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

<b>set </b><font color="Maroon">E</font> = <font color="Maroon">D</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span> ;<br/>
<a NAME="E1:81_1_1"/><i><font color="Green">E24</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E2:81_1_1"/>
<font color="Maroon">D</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:81_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1_1_1_1_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> or <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:81_1_1_1_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><a NAME="E2:81_1_1_1_1_1_1"/>
 <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T18" target="_self">Th18</a>, <a class="ref" href="zfmisc_1.html#T79">ZFMISC_1:79</a></i>;<br/><b>hence </b><a NAME="E3:81_1_1_1_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T19" target="_self">Th19</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:81_1_1_1_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:81_1_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 ;<br/><b>hence </b><a NAME="E3:81_1_1_1_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:81_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">E</font> = <font color="Maroon">D</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">XX</font> ;<br/>
<a NAME="E4:81_1_1"/>
<font color="Maroon">E</font> is    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1_1_2_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1"><b>case </b></a><a NAME="E1:81_1_1_2_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:81_1_1_2_1_1_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font> <a href="subset_1.html#K6">\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> implies  ex <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_1_1_2_1_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font> <a href="subset_1.html#K6">\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/>

<b>take </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span>
;<br/>

<b>thus </b><a NAME="E2:81_1_1_2_1_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>

<a NAME="E3:81_1_1_2_1_1_1_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font> &amp; not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:81_1_1_2_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div><b>given </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:81_1_1_2_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E3:81_1_1_2_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font><b> such that </b><br/><a NAME="E5:81_1_1_2_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E6:81_1_1_2_1_1_1"/><i><font color="Green">E37</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E7:81_1_1_2_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E8:81_1_1_2_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font> <a href="subset_1.html#K6">\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E3:81_1_1_2_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font> <a href="subset_1.html#K6">\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>thus </b> <a href="setfam_1.html#K5">union</a> <font color="Maroon">E</font> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span> </span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T96">ZFMISC_1:96</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font> <a href="subset_1.html#K6">\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span>
<b>by </b><i>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font>
<b>by </b><i><a class="ref" href="borsuk_1.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>
;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">XX</font>;<br/><b>assume </b><a NAME="E5:81_1_1_2_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1_1_2_1_1_3_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> or <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span>  )
 </span><b>by </b><i><a class="ref" href="borsuk_1.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:81_1_1_2_1_1_3_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:81_1_1_2_1_1_3_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:81_1_1_2_1_1_3_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:81_1_1_2_1_1_3_1_2"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 ;<br/><b>hence </b><a NAME="E3:81_1_1_2_1_1_3_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:81_1_1_2_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">XX</font>;<br/><b>assume </b><a NAME="E8:81_1_1_2_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2_1_1_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1_1_2_1_1_4_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> or <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span>  )
 </span><b>by </b><i><a class="ref" href="borsuk_1.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:81_1_1_2_1_1_4_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2_1_1_4_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1_4_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1_1_2_1_1_4_1_1_1_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> or <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1_4_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:81_1_1_2_1_1_4_1_1_1_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:81_1_1_2_1_1_4_1_1_1_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1_4_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:81_1_1_2_1_1_4_1_1_1_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:81_1_1_2_1_1_4_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> st <br/>( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 ;<br/><a NAME="E3:81_1_1_2_1_1_4_1_1_1_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/><b>hence </b><a NAME="E4:81_1_1_2_1_1_4_1_1_1_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:81_1_1_2_1_1_4_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:81_1_1_2_1_1_4_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:81_1_1_2_1_1_4_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
 ex <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 ;<br/><a NAME="E3:81_1_1_2_1_1_4_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_2_1_1_4_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1_4_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1_1_2_1_1_4_1_2_1_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> or <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1_4_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:81_1_1_2_1_1_4_1_2_1_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:81_1_1_2_1_1_4_1_2_1_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_1_4_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:81_1_1_2_1_1_4_1_2_1_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:81_1_1_2_1_1_4_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> st <br/>( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 ;<br/><b>hence </b><a NAME="E3:81_1_1_2_1_1_4_1_2_1_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T17">ZFMISC_1:17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:81_1_1_2_1_1_4_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:81_1_1_2_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_2_1_2"><b>case </b></a><a NAME="E1:81_1_1_2_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:81_1_1_2_1_2"/>
<font color="Maroon">E</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:81_1_1"/>
<font color="Maroon">D</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">p</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font> : not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> </span>}</span>  is  non <a href="xboole_0.html#V1">empty</a>  <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font>
 ;<br/>


</div>
