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<div class="add">

<b>let </b><font color="Maroon">IT</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E1:15_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:15_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ll</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#D5">CQC_LANG:def 5</a></i>;<br/>
<a NAME="E3:15_1_2"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">ll</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D8">QC_LANG1:def 8</a></i>;<br/>
<b>thus </b><a NAME="E4:15_1_2"/>
( <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">ll</font> implies (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">i</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">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">ll</font>
 ;<br/>

<a NAME="E2:15_1_2_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ll</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E4:15_1_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 ;<br/>

<a NAME="E5:15_1_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E6:15_1_2_1"/>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E5:15_1_2"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E6:15_1_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">i</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">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>
 ;<br/>

<a NAME="E7:15_1_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ll</font> &amp; <font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">v</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_2"><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>thus </b><a NAME="E1:15_1_2_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> implies ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ll</font> &amp; <font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">v</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 ;<br/>

<b>hence </b><a NAME="E2:15_1_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ll</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>

<a NAME="E3:15_1_2_2_1"/><b>then </b>
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ll</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:15_1_2_2_1"/>
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:15_1_2_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ll</font> &amp; <font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">v</font> )
 ;<br/>

<b>hence </b><a NAME="E3:15_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:15_1_2"/>
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="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> holds <br/><font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_3"><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="E1:15_1_2_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i</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="E3:15_1_2_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E4:15_1_2_3"/>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:15_1_2"/>
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">ll</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>


</div>
