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Type | Label | Description |
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Statement | ||
Theorem | retanclap 11701 | The closure of the tangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.) |
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Theorem | resincld 11702 | Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | recoscld 11703 | Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | retanclapd 11704 | Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | sinneg 11705 | The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.) |
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Theorem | cosneg 11706 | The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.) |
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Theorem | tannegap 11707 | The tangent of a negative is the negative of the tangent. (Contributed by David A. Wheeler, 23-Mar-2014.) |
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Theorem | sin0 11708 | Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.) |
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Theorem | cos0 11709 | Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.) |
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Theorem | tan0 11710 | The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.) |
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Theorem | efival 11711 | The exponential function in terms of sine and cosine. (Contributed by NM, 30-Apr-2005.) |
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Theorem | efmival 11712 | The exponential function in terms of sine and cosine. (Contributed by NM, 14-Jan-2006.) |
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Theorem | efeul 11713 | Eulerian representation of the complex exponential. (Suggested by Jeff Hankins, 3-Jul-2006.) (Contributed by NM, 4-Jul-2006.) |
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Theorem | efieq 11714 | The exponentials of two imaginary numbers are equal iff their sine and cosine components are equal. (Contributed by Paul Chapman, 15-Mar-2008.) |
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Theorem | sinadd 11715 | Addition formula for sine. Equation 14 of [Gleason] p. 310. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Revised by Mario Carneiro, 30-Apr-2014.) |
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Theorem | cosadd 11716 | Addition formula for cosine. Equation 15 of [Gleason] p. 310. (Contributed by NM, 15-Jan-2006.) (Revised by Mario Carneiro, 30-Apr-2014.) |
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Theorem | tanaddaplem 11717 | A useful intermediate step in tanaddap 11718 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.) (Revised by Jim Kingdon, 25-Dec-2022.) |
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Theorem | tanaddap 11718 | Addition formula for tangent. (Contributed by Mario Carneiro, 4-Apr-2015.) |
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Theorem | sinsub 11719 | Sine of difference. (Contributed by Paul Chapman, 12-Oct-2007.) |
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Theorem | cossub 11720 | Cosine of difference. (Contributed by Paul Chapman, 12-Oct-2007.) |
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Theorem | addsin 11721 | Sum of sines. (Contributed by Paul Chapman, 12-Oct-2007.) |
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Theorem | subsin 11722 | Difference of sines. (Contributed by Paul Chapman, 12-Oct-2007.) |
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Theorem | sinmul 11723 | Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 11716 and cossub 11720. (Contributed by David A. Wheeler, 26-May-2015.) |
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Theorem | cosmul 11724 | Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 11716 and cossub 11720. (Contributed by David A. Wheeler, 26-May-2015.) |
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Theorem | addcos 11725 | Sum of cosines. (Contributed by Paul Chapman, 12-Oct-2007.) |
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Theorem | subcos 11726 | Difference of cosines. (Contributed by Paul Chapman, 12-Oct-2007.) (Revised by Mario Carneiro, 10-May-2014.) |
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Theorem | sincossq 11727 | Sine squared plus cosine squared is 1. Equation 17 of [Gleason] p. 311. Note that this holds for non-real arguments, even though individually each term is unbounded. (Contributed by NM, 15-Jan-2006.) |
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Theorem | sin2t 11728 | Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.) |
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Theorem | cos2t 11729 | Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.) |
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Theorem | cos2tsin 11730 | Double-angle formula for cosine in terms of sine. (Contributed by NM, 12-Sep-2008.) |
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Theorem | sinbnd 11731 | The sine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.) |
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Theorem | cosbnd 11732 | The cosine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.) |
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Theorem | sinbnd2 11733 | The sine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.) |
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Theorem | cosbnd2 11734 | The cosine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.) |
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Theorem | ef01bndlem 11735* | Lemma for sin01bnd 11736 and cos01bnd 11737. (Contributed by Paul Chapman, 19-Jan-2008.) |
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Theorem | sin01bnd 11736 | Bounds on the sine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
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Theorem | cos01bnd 11737 | Bounds on the cosine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
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Theorem | cos1bnd 11738 | Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.) |
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Theorem | cos2bnd 11739 | Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.) |
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Theorem | sin01gt0 11740 | The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Wolf Lammen, 25-Sep-2020.) |
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Theorem | cos01gt0 11741 | The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) |
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Theorem | sin02gt0 11742 | The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) |
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Theorem | sincos1sgn 11743 | The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.) |
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Theorem | sincos2sgn 11744 | The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.) |
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Theorem | sin4lt0 11745 | The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.) |
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Theorem | cos12dec 11746 | Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.) |
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Theorem | absefi 11747 | The absolute value of the exponential of an imaginary number is one. Equation 48 of [Rudin] p. 167. (Contributed by Jason Orendorff, 9-Feb-2007.) |
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Theorem | absef 11748 | The absolute value of the exponential is the exponential of the real part. (Contributed by Paul Chapman, 13-Sep-2007.) |
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Theorem | absefib 11749 |
A complex number is real iff the exponential of its product with ![]() |
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Theorem | efieq1re 11750 | A number whose imaginary exponential is one is real. (Contributed by NM, 21-Aug-2008.) |
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Theorem | demoivre 11751 | De Moivre's Formula. Proof by induction given at http://en.wikipedia.org/wiki/De_Moivre's_formula, but restricted to nonnegative integer powers. See also demoivreALT 11752 for an alternate longer proof not using the exponential function. (Contributed by NM, 24-Jul-2007.) |
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Theorem | demoivreALT 11752 | Alternate proof of demoivre 11751. It is longer but does not use the exponential function. This is Metamath 100 proof #17. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Syntax | ctau 11753 |
Extend class notation to include the constant tau, ![]() |
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Definition | df-tau 11754 |
Define the circle constant tau, ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | eirraplem 11755* | Lemma for eirrap 11756. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 5-Jan-2022.) |
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Theorem | eirrap 11756 |
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Theorem | eirr 11757 |
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Theorem | egt2lt3 11758 |
Euler's constant ![]() |
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Theorem | epos 11759 |
Euler's constant ![]() |
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Theorem | epr 11760 |
Euler's constant ![]() |
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Theorem | ene0 11761 |
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Theorem | eap0 11762 |
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Theorem | ene1 11763 |
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Theorem | eap1 11764 |
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This part introduces elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory. | ||
Syntax | cdvds 11765 | Extend the definition of a class to include the divides relation. See df-dvds 11766. |
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Definition | df-dvds 11766* | Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | divides 11767* |
Define the divides relation. ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dvdsval2 11768 | One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.) |
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Theorem | dvdsval3 11769 | One nonzero integer divides another integer if and only if the remainder upon division is zero, see remark in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 15-Jul-2014.) |
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Theorem | dvdszrcl 11770 | Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
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Theorem | dvdsmod0 11771 | If a positive integer divides another integer, then the remainder upon division is zero. (Contributed by AV, 3-Mar-2022.) |
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Theorem | p1modz1 11772 | If a number greater than 1 divides another number, the second number increased by 1 is 1 modulo the first number. (Contributed by AV, 19-Mar-2022.) |
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Theorem | dvdsmodexp 11773 | If a positive integer divides another integer, this other integer is equal to its positive powers modulo the positive integer. (Formerly part of the proof for fermltl 12204). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by AV, 19-Mar-2022.) |
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Theorem | nndivdvds 11774 | Strong form of dvdsval2 11768 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | nndivides 11775* | Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.) |
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Theorem | dvdsdc 11776 | Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.) |
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Theorem | moddvds 11777 |
Two ways to say ![]() ![]() ![]() ![]() |
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Theorem | modm1div 11778 | An integer greater than one divides another integer minus one iff the second integer modulo the first integer is one. (Contributed by AV, 30-May-2023.) |
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Theorem | dvds0lem 11779 |
A lemma to assist theorems of ![]() |
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Theorem | dvds1lem 11780* |
A lemma to assist theorems of ![]() |
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Theorem | dvds2lem 11781* |
A lemma to assist theorems of ![]() |
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Theorem | iddvds 11782 | An integer divides itself. Theorem 1.1(a) in [ApostolNT] p. 14 (reflexive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | 1dvds 11783 | 1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvds0 11784 | Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | negdvdsb 11785 | An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsnegb 11786 | An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | absdvdsb 11787 | An integer divides another iff its absolute value does. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsabsb 11788 | An integer divides another iff it divides its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | 0dvds 11789 | Only 0 is divisible by 0. Theorem 1.1(h) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | zdvdsdc 11790 | Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.) |
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Theorem | dvdsmul1 11791 | An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsmul2 11792 | An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | iddvdsexp 11793 | An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
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Theorem | muldvds1 11794 | If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | muldvds2 11795 | If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdscmul 11796 | Multiplication by a constant maintains the divides relation. Theorem 1.1(d) in [ApostolNT] p. 14 (multiplication property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsmulc 11797 | Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdscmulr 11798 | Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsmulcr 11799 | Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | summodnegmod 11800 | The sum of two integers modulo a positive integer equals zero iff the first of the two integers equals the negative of the other integer modulo the positive integer. (Contributed by AV, 25-Jul-2021.) |
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