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<div class="add">

<b>thus </b><a NAME="E1:25_1_2"/>
for <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">G</font>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">g1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Olive">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="group_1.html#K3">"</a>  ) &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">g2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Olive">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font></span>)</span> <a href="group_1.html#K3">"</a>  ) holds <br/><font color="Olive">g1</font> <a href="hidden.html#R1">=</a> <font color="Olive">g2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g1</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">G</font>, <font color="Maroon">g2</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">G</font>;<br/>

<b>assume </b><a NAME="E1:25_1_2_1"/><i><font color="Green">E47</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="group_1.html#K3">"</a>  ) &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font></span>)</span> <a href="group_1.html#K3">"</a>  ) )
 ;<br/>

<a NAME="E2:25_1_2_1"/><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:25_1_2_1"/><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E4:25_1_2_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">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">g1</font> holds <br/><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:25_1_2_1_1"/><i><font color="Green">E54</font></i>: 
( 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">g1</font> )
 ;<br/>

<a NAME="E2:25_1_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font></span>)</span>
 
;<br/>
<a NAME="E3:25_1_2_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:25_1_2_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:25_1_2_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:25_1_2_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, </i>;<br/>
<a NAME="E7:25_1_2_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, </i>;<br/>
<a NAME="E8:25_1_2_1_1"/>
<font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K3">"</a> 
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E9:25_1_2_1_1"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:25_1_2_1"/>
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


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


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