int_n,N: n] :
( ( finite566214509_int_n @ ( minkow1468704075_n_int @ V ) @ N )
= ( ring_1_of_int_int @ ( finite566214509_int_n @ V @ N ) ) ) ).
% of_int_vec_nth
thf(fact_19_of__int__vec__nth,axiom,
! [V: finite964658038_int_n,N: n] :
( ( finite772340589real_n @ ( minkow1134813771n_real @ V ) @ N )
= ( ring_1_of_int_real @ ( finite566214509_int_n @ V @ N ) ) ) ).
% of_int_vec_nth
thf(fact_20_of__int__vec__nth,axiom,
! [V: finite2063899472l_num1,N: numeral_num1] :
( ( finite1106245191l_num1 @ ( minkow19073400691_real @ V ) @ N )
= ( ring_1_of_int_real @ ( finite1800583751l_num1 @ V @ N ) ) ) ).
% of_int_vec_nth
thf(fact_21_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( ( ring_1_of_int_int @ W )
= ( ring_1_of_int_int @ Z ) )
= ( W = Z ) ) ).
% of_int_eq_iff
thf(fact_22_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( ( ring_11737179077real_n @ W )
= ( ring_11737179077real_n @ Z ) )
= ( W = Z ) ) ).
% of_int_eq_iff
thf(fact_23_of__int__eq__iff,axiom,
! [W: int,Z: int] :
( ( ( ring_1_of_int_real @ W )
= ( ring_1_of_int_real @ Z ) )
= ( W = Z ) ) ).
% of_int_eq_iff
thf(fact_24_abs__abs,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_abs
thf(fact_25_abs__abs,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_abs
thf(fact_26_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_27_abs__idempotent,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_idempotent
thf(fact_28_abs__minus__commute,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] :
( ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ A @ B ) )
= ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_29_abs__minus__commute,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
= ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_30_abs__minus__commute,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
= ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_31_abs__minus__commute,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1] :
( ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ A @ B ) )
= ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_32_lambda__skolem,axiom,
! [P: n > real > $o] :
( ( ! [I2: n] :
? [X2: real] : ( P @ I2 @ X2 ) )
= ( ? [X3: finite1489363574real_n] :
! [I2: n] : ( P @ I2 @ ( finite772340589real_n @ X3 @ I2 ) ) ) ) ).
% lambda_skolem
thf(fact_33_lambda__skolem,axiom,
! [P: numeral_num1 > real > $o] :
( ( ! [I2: numeral_num1] :
? [X2: real] : ( P @ I2 @ X2 ) )
= ( ? [X3: finite1183840848l_num1] :
! [I2: numeral_num1] : ( P @ I2 @ ( finite1106245191l_num1 @ X3 @ I2 ) ) ) ) ).
% lambda_skolem
thf(fact_34_lambda__skolem,axiom,
! [P: numeral_num1 > int > $o] :
( ( ! [I2: numeral_num1] :
? [X2: int] : ( P @ I2 @ X2 ) )
= ( ? [X3: finite2063899472l_num1] :
! [I2: numeral_num1] : ( P @ I2 @ ( finite1800583751l_num1 @ X3 @ I2 ) ) ) ) ).
% lambda_skolem
thf(fact_35_lambda__skolem,axiom,
! [P: n > int > $o] :
( ( ! [I2: n] :
? [X2: int] : ( P @ I2 @ X2 ) )
= ( ? [X3: finite964658038_int_n] :
! [I2: n] : ( P @ I2 @ ( finite566214509_int_n @ X3 @ I2 ) ) ) ) ).
% lambda_skolem
thf(fact_36_vec__eq__iff,axiom,
( ( ^ [Y2: finite1489363574real_n,Z2: finite1489363574real_n] : Y2 = Z2 )
= ( ^ [X3: finite1489363574real_n,Y3: finite1489363574real_n] :
! [I2: n] :
( ( finite772340589real_n @ X3 @ I2 )
= ( finite772340589real_n @ Y3 @ I2 ) ) ) ) ).
% vec_eq_iff
thf(fact_37_vec__eq__iff,axiom,
( ( ^ [Y2: finite1183840848l_num1,Z2: finite1183840848l_num1] : Y2 = Z2 )
= ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] :
! [I2: numeral_num1] :
( ( finite1106245191l_num1 @ X3 @ I2 )
= ( finite1106245191l_num1 @ Y3 @ I2 ) ) ) ) ).
% vec_eq_iff
thf(fact_38_vec__eq__iff,axiom,
( ( ^ [Y2: finite2063899472l_num1,Z2: finite2063899472l_num1] : Y2 = Z2 )
= ( ^ [X3: finite2063899472l_num1,Y3: finite2063899472l_num1] :
! [I2: numeral_num1] :
( ( finite1800583751l_num1 @ X3 @ I2 )
= ( finite1800583751l_num1 @ Y3 @ I2 ) ) ) ) ).
% vec_eq_iff
thf(fact_39_vec__eq__iff,axiom,
( ( ^ [Y2: finite964658038_int_n,Z2: finite964658038_int_n] : Y2 = Z2 )
= ( ^ [X3: finite964658038_int_n,Y3: finite964658038_int_n] :
! [I2: n] :
( ( finite566214509_int_n @ X3 @ I2 )
= ( finite566214509_int_n @ Y3 @ I2 ) ) ) ) ).
% vec_eq_iff
thf(fact_40_assms,axiom,
! [V: finite1489363574real_n,I: n] :
( ( member1352538125real_n @ V @ a )
=> ( member_real @ ( finite772340589real_n @ V @ I ) @ ring_1_Ints_real ) ) ).
% assms
thf(fact_41_of__int__vec__eq__iff,axiom,
! [A: finite2063899472l_num1,B: finite2063899472l_num1] :
( ( ( minkow19073400691_real @ A )
= ( minkow19073400691_real @ B ) )
= ( A = B ) ) ).
% of_int_vec_eq_iff
thf(fact_42_of__int__vec__eq__iff,axiom,
! [A: finite964658038_int_n,B: finite964658038_int_n] :
( ( ( minkow1134813771n_real @ A )
= ( minkow1134813771n_real @ B ) )
= ( A = B ) ) ).
% of_int_vec_eq_iff
thf(fact_43_Ints__diff,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] :
( ( member1352538125real_n @ A @ ring_1723899781real_n )
=> ( ( member1352538125real_n @ B @ ring_1723899781real_n )
=> ( member1352538125real_n @ ( minus_1037315151real_n @ A @ B ) @ ring_1723899781real_n ) ) ) ).
% Ints_diff
thf(fact_44_Ints__diff,axiom,
! [A: real,B: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( member_real @ B @ ring_1_Ints_real )
=> ( member_real @ ( minus_minus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% Ints_diff
thf(fact_45_Ints__diff,axiom,
! [A: int,B: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( member_int @ B @ ring_1_Ints_int )
=> ( member_int @ ( minus_minus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% Ints_diff
thf(fact_46_Ints__diff,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1] :
( ( member1413569767l_num1 @ A @ ring_11529266783l_num1 )
=> ( ( member1413569767l_num1 @ B @ ring_11529266783l_num1 )
=> ( member1413569767l_num1 @ ( minus_249161513l_num1 @ A @ B ) @ ring_11529266783l_num1 ) ) ) ).
% Ints_diff
thf(fact_47_Ints__abs,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( member_int @ ( abs_abs_int @ A ) @ ring_1_Ints_int ) ) ).
% Ints_abs
thf(fact_48_Ints__abs,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( member_real @ ( abs_abs_real @ A ) @ ring_1_Ints_real ) ) ).
% Ints_abs
thf(fact_49_Ints__of__int,axiom,
! [Z: int] : ( member_real @ ( ring_1_of_int_real @ Z ) @ ring_1_Ints_real ) ).
% Ints_of_int
thf(fact_50_Ints__of__int,axiom,
! [Z: int] : ( member_int @ ( ring_1_of_int_int @ Z ) @ ring_1_Ints_int ) ).
% Ints_of_int
thf(fact_51_Ints__of__int,axiom,
! [Z: int] : ( member1352538125real_n @ ( ring_11737179077real_n @ Z ) @ ring_1723899781real_n ) ).
% Ints_of_int
thf(fact_52_Ints__induct,axiom,
! [Q: real,P: real > $o] :
( ( member_real @ Q @ ring_1_Ints_real )
=> ( ! [Z3: int] : ( P @ ( ring_1_of_int_real @ Z3 ) )
=> ( P @ Q ) ) ) ).
% Ints_induct
thf(fact_53_Ints__induct,axiom,
! [Q: int,P: int > $o] :
( ( member_int @ Q @ ring_1_Ints_int )
=> ( ! [Z3: int] : ( P @ ( ring_1_of_int_int @ Z3 ) )
=> ( P @ Q ) ) ) ).
% Ints_induct
thf(fact_54_Ints__induct,axiom,
! [Q: finite1489363574real_n,P: finite1489363574real_n > $o] :
( ( member1352538125real_n @ Q @ ring_1723899781real_n )
=> ( ! [Z3: int] : ( P @ ( ring_11737179077real_n @ Z3 ) )
=> ( P @ Q ) ) ) ).
% Ints_induct
thf(fact_55_Ints__cases,axiom,
! [Q: real] :
( ( member_real @ Q @ ring_1_Ints_real )
=> ~ ! [Z3: int] :
( Q
!= ( ring_1_of_int_real @ Z3 ) ) ) ).
% Ints_cases
thf(fact_56_Ints__cases,axiom,
! [Q: int] :
( ( member_int @ Q @ ring_1_Ints_int )
=> ~ ! [Z3: int] :
( Q
!= ( ring_1_of_int_int @ Z3 ) ) ) ).
% Ints_cases
thf(fact_57_Ints__cases,axiom,
! [Q: finite1489363574real_n] :
( ( member1352538125real_n @ Q @ ring_1723899781real_n )
=> ~ ! [Z3: int] :
( Q
!= ( ring_11737179077real_n @ Z3 ) ) ) ).
% Ints_cases
thf(fact_58_diff__right__commute,axiom,
! [A: finite1489363574real_n,C: finite1489363574real_n,B: finite1489363574real_n] :
( ( minus_1037315151real_n @ ( minus_1037315151real_n @ A @ C ) @ B )
= ( minus_1037315151real_n @ ( minus_1037315151real_n @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_59_diff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_60_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_61_diff__right__commute,axiom,
! [A: finite1183840848l_num1,C: finite1183840848l_num1,B: finite1183840848l_num1] :
( ( minus_249161513l_num1 @ ( minus_249161513l_num1 @ A @ C ) @ B )
= ( minus_249161513l_num1 @ ( minus_249161513l_num1 @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_62_diff__eq__diff__eq,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n,D: finite1489363574real_n] :
( ( ( minus_1037315151real_n @ A @ B )
= ( minus_1037315151real_n @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_63_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_64_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_65_diff__eq__diff__eq,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1,D: finite1183840848l_num1] :
( ( ( minus_249161513l_num1 @ A @ B )
= ( minus_249161513l_num1 @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_66_vec__nth__inject,axiom,
! [X: finite1489363574real_n,Y: finite1489363574real_n] :
( ( ( finite772340589real_n @ X )
= ( finite772340589real_n @ Y ) )
= ( X = Y ) ) ).
% vec_nth_inject
thf(fact_67_vec__nth__inject,axiom,
! [X: finite1183840848l_num1,Y: finite1183840848l_num1] :
( ( ( finite1106245191l_num1 @ X )
= ( finite1106245191l_num1 @ Y ) )
= ( X = Y ) ) ).
% vec_nth_inject
thf(fact_68_vec__nth__inject,axiom,
! [X: finite2063899472l_num1,Y: finite2063899472l_num1] :
( ( ( finite1800583751l_num1 @ X )
= ( finite1800583751l_num1 @ Y ) )
= ( X = Y ) ) ).
% vec_nth_inject
thf(fact_69_vec__nth__inject,axiom,
! [X: finite964658038_int_n,Y: finite964658038_int_n] :
( ( ( finite566214509_int_n @ X )
= ( finite566214509_int_n @ Y ) )
= ( X = Y ) ) ).
% vec_nth_inject
thf(fact_70_cond__component,axiom,
! [B: $o,X: finite1489363574real_n,Y: finite1489363574real_n,I: n] :
( ( B
=> ( ( finite772340589real_n @ ( if_Fin127821360real_n @ B @ X @ Y ) @ I )
= ( finite772340589real_n @ X @ I ) ) )
& ( ~ B
=> ( ( finite772340589real_n @ ( if_Fin127821360real_n @ B @ X @ Y ) @ I )
= ( finite772340589real_n @ Y @ I ) ) ) ) ).
% cond_component
thf(fact_71_cond__component,axiom,
! [B: $o,X: finite1183840848l_num1,Y: finite1183840848l_num1,I: numeral_num1] :
( ( B
=> ( ( finite1106245191l_num1 @ ( if_Fin1668193290l_num1 @ B @ X @ Y ) @ I )
= ( finite1106245191l_num1 @ X @ I ) ) )
& ( ~ B
=> ( ( finite1106245191l_num1 @ ( if_Fin1668193290l_num1 @ B @ X @ Y ) @ I )
= ( finite1106245191l_num1 @ Y @ I ) ) ) ) ).
% cond_component
thf(fact_72_cond__component,axiom,
! [B: $o,X: finite2063899472l_num1,Y: finite2063899472l_num1,I: numeral_num1] :
( ( B
=> ( ( finite1800583751l_num1 @ ( if_Fin1570437642l_num1 @ B @ X @ Y ) @ I )
= ( finite1800583751l_num1 @ X @ I ) ) )
& ( ~ B
=> ( ( finite1800583751l_num1 @ ( if_Fin1570437642l_num1 @ B @ X @ Y ) @ I )
= ( finite1800583751l_num1 @ Y @ I ) ) ) ) ).
% cond_component
thf(fact_73_cond__component,axiom,
! [B: $o,X: finite964658038_int_n,Y: finite964658038_int_n,I: n] :
( ( B
=> ( ( finite566214509_int_n @ ( if_Fin1767949360_int_n @ B @ X @ Y ) @ I )
= ( finite566214509_int_n @ X @ I ) ) )
& ( ~ B
=> ( ( finite566214509_int_n @ ( if_Fin1767949360_int_n @ B @ X @ Y ) @ I )
= ( finite566214509_int_n @ Y @ I ) ) ) ) ).
% cond_component
thf(fact_74_dist__of__int,axiom,
! [M2: int,N: int] :
( ( real_V1934908667t_real @ ( ring_1_of_int_real @ M2 ) @ ( ring_1_of_int_real @ N ) )
= ( ring_1_of_int_real @ ( abs_abs_int @ ( minus_minus_int @ M2 @ N ) ) ) ) ).
% dist_of_int
thf(fact_75_vec__sub,axiom,
! [X: real,Y: real] :
( ( finite667548811real_n @ ( minus_minus_real @ X @ Y ) )
= ( minus_1037315151real_n @ ( finite667548811real_n @ X ) @ ( finite667548811real_n @ Y ) ) ) ).
% vec_sub
thf(fact_76_vec__sub,axiom,
! [X: real,Y: real] :
( ( finite1329045861l_num1 @ ( minus_minus_real @ X @ Y ) )
= ( minus_249161513l_num1 @ ( finite1329045861l_num1 @ X ) @ ( finite1329045861l_num1 @ Y ) ) ) ).
% vec_sub
thf(fact_77_vector__sub__rdistrib,axiom,
! [A: real,B: real,X: finite1489363574real_n] :
( ( finite2053577756real_n @ ( minus_minus_real @ A @ B ) @ X )
= ( minus_1037315151real_n @ ( finite2053577756real_n @ A @ X ) @ ( finite2053577756real_n @ B @ X ) ) ) ).
% vector_sub_rdistrib
thf(fact_78_vector__sub__rdistrib,axiom,
! [A: real,B: real,X: finite1183840848l_num1] :
( ( finite1961675766l_num1 @ ( minus_minus_real @ A @ B ) @ X )
= ( minus_249161513l_num1 @ ( finite1961675766l_num1 @ A @ X ) @ ( finite1961675766l_num1 @ B @ X ) ) ) ).
% vector_sub_rdistrib
thf(fact_79__C1_C_I3_J,axiom,
ord_less_real @ ( real_V783965509real_n @ y @ x ) @ one_one_real ).
% "1"(3)
thf(fact_80_norm__of__int,axiom,
! [Z: int] :
( ( real_V646646907m_real @ ( ring_1_of_int_real @ Z ) )
= ( abs_abs_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% norm_of_int
thf(fact_81_ceiling__diff__of__int,axiom,
! [X: real,Z: int] :
( ( archim1371465213g_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
= ( minus_minus_int @ ( archim1371465213g_real @ X ) @ Z ) ) ).
% ceiling_diff_of_int
thf(fact_82_axis__nth,axiom,
! [I: n,X: real] :
( ( finite772340589real_n @ ( finite703491444n_real @ I @ X ) @ I )
= X ) ).
% axis_nth
thf(fact_83_axis__nth,axiom,
! [I: numeral_num1,X: real] :
( ( finite1106245191l_num1 @ ( finite15144922381_real @ I @ X ) @ I )
= X ) ).
% axis_nth
thf(fact_84_axis__nth,axiom,
! [I: numeral_num1,X: int] :
( ( finite1800583751l_num1 @ ( finite627713486m1_int @ I @ X ) @ I )
= X ) ).
% axis_nth
thf(fact_85_axis__nth,axiom,
! [I: n,X: int] :
( ( finite566214509_int_n @ ( finite1888777460_n_int @ I @ X ) @ I )
= X ) ).
% axis_nth
thf(fact_86_vec_Oscale__left__diff__distrib,axiom,
! [A: real,B: real,X: finite1489363574real_n] :
( ( finite2053577756real_n @ ( minus_minus_real @ A @ B ) @ X )
= ( minus_1037315151real_n @ ( finite2053577756real_n @ A @ X ) @ ( finite2053577756real_n @ B @ X ) ) ) ).
% vec.scale_left_diff_distrib
thf(fact_87_vec_Oscale__left__diff__distrib,axiom,
! [A: real,B: real,X: finite1183840848l_num1] :
( ( finite1961675766l_num1 @ ( minus_minus_real @ A @ B ) @ X )
= ( minus_249161513l_num1 @ ( finite1961675766l_num1 @ A @ X ) @ ( finite1961675766l_num1 @ B @ X ) ) ) ).
% vec.scale_left_diff_distrib
thf(fact_88_floor__diff__of__int,axiom,
! [X: real,Z: int] :
( ( archim1031974863r_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
= ( minus_minus_int @ ( archim1031974863r_real @ X ) @ Z ) ) ).
% floor_diff_of_int
thf(fact_89_norm__axis__1,axiom,
! [M2: numeral_num1] :
( ( real_V400292255l_num1 @ ( finite15144922381_real @ M2 @ one_one_real ) )
= one_one_real ) ).
% norm_axis_1
thf(fact_90_abs__1,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
% abs_1
thf(fact_91_abs__1,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_1
thf(fact_92_mem__Collect__eq,axiom,
! [A: finite1489363574real_n,P: finite1489363574real_n > $o] :
( ( member1352538125real_n @ A @ ( collec321817931real_n @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_93_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_94_Collect__mem__eq,axiom,
! [A2: set_Fi1058188332real_n] :
( ( collec321817931real_n
@ ^ [X3: finite1489363574real_n] : ( member1352538125real_n @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_95_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_96_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_97_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_98_one__index,axiom,
! [I: n] :
( ( finite772340589real_n @ one_on1253059131real_n @ I )
= one_one_real ) ).
% one_index
thf(fact_99_one__index,axiom,
! [I: numeral_num1] :
( ( finite1106245191l_num1 @ one_on1961019669l_num1 @ I )
= one_one_real ) ).
% one_index
thf(fact_100_one__index,axiom,
! [I: numeral_num1] :
( ( finite1800583751l_num1 @ one_on601734293l_num1 @ I )
= one_one_int ) ).
% one_index
thf(fact_101_one__index,axiom,
! [I: n] :
( ( finite566214509_int_n @ one_on1516019131_int_n @ I )
= one_one_int ) ).
% one_index
thf(fact_102_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_real @ Z )
= one_one_real )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_103_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= one_one_int )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_104_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_11737179077real_n @ Z )
= one_on1253059131real_n )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_105_of__int__1,axiom,
( ( ring_1_of_int_real @ one_one_int )
= one_one_real ) ).
% of_int_1
thf(fact_106_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_107_of__int__1,axiom,
( ( ring_11737179077real_n @ one_one_int )
= one_on1253059131real_n ) ).
% of_int_1
thf(fact_108_abs__norm__cancel,axiom,
! [A: real] :
( ( abs_abs_real @ ( real_V646646907m_real @ A ) )
= ( real_V646646907m_real @ A ) ) ).
% abs_norm_cancel
thf(fact_109_abs__norm__cancel,axiom,
! [A: finite1183840848l_num1] :
( ( abs_abs_real @ ( real_V400292255l_num1 @ A ) )
= ( real_V400292255l_num1 @ A ) ) ).
% abs_norm_cancel
thf(fact_110_floor__one,axiom,
( ( archim1031974863r_real @ one_one_real )
= one_one_int ) ).
% floor_one
thf(fact_111_ceiling__one,axiom,
( ( archim1371465213g_real @ one_one_real )
= one_one_int ) ).
% ceiling_one
thf(fact_112_floor__of__int,axiom,
! [Z: int] :
( ( archim1031974863r_real @ ( ring_1_of_int_real @ Z ) )
= Z ) ).
% floor_of_int
thf(fact_113_ceiling__of__int,axiom,
! [Z: int] :
( ( archim1371465213g_real @ ( ring_1_of_int_real @ Z ) )
= Z ) ).
% ceiling_of_int
thf(fact_114_vec__component,axiom,
! [X: real,I: n] :
( ( finite772340589real_n @ ( finite667548811real_n @ X ) @ I )
= X ) ).
% vec_component
thf(fact_115_vec__component,axiom,
! [X: real,I: numeral_num1] :
( ( finite1106245191l_num1 @ ( finite1329045861l_num1 @ X ) @ I )
= X ) ).
% vec_component
thf(fact_116_vec__component,axiom,
! [X: int,I: numeral_num1] :
( ( finite1800583751l_num1 @ ( finite1470207077l_num1 @ X ) @ I )
= X ) ).
% vec_component
thf(fact_117_vec__component,axiom,
! [X: int,I: n] :
( ( finite566214509_int_n @ ( finite55766923_int_n @ X ) @ I )
= X ) ).
% vec_component
thf(fact_118_vector__ssub__ldistrib,axiom,
! [C: real,X: finite1489363574real_n,Y: finite1489363574real_n] :
( ( finite2053577756real_n @ C @ ( minus_1037315151real_n @ X @ Y ) )
= ( minus_1037315151real_n @ ( finite2053577756real_n @ C @ X ) @ ( finite2053577756real_n @ C @ Y ) ) ) ).
% vector_ssub_ldistrib
thf(fact_119_vector__ssub__ldistrib,axiom,
! [C: real,X: finite1183840848l_num1,Y: finite1183840848l_num1] :
( ( finite1961675766l_num1 @ C @ ( minus_249161513l_num1 @ X @ Y ) )
= ( minus_249161513l_num1 @ ( finite1961675766l_num1 @ C @ X ) @ ( finite1961675766l_num1 @ C @ Y ) ) ) ).
% vector_ssub_ldistrib
thf(fact_120_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_121_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_122_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_123_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_124_norm__one,axiom,
( ( real_V646646907m_real @ one_one_real )
= one_one_real ) ).
% norm_one
thf(fact_125_dist__diff_I2_J,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] :
( ( real_V783965509real_n @ ( minus_1037315151real_n @ A @ B ) @ A )
= ( real_V739724485real_n @ B ) ) ).
% dist_diff(2)
thf(fact_126_dist__diff_I2_J,axiom,
! [A: real,B: real] :
( ( real_V1934908667t_real @ ( minus_minus_real @ A @ B ) @ A )
= ( real_V646646907m_real @ B ) ) ).
% dist_diff(2)
thf(fact_127_dist__diff_I2_J,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1] :
( ( real_V488409631l_num1 @ ( minus_249161513l_num1 @ A @ B ) @ A )
= ( real_V400292255l_num1 @ B ) ) ).
% dist_diff(2)
thf(fact_128_dist__diff_I1_J,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] :
( ( real_V783965509real_n @ A @ ( minus_1037315151real_n @ A @ B ) )
= ( real_V739724485real_n @ B ) ) ).
% dist_diff(1)
thf(fact_129_dist__diff_I1_J,axiom,
! [A: real,B: real] :
( ( real_V1934908667t_real @ A @ ( minus_minus_real @ A @ B ) )
= ( real_V646646907m_real @ B ) ) ).
% dist_diff(1)
thf(fact_130_dist__diff_I1_J,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1] :
( ( real_V488409631l_num1 @ A @ ( minus_249161513l_num1 @ A @ B ) )
= ( real_V400292255l_num1 @ B ) ) ).
% dist_diff(1)
thf(fact_131_floor__less__one,axiom,
! [X: real] :
( ( ord_less_int @ ( archim1031974863r_real @ X ) @ one_one_int )
= ( ord_less_real @ X @ one_one_real ) ) ).
% floor_less_one
thf(fact_132_one__less__ceiling,axiom,
! [X: real] :
( ( ord_less_int @ one_one_int @ ( archim1371465213g_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ).
% one_less_ceiling
thf(fact_133_floor__diff__one,axiom,
! [X: real] :
( ( archim1031974863r_real @ ( minus_minus_real @ X @ one_one_real ) )
= ( minus_minus_int @ ( archim1031974863r_real @ X ) @ one_one_int ) ) ).
% floor_diff_one
thf(fact_134_ceiling__diff__one,axiom,
! [X: real] :
( ( archim1371465213g_real @ ( minus_minus_real @ X @ one_one_real ) )
= ( minus_minus_int @ ( archim1371465213g_real @ X ) @ one_one_int ) ) ).
% ceiling_diff_one
thf(fact_135_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_136_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_137_one__reorient,axiom,
! [X: numeral_num1] :
( ( one_one_Numeral_num1 = X )
= ( X = one_one_Numeral_num1 ) ) ).
% one_reorient
thf(fact_138_floor__less__iff,axiom,
! [X: real,Z: int] :
( ( ord_less_int @ ( archim1031974863r_real @ X ) @ Z )
= ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% floor_less_iff
thf(fact_139_less__ceiling__iff,axiom,
! [Z: int,X: real] :
( ( ord_less_int @ Z @ ( archim1371465213g_real @ X ) )
= ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% less_ceiling_iff
thf(fact_140_dist__norm,axiom,
( real_V783965509real_n
= ( ^ [X3: finite1489363574real_n,Y3: finite1489363574real_n] : ( real_V739724485real_n @ ( minus_1037315151real_n @ X3 @ Y3 ) ) ) ) ).
% dist_norm
thf(fact_141_dist__norm,axiom,
( real_V1934908667t_real
= ( ^ [X3: real,Y3: real] : ( real_V646646907m_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ).
% dist_norm
thf(fact_142_dist__norm,axiom,
( real_V488409631l_num1
= ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] : ( real_V400292255l_num1 @ ( minus_249161513l_num1 @ X3 @ Y3 ) ) ) ) ).
% dist_norm
thf(fact_143_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_144_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_145_dist__commute__lessI,axiom,
! [Y: finite1489363574real_n,X: finite1489363574real_n,E: real] :
( ( ord_less_real @ ( real_V783965509real_n @ Y @ X ) @ E )
=> ( ord_less_real @ ( real_V783965509real_n @ X @ Y ) @ E ) ) ).
% dist_commute_lessI
thf(fact_146_dist__commute__lessI,axiom,
! [Y: real,X: real,E: real] :
( ( ord_less_real @ ( real_V1934908667t_real @ Y @ X ) @ E )
=> ( ord_less_real @ ( real_V1934908667t_real @ X @ Y ) @ E ) ) ).
% dist_commute_lessI
thf(fact_147_dist__commute__lessI,axiom,
! [Y: finite1183840848l_num1,X: finite1183840848l_num1,E: real] :
( ( ord_less_real @ ( real_V488409631l_num1 @ Y @ X ) @ E )
=> ( ord_less_real @ ( real_V488409631l_num1 @ X @ Y ) @ E ) ) ).
% dist_commute_lessI
thf(fact_148_dist__commute,axiom,
( real_V783965509real_n
= ( ^ [X3: finite1489363574real_n,Y3: finite1489363574real_n] : ( real_V783965509real_n @ Y3 @ X3 ) ) ) ).
% dist_commute
thf(fact_149_dist__commute,axiom,
( real_V1934908667t_real
= ( ^ [X3: real,Y3: real] : ( real_V1934908667t_real @ Y3 @ X3 ) ) ) ).
% dist_commute
thf(fact_150_dist__commute,axiom,
( real_V488409631l_num1
= ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] : ( real_V488409631l_num1 @ Y3 @ X3 ) ) ) ).
% dist_commute
thf(fact_151_ceiling__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_int @ ( archim1371465213g_real @ X ) @ ( archim1371465213g_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% ceiling_less_cancel
thf(fact_152_floor__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_int @ ( archim1031974863r_real @ X ) @ ( archim1031974863r_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% floor_less_cancel
thf(fact_153_norm__minus__commute,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] :
( ( real_V739724485real_n @ ( minus_1037315151real_n @ A @ B ) )
= ( real_V739724485real_n @ ( minus_1037315151real_n @ B @ A ) ) ) ).
% norm_minus_commute
thf(fact_154_norm__minus__commute,axiom,
! [A: real,B: real] :
( ( real_V646646907m_real @ ( minus_minus_real @ A @ B ) )
= ( real_V646646907m_real @ ( minus_minus_real @ B @ A ) ) ) ).
% norm_minus_commute
thf(fact_155_norm__minus__commute,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1] :
( ( real_V400292255l_num1 @ ( minus_249161513l_num1 @ A @ B ) )
= ( real_V400292255l_num1 @ ( minus_249161513l_num1 @ B @ A ) ) ) ).
% norm_minus_commute
thf(fact_156_real__norm__def,axiom,
real_V646646907m_real = abs_abs_real ).
% real_norm_def
thf(fact_157_ex__less__of__int,axiom,
! [X: real] :
? [Z3: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% ex_less_of_int
thf(fact_158_ex__of__int__less,axiom,
! [X: real] :
? [Z3: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X ) ).
% ex_of_int_less
thf(fact_159_norm__bound__component__lt__cart,axiom,
! [X: finite1489363574real_n,E: real,I: n] :
( ( ord_less_real @ ( real_V739724485real_n @ X ) @ E )
=> ( ord_less_real @ ( abs_abs_real @ ( finite772340589real_n @ X @ I ) ) @ E ) ) ).
% norm_bound_component_lt_cart
thf(fact_160_norm__bound__component__lt__cart,axiom,
! [X: finite1183840848l_num1,E: real,I: numeral_num1] :
( ( ord_less_real @ ( real_V400292255l_num1 @ X ) @ E )
=> ( ord_less_real @ ( abs_abs_real @ ( finite1106245191l_num1 @ X @ I ) ) @ E ) ) ).
% norm_bound_component_lt_cart
thf(fact_161_diff__strict__mono,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,D: finite1489363574real_n,C: finite1489363574real_n] :
( ( ord_le1017090250real_n @ A @ B )
=> ( ( ord_le1017090250real_n @ D @ C )
=> ( ord_le1017090250real_n @ ( minus_1037315151real_n @ A @ C ) @ ( minus_1037315151real_n @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_162_diff__strict__mono,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1,D: finite1183840848l_num1,C: finite1183840848l_num1] :
( ( ord_le2035778724l_num1 @ A @ B )
=> ( ( ord_le2035778724l_num1 @ D @ C )
=> ( ord_le2035778724l_num1 @ ( minus_249161513l_num1 @ A @ C ) @ ( minus_249161513l_num1 @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_163_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_164_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_165_diff__eq__diff__less,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n,D: finite1489363574real_n] :
( ( ( minus_1037315151real_n @ A @ B )
= ( minus_1037315151real_n @ C @ D ) )
=> ( ( ord_le1017090250real_n @ A @ B )
= ( ord_le1017090250real_n @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_166_diff__eq__diff__less,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1,D: finite1183840848l_num1] :
( ( ( minus_249161513l_num1 @ A @ B )
= ( minus_249161513l_num1 @ C @ D ) )
=> ( ( ord_le2035778724l_num1 @ A @ B )
= ( ord_le2035778724l_num1 @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_167_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_168_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_169_diff__strict__left__mono,axiom,
! [B: finite1489363574real_n,A: finite1489363574real_n,C: finite1489363574real_n] :
( ( ord_le1017090250real_n @ B @ A )
=> ( ord_le1017090250real_n @ ( minus_1037315151real_n @ C @ A ) @ ( minus_1037315151real_n @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_170_diff__strict__left__mono,axiom,
! [B: finite1183840848l_num1,A: finite1183840848l_num1,C: finite1183840848l_num1] :
( ( ord_le2035778724l_num1 @ B @ A )
=> ( ord_le2035778724l_num1 @ ( minus_249161513l_num1 @ C @ A ) @ ( minus_249161513l_num1 @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_171_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_172_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_173_diff__strict__right__mono,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n] :
( ( ord_le1017090250real_n @ A @ B )
=> ( ord_le1017090250real_n @ ( minus_1037315151real_n @ A @ C ) @ ( minus_1037315151real_n @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_174_diff__strict__right__mono,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1] :
( ( ord_le2035778724l_num1 @ A @ B )
=> ( ord_le2035778724l_num1 @ ( minus_249161513l_num1 @ A @ C ) @ ( minus_249161513l_num1 @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_175_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_176_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_177_abs__one,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
% abs_one
thf(fact_178_abs__one,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_one
thf(fact_179_dist__real__def,axiom,
( real_V1934908667t_real
= ( ^ [X3: real,Y3: real] : ( abs_abs_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ).
% dist_real_def
thf(fact_180_Ints__1,axiom,
member1352538125real_n @ one_on1253059131real_n @ ring_1723899781real_n ).
% Ints_1
thf(fact_181_Ints__1,axiom,
member_int @ one_one_int @ ring_1_Ints_int ).
% Ints_1
thf(fact_182_Ints__1,axiom,
member_real @ one_one_real @ ring_1_Ints_real ).
% Ints_1
thf(fact_183_vec_Oscale__right__diff__distrib,axiom,
! [A: real,X: finite1489363574real_n,Y: finite1489363574real_n] :
( ( finite2053577756real_n @ A @ ( minus_1037315151real_n @ X @ Y ) )
= ( minus_1037315151real_n @ ( finite2053577756real_n @ A @ X ) @ ( finite2053577756real_n @ A @ Y ) ) ) ).
% vec.scale_right_diff_distrib
thf(fact_184_vec_Oscale__right__diff__distrib,axiom,
! [A: real,X: finite1183840848l_num1,Y: finite1183840848l_num1] :
( ( finite1961675766l_num1 @ A @ ( minus_249161513l_num1 @ X @ Y ) )
= ( minus_249161513l_num1 @ ( finite1961675766l_num1 @ A @ X ) @ ( finite1961675766l_num1 @ A @ Y ) ) ) ).
% vec.scale_right_diff_distrib
thf(fact_185_Ints__eq__abs__less1,axiom,
! [X: real,Y: real] :
( ( member_real @ X @ ring_1_Ints_real )
=> ( ( member_real @ Y @ ring_1_Ints_real )
=> ( ( X = Y )
= ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ one_one_real ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_186_Ints__eq__abs__less1,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ ring_1_Ints_int )
=> ( ( member_int @ Y @ ring_1_Ints_int )
=> ( ( X = Y )
= ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y ) ) @ one_one_int ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_187_of__int__ceiling__cancel,axiom,
! [X: real] :
( ( ( ring_1_of_int_real @ ( archim1371465213g_real @ X ) )
= X )
= ( ? [N2: int] :
( X
= ( ring_1_of_int_real @ N2 ) ) ) ) ).
% of_int_ceiling_cancel
thf(fact_188_of__int__floor__cancel,axiom,
! [X: real] :
( ( ( ring_1_of_int_real @ ( archim1031974863r_real @ X ) )
= X )
= ( ? [N2: int] :
( X
= ( ring_1_of_int_real @ N2 ) ) ) ) ).
% of_int_floor_cancel
thf(fact_189_real__of__int__floor__gt__diff__one,axiom,
! [R: real] : ( ord_less_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim1031974863r_real @ R ) ) ) ).
% real_of_int_floor_gt_diff_one
thf(fact_190_real__abs__dist,axiom,
! [X: finite1489363574real_n,Y: finite1489363574real_n] :
( ( abs_abs_real @ ( real_V783965509real_n @ X @ Y ) )
= ( real_V783965509real_n @ X @ Y ) ) ).
% real_abs_dist
thf(fact_191_real__abs__dist,axiom,
! [X: real,Y: real] :
( ( abs_abs_real @ ( real_V1934908667t_real @ X @ Y ) )
= ( real_V1934908667t_real @ X @ Y ) ) ).
% real_abs_dist
thf(fact_192_real__abs__dist,axiom,
! [X: finite1183840848l_num1,Y: finite1183840848l_num1] :
( ( abs_abs_real @ ( real_V488409631l_num1 @ X @ Y ) )
= ( real_V488409631l_num1 @ X @ Y ) ) ).
% real_abs_dist
thf(fact_193_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_194_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_195_le__ceiling__iff,axiom,
! [Z: int,X: real] :
( ( ord_less_eq_int @ Z @ ( archim1371465213g_real @ X ) )
= ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).
% le_ceiling_iff
thf(fact_196_ceiling__split,axiom,
! [P: int > $o,T: real] :
( ( P @ ( archim1371465213g_real @ T ) )
= ( ! [I2: int] :
( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) @ T )
& ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I2 ) ) )
=> ( P @ I2 ) ) ) ) ).
% ceiling_split
thf(fact_197_ceiling__eq__iff,axiom,
! [X: real,A: int] :
( ( ( archim1371465213g_real @ X )
= A )
= ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
& ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).
% ceiling_eq_iff
thf(fact_198_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_199_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_200_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_201_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_202_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_203_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_204_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_205_one__le__floor,axiom,
! [X: real] :
( ( ord_less_eq_int @ one_one_int @ ( archim1031974863r_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ).
% one_le_floor
thf(fact_206_ceiling__le__one,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ one_one_int )
= ( ord_less_eq_real @ X @ one_one_real ) ) ).
% ceiling_le_one
thf(fact_207_ceiling__mono,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ord_less_eq_int @ ( archim1371465213g_real @ Y ) @ ( archim1371465213g_real @ X ) ) ) ).
% ceiling_mono
thf(fact_208_floor__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_int @ ( archim1031974863r_real @ X ) @ ( archim1031974863r_real @ Y ) ) ) ).
% floor_mono
thf(fact_209_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% less_eq_real_def
thf(fact_210_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_211_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_212_less__eq__vec__def,axiom,
( ord_le866583254real_n
= ( ^ [X3: finite1489363574real_n,Y3: finite1489363574real_n] :
! [I2: n] : ( ord_less_eq_real @ ( finite772340589real_n @ X3 @ I2 ) @ ( finite772340589real_n @ Y3 @ I2 ) ) ) ) ).
% less_eq_vec_def
thf(fact_213_less__eq__vec__def,axiom,
( ord_le1724070576l_num1
= ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] :
! [I2: numeral_num1] : ( ord_less_eq_real @ ( finite1106245191l_num1 @ X3 @ I2 ) @ ( finite1106245191l_num1 @ Y3 @ I2 ) ) ) ) ).
% less_eq_vec_def
thf(fact_214_less__eq__vec__def,axiom,
( ord_le1202326448l_num1
= ( ^ [X3: finite2063899472l_num1,Y3: finite2063899472l_num1] :
! [I2: numeral_num1] : ( ord_less_eq_int @ ( finite1800583751l_num1 @ X3 @ I2 ) @ ( finite1800583751l_num1 @ Y3 @ I2 ) ) ) ) ).
% less_eq_vec_def
thf(fact_215_less__eq__vec__def,axiom,
( ord_le1908593622_int_n
= ( ^ [X3: finite964658038_int_n,Y3: finite964658038_int_n] :
! [I2: n] : ( ord_less_eq_int @ ( finite566214509_int_n @ X3 @ I2 ) @ ( finite566214509_int_n @ Y3 @ I2 ) ) ) ) ).
% less_eq_vec_def
thf(fact_216_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_217_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_218_ceiling__le,axiom,
! [X: real,A: int] :
( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
=> ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ A ) ) ).
% ceiling_le
thf(fact_219_le__floor__iff,axiom,
! [Z: int,X: real] :
( ( ord_less_eq_int @ Z @ ( archim1031974863r_real @ X ) )
= ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% le_floor_iff
thf(fact_220_ceiling__le__iff,axiom,
! [X: real,Z: int] :
( ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ Z )
= ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% ceiling_le_iff
thf(fact_221_diff__eq__diff__less__eq,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n,D: finite1489363574real_n] :
( ( ( minus_1037315151real_n @ A @ B )
= ( minus_1037315151real_n @ C @ D ) )
=> ( ( ord_le866583254real_n @ A @ B )
= ( ord_le866583254real_n @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_222_diff__eq__diff__less__eq,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1,D: finite1183840848l_num1] :
( ( ( minus_249161513l_num1 @ A @ B )
= ( minus_249161513l_num1 @ C @ D ) )
=> ( ( ord_le1724070576l_num1 @ A @ B )
= ( ord_le1724070576l_num1 @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_223_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_224_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_225_diff__right__mono,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n] :
( ( ord_le866583254real_n @ A @ B )
=> ( ord_le866583254real_n @ ( minus_1037315151real_n @ A @ C ) @ ( minus_1037315151real_n @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_226_diff__right__mono,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1] :
( ( ord_le1724070576l_num1 @ A @ B )
=> ( ord_le1724070576l_num1 @ ( minus_249161513l_num1 @ A @ C ) @ ( minus_249161513l_num1 @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_227_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_228_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_229_diff__left__mono,axiom,
! [B: finite1489363574real_n,A: finite1489363574real_n,C: finite1489363574real_n] :
( ( ord_le866583254real_n @ B @ A )
=> ( ord_le866583254real_n @ ( minus_1037315151real_n @ C @ A ) @ ( minus_1037315151real_n @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_230_diff__left__mono,axiom,
! [B: finite1183840848l_num1,A: finite1183840848l_num1,C: finite1183840848l_num1] :
( ( ord_le1724070576l_num1 @ B @ A )
=> ( ord_le1724070576l_num1 @ ( minus_249161513l_num1 @ C @ A ) @ ( minus_249161513l_num1 @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_231_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_232_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_233_diff__mono,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,D: finite1489363574real_n,C: finite1489363574real_n] :
( ( ord_le866583254real_n @ A @ B )
=> ( ( ord_le866583254real_n @ D @ C )
=> ( ord_le866583254real_n @ ( minus_1037315151real_n @ A @ C ) @ ( minus_1037315151real_n @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_234_diff__mono,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1,D: finite1183840848l_num1,C: finite1183840848l_num1] :
( ( ord_le1724070576l_num1 @ A @ B )
=> ( ( ord_le1724070576l_num1 @ D @ C )
=> ( ord_le1724070576l_num1 @ ( minus_249161513l_num1 @ A @ C ) @ ( minus_249161513l_num1 @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_235_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_236_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_237_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_238_abs__ge__self,axiom,
! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% abs_ge_self
thf(fact_239_abs__le__D1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% abs_le_D1
thf(fact_240_abs__le__D1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% abs_le_D1
thf(fact_241_norm__vector__1,axiom,
( real_V316032763l_num1
= ( ^ [X3: finite463740460l_num1] : ( real_V400292255l_num1 @ ( finite1449546019l_num1 @ X3 @ one_one_Numeral_num1 ) ) ) ) ).
% norm_vector_1
thf(fact_242_norm__vector__1,axiom,
( real_V400292255l_num1
= ( ^ [X3: finite1183840848l_num1] : ( real_V646646907m_real @ ( finite1106245191l_num1 @ X3 @ one_one_Numeral_num1 ) ) ) ) ).
% norm_vector_1
thf(fact_243_ex__le__of__int,axiom,
! [X: real] :
? [Z3: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% ex_le_of_int
thf(fact_244_dist__vector__1,axiom,
( real_V845814485l_num1
= ( ^ [X3: finite121157254l_num1,Y3: finite121157254l_num1] : ( real_V783965509real_n @ ( finite1503577981l_num1 @ X3 @ one_one_Numeral_num1 ) @ ( finite1503577981l_num1 @ Y3 @ one_one_Numeral_num1 ) ) ) ) ).
% dist_vector_1
thf(fact_245_dist__vector__1,axiom,
( real_V1974343547l_num1
= ( ^ [X3: finite463740460l_num1,Y3: finite463740460l_num1] : ( real_V488409631l_num1 @ ( finite1449546019l_num1 @ X3 @ one_one_Numeral_num1 ) @ ( finite1449546019l_num1 @ Y3 @ one_one_Numeral_num1 ) ) ) ) ).
% dist_vector_1
thf(fact_246_dist__vector__1,axiom,
( real_V488409631l_num1
= ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] : ( real_V1934908667t_real @ ( finite1106245191l_num1 @ X3 @ one_one_Numeral_num1 ) @ ( finite1106245191l_num1 @ Y3 @ one_one_Numeral_num1 ) ) ) ) ).
% dist_vector_1
thf(fact_247_norm__real,axiom,
( real_V400292255l_num1
= ( ^ [X3: finite1183840848l_num1] : ( abs_abs_real @ ( finite1106245191l_num1 @ X3 @ one_one_Numeral_num1 ) ) ) ) ).
% norm_real
thf(fact_248_ceiling__diff__floor__le__1,axiom,
! [X: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim1371465213g_real @ X ) @ ( archim1031974863r_real @ X ) ) @ one_one_int ) ).
% ceiling_diff_floor_le_1
thf(fact_249_ceiling__less__iff,axiom,
! [X: real,Z: int] :
( ( ord_less_int @ ( archim1371465213g_real @ X ) @ Z )
= ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% ceiling_less_iff
thf(fact_250_abs__triangle__ineq2__sym,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] : ( ord_le866583254real_n @ ( minus_1037315151real_n @ ( abs_ab1599551059real_n @ A ) @ ( abs_ab1599551059real_n @ B ) ) @ ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_251_abs__triangle__ineq2__sym,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1] : ( ord_le1724070576l_num1 @ ( minus_249161513l_num1 @ ( abs_ab948591917l_num1 @ A ) @ ( abs_ab948591917l_num1 @ B ) ) @ ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_252_abs__triangle__ineq2__sym,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_253_abs__triangle__ineq2__sym,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_254_abs__triangle__ineq3,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] : ( ord_le866583254real_n @ ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ ( abs_ab1599551059real_n @ A ) @ ( abs_ab1599551059real_n @ B ) ) ) @ ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_255_abs__triangle__ineq3,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1] : ( ord_le1724070576l_num1 @ ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ ( abs_ab948591917l_num1 @ A ) @ ( abs_ab948591917l_num1 @ B ) ) ) @ ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_256_abs__triangle__ineq3,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_257_abs__triangle__ineq3,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_258_abs__triangle__ineq2,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] : ( ord_le866583254real_n @ ( minus_1037315151real_n @ ( abs_ab1599551059real_n @ A ) @ ( abs_ab1599551059real_n @ B ) ) @ ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_259_abs__triangle__ineq2,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1] : ( ord_le1724070576l_num1 @ ( minus_249161513l_num1 @ ( abs_ab948591917l_num1 @ A ) @ ( abs_ab948591917l_num1 @ B ) ) @ ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_260_abs__triangle__ineq2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_261_abs__triangle__ineq2,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_262_norm__le__componentwise__cart,axiom,
! [X: finite337957458num1_n,Y: finite1489363574real_n] :
( ! [I3: n] : ( ord_less_eq_real @ ( real_V400292255l_num1 @ ( finite1639785033num1_n @ X @ I3 ) ) @ ( real_V646646907m_real @ ( finite772340589real_n @ Y @ I3 ) ) )
=> ( ord_less_eq_real @ ( real_V1121707553num1_n @ X ) @ ( real_V739724485real_n @ Y ) ) ) ).
% norm_le_componentwise_cart
thf(fact_263_norm__le__componentwise__cart,axiom,
! [X: finite463740460l_num1,Y: finite1183840848l_num1] :
( ! [I3: numeral_num1] : ( ord_less_eq_real @ ( real_V400292255l_num1 @ ( finite1449546019l_num1 @ X @ I3 ) ) @ ( real_V646646907m_real @ ( finite1106245191l_num1 @ Y @ I3 ) ) )
=> ( ord_less_eq_real @ ( real_V316032763l_num1 @ X ) @ ( real_V400292255l_num1 @ Y ) ) ) ).
% norm_le_componentwise_cart
thf(fact_264_norm__le__componentwise__cart,axiom,
! [X: finite1489363574real_n,Y: finite337957458num1_n] :
( ! [I3: n] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite772340589real_n @ X @ I3 ) ) @ ( real_V400292255l_num1 @ ( finite1639785033num1_n @ Y @ I3 ) ) )
=> ( ord_less_eq_real @ ( real_V739724485real_n @ X ) @ ( real_V1121707553num1_n @ Y ) ) ) ).
% norm_le_componentwise_cart
thf(fact_265_norm__le__componentwise__cart,axiom,
! [X: finite1489363574real_n,Y: finite1489363574real_n] :
( ! [I3: n] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite772340589real_n @ X @ I3 ) ) @ ( real_V646646907m_real @ ( finite772340589real_n @ Y @ I3 ) ) )
=> ( ord_less_eq_real @ ( real_V739724485real_n @ X ) @ ( real_V739724485real_n @ Y ) ) ) ).
% norm_le_componentwise_cart
thf(fact_266_norm__le__componentwise__cart,axiom,
! [X: finite1183840848l_num1,Y: finite463740460l_num1] :
( ! [I3: numeral_num1] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite1106245191l_num1 @ X @ I3 ) ) @ ( real_V400292255l_num1 @ ( finite1449546019l_num1 @ Y @ I3 ) ) )
=> ( ord_less_eq_real @ ( real_V400292255l_num1 @ X ) @ ( real_V316032763l_num1 @ Y ) ) ) ).
% norm_le_componentwise_cart
thf(fact_267_norm__le__componentwise__cart,axiom,
! [X: finite1183840848l_num1,Y: finite1183840848l_num1] :
( ! [I3: numeral_num1] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite1106245191l_num1 @ X @ I3 ) ) @ ( real_V646646907m_real @ ( finite1106245191l_num1 @ Y @ I3 ) ) )
=> ( ord_less_eq_real @ ( real_V400292255l_num1 @ X ) @ ( real_V400292255l_num1 @ Y ) ) ) ).
% norm_le_componentwise_cart
thf(fact_268_Finite__Cartesian__Product_Onorm__nth__le,axiom,
! [X: finite1489363574real_n,I: n] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite772340589real_n @ X @ I ) ) @ ( real_V739724485real_n @ X ) ) ).
% Finite_Cartesian_Product.norm_nth_le
thf(fact_269_Finite__Cartesian__Product_Onorm__nth__le,axiom,
! [X: finite1183840848l_num1,I: numeral_num1] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite1106245191l_num1 @ X @ I ) ) @ ( real_V400292255l_num1 @ X ) ) ).
% Finite_Cartesian_Product.norm_nth_le
thf(fact_270_real__of__int__floor__ge__diff__one,axiom,
! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim1031974863r_real @ R ) ) ) ).
% real_of_int_floor_ge_diff_one
thf(fact_271_of__int__floor__le,axiom,
! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim1031974863r_real @ X ) ) @ X ) ).
% of_int_floor_le
thf(fact_272_le__of__int__ceiling,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim1371465213g_real @ X ) ) ) ).
% le_of_int_ceiling
thf(fact_273_norm__bound__component__le__cart,axiom,
! [X: finite1489363574real_n,E: real,I: n] :
( ( ord_less_eq_real @ ( real_V739724485real_n @ X ) @ E )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( finite772340589real_n @ X @ I ) ) @ E ) ) ).
% norm_bound_component_le_cart
thf(fact_274_norm__bound__component__le__cart,axiom,
! [X: finite1183840848l_num1,E: real,I: numeral_num1] :
( ( ord_less_eq_real @ ( real_V400292255l_num1 @ X ) @ E )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( finite1106245191l_num1 @ X @ I ) ) @ E ) ) ).
% norm_bound_component_le_cart
thf(fact_275_component__le__norm__cart,axiom,
! [X: finite1489363574real_n,I: n] : ( ord_less_eq_real @ ( abs_abs_real @ ( finite772340589real_n @ X @ I ) ) @ ( real_V739724485real_n @ X ) ) ).
% component_le_norm_cart
thf(fact_276_component__le__norm__cart,axiom,
! [X: finite1183840848l_num1,I: numeral_num1] : ( ord_less_eq_real @ ( abs_abs_real @ ( finite1106245191l_num1 @ X @ I ) ) @ ( real_V400292255l_num1 @ X ) ) ).
% component_le_norm_cart
thf(fact_277_dist__real,axiom,
( real_V488409631l_num1
= ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] : ( abs_abs_real @ ( minus_minus_real @ ( finite1106245191l_num1 @ X3 @ one_one_Numeral_num1 ) @ ( finite1106245191l_num1 @ Y3 @ one_one_Numeral_num1 ) ) ) ) ) ).
% dist_real
thf(fact_278_floor__le__ceiling,axiom,
! [X: real] : ( ord_less_eq_int @ ( archim1031974863r_real @ X ) @ ( archim1371465213g_real @ X ) ) ).
% floor_le_ceiling
thf(fact_279_of__int__ceiling__diff__one__le,axiom,
! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim1371465213g_real @ R ) ) @ one_one_real ) @ R ) ).
% of_int_ceiling_diff_one_le
thf(fact_280_norm__triangle__ineq2,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V739724485real_n @ A ) @ ( real_V739724485real_n @ B ) ) @ ( real_V739724485real_n @ ( minus_1037315151real_n @ A @ B ) ) ) ).
% norm_triangle_ineq2
thf(fact_281_norm__triangle__ineq2,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V646646907m_real @ A ) @ ( real_V646646907m_real @ B ) ) @ ( real_V646646907m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% norm_triangle_ineq2
thf(fact_282_norm__triangle__ineq2,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V400292255l_num1 @ A ) @ ( real_V400292255l_num1 @ B ) ) @ ( real_V400292255l_num1 @ ( minus_249161513l_num1 @ A @ B ) ) ) ).
% norm_triangle_ineq2
thf(fact_283_abs__dist__diff__le,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V783965509real_n @ A @ B ) @ ( real_V783965509real_n @ B @ C ) ) ) @ ( real_V783965509real_n @ A @ C ) ) ).
% abs_dist_diff_le
thf(fact_284_abs__dist__diff__le,axiom,
! [A: real,B: real,C: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1934908667t_real @ A @ B ) @ ( real_V1934908667t_real @ B @ C ) ) ) @ ( real_V1934908667t_real @ A @ C ) ) ).
% abs_dist_diff_le
thf(fact_285_abs__dist__diff__le,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V488409631l_num1 @ A @ B ) @ ( real_V488409631l_num1 @ B @ C ) ) ) @ ( real_V488409631l_num1 @ A @ C ) ) ).
% abs_dist_diff_le
thf(fact_286_metric__eq__thm,axiom,
! [X: finite1489363574real_n,S: set_Fi1058188332real_n,Y: finite1489363574real_n] :
( ( member1352538125real_n @ X @ S )
=> ( ( member1352538125real_n @ Y @ S )
=> ( ( X = Y )
= ( ! [X3: finite1489363574real_n] :
( ( member1352538125real_n @ X3 @ S )
=> ( ( real_V783965509real_n @ X @ X3 )
= ( real_V783965509real_n @ Y @ X3 ) ) ) ) ) ) ) ).
% metric_eq_thm
thf(fact_287_metric__eq__thm,axiom,
! [X: real,S: set_real,Y: real] :
( ( member_real @ X @ S )
=> ( ( member_real @ Y @ S )
=> ( ( X = Y )
= ( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ( real_V1934908667t_real @ X @ X3 )
= ( real_V1934908667t_real @ Y @ X3 ) ) ) ) ) ) ) ).
% metric_eq_thm
thf(fact_288_metric__eq__thm,axiom,
! [X: finite1183840848l_num1,S: set_Fi1257234438l_num1,Y: finite1183840848l_num1] :
( ( member1413569767l_num1 @ X @ S )
=> ( ( member1413569767l_num1 @ Y @ S )
=> ( ( X = Y )
= ( ! [X3: finite1183840848l_num1] :
( ( member1413569767l_num1 @ X3 @ S )
=> ( ( real_V488409631l_num1 @ X @ X3 )
= ( real_V488409631l_num1 @ Y @ X3 ) ) ) ) ) ) ) ).
% metric_eq_thm
thf(fact_289_dist__vec__nth__le,axiom,
! [X: finite1489363574real_n,I: n,Y: finite1489363574real_n] : ( ord_less_eq_real @ ( real_V1934908667t_real @ ( finite772340589real_n @ X @ I ) @ ( finite772340589real_n @ Y @ I ) ) @ ( real_V783965509real_n @ X @ Y ) ) ).
% dist_vec_nth_le
thf(fact_290_dist__vec__nth__le,axiom,
! [X: finite1183840848l_num1,I: numeral_num1,Y: finite1183840848l_num1] : ( ord_less_eq_real @ ( real_V1934908667t_real @ ( finite1106245191l_num1 @ X @ I ) @ ( finite1106245191l_num1 @ Y @ I ) ) @ ( real_V488409631l_num1 @ X @ Y ) ) ).
% dist_vec_nth_le
thf(fact_291_norm__triangle__ineq3,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V739724485real_n @ A ) @ ( real_V739724485real_n @ B ) ) ) @ ( real_V739724485real_n @ ( minus_1037315151real_n @ A @ B ) ) ) ).
% norm_triangle_ineq3
thf(fact_292_norm__triangle__ineq3,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V646646907m_real @ A ) @ ( real_V646646907m_real @ B ) ) ) @ ( real_V646646907m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% norm_triangle_ineq3
thf(fact_293_norm__triangle__ineq3,axiom,
! [A: finite1183840848l_num1,B: finite1183840848l_num1] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V400292255l_num1 @ A ) @ ( real_V400292255l_num1 @ B ) ) ) @ ( real_V400292255l_num1 @ ( minus_249161513l_num1 @ A @ B ) ) ) ).
% norm_triangle_ineq3
thf(fact_294_ceiling__correct,axiom,
! [X: real] :
( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim1371465213g_real @ X ) ) @ one_one_real ) @ X )
& ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim1371465213g_real @ X ) ) ) ) ).
% ceiling_correct
thf(fact_295_ceiling__unique,axiom,
! [Z: int,X: real] :
( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) )
=> ( ( archim1371465213g_real @ X )
= Z ) ) ) ).
% ceiling_unique
thf(fact_296_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_297_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_298_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ C2 ) )
=> ( P @ X4 ) )
& ! [D2: int] :
( ! [X5: int] :
( ( ( ord_less_eq_int @ A @ X5 )
& ( ord_less_int @ X5 @ D2 ) )
=> ( P @ X5 ) )
=> ( ord_less_eq_int @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_299_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B )
& ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_real @ X4 @ C2 ) )
=> ( P @ X4 ) )
& ! [D2: real] :
( ! [X5: real] :
( ( ( ord_less_eq_real @ A @ X5 )
& ( ord_less_real @ X5 @ D2 ) )
=> ( P @ X5 ) )
=> ( ord_less_eq_real @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_300_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ~ ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_301_order_Onot__eq__order__implies__strict,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_302_order_Onot__eq__order__implies__strict,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_303_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_304_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_305_vector__one__nth,axiom,
! [X: finite1183840848l_num1] :
( ( finite1329045861l_num1 @ ( finite1106245191l_num1 @ X @ one_one_Numeral_num1 ) )
= X ) ).
% vector_one_nth
thf(fact_306_vector__one__nth,axiom,
! [X: finite2063899472l_num1] :
( ( finite1470207077l_num1 @ ( finite1800583751l_num1 @ X @ one_one_Numeral_num1 ) )
= X ) ).
% vector_one_nth
thf(fact_307_complete__real,axiom,
! [S2: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S2 )
=> ( ? [Z4: real] :
! [X5: real] :
( ( member_real @ X5 @ S2 )
=> ( ord_less_eq_real @ X5 @ Z4 ) )
=> ? [Y4: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Y4 ) )
& ! [Z4: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S2 )
=> ( ord_less_eq_real @ X5 @ Z4 ) )
=> ( ord_less_eq_real @ Y4 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_308_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_309_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_310_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: int,Z2: int] : Y2 = Z2 )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A3 )
& ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_311_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: real,Z2: real] : Y2 = Z2 )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_312_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_313_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_314_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_315_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_316_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_317_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_318_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_319_order__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X @ Z ) ) ) ).
% order_trans
thf(fact_320_order__class_Oorder_Oantisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_321_order__class_Oorder_Oantisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_322_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_323_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_324_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_325_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_326_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: int,Z2: int] : Y2 = Z2 )
= ( ^ [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
& ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_327_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: real,Z2: real] : Y2 = Z2 )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ A3 @ B2 )
& ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_328_antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_329_antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_330_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_331_le__cases3,axiom,
! [X: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z ) )
=> ( ( ( ord_less_eq_real @ X @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X ) )
=> ~ ( ( ord_less_eq_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_332_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_333_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_334_le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% le_cases
thf(fact_335_le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% le_cases
thf(fact_336_eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% eq_refl
thf(fact_337_eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% eq_refl
thf(fact_338_linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linear
thf(fact_339_linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linear
thf(fact_340_antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_341_antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_342_eq__iff,axiom,
( ( ^ [Y2: int,Z2: int] : Y2 = Z2 )
= ( ^ [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
& ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_343_eq__iff,axiom,
( ( ^ [Y2: real,Z2: real] : Y2 = Z2 )
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_344_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_345_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_346_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: real,Y4: real] :
( ( ord_less_eq_real @ X5 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_347_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: real,Y4: real] :
( ( ord_less_eq_real @ X5 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_348_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_349_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_350_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X5: real,Y4: real] :
( ( ord_less_eq_real @ X5 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_351_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X5: real,Y4: real] :
( ( ord_less_eq_real @ X5 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_352_forall__1,axiom,
( ( ^ [P2: numeral_num1 > $o] :
! [X2: numeral_num1] : ( P2 @ X2 ) )
= ( ^ [P3: numeral_num1 > $o] : ( P3 @ one_one_Numeral_num1 ) ) ) ).
% forall_1
thf(fact_353_ex__1,axiom,
( ( ^ [P2: numeral_num1 > $o] :
? [X2: numeral_num1] : ( P2 @ X2 ) )
= ( ^ [P3: numeral_num1 > $o] : ( P3 @ one_one_Numeral_num1 ) ) ) ).
% ex_1
% Helper facts (9)
thf(help_If_2_1_If_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J_T,axiom,
! [X: finite964658038_int_n,Y: finite964658038_int_n] :
( ( if_Fin1767949360_int_n @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J_T,axiom,
! [X: finite964658038_int_n,Y: finite964658038_int_n] :
( ( if_Fin1767949360_int_n @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_T,axiom,
! [X: finite1489363574real_n,Y: finite1489363574real_n] :
( ( if_Fin127821360real_n @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_T,axiom,
! [X: finite1489363574real_n,Y: finite1489363574real_n] :
( ( if_Fin127821360real_n @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mt__Numeral____Type__Onum1_J_T,axiom,
! [X: finite2063899472l_num1,Y: finite2063899472l_num1] :
( ( if_Fin1570437642l_num1 @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mt__Numeral____Type__Onum1_J_T,axiom,
! [X: finite2063899472l_num1,Y: finite2063899472l_num1] :
( ( if_Fin1570437642l_num1 @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_T,axiom,
! [X: finite1183840848l_num1,Y: finite1183840848l_num1] :
( ( if_Fin1668193290l_num1 @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_T,axiom,
! [X: finite1183840848l_num1,Y: finite1183840848l_num1] :
( ( if_Fin1668193290l_num1 @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ring_1_of_int_real @ ( abs_abs_int @ m ) )
= ( abs_abs_real @ ( finite772340589real_n @ ( minus_1037315151real_n @ x @ y ) @ i ) ) ) ).
%------------------------------------------------------------------------------