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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L2">ZeroStr</a> ;<br/><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><a NAME="E1:24"/><i><font color="Green">E26</font></i>: 
(  <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> &amp; ( 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> ) )
 ;<br/>

<a NAME="E2:24"/><i><font color="Green">E27</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:24"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:24_1"/>
<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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:24_1_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> implies <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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:24_1_1_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &amp; <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:24_1_1_1"/>
<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>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:24_1_1"/>
<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/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:24_1"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:24"/>
<font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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