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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="midsp_1.html#NM1">MidSp</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>





<b>assume </b><a NAME="E1:37"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="midsp_1.html#R2" target="_self">##</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:37"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="midsp_1.html#K8" target="_self">~</a>  iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="midsp_1.html#K8" target="_self">~</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:37_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="midsp_1.html#K8" target="_self">~</a>  implies <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="midsp_1.html#K8" target="_self">~</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="midsp_1.html#K8" target="_self">~</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E3:37_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#R2" target="_self">##</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="midsp_1.html#E1:37_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="midsp_1.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E4:37_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#R2" target="_self">##</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="midsp_1.html#E1:37"><i><font color="Green">E24</font></i></a>, <a class="ref" href="midsp_1.html#T35" target="_self">Th35</a></i>;<br/>
<b>hence </b><a NAME="E5:37_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="midsp_1.html#K8" target="_self">~</a> 
 ;<br/>


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

<b>thus </b><a NAME="E2:37_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="midsp_1.html#K8" target="_self">~</a>  implies <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="midsp_1.html#K8" target="_self">~</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="midsp_1.html#K8" target="_self">~</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E3:37_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#R2" target="_self">##</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="midsp_1.html#E1:37_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="midsp_1.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E4:37_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#R2" target="_self">##</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="midsp_1.html#E1:37"><i><font color="Green">E24</font></i></a>, <a class="ref" href="midsp_1.html#T35" target="_self">Th35</a></i>;<br/>
<b>hence </b><a NAME="E5:37_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="midsp_1.html#K8" target="_self">~</a> 
 ;<br/>


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

<b>thus </b><a NAME="E3:37_1"/>
verum
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:37"/>
<font color="Maroon">c<sub>2</sub></font> <a href="midsp_1.html#K8" target="_self">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <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>
