<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">x0</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="fdiff_3.html#R4" target="_self">is_left_differentiable_in</a> <font color="Maroon">x0</font>
 ;<br/>

<b>given </b><font color="Maroon">r0</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E2:18"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">r0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K1">.]</a></span> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 ;<br/>

<b>consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:18"/><i><font color="Green">E45</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> &amp; <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">r3</font> =  <a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font>;<br/>
<a NAME="E5:18"/><i><font color="Green">E46</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>
<a NAME="E6:18"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T45">XREAL_1:45</a></i>;<br/>
<a NAME="E7:18"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font></span>)</span></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E8:18"/>
 <a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r0</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E9:18"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E10:18"/><b>then </b>
<font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E11:18"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E12:18"/>
<font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:18"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E14:18"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font></span>)</span></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r0</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/>
<a NAME="E15:18"/>
 <a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E16:18"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E17:18"/><b>then </b>
<font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E18:18"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E19:18"/>
<font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:18"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E21:18"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font></span>)</span></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/>
<a NAME="E22:18"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font></span>)</span></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K1">.]</a></span> <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="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E23:18"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font></span>)</span></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/>
not <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font></span>)</span></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r0</font>,<font color="Maroon">r2</font></span>)</span></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E5:18_1"/><i><font color="Green">E62</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rfunct_1.html#D8">RFUNCT_1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_1"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_1_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:18_1_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E3:18_1_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:18_1_1_1_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:18_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E24:18"/>
for <font color="Olive">h</font> being <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a><br/> for <font color="Olive">c</font> being <a href="funct_1.html#V3">constant</a> <a href="seq_1.html#NM1">Real_Sequence</a>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">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> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K9">+</a> <font color="Olive">c</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> &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="Olive">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> 0 ) holds <br/>( <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</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="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K9">+</a> <font color="Olive">c</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">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Olive">c</font></span>)</span></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="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <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="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K9">+</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Olive">c</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_3.html#K1" target="_self">Ldiff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">h</font> be  <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">c</font> be  <a href="funct_1.html#V3">constant</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:18_2"/><i><font color="Green">E63</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>
 <b>and </b><br/><a NAME="E2:18_2"/><i><font color="Green">E64</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span>
 <b>and </b><br/><a NAME="E3:18_2"/><i><font color="Green">E65</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">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/>

<a NAME="E4:18_2"/><i><font color="Green">E66</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rfunct_1.html#D8">RFUNCT_1:def 8</a></i>;<br/>
<a NAME="E5:18_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">h</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">h</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a>  )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/>
<a NAME="E6:18_2"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span> <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="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E7:18_2"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <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="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:18_2"/>
 <a href="fdiff_3.html#K1" target="_self">Ldiff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="fdiff_3.html#K1" target="_self">Ldiff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font>
 
;<br/>
<a NAME="E9:18_2"/><b>then </b><i><font color="Green">E75</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"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fdiff_3.html#K1" target="_self">Ldiff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> )
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E10:18_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T26">RFUNCT_2:26</a></i>;<br/>
<a NAME="E11:18_2"/><i><font color="Green">E77</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>  holds  <font color="Maroon">c</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>proof </b></a><div class="add">

<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/>

<a NAME="E1:18_2_1"/>
<font color="Maroon">c</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</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="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/>
<b>hence </b><a NAME="E2:18_2_1"/>
<font color="Maroon">c</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</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/>


</div><b>end;</b></div>
<a NAME="E12:18_2"/>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:18_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D8">RFUNCT_1:def 8</a></i>;<br/>
<a NAME="E14:18_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:18_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c</font>
 ;<br/>

<b>hence </b><a NAME="E2:18_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i>, <a class="txt" href="fdiff_3.html#E1:18"><i><font color="Green">E42</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:18_2"/><b>then </b><i><font color="Green">E79</font></i>: 
 <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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D8">RFUNCT_1:def 8</a></i>;<br/>
<a NAME="E16:18_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T26">RFUNCT_2:26</a></i>;<br/>
<a NAME="E17:18_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 
<b>by </b><i>, <a class="ref" href="seq_1.html#T43">SEQ_1:43</a></i>;<br/>
<a NAME="E18:18_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D8">RFUNCT_1:def 8</a></i>;<br/>
<a NAME="E19:18_2"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <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="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E20:18_2"/><b>then </b><i><font color="Green">E82</font></i>: 
 <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="fdiff_3.html#E2:18"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E21:18_2"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">g</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</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="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:18_2_3"/><i><font color="Green">E84</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</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="E2:18_2_3"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

 <a href="complex1.html#K18">abs</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="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E5:18"><i><font color="Green">E46</font></i></a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fdiff_3.html#T1" target="_self">Th1</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E4:18_2_3"/>
 <a href="complex1.html#K18">abs</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="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="fdiff_3.html#E7:18"><i><font color="Green">E48</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:18_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E23:18_2"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E6:18"><i><font color="Green">E47</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E24:18_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">h</font> <a href="seq_1.html#K11">(#)</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, , <a class="ref" href="seq_2.html#T28">SEQ_2:28</a></i>;<br/>
<a NAME="E25:18_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">h</font> <a href="seq_1.html#K11">(#)</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_4_1"><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="E1:18_2_4_1"/><i><font color="Green">E89</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/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K11">(#)</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"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</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="seq_1.html#T22">SEQ_1:22</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#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span></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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></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#T12">SEQ_1:12</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#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</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#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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></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#T12">SEQ_1:12</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#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><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> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</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="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></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/>.= 
1 <a href="real_1.html#K4">*</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="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E14:18"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_4"/>
<font color="Maroon">h</font> <a href="seq_1.html#K11">(#)</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</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><b>end;</b></div>
<a NAME="E26:18_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_5_1"><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"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <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"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></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#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <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"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</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"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6">RFUNCT_2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_5"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</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><b>end;</b></div>
<a NAME="E27:18_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E7:18"><i><font color="Green">E48</font></i></a>, <a class="txt" href="fdiff_3.html#E11:18"><i><font color="Green">E50</font></i></a>, <a class="txt" href="fdiff_3.html#E13:18"><i><font color="Green">E51</font></i></a>, <a class="ref" href="seq_2.html#T19">SEQ_2:19</a></i>;<br/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K11">(#)</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"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">h</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</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"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="seq_2.html#T29">SEQ_2:29</a></i>
<br/>.= 
0
<b>by </b><i/>
;<br/>
<a NAME="E29:18_2"/><b>then </b><i><font color="Green">E98</font></i>: 
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span></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>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E7:18"><i><font color="Green">E48</font></i></a>, <a class="txt" href="fdiff_3.html#E9:18"><i><font color="Green">E49</font></i></a>, <a class="txt" href="fdiff_3.html#E13:18"><i><font color="Green">E51</font></i></a>, <a class="txt" href="fdiff_3.html#E14:18"><i><font color="Green">E52</font></i></a>, <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>;<br/>
<a NAME="E30:18_2"/><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E7:18"><i><font color="Green">E48</font></i></a>, <a class="txt" href="fdiff_3.html#E14:18"><i><font color="Green">E52</font></i></a>, <a class="ref" href="seq_2.html#T28">SEQ_2:28</a></i>;<br/>
<a NAME="E31:18_2"/><i><font color="Green">E100</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E7:18"><i><font color="Green">E48</font></i></a>, <a class="txt" href="fdiff_3.html#E9:18"><i><font color="Green">E49</font></i></a>, <a class="txt" href="fdiff_3.html#E14:18"><i><font color="Green">E52</font></i></a>, <a class="txt" href="fdiff_3.html#E16:18"><i><font color="Green">E53</font></i></a>, <a class="ref" href="seq_2.html#T29">SEQ_2:29</a></i>;<br/>
<a NAME="E32:18_2"/><i><font color="Green">E101</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2_7"/>
not  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:18_2_7"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="fdiff_3.html#T2" target="_self">Th2</a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<b>hence </b><a NAME="E3:18_2_7"/>
contradiction
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E33:18_2"/><i><font color="Green">E102</font></i>: 
 <a href="seq_1.html#K17">-</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T23">SEQ_2:23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_8"><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="E1:18_2_8"/><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <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="E2:18_2_8"/><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <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/><b>thus </b><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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<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> <a href="real_1.html#K4">*</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">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></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#T12">SEQ_1:12</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#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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</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">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6">RFUNCT_2:6</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#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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K18">"</a> </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">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T27">RFUNCT_2:27</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#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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K18">"</a> </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">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E2:18"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rfunct_2.html#T27">RFUNCT_2:27</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#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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </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">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></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">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> <a href="real_1.html#K4">*</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">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </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">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></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">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> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</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="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</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#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">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</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="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</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#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"><span class="p3">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></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="p3">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <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="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></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="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E20:18"><i><font color="Green">E58</font></i></a>, <a class="txt" href="fdiff_3.html#E22:18"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xcmplx_1.html#T131">XCMPLX_1:131</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#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"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</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="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <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="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</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#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"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</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#K6">/</a> <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="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6">RFUNCT_2:6</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#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T188">XCMPLX_1:188</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</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#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="real_1.html#K1">-</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#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</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#K6">/</a> <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="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <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/>.= 
 <a href="real_1.html#K1">-</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"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</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"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></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="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</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"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></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="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</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"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span></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#T12">SEQ_1:12</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</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"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></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#D9">SEQ_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_1.html#K17">-</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"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></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="seq_1.html#T14">SEQ_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K17">-</a> <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"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</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="seq_1.html#T56">SEQ_1:56</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E35:18_2"/><b>then </b><i><font color="Green">E107</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</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">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</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"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<b>then </b> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="seq_1.html#K17">-</a> <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"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E4:18"><i><font color="Green">E45</font></i></a>, <a class="txt" href="fdiff_3.html#E18:18"><i><font color="Green">E54</font></i></a>, <a class="ref" href="fdiff_3.html#T2" target="_self">Th2</a>, <a class="ref" href="fdiff_3.html#T3" target="_self">Th3</a>, , <a class="ref" href="seq_2.html#T38">SEQ_2:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="fdiff_3.html#K1" target="_self">Ldiff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="seq_2.html#T24">SEQ_2:24</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_3.html#K1" target="_self">Ldiff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T188">XCMPLX_1:188</a></i>
;<br/>
<b>hence </b><a NAME="E37:18_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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</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">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_3.html#K1" target="_self">Ldiff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="fdiff_3.html#E4:18"><i><font color="Green">E45</font></i></a>, <a class="txt" href="fdiff_3.html#E18:18"><i><font color="Green">E54</font></i></a>, <a class="ref" href="fdiff_3.html#T3" target="_self">Th3</a>, , <a class="txt" href="fdiff_3.html#E20:18"><i><font color="Green">E58</font></i></a>, <a class="ref" href="seq_2.html#T37">SEQ_2:37</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E25:18"/>
( <font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a>  <a href="fdiff_3.html#R4" target="_self">is_left_differentiable_in</a> <font color="Maroon">x0</font> &amp;  <a href="fdiff_3.html#K1" target="_self">Ldiff</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_3.html#K1" target="_self">Ldiff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> )
 <b>by </b><i>, , </i>;<br/>


</div>
