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

<b>let </b><font color="Maroon">M</font> be   <a href="midsp_1.html#NM1">MidSp</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/>



<a NAME="E1:43"/>
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>  <a href="midsp_1.html#K9" target="_self">ID</a> <font color="Maroon">M</font> iff <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="midsp_1.html#K8" target="_self">~</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <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/>

<b>thus </b><a NAME="E1:43_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="midsp_1.html#K9" target="_self">ID</a> <font color="Maroon">M</font> implies <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="midsp_1.html#K8" target="_self">~</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="midsp_1.html#K9" target="_self">ID</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:43_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</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">q</font> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="midsp_1.html#K6" target="_self">`2</a>  </span>}</span> 
 
;<br/>
<a NAME="E3:43_1_1"/><b>then </b>
 ex <font color="Olive">q</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">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="midsp_1.html#K6" target="_self">`2</a>  )
 
;<br/>
<a NAME="E4:43_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">p</font> <a href="midsp_1.html#K5" target="_self">`1</a> ,<font color="Maroon">p</font> <a href="midsp_1.html#K6" target="_self">`2</a>  <a href="midsp_1.html#R1" target="_self">@@</a> <font color="Maroon">b</font>,<font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:43_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="midsp_1.html#K6" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E6:43_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="midsp_1.html#R2" target="_self">##</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:43_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="midsp_1.html#K8" target="_self">~</a> 
 ;<br/>


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

<b>thus </b><a NAME="E2:43_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="midsp_1.html#K8" target="_self">~</a>  implies <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="midsp_1.html#K9" target="_self">ID</a> <font color="Maroon">M</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="midsp_1.html#K8" target="_self">~</a> 
 ;<br/>

<a NAME="E2:43_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">p</font> <a href="midsp_1.html#R2" target="_self">##</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:43_1_2"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="midsp_1.html#K6" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E4:43_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="midsp_1.html#K5" target="_self">`1</a> ,<font color="Maroon">p</font> <a href="midsp_1.html#K6" target="_self">`2</a>  <a href="midsp_1.html#R1" target="_self">@@</a> <font color="Maroon">b</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:43_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="midsp_1.html#K6" target="_self">`2</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:43_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</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">q</font> <a href="midsp_1.html#K5" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="midsp_1.html#K6" target="_self">`2</a>  </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E7:43_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="midsp_1.html#K9" target="_self">ID</a> <font color="Maroon">M</font>
 ;<br/>


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

<b>thus </b><a NAME="E3:43_1"/>
verum
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:43"/><b>then </b>
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="midsp_1.html#K9" target="_self">ID</a> <font color="Maroon">M</font> iff <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="midsp_1.html#K8" target="_self">~</a>  )
 
;<br/>
<b>hence </b><a NAME="E3:43"/>
 <a href="midsp_1.html#K9" target="_self">ID</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="midsp_1.html#K8" target="_self">~</a> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
