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<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="midsp_1.html#NM1" title="MIDSP_1:NM.1">MidSp</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>





<b>assume </b><a NAME="E1:34"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="midsp_1.html#R2" title="MIDSP_1:pred.2">##</a> <font color="Maroon" title="c3">q</font>
 ;<br/>

<a NAME="E2:34"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">p</font> <a href="midsp_1.html#K6" title="MIDSP_1:func.6">~</a>  iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">q</font> <a href="midsp_1.html#K6" title="MIDSP_1:func.6">~</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:34_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">p</font> <a href="midsp_1.html#K6" title="MIDSP_1:func.6">~</a>  implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">q</font> <a href="midsp_1.html#K6" title="MIDSP_1:func.6">~</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1_1"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">p</font> <a href="midsp_1.html#K6" title="MIDSP_1:func.6">~</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">r</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/>
<a NAME="E3:34_1_1"/>
<font color="Maroon" title="c5">r</font> <a href="midsp_1.html#R2" title="MIDSP_1:pred.2">##</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="midsp_1.html#E1:34_1_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="midsp_1.html#T41" target="_self" title="MIDSP_1:th.41">Th41</a></i>;<br/>
<a NAME="E4:34_1_1"/><b>then </b>
<font color="Maroon" title="c5">r</font> <a href="midsp_1.html#R2" title="MIDSP_1:pred.2">##</a> <font color="Maroon" title="c3">q</font>
 
<b>by </b><i><a class="txt" href="midsp_1.html#E1:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="midsp_1.html#T35" target="_self" title="MIDSP_1:th.35">Th35</a></i>;<br/>
<b>hence </b><a NAME="E5:34_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">q</font> <a href="midsp_1.html#K6" title="MIDSP_1:func.6">~</a> 
 ;<br/>


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

<b>thus </b><a NAME="E2:34_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">q</font> <a href="midsp_1.html#K6" title="MIDSP_1:func.6">~</a>  implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">p</font> <a href="midsp_1.html#K6" title="MIDSP_1:func.6">~</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1_2"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">q</font> <a href="midsp_1.html#K6" title="MIDSP_1:func.6">~</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">r</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/>
<a NAME="E3:34_1_2"/>
<font color="Maroon" title="c5">r</font> <a href="midsp_1.html#R2" title="MIDSP_1:pred.2">##</a> <font color="Maroon" title="c3">q</font>
 
<b>by </b><i><a class="txt" href="midsp_1.html#E1:34_1_2"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="midsp_1.html#T41" target="_self" title="MIDSP_1:th.41">Th41</a></i>;<br/>
<a NAME="E4:34_1_2"/><b>then </b>
<font color="Maroon" title="c5">r</font> <a href="midsp_1.html#R2" title="MIDSP_1:pred.2">##</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="midsp_1.html#E1:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="midsp_1.html#T35" target="_self" title="MIDSP_1:th.35">Th35</a></i>;<br/>
<b>hence </b><a NAME="E5:34_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">p</font> <a href="midsp_1.html#K6" title="MIDSP_1:func.6">~</a> 
 ;<br/>


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

<b>thus </b><a NAME="E3:34_1"/>
verum
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:34"/>
<font color="Maroon" title="c2">p</font> <a href="midsp_1.html#K6" title="MIDSP_1:func.6">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="midsp_1.html#K6" title="MIDSP_1:func.6">~</a> 
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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