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<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>now </b></a><div class="add"><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/><a NAME="E1:44_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K1" target="_self">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Nseq</font> <a href="seqm_3.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:44_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K1" target="_self">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_2"><b>now </b></a><div class="add"><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/><a NAME="E1:44_1_1_2"/>
<font color="Maroon">Nseq</font> <a href="seqm_3.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">Nseq</font> <a href="seqm_3.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E2:44_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K1" target="_self">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">Nseq</font> <a href="seqm_3.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:44_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K1" target="_self">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">Nseq</font> <a href="seqm_3.html#K1" target="_self">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E3:44_1_1"/><b>then </b>
<font color="Maroon">Nseq</font> <a href="seqm_3.html#K1" target="_self">^\</a> <font color="Maroon">k</font> is <a href="seqm_3.html#V1" target="_self">increasing</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:44_1_1"/>
( <font color="Maroon">Nseq</font> <a href="seqm_3.html#K1" target="_self">^\</a> <font color="Maroon">k</font> is <a href="seqm_3.html#V1" target="_self">increasing</a> &amp; <font color="Maroon">Nseq</font> <a href="seqm_3.html#K1" target="_self">^\</a> <font color="Maroon">k</font> is <a href="seqm_3.html#V6" target="_self">natural-yielding</a> )
 <b>by </b><i>, <a class="txt" href="seqm_3.html#E2"><i><font color="Green">Lemma35</font></i></a></i>;<br/>


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