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

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



<b>assume </b><a NAME="E1:3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 ;<br/>

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

<b>assume </b><b>that </b><br/><a NAME="E2:3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E3:3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">f</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">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<a NAME="E4:3"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="fcont_1.html#D2">FCONT_1:def 2</a></i>;<br/>
<a NAME="E5:3"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T11">RCOMP_1:11</a></i>;<br/>
<a NAME="E6:3"/><b>then </b>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E7:3"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K2">.[</a></span> as  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E9:3"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E10:3"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:3"/><i><font color="Green">E46</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="real_1.html#NM1">Real</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">f1</font> being   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E13:3"/><i><font color="Green">E47</font></i>: 
( ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) ) &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> holds <br/><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) ) )
 <b>from </b><i><a class="ref" href="seq_1.html#S3">SEQ_1:sch 3</a>();<br/></i>
<a NAME="E14:3"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><a NAME="E2:3_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i/>;<br/><b>hence </b><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span></span>)</span>

;<br/><br/></div><b>end;</b></div>
<a NAME="E16:3"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">f1</font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="fcont_1.html#T48">FCONT_1:48</a></i>;<br/>
<a NAME="E17:3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:3_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><a NAME="E2:3_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span></span>)</span>

;<br/><br/></div><b>end;</b></div>
<a NAME="E19:3"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">f1</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">Z</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> ) )
 
<b>by </b><i>, <a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a></i>;<br/>
<b>set </b><font color="Maroon">f2</font> = <font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f1</font>;<br/>
<a NAME="E20:3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f1</font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, , <a class="ref" href="fcont_1.html#T19">FCONT_1:19</a></i>;<br/>
<a NAME="E21:3"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E22:3"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>;<br/>
<a NAME="E23:3"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f1</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">Z</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f1</font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f1</font>,<font color="Olive">x</font></span>)</span> ) )
 
<b>by </b><i>, , <a class="ref" href="fdiff_1.html#T27">FDIFF_1:27</a></i>;<br/>
<a NAME="E24:3"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E25:3"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E26:3"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>;<br/>
<b>then </b><i><font color="Green">E58</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</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"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>

;<br/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</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"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</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">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
<b>by </b><i/>
;<br/>
<b>then consider </b><font color="Maroon">x0</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E30:3"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp;  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f1</font></span>)</span>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, , , </i>;<br/>
<b>take </b>
<font color="Maroon">x0</font>
;<br/>

 <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f1</font></span>)</span>,<font color="Maroon">x0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K7">-</a> <font color="Maroon">f1</font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>
<b>by </b><i>, , <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x0</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="fdiff_1.html#T27">FDIFF_1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</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">f1</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E32:3"/>
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a></i>;<br/>


</div>
