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



<b>assume </b><a NAME="E1:4"/><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t</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="E2:4"/><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">x</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E3:4"/><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">x</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<a NAME="E4:4"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></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="E6:4"/><i><font color="Green">E41</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">x</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t</font></span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/>
<b>take </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">t</font>;<br/>

<a NAME="E7:4"/>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/>
<a NAME="E8:4"/><b>then </b><i><font color="Green">E43</font></i>: 
 ex <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t</font> )
 
;<br/>
<a NAME="E9:4"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E10:4"/><b>then </b>
0 <a href="real_1.html#K6">/</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>hence </b><a NAME="E11:4"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E12:4"/>
<font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E13:4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t</font> <a href="real_1.html#K6">/</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>hence </b><a NAME="E14:4"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/>

<a NAME="E15:4"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E16:4"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>hence </b><a NAME="E17:4"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


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