<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a> ;<br/>





<b>thus </b><a NAME="E1:17"/>
( <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  implies <span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_4.html#K2">*&gt;</a></span>,<font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>

<b>hence </b><span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_4.html#K2">*&gt;</a></span>,<font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><br/>


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

<a NAME="E2:17"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:17"/><b>then </b><i><font color="Green">E31</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T29">FINSEQ_3:29</a></i>;<br/>
<b>thus </b><a NAME="E4:17"/>
( <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  implies <span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_4.html#K2">*&gt;</a></span>,<font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_2"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/>

<b>hence </b><span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_4.html#K2">*&gt;</a></span>,<font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><br/>


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

<b>thus </b><a NAME="E5:17"/>
( <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  implies <span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_4.html#K2">*&gt;</a></span>,<font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_3"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>

<b>hence </b><span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_4.html#K2">*&gt;</a></span>,<font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> 2 = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><br/>


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

<a NAME="E6:17"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:17"/><b>then </b><i><font color="Green">E32</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T29">FINSEQ_3:29</a></i>;<br/>
<b>assume </b><a NAME="E8:17"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/>

<b>hence </b><span class="p1">(<span class="default"><a href="yellow15.html#K1" target="_self">MergeSequence</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_4.html#K2">*&gt;</a></span>,<font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> 2 = 
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><br/>


</div>
