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<b>let </b><font color="Maroon">x0</font> be   <a href="real_1.html#NM1">Real</a>;<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>given </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E1:32"/><i><font color="Green">E30</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">f</font> <a href="rfunct_2.html#R1">is_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; not <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:32"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> holds <br/> ex <font color="Olive">g</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x0</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E3:32"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#R3">is_non_decreasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">x0</font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T55">RFUNCT_2:55</a></i>;<br/>
<b>hence </b><a NAME="E4:32"/>
<font color="Maroon">f</font> <a href="limfunc2.html#R2" target="_self">is_left_divergent_to+infty_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, , </i>;<br/>


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