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

<b>consider </b><font color="Maroon">u</font> being    <a href="midsp_1.html#M1" target="_self">Vector</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E2:42_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> iff <font color="Olive">p</font> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="midsp_1.html#K6" target="_self">`2</a>  )
 <b>by </b><i/>;<br/>
<a NAME="E3:42_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Olive">p</font> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="midsp_1.html#K6" target="_self">`2</a>  </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:42_1_1_1"/>
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> <font color="Maroon">u</font> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Olive">p</font> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="midsp_1.html#K6" target="_self">`2</a>  </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_1"><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="E1:42_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Olive">p</font> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="midsp_1.html#K6" target="_self">`2</a>  </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E3:42_1_1_1_1_1"/>
<font color="Maroon">r</font> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="midsp_1.html#K6" target="_self">`2</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:42_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Olive">p</font> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="midsp_1.html#K6" target="_self">`2</a>  </span>}</span> 
 ;<br/>


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

<b>thus </b><a NAME="E2:42_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Olive">p</font> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="midsp_1.html#K6" target="_self">`2</a>  </span>}</span>  implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Olive">p</font> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="midsp_1.html#K6" target="_self">`2</a>  </span>}</span> 
 ;<br/>

<a NAME="E2:42_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="midsp_1.html#K6" target="_self">`2</a>  )
 
;<br/>
<b>hence </b><a NAME="E3:42_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font>
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:42_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Olive">p</font> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="midsp_1.html#K6" target="_self">`2</a>  </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:42_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span> : <font color="Olive">p</font> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="midsp_1.html#K6" target="_self">`2</a>  </span>}</span>  is    <a href="midsp_1.html#M1" target="_self">Vector</a> of <font color="Maroon">M</font>
 ;<br/>


</div>
