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<b>let </b><font color="Maroon">l1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="qc_lang1.html#K1">QC-variables</a> ;<br/><b>let </b><font color="Maroon">l2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="qc_lang1.html#K1">QC-variables</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:9_1_2"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font>
 <b>and </b><br/><a NAME="E2:9_1_2"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> holds <br/>( ( <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> implies <font color="Maroon">l1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> ) &amp; ( not <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> implies <font color="Maroon">l1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> ) )
 <b>and </b><br/><a NAME="E3:9_1_2"/><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">l2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font>
 <b>and </b><br/><a NAME="E4:9_1_2"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> holds <br/>( ( <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> implies <font color="Maroon">l2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span> ) &amp; ( not <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> implies <font color="Maroon">l2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1"><b>now </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="E1:9_1_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> )
 ;<br/><a NAME="E2:9_1_2_1"/><b>then </b>
( ( <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> implies <font color="Maroon">l1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> ) &amp; ( not <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> implies <font color="Maroon">l1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> ) &amp; ( <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> implies <font color="Maroon">l2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> ) &amp; ( not <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> implies <font color="Maroon">l2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:9_1_2_1"/>
<font color="Maroon">l1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:9_1_2"/>
<font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


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