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



<b>assume </b><a NAME="E1:27"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">f</font> <a href="fdiff_3.html#R3" target="_self">is_right_differentiable_in</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E2:27"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="fdiff_3.html#R2" target="_self">is_Rcontinuous_in</a> <font color="Maroon">x0</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:27"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Olive">r</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:27"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 <b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">r1</font> = <font color="Maroon">r1</font>;<br/><b>thus </b><a NAME="E1:27_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E2:27_1"/>
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/><b>hence </b><a NAME="E3:27_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:27"/>
( <font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a>  <a href="fdiff_3.html#R3" target="_self">is_right_differentiable_in</a> <font color="Maroon">x0</font> &amp;  <a href="fdiff_3.html#K2" target="_self">Rdiff</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_1.html#K4">^</a> </span>)</span>,<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_3.html#K2" target="_self">Rdiff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/>


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