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<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">m</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">g</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/>







<div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:3_1"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E2:3_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">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">g</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E3:3_1"/>
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font></span>)</span> &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 <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>proof </b></a><div class="add">

<b>thus </b> <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> = 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T68">FINSEQ_2:68</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="E2:3_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E3:3_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">a</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_2.html#T71">FINSEQ_2:71</a></i>
;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:3_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E3:3"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T68">FINSEQ_2:68</a></i>;<br/>
<b>hence </b><a NAME="E4:3"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">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">g</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a>, <a class="ref" href="finseq_2.html#T71">FINSEQ_2:71</a></i>;<br/>


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