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<b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:55"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">seq</font> is <a href="seqm_3.html#V4">non-increasing</a> &amp; not <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a> )
 ;<br/>

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="limfunc1.html#D5" target="_self">LIMFUNC1:def 5</a><br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:55"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:55"><i><font color="Green">E34</font></i></a>, <a class="ref" href="seq_2.html#D4">SEQ_2:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<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> ;<br/>

<b>assume </b><a NAME="E4:55"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E5:55"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="limfunc1.html#E1:55"><i><font color="Green">E34</font></i></a>, <a class="ref" href="seqm_3.html#T14">SEQM_3:14</a></i>;<br/>
<a NAME="E6:55"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="limfunc1.html#E3"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:55"/>
<font color="Maroon">seq</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">r</font>
 <b>by </b><i/>;<br/>


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