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Theorem List for Metamath Proof Explorer - 27701-27800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrefsumcn 27701* A finite sum of continuous real functions, from a common topological space, is continuous. The class expression for B normally contains free variables k and x to index it. See fsumcn 18374 for the analogous theorem on continuous complex functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
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Theoremrfcnpre2 27702 If is a continuous function with respect to the standard topology, then the preimage A of the values smaller than a given extended real , is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremcncmpmax 27703* When the hypothesis for the extreme value theorem hold, then the sup of the range of the function belongs to the range, it is real and it an upper bound of the range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremrfcnpre3 27704* If F is a continuous function with respect to the standard topology, then the preimage A of the values greater or equal than a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremrfcnpre4 27705* If F is a continuous function with respect to the standard topology, then the preimage A of the values smaller or equal than a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremsumpair 27706* Sum of two distinct complex values. The class expression for and normally contain free variable to index it. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremrfcnnnub 27707* Given a real continuous function defined on a compact topological space, there is always a natural number that is a strict upper bound of it's range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremrefsum2cnlem1 27708* This is the core Lemma for refsum2cn 27709: the sum of two continuous real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
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Theoremrefsum2cn 27709* The sum of two continuus real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
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18.20.2  Finite multiplication of numbers and finite multiplication of functions

Theoremfmul01 27710* Multiplying a finite number of values in [ 0 , 1 ] , gives the final product itself a number in [ 0 , 1 ]. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremfmulcl 27711* If ' Y ' is closed under the multiplication of two functions, then Y is closed under the multiplication ( ' X ' ) of a finite number of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremfmuldfeqlem1 27712* induction step for the proof of fmuldfeq 27713. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremfmuldfeq 27713* X and Z are two equivalent definitions of the finite product of real functions. Y is a set of real functions from a common domain T, Y is closed under function multiplication and U is a finite sequence of functions in Y. M is the number of functions multiplied together. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremfmul01lt1lem1 27714* Given a finite multiplication of values betweeen 0 and 1, a value larger than its frist element is larger the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremfmul01lt1lem2 27715* Given a finite multiplication of values betweeen 0 and 1, a value larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremfmul01lt1 27716* Given a finite multiplication of values betweeen 0 and 1, a value E larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremcncfmptss 27717* A continuous complex function restricted to a subset is continuous, using "map to" notation. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremfmptdf 27718* A version of fmptd 5684 using bound-variable hypothesis instead of a distinct variable condition for . (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremrrpsscn 27719 The positive reals are a subset of the complex numbers. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremmulcncf 27720* The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremmulc1cncfg 27721* A version of mulc1cncf 18409 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 30-Jun-2017.)

Theoreminfrglb 27722* The infimum of a non-empty bounded set of reals is the greatest lower bound. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremexpcncf 27723* The power function on complex numbers, for fixed exponent N, is continuous. Similar to expcn 18376. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremeluzelcn 27724 A member of a set of upper integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremm1expeven 27725 Exponentiation of negative one to an even power. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremexpcnfg 27726* If is a complex continuous function and N is a fixed number, then F^N is continuous too. A generalization of expcncf 27723. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

18.20.3  Limits

Theoremclim1fr1 27727* A class of sequences of fractions that converge to 1 (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremisumneg 27728* Negation of a converging sum. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremclimrec 27729* Limit of the reciprocal of a converging sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremclimmulf 27730* A version of climmul 12106 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremclimexp 27731* The limit of natural powers, is the natural power of the limit. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremcliminf 27732* A bounded monotonic non increasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremclimsuselem1 27733* The subsequence index has the expected properties: it belongs to the same upper integers as the original index, and it is always larger or equal than the original index. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremclimsuse 27734* A subsequence of a converging sequence , converges to the same limit. is the strictly increasing and it is used to index the subsequence (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremclimrecf 27735* A version of climrec 27729 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremclimneg 27736* Complex limit of the negative of a sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremcliminff 27737* A version of climinf 27732 using bound-variable hypotheses instead of distinct variable conditions (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremclimdivf 27738* Limit of the ratio of two converging sequences. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremclimreeq 27739 If is a real function, then converges to with respect to the standard topology on the reals if and only if it converges to with respect to the standard topology on complex numbers. In the theorem, is defined to be convergence w.r.t. the standard topology on the reals and then represents the statement " converges to , with respect to the standard topology on the reals". Notice that there is no need for the hypothesis that is a real number. (Contributed by Glauco Siliprandi, 2-Jul-2017.)

18.20.4  Derivatives

Theoremdvsinexp 27740* The derivative of sin^N . (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremdvcosre 27741 The real derivative of the cosine (Contributed by Glauco Siliprandi, 29-Jun-2017.)

18.20.5  Integrals

Theoremioovolcl 27742 An open real interval has finite volume. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremvolioo 27743 The measure of an open interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremitgsin0pilem1 27744* Calculation of the integral for sine on the (0,π) interval (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremibliccsinexp 27745* sin^n on a closed interval is integrable. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremitgsin0pi 27746 Calculation of the integral for sine on the (0,π) interval (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremiblioosinexp 27747* sin^n on an open integral is integrable (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremitgsinexplem1 27748* Integration by parts is applied to integrate sin^(N+1) (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremitgsinexp 27749* A recursive formula for the integral of sin^N on the interval (0,π) .

(Contributed by Glauco Siliprandi, 29-Jun-2017.)

18.20.6  Stone Weierstrass theorem - real version

Theoremstoweidlem1 27750 Lemma for stoweid 27812. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90; the key step uses Bernoulli's inequality bernneq 11227. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem2 27751* lemma for stoweid 27812: here we prove that the subalgebra of continuous functions, which contains constant functions, is closed under scaling. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem3 27752* Lemma for stoweid 27812: if is positive and all terms of a finite product are larger than , then the finite product is larger than A^M. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem4 27753* Lemma for stoweid 27812: a class variable replaces a set variable, for constant functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem5 27754* There exists a δ as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: 0 < δ < 1 , p >= δ on . Here is used to represent δ in the paper and to represent in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem6 27755* Lemma for stoweid 27812: two class variables replace two set variables, for multiplication of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem7 27756* This lemma is used to prove that qn as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91, (at the top of page 91), is such that qn < ε on , and qn > 1 - ε on . Here it is proven that, for large enough, 1-(k*δ/2)^n > 1 - ε , and 1/(k*δ)^n < ε. The variable is used to represent (k*δ) in the paper, and is used to represent (k*δ/2). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem8 27757* Lemma for stoweid 27812: two class variables replace two set variables, for the sum of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem9 27758* Lemma for stoweid 27812: here the Stone Weierstrass theorem is proven for the trivial case, T is the empty set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem10 27759 Lemma for stoweid 27812. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90, this lemma is an application of Bernoulli's inequality. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem11 27760* This lemma is used to prove that there is a function as in the proof of [BrosowskiDeutsh] p. 92, (at the top of page 92): this lemma proves that g(t) < ( j + 1 / 3 ) * ε. Here is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem12 27761* Lemma for stoweid 27812. This Lemma is used by other three Lemmas. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem13 27762 Lemma for stoweid 27812. This lemma is used to prove the statement abs( f(t) - g(t) ) < 2 epsilon , in [BrosowskiDeutsh] p. 92, the last step of the proof. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem14 27763* There exists a as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: is an integer and 1 < k * δ < 2. is used to represent δ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem15 27764* This lemma is used to prove the existence of a function as in Lemma 1 from [BrosowskiDeutsh] p. 90: is in the subalgebra, such that 0 ≤ p ≤ 1, p(t_0) = 0, and p > 0 on T - U. Here is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem16 27765* Lemma for stoweid 27812. The subset of functions in the algebra , with values in [ 0 , 1 ], is closed under multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem17 27766* This lemma proves that the function (as defined in [BrosowskiDeutsh] p. 91, at the end of page 91) belongs to the subalgebra. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem18 27767* This theorem proves Lemma 2 in [BrosowskiDeutsh] p. 92 when A is empty, the trivial case. Here D is used to denote the set A of Lemma 2, because the variable A is used for the subalgebra. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem19 27768* If a set of real functions is closed under multiplication and it contains constants, then it is closed under finite exponentiation. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem20 27769* If a set A of real functions from a common domain T is closed under the sum of two functions, then it is closed under the sum of a finite number of functions, indexed by G. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem21 27770* Once the Stone Weierstrass theorem has been proven for approximating nonnegative functions, then this lemma is used to extend the result to functions with (possibly) negative values. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem22 27771* If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem23 27772* This lemma is used to prove the existence of a function pt as in the beginning of Lemma 1 [BrosowskiDeutsh] p. 90: for all t in T - U, there exists a function p in the subalgebra, such that pt ( t0 ) = 0 , pt ( t ) > 0, and 0 <= pt <= 1. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem24 27773* This lemma proves that for sufficiently large, qn( t ) > ( 1 - epsilon ), for all in : see Lemma 1 [BrosowskiDeutsh] p. 90, (at the bottom of page 90). is used to represent qn in the paper, to represent in the paper, to represent , to represent δ, and to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem25 27774* This lemma proves that for n sufficiently large, qn( t ) < ε, for all in : see Lemma 1 [BrosowskiDeutsh] p. 91 (at the top of page 91). is used to represent qn in the paper, to represent n in the paper, to represent k, to represent δ, to represent p, and to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem26 27775* This lemma is used to prove that there is a function as in the proof of [BrosowskiDeutsh] p. 92: this lemma proves that g(t) > ( j - 4 / 3 ) * ε. Here is used to represnt j in the paper, is used to represent A in the paper, is used to represent t, and is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem27 27776* This lemma is used to prove the existence of a function p as in Lemma 1 [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Here is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem28 27777* There exists a δ as in Lemma 1 [BrosowskiDeutsh] p. 90: 0 < delta < 1 and p >= delta on . Here is used to represent δ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem29 27778* When the hypothesis for the extreme value theorem hold, then the inf of the range of the function belongs to the range, it is real and it a lower bound of the range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem30 27779* This lemma is used to prove the existence of a function p as in Lemma 1 [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Z is used for t0, P is used for p, is used for p(t_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem31 27780* This lemma is used to prove that there exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91: assuming that is a finite subset of , indexes a finite set of functions in the subalgebra (of the Stone Weierstrass theorem), such that for all ranging in the finite indexing set, 0 ≤ xi ≤ 1, xi < ε / m on V(ti), and xi > 1 - ε / m on . Here M is used to represent m in the paper, is used to represent ε in the paper, vi is used to represent V(ti). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem32 27781* If a set A of real functions from a common domain T is a subalgebra and it contains constants, then it is closed under the sum of a finite number of functions, indexed by G and finally scaled by a real Y. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem33 27782* If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem34 27783* This lemma proves that for all in there is a as in the proof of [BrosowskiDeutsh] p. 91 (at the bottom of page 91 and at the top of page 92): (j-4/3) * ε < f(t) <= (j-1/3) * ε , g(t) < (j+1/3) * ε, and g(t) > (j-4/3) * ε. Here is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem35 27784* This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Here is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem36 27785* This lemma is used to prove the existence of a function pt as in Lemma 1 of [BrosowskiDeutsh] p. 90 (at the beginning of Lemma 1): for all t in T - U, there exists a function p in the subalgebra, such that pt ( t0 ) = 0 , pt ( t ) > 0, and 0 <= pt <= 1. Z is used for t0 , S is used for t e. T - U , h is used for pt . G is used for (ht)^2 and the final h is a normalized version of G ( divided by it's norm, see the variable N ). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem37 27786* This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Z is used for t0, P is used for p, is used for p(t_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem38 27787* This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Z is used for t0, P is used for p, is used for p(t_i). (Contributed by GlaucoSiliprandi, 20-Apr-2017.)

Theoremstoweidlem39 27788* This lemma is used to prove that there exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91: assuming that is a finite subset of , indexes a finite set of functions in the subalgebra (of the Stone Weierstrass theorem), such that for all i ranging in the finite indexing set, 0 ≤ xi ≤ 1, xi < ε / m on V(ti), and xi > 1 - ε / m on . Here is used to represent A in the paper's Lemma 2 (because is used for the subalgebra), is used to represent m in the paper, is used to represent ε, and vi is used to represent V(ti). is just a local definition, used to shorten statements. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem40 27789* This lemma proves that qn is in the subalgebra, as in the prove of Lemma 1 in [BrosowskiDeutsh] p. 90. Q is used to represent qn in the paper, N is used to represent n in the paper, and M is used to represent k^n in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem41 27790* This lemma is used to prove that there exists x as in Lemma 1 of [BrosowskiDeutsh] p. 90: 0 <= x(t) <= 1 for all t in T, x(t) < epsilon for all t in V, x(t) > 1 - epsilon for all t in T \ U. Here we prove the very last step of the proof of Lemma 1: "The result follows from taking x = 1 - qn";. Here is used to represent ε in the paper, and to represent qn in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem42 27791* This lemma is used to prove that built as in Lemma 2 of [BrosowskiDeutsh] p. 91, is such that x > 1 - ε on B. Here is used to represent in the paper, and E is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem43 27792* This lemma is used to prove the existence of a function pt as in Lemma 1 of [BrosowskiDeutsh] p. 90 (at the beginning of Lemma 1): for all t in T - U, there exists a function pt in the subalgebra, such that pt( t0 ) = 0 , pt ( t ) > 0, and 0 <= pt <= 1. Hera Z is used for t0 , S is used for t e. T - U , h is used for pt. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem44 27793* This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Z is used to represent t0 in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem45 27794* This lemma proves that, given an appropriate (in another theorem we prove such a exists), there exists a function qn as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91 ( at the top of page 91): 0 <= qn <= 1 , qn < ε on T \ U, and qn > 1 - ε on . We use y to represent the final qn in the paper (the one with n large enough), to represent in the paper, to represent , to represent δ, to represent ε, and to represent . (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem46 27795* This lemma proves that sets U(t) as defined in Lemma 1 of [BrosowskiDeutsh] p. 90, are a cover of T \ U. Using this lemma, in a later theorem we will prove that a finite subcover exists. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem47 27796* Subtracting a constant from a real continuous function gives another continuous function. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem48 27797* This lemma is used to prove that built as in Lemma 2 of [BrosowskiDeutsh] p. 91, is such that x < ε on . Here is used to represent in the paper, is used to represent ε in the paper, and is used to represent in the paper (because is always used to represent the subalgebra). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem49 27798* There exists a function qn as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91 (at the top of page 91): 0 <= qn <= 1 , qn < ε on , and qn > 1 - ε on . Here y is used to represent the final qn in the paper (the one with n large enough), represents in the paper, represents , represents δ, represents ε, and represents . (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem50 27799* This lemma proves that sets U(t) as defined in Lemma 1 of [BrosowskiDeutsh] p. 90, contain a finite subcover of T \ U. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem51 27800* There exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. Here is used to represent in the paper, because here is used for the subalgebra of functions. is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

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