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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>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">f</font> <a href="fdiff_1.html#R2">is_differentiable_on</a>  <a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a> 
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green">E42</font></i>: 
( for <font color="Olive">x0</font> being  <a href="real_1.html#NM1">Real</a>  holds 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Olive">x0</font> or for <font color="Olive">x0</font> being  <a href="real_1.html#NM1">Real</a>  holds  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Olive">x0</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 ;<br/>

<a NAME="E3:49"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a>  <a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="fdiff_1.html#T33">FDIFF_1:33</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1"/>
( for <font color="Olive">x0</font> being  <a href="real_1.html#NM1">Real</a>  holds 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Olive">x0</font> or for <font color="Olive">x0</font> being  <a href="real_1.html#NM1">Real</a>  holds  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Olive">x0</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1"/>
for <font color="Olive">x0</font> being  <a href="real_1.html#NM1">Real</a>  holds 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Olive">x0</font>
 ;<br/><div class="add"><a NAME="E2:49_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="rfunct_2.html#R1">is_increasing_on</a>  <a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:49_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="rcomp_1.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="fcont_3.html#T34">FCONT_3:34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_2"/>
for <font color="Olive">x0</font> being  <a href="real_1.html#NM1">Real</a>  holds  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Olive">x0</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:49_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="rfunct_2.html#R2">is_decreasing_on</a>  <a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:49_1_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="rcomp_1.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="fcont_3.html#T34">FCONT_3:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:49"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="rcomp_1.html#V3">open</a>
 ;<br/>


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