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

<b>let </b><font color="Maroon" title="c1">x0</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>given </b><font color="Maroon" title="c3">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><a NAME="E1:26"/><i><font color="Green" title="E21">A1</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">r</font> &amp; <font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#R1" title="RFUNCT_2:pred.1">is_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c1">x0</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#R2" title="RFUNCT_2:pred.2">is_decreasing_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; not <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c1">x0</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; not <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:26"/><i><font color="Green" title="E22">A2</font></i>: 
for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/> ex <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">g1</font> &amp; <font color="Olive" title="b3">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b4">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b4">g2</font> &amp; <font color="Olive" title="b4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> )
 ;<br/>

<a NAME="E3:26"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c1">x0</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E1:26"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="rfunct_2.html#T55" title="RFUNCT_2:th.55">RFUNCT_2:55</a></i>;<br/>
<a NAME="E4:26"/>
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#R4" title="RFUNCT_2:pred.4">is_non_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E1:26"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="rfunct_2.html#T56" title="RFUNCT_2:th.56">RFUNCT_2:56</a></i>;<br/>
<b>hence </b><a NAME="E5:26"/>
<font color="Maroon" title="c2">f</font> <a href="limfunc3.html#R2" title="LIMFUNC3:pred.2">is_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E1:26"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="limfunc3.html#E2:26"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="limfunc3.html#E3:26"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="limfunc3.html#T23" target="_self" title="LIMFUNC3:th.23">Th23</a></i>;<br/>


</div>
