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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#K5">NAT</a> , <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:28_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">f</font> is <a href="seqm_3.html#V1" target="_self">increasing</a>
 ;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="seqm_3.html#D3" target="_self">SEQM_3:def 3</a><br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:28_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 <b>and </b><br/><a NAME="E3:28_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="seqm_3.html#E1"><i><font color="Green">Lemma30</font></i></a>, <a class="txt" href="seqm_3.html#E1:28_1_1"><i><font color="Green">E31</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:28_1_1_1_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:28_1_1_1_2"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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