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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="funct_1.html#V2">one-to-one</a> <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:53"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>and </b><br/><a NAME="E2:53"/><i><font color="Green">E31</font></i>: 
( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> or for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 ;<br/>

<a NAME="E3:53"/><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> is <a href="rcomp_1.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E1:53"><i><font color="Green">E30</font></i></a>, <a class="txt" href="fdiff_2.html#E2:53"><i><font color="Green">E31</font></i></a>, <a class="ref" href="fdiff_2.html#T41" target="_self">Th41</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>thus </b><a NAME="E4:53"/>
<font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<a NAME="E5:53"/><i><font color="Green">E33</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E6:53"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:53_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="fdiff_2.html#E5:53"><i><font color="Green">E33</font></i></a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:53_1"/><i><font color="Green">E37</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>6</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E3:53"><i><font color="Green">E32</font></i></a>, <a class="txt" href="fdiff_2.html#E5:53"><i><font color="Green">E33</font></i></a>, <a class="txt" href="fdiff_2.html#E1:53_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rcomp_1.html#T39">RCOMP_1:39</a></i>;<br/>
<a NAME="E5:53_1"/>
for <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></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">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">b<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K9">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></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<sub>9</sub></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="E1:53_1_1"/><i><font color="Green">E38</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <a href="numbers.html#K5">NAT</a>  <a href="partfun2.html#K4">--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E3:53_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  <a href="funct_1.html#V3">constant</a> <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<a NAME="E5:53_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E2:53_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/>
<a NAME="E3:53_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:53_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>12</sub></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="E2:53_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:53_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:53_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:53_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:53_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><b>hence </b><a NAME="E2:53_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="fdiff_2.html#E1:53_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>, <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> );<br/>
<a NAME="E7:53_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:53_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></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<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:53_1_1_3"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="fdiff_2.html#E5:53"><i><font color="Green">E33</font></i></a>, <a class="txt" href="fdiff_2.html#E1:53_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<i><font color="Green">E44</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="fdiff_2.html#E3:53_1_1"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
;<br/>
<b>take </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font>;<br/>

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></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="E5:53_1_1_3"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/>

<b>hence </b><a NAME="E6:53_1_1_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="fdiff_2.html#E4:53_1_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>

<a NAME="E7:53_1_1_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1_1_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<b>hence </b><a NAME="E8:53_1_1_3"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="fdiff_2.html#E4:53_1_1_3"><i><font color="Green">E44</font></i></a>, <a class="txt" href="fdiff_2.html#E5:53_1_1_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E9:53_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="rcomp_1.html#S1">RCOMP_1:sch 1</a>(<a class="txt" href="fdiff_2.html#E7:53_1_1"><i><font color="Green">E42</font></i></a>);<br/></i>
<a NAME="E10:53_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#NR1" target="_self">is_not_0</a>  &amp; <font color="Maroon">c<sub>8</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>8</sub></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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:53_1_1_4"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><i><font color="Green">E46</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E6:53_1_1"><i><font color="Green">E41</font></i></a></i>
;<br/><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="fdiff_2.html#E3:53_1_1"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
;<br/><a NAME="E4:53_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="fdiff_2.html#E1:53_1_1_4"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E5:53_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E9:53_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="fdiff_2.html#E2:53_1_1_4"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E6:53_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E7:53_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="fdiff_2.html#E10:53_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:53_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>
<a NAME="E13:53_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E1:53_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="fdiff_2.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E14:53_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#R1">is_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#R2">is_decreasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_5_1_1"/>
( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> or for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="fdiff_2.html#E2:53"><i><font color="Green">E31</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_5_1_1_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_5_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_5_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_5_1_1_1_1_1"/>
( <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_5_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_5_1_1_1_1_1_1"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:53_1_1_5_1_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:53_1_1_5_1_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:53_1_1_5_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#R1">is_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#R2">is_decreasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="ref" href="rfunct_2.html#T54">RFUNCT_2:54</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_5_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_5_1_1_1_1_1_2"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:53_1_1_5_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T12">RCOMP_1:12</a></i>;<br/><b>hence </b><a NAME="E3:53_1_1_5_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#R1">is_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#R2">is_decreasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="txt" href="fdiff_2.html#E1:53"><i><font color="Green">E30</font></i></a>, <a class="txt" href="fdiff_2.html#E1:53_1_1_5_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="rolle.html#T9">ROLLE:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:53_1_1_5_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#R1">is_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#R2">is_decreasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_5_1_1_2"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_5_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_5_1_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="E1:53_1_1_5_1_1_2_1_1"/>
( <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_5_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_5_1_1_2_1_1_1"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:53_1_1_5_1_1_2_1_1_1"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:53_1_1_5_1_1_2_1_1_1"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:53_1_1_5_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#R1">is_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#R2">is_decreasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="ref" href="rfunct_2.html#T54">RFUNCT_2:54</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_5_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_5_1_1_2_1_1_2"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:53_1_1_5_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T12">RCOMP_1:12</a></i>;<br/><b>hence </b><a NAME="E3:53_1_1_5_1_1_2_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#R1">is_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#R2">is_decreasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="txt" href="fdiff_2.html#E1:53"><i><font color="Green">E30</font></i></a>, <a class="txt" href="fdiff_2.html#E1:53_1_1_5_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="rolle.html#T10">ROLLE:10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:53_1_1_5_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#R1">is_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#R2">is_decreasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:53_1_1_5"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#R1">is_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#R2">is_decreasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:53_1_1"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E1:53"><i><font color="Green">E30</font></i></a>, <a class="ref" href="fdiff_1.html#D7">FDIFF_1:def 7</a></i>;<br/>
<a NAME="E16:53_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a>  <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E14:53_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="fcont_3.html#T25">FCONT_3:25</a></i>;<br/>
<a NAME="E17:53_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a>  <a href="fcont_1.html#R2">is_continuous_on</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>;<br/>
<a NAME="E18:53_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span></span>)</span> <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E5:53"><i><font color="Green">E33</font></i></a>, <a class="txt" href="fdiff_2.html#E1:53_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="fcont_1.html#D2">FCONT_1:def 2</a></i>;<br/>
<a NAME="E19:53_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a>  <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E5:53"><i><font color="Green">E33</font></i></a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/>
<a NAME="E20:53_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <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">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E1:53_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="fdiff_2.html#E13:53_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="fcont_1.html#D1">FCONT_1:def 1</a></i>;<br/>
<a NAME="E21:53_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/>
<i><font color="Green">E50</font></i>:  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>11</sub></font> = 
<font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="seq_4.html#T41">SEQ_4:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1_1"><i><font color="Green">E39</font></i></a></i>
;<br/>
<a NAME="E23:53_1_1"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>11</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E20:53_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="fdiff_2.html#E21:53_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="seq_2.html#T25">SEQ_2:25</a></i>;<br/>
<i><font color="Green">E52</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E20:53_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="fdiff_2.html#E21:53_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="fdiff_2.html#E22:53_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>
<br/>.= 
0

;<br/>
<a NAME="E25:53_1_1"/><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_1_1_8"/>
<font color="Maroon">c<sub>13</sub></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<sub>12</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:53_1_1_8"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/>
<a NAME="E4:53_1_1_8"/><i><font color="Green">E55</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> )
 
;<br/>
<a NAME="E5:53_1_1_8"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1_1_8"><i><font color="Green">E54</font></i></a>, <a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>
<b>hence </b><a NAME="E6:53_1_1_8"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E9:53_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="fdiff_2.html#E4:53_1_1_8"><i><font color="Green">E55</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:53_1_1_9"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><a NAME="E2:53_1_1_9"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E9:53_1_1"><i><font color="Green">E43</font></i></a></i>;<br/><b>thus </b><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">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<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">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6">RFUNCT_2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E1:53_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E9:53_1_1"><i><font color="Green">E43</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E2:53_1_1_9"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E2:53_1_1_9"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font>

;<br/></div><b>end;</b></div>
<a NAME="E27:53_1_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E23:53_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="fdiff_2.html#E24:53_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></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="txt" href="fdiff_2.html#E12:53_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/>
<a NAME="E29:53_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_1_1_10"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E2:53_1_1_10"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E5:53_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:53_1_1_10"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/>
<a NAME="E4:53_1_1_10"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<b>hence </b><a NAME="E5:53_1_1_10"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:53_1_1_11"/>
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E2:53_1_1_11"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E1:53_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:53_1_1_11"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E1:53_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E29:53_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1_1"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E5:53"><i><font color="Green">E33</font></i></a>, <a class="txt" href="fdiff_2.html#E3:53_1_1_11"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="fdiff_2.html#E5:53"><i><font color="Green">E33</font></i></a>, <a class="txt" href="fdiff_2.html#E1:53_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="fdiff_2.html#E2:53_1_1_11"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E31:53_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:53_1_1_12"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E9:53_1_1"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E2:53_1_1_12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/><b>hence </b><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E31:53_1_1"><i><font color="Green">E55</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E25:53_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6">RFUNCT_2:6</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E33:53_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E34:53_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/>
<a NAME="E35:53_1_1"/><i><font color="Green">E58</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_1_1_13"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:53_1_1_13"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E1:53_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="fdiff_2.html#E1:53_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E36:53_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K18">"</a>  <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E12:53_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="seq_1.html#T41">SEQ_1:41</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:53_1_1_14"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> )
 ;<br/><a NAME="E2:53_1_1_14"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E9:53_1_1"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E3:53_1_1_14"/>
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E4:53_1_1_14"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E1:53_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6">RFUNCT_2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E1:53_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E9:53_1_1"><i><font color="Green">E43</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E2:53_1_1_14"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E2:53_1_1_14"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E35:53_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_2.html#E3:53_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="fdiff_2.html#E4:53_1_1_14"><i><font color="Green">E61</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T42">SEQ_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>13</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="fdiff_2.html#E33:53_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="seq_1.html#T44">SEQ_1:44</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E38:53_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K18">"</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E39:53_1_1"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E34:53_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="fdiff_2.html#E36:53_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="seq_1.html#T43">SEQ_1:43</a></i>;<br/>
<a NAME="E40:53_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E5:53"><i><font color="Green">E33</font></i></a>, <a class="txt" href="fdiff_2.html#E1:53_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E41:53_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E42:53_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E43:53_1_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E1:53"><i><font color="Green">E30</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E44:53_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>11</sub></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">c<sub>13</sub></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">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E5:53_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="fdiff_2.html#E25:53_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="fdiff_2.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E45:53_1_1"/><i><font color="Green">E64</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_2.html#E2:53"><i><font color="Green">E31</font></i></a>, <a class="txt" href="fdiff_2.html#E42:53_1_1"><i><font color="Green">E62</font></i></a></i>;<br/>
<b>hence </b><a NAME="E46:53_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E38:53_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="fdiff_2.html#E39:53_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="fdiff_2.html#E44:53_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="seq_2.html#T35">SEQ_2:35</a></i>;<br/>

<b>thus </b> <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K9">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p0"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> 
<b>by </b><i><a class="txt" href="fdiff_2.html#E38:53_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="fdiff_2.html#E39:53_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="fdiff_2.html#E44:53_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="fdiff_2.html#E45:53_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="seq_2.html#T36">SEQ_2:36</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:53_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="fdiff_2.html#E4:53_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="fdiff_2.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:53"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E8:53"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a>  <a href="fdiff_1.html#R2">is_differentiable_on</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E3:53"><i><font color="Green">E32</font></i></a>, <a class="txt" href="fdiff_2.html#E5:53"><i><font color="Green">E33</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E9:53"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span>
 ;<br/>

<b>hence </b><a NAME="E10:53"/>
 <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_2.html#E6:53"><i><font color="Green">E34</font></i></a></i>;<br/>


</div>
