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<b>let </b><font color="Maroon">T</font> be  non <a href="realset2.html#V3">trivial</a> <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="realset2.html#V3">trivial</a> <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/>







<b>assume </b><a NAME="E1:64"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="ndiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">x0</font>
 ;<br/>

<b>then consider </b><font color="Maroon">N</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E3:64"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">L</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> ex <font color="Olive">R</font> being  <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> st <br/>for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">L</font> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E4:64"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="nfcont_1.html#T4">NFCONT_1:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:64_1"/><i><font color="Green">E47</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">s1</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> ) )
 ;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:64_1"/><i><font color="Green">E48</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> &amp; <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font> )
 <b>by </b><i><a class="ref" href="nfcont_1.html#D3">NFCONT_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">s2</font> = <a href="numbers.html#K5">NAT</a>  <a href="partfun2.html#K4">--&gt;</a> <font color="Maroon">x0</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/><a NAME="E5:64_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> = <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">s3</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E7:64_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">s3</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i><a NAME="E8:64_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">s2</font> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i/>;<br/><i><font color="Green">E54</font></i>:  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s2</font> = 
<font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> 0
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x0</font>
<b>by </b><i/>
;<br/><a NAME="E10:64_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">s3</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">s1</font> <a href="normsp_1.html#K4">-</a> <font color="Maroon">s2</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E5:64_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="normsp_1.html#D6">NORMSP_1:def 6</a></i>;<br/><a NAME="E11:64_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">s3</font> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:64_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="ndiff_1.html#E7:64_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="normsp_1.html#T35">NORMSP_1:35</a></i>;<br/><i><font color="Green">E61</font></i>:  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s3</font> = 
<font color="Maroon">x0</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E1:64_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="ndiff_1.html#E7:64_1"><i><font color="Green">E51</font></i></a>, , <a class="txt" href="ndiff_1.html#E8:64_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="normsp_1.html#T43">NORMSP_1:43</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
;<br/><div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:64_1_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">s3</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:64_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E5:64_1"><i><font color="Green">E50</font></i></a>, </i>;<br/><a NAME="E3:64_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/><a NAME="E4:64_1_3"/><b>then </b>
<font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T35">RLVECT_1:35</a></i>;<br/><b>hence </b><a NAME="E5:64_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:64_1"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E15:64_1"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:64_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="ndiff_1.html#E3:64_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/><a NAME="E16:64_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">s3</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">l</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s3</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="txt" href="ndiff_1.html#E9:64_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="ndiff_1.html#E10:64_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="lopban_3.html#T14">LOPBAN_3:14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:64_1_4"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s3</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">l</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:64_1_4"/>
<font color="Maroon">s3</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="lopban_3.html#D6">LOPBAN_3:def 6</a></i>;<br/><b>hence </b><a NAME="E3:64_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="ndiff_1.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><a NAME="E18:64_1"/><b>then </b>
<font color="Maroon">s3</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">l</font> <a href="ndiff_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">s3</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">l</font> as  <a href="ndiff_1.html#V2" target="_self">convergent_to_0</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font> <b>by </b><i>, </i>;<br/><a NAME="E20:64_1"/>
<font color="Maroon">s2</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">l</font> is    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">s2</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">s2</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">l</font> as  <a href="funct_1.html#V3">V6</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E22:64_1"/><i><font color="Green">E68</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:64_1_5"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:64_1_5_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:64_1_5_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">l</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T6">NFCONT_1:6</a></i>;<br/>
<a NAME="E4:64_1_5_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E11:64_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="lopban_3.html#D6">LOPBAN_3:def 6</a></i>;<br/>
<a NAME="E5:64_1_5_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:64_1_5_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:64_1_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:64_1_5"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> 0 = 
<font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6">LOPBAN_3:def 6</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="txt" href="ndiff_1.html#E11:64_1"><i><font color="Green">E60</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:64_1_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T6">NFCONT_1:6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E23:64_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:64_1_6"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:64_1_6"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T6">NFCONT_1:6</a></i>;<br/>
<i><font color="Green">E118</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K4">-</a> <font color="Maroon">s2</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">l</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E8:64_1"><i><font color="Green">E53</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K4">-</a> <font color="Maroon">s2</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">s2</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6">LOPBAN_3:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K4">-</a> <font color="Maroon">s2</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D5">NORMSP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D6">NORMSP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
;<br/>
<a NAME="E5:64_1_6"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E6:64_1_6"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="txt" href="ndiff_1.html#E13:64_1"><i><font color="Green">E62</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="ndiff_1.html#E11:64_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E8:64_1_6"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">z</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E12:64_1"><i><font color="Green">E61</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E9:64_1_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E3:64_1"><i><font color="Green">E48</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E24:64_1"/><b>then </b><i><font color="Green">E120</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font> )
 <b>by </b><i>, <a class="txt" href="ndiff_1.html#E3:64"><i><font color="Green">E45</font></i></a>, , </i>;<br/><div><i><font color="Green">E121</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:64_1_7"/><i><font color="Green">E122</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/><b>take </b><font color="Maroon">n</font> = 0;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E2:64_1_7"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E3:64_1_7"/>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T4">NFCONT_1:4</a></i>;<br/><a NAME="E4:64_1_7"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E3:64"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>then </b><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="nfcont_1.html#T8">NFCONT_1:8</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6">LOPBAN_3:def 6</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="normsp_1.html#T5">NORMSP_1:5</a></i>
;<br/><b>hence </b><a NAME="E6:64_1_7"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E13:64_1"><i><font color="Green">E62</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E26:64_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="normsp_1.html#D9">NORMSP_1:def 9</a></i>;<br/><a NAME="E27:64_1"/><b>then </b><i><font color="Green">E124</font></i>: 
 <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E12:64_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/><a NAME="E28:64_1"/><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E11:64_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="ndiff_1.html#E13:64_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="normsp_1.html#T34">NORMSP_1:34</a></i>;<br/><i><font color="Green">E126</font></i>:  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E11:64_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="ndiff_1.html#E13:64_1"><i><font color="Green">E62</font></i></a>, , <a class="ref" href="normsp_1.html#T42">NORMSP_1:42</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D5">NORMSP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D6">NORMSP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
;<br/></div><b>end;</b></div><a NAME="E31:64_1"/><b>then </b><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K4">-</a> <font color="Maroon">s2</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">l</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E8:64_1"><i><font color="Green">E53</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K4">-</a> <font color="Maroon">s2</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">s2</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6">LOPBAN_3:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K4">-</a> <font color="Maroon">s2</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D5">NORMSP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D6">NORMSP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">l</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="lopban_3.html#D6">LOPBAN_3:def 6</a></i>
;<br/></div><b>end;</b></div><b>then </b><i><font color="Green">E128</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> = 
<font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E16:64_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s1</font></span>)</span> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">l</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E1:64_1"><i><font color="Green">E47</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E34:64_1"/>
<font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s1</font> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i>, <a class="ref" href="lopban_3.html#T15">LOPBAN_3:15</a></i>;<br/><b>thus </b><a NAME="E35:64_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s1</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="ndiff_1.html#E15:64_1"><i><font color="Green">E65</font></i></a>, , <a class="ref" href="lopban_3.html#T16">LOPBAN_3:16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:64"/>
<font color="Maroon">f</font> <a href="nfcont_1.html#R1">is_continuous_in</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E3:64"><i><font color="Green">E45</font></i></a>, <a class="txt" href="ndiff_1.html#E4:64"><i><font color="Green">E46</font></i></a>, </i>;<br/>


</div>
