<?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="E1:28"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">f</font> <a href="fdiff_3.html#R3" target="_self">is_right_differentiable_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">f</font> <a href="fdiff_3.html#R4" target="_self">is_left_differentiable_in</a> <font color="Maroon">x0</font> &amp;  <a href="fdiff_3.html#K2" target="_self">Rdiff</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="E2:28"/><i><font color="Green">E43</font></i>: 
 ex <font color="Olive">N</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> st <font color="Olive">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:28_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &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">r1</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>consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:28_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">r2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span></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">r</font> =  <a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>;<br/>
<a NAME="E5:28_1"/>
 <a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">N</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> as    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> <b>by </b><i><a class="ref" href="rcomp_1.html#D7">RCOMP_1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon">N</font>
;<br/>

<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="E7:28_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E8:28_1"/><b>then </b>
<font color="Maroon">x</font> <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> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <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> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:28_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <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">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_1"/>
( <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> or <font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_1_1"/>
 <a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/><a NAME="E3:28_1_1_1_1"/><b>then </b>
<font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</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">r1</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="E4:28_1_1_1_1"/><b>then </b>
<font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:28_1_1_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</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">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">g1</font> &amp; <font color="Olive">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E6:28_1_1_1_1"/><b>then </b>
<font color="Maroon">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">r1</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/><b>hence </b><a NAME="E7:28_1_1_1_1"/>
<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>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x0</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_1_2"/>
 <a href="square_1.html#K1">min</a> <font color="Maroon">r1</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="E3:28_1_1_1_2"/><b>then </b>
<font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:28_1_1_1_2"/><b>then </b>
<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:28_1_1_1_2"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">g1</font> &amp; <font color="Olive">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E6:28_1_1_1_2"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r2</font></span>)</span></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/><b>hence </b><a NAME="E7:28_1_1_1_2"/>
<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>
 <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="E12:28_1"/>
<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><b>end;</b></div>
<a NAME="E3:28"/>
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> <font color="Maroon">f</font> 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"><font color="Maroon">f</font> <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"><font color="Maroon">f</font> <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"><font color="Maroon">f</font> <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"><font color="Maroon">f</font> <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="fdiff_3.html#K1" target="_self">Ldiff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_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><a NAME="E1:28_2"/><i><font color="Green">E50</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> &amp;  <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> )
 ;<br/>

<a NAME="E2:28_2"/><i><font color="Green">E51</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">S</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E2:28_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">S</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/>;<br/>
<a NAME="E3:28_2_1"/>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T37">RCOMP_1:37</a></i>;<br/>
<b>hence </b><a NAME="E4:28_2_1"/>
 <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="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_2_1"/>
(  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">n</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#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or for <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> &amp; <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 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_1_1"><b>suppose </b></a><a NAME="E1:28_2_2_1_1"/>
 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">n</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#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">N</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:28_2_2_1_1"/><i><font color="Green">E53</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>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> holds <br/><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>set </b><font color="Maroon">h1</font> = <font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font>;<br/><b>set </b><font color="Maroon">c1</font> = <font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font>;<br/><a NAME="E4:28_2_2_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:28_2_2_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T7">RFUNCT_2:7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<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="E2:28_2_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:28_2_2_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">x</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/>
<i><font color="Green">E59</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
<font color="Maroon">c</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
<font color="Maroon">c</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">N</font>

;<br/>
<a NAME="E5:28_2_2_1_1_1"/>
<font color="Maroon">c</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">N</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/>
<a NAME="E6:28_2_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:28_2_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:28_2_2_1_1"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</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="seqm_3.html#K1">^\</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#T37">SEQM_3:37</a></i>;<br/><a NAME="E6:28_2_2_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <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>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T7">RFUNCT_2:7</a></i>;<br/><a NAME="E7:28_2_2_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</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#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:28_2_2_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:28_2_2_1_1_2"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/>
<a NAME="E2:28_2_2_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E3:28_2_2_1_1_2"/><b>then </b>
<font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E4:28_2_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E9:28_2_2_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span></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"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <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"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span></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> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fdiff_3.html#K2" target="_self">Rdiff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> )
 <b>by </b><i>, , , </i>;<br/><i><font color="Green">E64</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <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_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="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></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> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="rfunct_2.html#T22">RFUNCT_2:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <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_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="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></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_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T22">RFUNCT_2:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <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_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="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T39">SEQM_3:39</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="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></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_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="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T41">SEQM_3:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><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="seqm_3.html#K1">^\</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="seqm_3.html#T42">SEQM_3:42</a></i>
;<br/><b>hence </b><a NAME="E11:28_2_2_1_1"/>
<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>
 <b>by </b><i>, <a class="ref" href="seq_4.html#T35">SEQ_4:35</a></i>;<br/><b>thus </b><a NAME="E12:28_2_2_1_1"/>
 <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>, , , <a class="ref" href="seq_4.html#T36">SEQ_4:36</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_1_2"><b>suppose </b></a><a NAME="E1:28_2_2_1_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>   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/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> &amp; <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/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_2_1_2_1_1"/>
(  ex <font color="Olive">M</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">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">M</font> holds <br/><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;</a> 0 or for <font color="Olive">M</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">M</font> &amp; <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:28_2_2_1_2_1_1_1"/>
 ex <font color="Olive">M</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">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">M</font> holds <br/><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">M</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:28_2_2_1_2_1_1_1"/><i><font color="Green">E66</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>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">M</font> holds <br/><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/><b>set </b><font color="Maroon">h1</font> = <font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font>;<br/><b>set </b><font color="Maroon">c1</font> = <font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font>;<br/><a NAME="E4:28_2_2_1_2_1_1_1"/><i><font color="Green">E67</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:28_2_2_1_2_1_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T7">RFUNCT_2:7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<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="E2:28_2_2_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:28_2_2_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</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/>
<i><font color="Green">E75</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
<font color="Maroon">c</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <font color="Maroon">M</font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
<font color="Maroon">c</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">M</font>

;<br/>
<a NAME="E5:28_2_2_1_2_1_1_1_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/>
<a NAME="E6:28_2_2_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:28_2_2_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:28_2_2_1_2_1_1_1"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</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="seqm_3.html#K1">^\</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#T37">SEQM_3:37</a></i>;<br/><a NAME="E6:28_2_2_1_2_1_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <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>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T7">RFUNCT_2:7</a></i>;<br/><a NAME="E7:28_2_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</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#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:28_2_2_1_2_1_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2_1_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:28_2_2_1_2_1_1_1_2"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/>
<a NAME="E2:28_2_2_1_2_1_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E3:28_2_2_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">M</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E4:28_2_2_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E9:28_2_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span></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"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <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"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span></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> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span></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/><i><font color="Green">E80</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <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_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="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></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> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="rfunct_2.html#T22">RFUNCT_2:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <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_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="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></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_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T22">RFUNCT_2:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span> <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_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="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T39">SEQM_3:39</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="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></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_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="seqm_3.html#K1">^\</a> <font color="Maroon">M</font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T41">SEQM_3:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><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="seqm_3.html#K1">^\</a> <font color="Maroon">M</font>
<b>by </b><i><a class="ref" href="seqm_3.html#T42">SEQM_3:42</a></i>
;<br/><b>hence </b><a NAME="E11:28_2_2_1_2_1_1_1"/>
<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>
 <b>by </b><i>, <a class="ref" href="seq_4.html#T35">SEQ_4:35</a></i>;<br/><b>thus </b><a NAME="E12:28_2_2_1_2_1_1_1"/>
 <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>, , <a class="ref" href="seq_4.html#T36">SEQ_4:36</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:28_2_2_1_2_1_1_2"/><i><font color="Green">E81</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>   ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">M</font> &amp; <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:28_2_2_1_2_1_1_2"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">h</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a>  &amp; <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 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[ <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0;<br/><a NAME="E3:28_2_2_1_2_1_1_2"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2_1_2_1_1_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/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:28_2_2_1_2_1_1_2_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E3:28_2_2_1_2_1_1_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/>

<a NAME="E4:28_2_2_1_2_1_1_2_1"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>
<b>hence </b><a NAME="E5:28_2_2_1_2_1_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:28_2_2_1_2_1_1_2"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2_1_2_1_1_2_2"><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/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:28_2_2_1_2_1_1_2_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E3:28_2_2_1_2_1_1_2_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/>

<a NAME="E4:28_2_2_1_2_1_1_2_2"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>
<b>hence </b><a NAME="E5:28_2_2_1_2_1_1_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">q1</font> being  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E6:28_2_2_1_2_1_1_2"/><i><font color="Green">E87</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">S<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>] ) &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] ) holds <br/> ex <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="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> ) )
 <b>from </b><i><a class="ref" href="fdiff_2.html#S1">FDIFF_2:sch 1</a>();<br/></i><b>consider </b><font color="Maroon">q2</font> being  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E8:28_2_2_1_2_1_1_2"/><i><font color="Green">E88</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">S<sub>2</sub></font>[<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>] ) &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">r</font>] ) holds <br/> ex <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="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> ) )
 <b>from </b><i><a class="ref" href="fdiff_2.html#S1">FDIFF_2:sch 1</a>();<br/></i><b>set </b><font color="Maroon">h1</font> = <font color="Maroon">h</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q1</font>;<br/><b>set </b><font color="Maroon">h2</font> = <font color="Maroon">h</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q2</font>;<br/><b>set </b><font color="Maroon">c1</font> = <font color="Maroon">c</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q1</font>;<br/><b>set </b><font color="Maroon">c2</font> = <font color="Maroon">c</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q2</font>;<br/><a NAME="E9:28_2_2_1_2_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">h</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q1</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/><a NAME="E10:28_2_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">h</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q1</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, <a class="ref" href="seq_4.html#T29">SEQ_4:29</a></i>;<br/><a NAME="E11:28_2_2_1_2_1_1_2"/><i><font color="Green">E98</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="seq_4.html#T30">SEQ_4:30</a></i>;<br/><a NAME="E12:28_2_2_1_2_1_1_2"/>
<font color="Maroon">h</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q1</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i>, , <a class="ref" href="rfunct_2.html#T12">RFUNCT_2:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">h</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q1</font> as  <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i>, , <a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/><a NAME="E14:28_2_2_1_2_1_1_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">h</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q2</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/><a NAME="E15:28_2_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">h</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q2</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, <a class="ref" href="seq_4.html#T29">SEQ_4:29</a></i>;<br/><a NAME="E16:28_2_2_1_2_1_1_2"/><i><font color="Green">E101</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q2</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="seq_4.html#T30">SEQ_4:30</a></i>;<br/><a NAME="E17:28_2_2_1_2_1_1_2"/>
<font color="Maroon">h</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q2</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i>, , <a class="ref" href="rfunct_2.html#T12">RFUNCT_2:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">h2</font> = <font color="Maroon">h</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q2</font> as  <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i>, , <a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/><a NAME="E19:28_2_2_1_2_1_1_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q1</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/><a NAME="E20:28_2_2_1_2_1_1_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q2</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/><a NAME="E21:28_2_2_1_2_1_1_2"/><i><font color="Green">E104</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q1</font></span>)</span> <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>by </b><i>, , <a class="ref" href="seqm_3.html#T55">SEQM_3:55</a></i>;<br/><b>reconsider </b><font color="Maroon">c1</font> = <font color="Maroon">c</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q1</font> as  <a href="funct_1.html#V3">constant</a> <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i/>;<br/><a NAME="E23:28_2_2_1_2_1_1_2"/>
<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="seqm_3.html#K3">*</a> <font color="Maroon">q1</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/><a NAME="E24:28_2_2_1_2_1_1_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><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> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <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>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T11">RFUNCT_2:11</a></i>;<br/><a NAME="E25:28_2_2_1_2_1_1_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><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> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q1</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="E26:28_2_2_1_2_1_1_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c1</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="rfunct_2.html#T13">RFUNCT_2:13</a></i>;<br/><a NAME="E27:28_2_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h1</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">h1</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c1</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">c1</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">h1</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">h1</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c1</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">c1</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="E28:28_2_2_1_2_1_1_2"/><i><font color="Green">E108</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q2</font></span>)</span> <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>by </b><i>, , <a class="ref" href="seqm_3.html#T55">SEQM_3:55</a></i>;<br/><b>reconsider </b><font color="Maroon">c2</font> = <font color="Maroon">c</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q2</font> as  <a href="funct_1.html#V3">constant</a> <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i/>;<br/><a NAME="E30:28_2_2_1_2_1_1_2"/>
<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="seqm_3.html#K3">*</a> <font color="Maroon">q2</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">h</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/><a NAME="E31:28_2_2_1_2_1_1_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><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> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <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>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T11">RFUNCT_2:11</a></i>;<br/><a NAME="E32:28_2_2_1_2_1_1_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><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> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q2</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="E33:28_2_2_1_2_1_1_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c2</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="rfunct_2.html#T13">RFUNCT_2:13</a></i>;<br/><a NAME="E34:28_2_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h2</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">h2</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c2</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">c2</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">h2</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">h2</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c2</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">c2</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>, , <a class="txt" href="fdiff_3.html#E1:28"><i><font color="Green">E42</font></i></a>, </i>;<br/><i><font color="Green">E110</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h1</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">h1</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c1</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">c1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">h1</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"><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> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q1</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">c1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T13">RFUNCT_2:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h1</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_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="seqm_3.html#K3">*</a> <font color="Maroon">q1</font></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">c1</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T28">RFUNCT_2:28</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="seqm_3.html#K3">*</a> <font color="Maroon">q1</font></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_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="seqm_3.html#K3">*</a> <font color="Maroon">q1</font></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">c1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T16">RFUNCT_2:16</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="seqm_3.html#K3">*</a> <font color="Maroon">q1</font></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_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="seqm_3.html#K3">*</a> <font color="Maroon">q1</font></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_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q1</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T28">RFUNCT_2:28</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="seqm_3.html#K3">*</a> <font color="Maroon">q1</font></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_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="seqm_3.html#K3">*</a> <font color="Maroon">q1</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T13">RFUNCT_2:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><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="seqm_3.html#K3">*</a> <font color="Maroon">q1</font>
<b>by </b><i><a class="ref" href="rfunct_2.html#T13">RFUNCT_2:13</a></i>
;<br/><i><font color="Green">E111</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h2</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">h2</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c2</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">c2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">h2</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"><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> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q2</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">c2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T13">RFUNCT_2:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h2</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_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="seqm_3.html#K3">*</a> <font color="Maroon">q2</font></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">c2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T28">RFUNCT_2:28</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="seqm_3.html#K3">*</a> <font color="Maroon">q2</font></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_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="seqm_3.html#K3">*</a> <font color="Maroon">q2</font></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">c2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T16">RFUNCT_2:16</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="seqm_3.html#K3">*</a> <font color="Maroon">q2</font></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_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="seqm_3.html#K3">*</a> <font color="Maroon">q2</font></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_2.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">q2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T28">RFUNCT_2:28</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="seqm_3.html#K3">*</a> <font color="Maroon">q2</font></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_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="seqm_3.html#K3">*</a> <font color="Maroon">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T13">RFUNCT_2:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><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="seqm_3.html#K3">*</a> <font color="Maroon">q2</font>
<b>by </b><i><a class="ref" href="rfunct_2.html#T13">RFUNCT_2:13</a></i>
;<br/><b>set </b><font color="Maroon">s</font> = <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>;<br/><a NAME="E37:28_2_2_1_2_1_1_2"/><i><font color="Green">E112</font></i>: 
for <font color="Olive">g1</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">g1</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"><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="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</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="xxreal_0.html#R1">&lt;</a> <font color="Olive">g1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2_1_2_1_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g1</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="E1:28_2_2_1_2_1_1_2_5"/><i><font color="Green">E113</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:28_2_2_1_2_1_1_2_5"/><i><font color="Green">E114</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n1</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"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p0">(<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="seqm_3.html#K3">*</a> <font color="Maroon">q1</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"><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="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="ref" href="fdiff_3.html#T1" target="_self">Th1</a>, , <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:28_2_2_1_2_1_1_2_5"/><i><font color="Green">E115</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n2</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"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p0">(<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="seqm_3.html#K3">*</a> <font color="Maroon">q2</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"><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="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="fdiff_3.html#E2:28"><i><font color="Green">E43</font></i></a>, , , <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<b>take </b><font color="Maroon">n</font> =  <a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n2</font></span>)</span>;<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="E6:28_2_2_1_2_1_1_2_5"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E7:28_2_2_1_2_1_1_2_5"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">q1</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">q2</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n2</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2_1_2_1_1_2_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_1_2_1_1_2_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_2_1_2_1_1_2_5_1_1"/>
( <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_1_2_1_1_2_5_1_1_1"><b>suppose </b></a><a NAME="E1:28_2_2_1_2_1_1_2_5_1_1_1"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:28_2_2_1_2_1_1_2_5_1_1_1"/><b>then </b>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:28_2_2_1_2_1_1_2_5_1_1_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E5:28_2_2_1_2_1_1_2_5_1_1_1"/>
<font color="Maroon">q2</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">q2</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="txt" href="fdiff_3.html#E1:28_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="fdiff_3.html#E2:28_2"><i><font color="Green">E51</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:28_2_2_1_2_1_1_2_5_1_1_1"/><b>then </b>
not <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#T7">SEQM_3:7</a></i>;<br/><a NAME="E7:28_2_2_1_2_1_1_2_5_1_1_1"/><b>then </b>
 <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"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p0">(<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="seqm_3.html#K3">*</a> <font color="Maroon">q2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</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="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:28_2_2_1_2_1_1_2_5_1_1_1"/>
 <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"><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">m</font></span>)</span> <a href="real_1.html#K5">-</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="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font>
 <b>by </b><i>, <a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2_1_2_1_1_2_5_1_1_2"><b>suppose </b></a><a NAME="E1:28_2_2_1_2_1_1_2_5_1_1_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:28_2_2_1_2_1_1_2_5_1_1_2"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</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="E3:28_2_2_1_2_1_1_2_5_1_1_2"/><b>then </b>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:28_2_2_1_2_1_1_2_5_1_1_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E6:28_2_2_1_2_1_1_2_5_1_1_2"/>
<font color="Maroon">q1</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">q1</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="txt" href="fdiff_3.html#E1:28_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="fdiff_3.html#E2:28_2"><i><font color="Green">E51</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:28_2_2_1_2_1_1_2_5_1_1_2"/><b>then </b>
not <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#T7">SEQM_3:7</a></i>;<br/><a NAME="E8:28_2_2_1_2_1_1_2_5_1_1_2"/><b>then </b>
 <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"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><span class="p0">(<span class="default"><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></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p0">(<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="seqm_3.html#K3">*</a> <font color="Maroon">q1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</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="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:28_2_2_1_2_1_1_2_5_1_1_2"/>
 <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"><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">m</font></span>)</span> <a href="real_1.html#K5">-</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="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font>
 <b>by </b><i>, <a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:28_2_2_1_2_1_1_2_5"/>
 <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"><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">m</font></span>)</span> <a href="real_1.html#K5">-</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="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g1</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E38:28_2_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><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>
 <b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/><b>hence </b><a NAME="E39:28_2_2_1_2_1_1_2"/>
 <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>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:28_2_2_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><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> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:28_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"><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> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:28"/>
( <font color="Maroon">f</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x0</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="fdiff_3.html#K2" target="_self">Rdiff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> &amp;  <a href="fdiff_1.html#K1">diff</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> )
 <b>by </b><i>, , <a class="ref" href="fdiff_2.html#T12">FDIFF_2:12</a></i>;<br/>


</div>
