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<a NAME="E1:76"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">bq</font> is <a href="seqm_3.html#V1">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<a NAME="E1:76_1"/>
for <font color="Olive">n</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">bq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">bq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>proof </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/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:76_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:76_1_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E3:76_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E4:76_1_1"/><b>then </b>
<font color="Maroon">bq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:76_1_1"/>
<font color="Maroon">bq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">bq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:76_1"/>
<font color="Maroon">bq</font> is <a href="seqm_3.html#V1">increasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#T7">SEQM_3:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:76"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">bq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2"><b>proof </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="E1:76_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<b>hence </b><a NAME="E2:76_2"/>
<font color="Maroon">bq</font> <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> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:76"/>
<font color="Maroon">bq</font> is  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a>
 <b>by </b><i>, <a class="ref" href="seqm_3.html#T29">SEQM_3:29</a></i>;<br/>


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