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<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:19_1_2"/><i><font color="Green">E67</font></i>: 
 ex <font color="Olive">N</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> st <br/>( <font color="Olive">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="fdiff_1.html#NM2">LINEAR</a> ex <font color="Olive">R</font> being  <a href="fdiff_1.html#NM1">REST</a> st <br/>( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">L</font> <a href="seq_1.html#K2">.</a> 1 &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</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="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Olive">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> ) ) )
 <b>and </b><br/><a NAME="E2:19_1_2"/><i><font color="Green">E70</font></i>: 
 ex <font color="Olive">N</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> st <br/>( <font color="Olive">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="fdiff_1.html#NM2">LINEAR</a> ex <font color="Olive">R</font> being  <a href="fdiff_1.html#NM1">REST</a> st <br/>( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">L</font> <a href="seq_1.html#K2">.</a> 1 &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</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="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Olive">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> ) ) )
 ;<br/>

<b>consider </b><font color="Maroon">N</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E4:19_1_2"/><i><font color="Green">E71</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="fdiff_1.html#NM2">LINEAR</a> ex <font color="Olive">R</font> being  <a href="fdiff_1.html#NM1">REST</a> st <br/>( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">L</font> <a href="seq_1.html#K2">.</a> 1 &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  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="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</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="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Olive">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> ) ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">L</font> being   <a href="fdiff_1.html#NM2">LINEAR</a>, <font color="Maroon">R</font> being   <a href="fdiff_1.html#NM1">REST</a><b> such that </b><br/><a NAME="E6:19_1_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> 1 &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  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="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">N1</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E8:19_1_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">N1</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="fdiff_1.html#NM2">LINEAR</a> ex <font color="Olive">R</font> being  <a href="fdiff_1.html#NM1">REST</a> st <br/>( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">L</font> <a href="seq_1.html#K2">.</a> 1 &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</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="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Olive">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> ) ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">L1</font> being   <a href="fdiff_1.html#NM2">LINEAR</a>, <font color="Maroon">R1</font> being   <a href="fdiff_1.html#NM1">REST</a><b> such that </b><br/><a NAME="E10:19_1_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="seq_1.html#K2">.</a> 1 &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E12:19_1_2"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <font color="Olive">p</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">p1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E14:19_1_2"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">L1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="real_1.html#K4">*</a> <font color="Olive">p</font>
 <b>by </b><i/>;<br/>
<a NAME="E15:19_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:19_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="real_1.html#K4">*</a> 1
 
<b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:19_1_2_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> )
 ;<br/><a NAME="E2:19_1_2_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:19_1_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E4:19_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E5:19_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:19_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">N0</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E19:19_1_2"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">N0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font> &amp; <font color="Maroon">N0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#T38">RCOMP_1:38</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E21:19_1_2"/><i><font color="Green">E92</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">N0</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#D7">RCOMP_1:def 7</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="hidden.html#M1">set</a>  = <font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>;<br/>
<b>consider </b><font color="Maroon">s1</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E23:19_1_2"/><i><font color="Green">E93</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">s1</font> <a href="seq_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="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_2"><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/><a NAME="E1:19_1_2_2"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><b>hence </b><a NAME="E2:19_1_2_2"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E25:19_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">s1</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>
<a NAME="E26:19_1_2"/>
( <font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T46">SEQ_4:46</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">s1</font> as  <a href="fdiff_1.html#V1" target="_self">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i>, </i>;<br/>
<a NAME="E28:19_1_2"/><i><font color="Green">E95</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>   ex <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> &amp; <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_3"><b>proof </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/>

<a NAME="E1:19_1_2_3"/><i><font color="Green">E96</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/>
<a NAME="E2:19_1_2_3"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:19_1_2_3"/><b>then </b>
<font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T78">XREAL_1:78</a></i>;<br/>
<a NAME="E4:19_1_2_3"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:19_1_2_3"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E6:19_1_2_3"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E7:19_1_2_3"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:19_1_2_3"/><b>then </b>
<font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">g</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:19_1_2_3"/><b>then </b>
<font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p2</font> where <font color="Olive">p2</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p2</font> &amp; <font color="Olive">p2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:19_1_2_3"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/>
<b>take </b><font color="Maroon">x</font> = <font color="Maroon">x0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>;<br/>

<a NAME="E11:19_1_2_3"/>
<font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:19_1_2_3"/>
( <font color="Maroon">x</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> &amp; <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E100</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_4"><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/><a NAME="E1:19_1_2_4"/>
 ex <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> &amp; <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x0</font> )
 <b>by </b><i/>;<br/><a NAME="E2:19_1_2_4"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:19_1_2_4"/>
<font color="Maroon">h</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i/>;<br/><a NAME="E4:19_1_2_4"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/><a NAME="E5:19_1_2_4"/><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>;<br/><i><font color="Green">E104</font></i>: <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> = 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> 1
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon">r</font>

;<br/><span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> = 
<font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="real_1.html#K4">*</a> 1
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon">s</font>

;<br/><a NAME="E8:19_1_2_4"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>;<br/><a NAME="E9:19_1_2_4"/>
<font color="Maroon">R</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i/>;<br/><a NAME="E10:19_1_2_4"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/><a NAME="E11:19_1_2_4"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font>
 ;<br/><a NAME="E12:19_1_2_4"/>
<font color="Maroon">R1</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i/>;<br/><a NAME="E13:19_1_2_4"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/><a NAME="E14:19_1_2_4"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">R1</font>
 ;<br/><i><font color="Green">E108</font></i>: <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
;<br/><a NAME="E17:19_1_2_4"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K3">+</a> <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="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R1</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E18:19_1_2_4"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="real_1.html#K3">+</a> <span class="p1">(<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="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R1</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <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="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E19:19_1_2_4"/>
<font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">R1</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T6">RFUNCT_2:6</a></i>;<br/></div><b>end;</b></div>
<a NAME="E30:19_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> is <a href="funct_1.html#V3">constant</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D6">SEQM_3:def 6</a></i>;<br/>
<a NAME="E31:19_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">R1</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E32:19_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">R1</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T40">SEQ_4:40</a></i>;<br/>
<a NAME="E33:19_1_2"/><i><font color="Green">E111</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E34:19_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E35:19_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0 <a href="real_1.html#K5">-</a> 0
 
<b>by </b><i>, , <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>;<br/>
<b>hence </b><a NAME="E36:19_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>


</div>
