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<b>let </b><font color="Maroon">p</font> be   <a href="algseq_1.html#NM1">AlgSequence</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="algseq_1.html#NM1">AlgSequence</a> of <font color="Maroon">R</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:26_1_2"/><i><font color="Green">E27</font></i>: 
(  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>and </b><br/><a NAME="E2:26_1_2"/><i><font color="Green">E28</font></i>: 
(  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 ;<br/>

<a NAME="E3:26_1_2"/><i><font color="Green">E29</font></i>: 
1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 ;<br/><a NAME="E2:26_1_2_1"/><b>then </b>
(  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp;  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 <b>by </b><i>, , , </i>;<br/><a NAME="E3:26_1_2_1"/><b>then </b>
(  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:26_1_2_1"/><b>then </b>
(  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>hence </b><a NAME="E5:26_1_2_1"/>
 <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 ;<br/><a NAME="E2:26_1_2_2"/><b>then </b>
(  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i>, , </i>;<br/><a NAME="E3:26_1_2_2"/><b>then </b>
(  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><b>hence </b><a NAME="E4:26_1_2_2"/>
 <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E6:26_1_2"/>
for <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">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2_3"><b>proof </b></a><div class="add">

<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="E1:26_1_2_3"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:26_1_2_3"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:26_1_2_3"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:26_1_2_3"/>
<font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:26_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i>, , </i>;<br/>


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