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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5" target="_self">:]</a></span>;<br/>





<a NAME="E1:64"/><i><font color="Green">E24</font></i>: 
(  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
;<br/>
<b>assume </b><a NAME="E2:64"/>
<font color="Maroon">D</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>then consider </b><font color="Maroon">H</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5" target="_self">:]</a></span><b> such that </b><br/><a NAME="E4:64"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E5:64"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> holds <br/> ex <font color="Olive">X1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y1</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">K</font> = <font color="Maroon">H</font> as   <a href="setfam_1.html#NM1">Subset-Family</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">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">X1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">Y1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/>



<b>thus </b><a NAME="E7:64"/>
( <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font> implies <font color="Maroon">X1</font> is <a href="pre_topc.html#V3">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">p</font> =  <a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>;<br/>
<b>assume </b><a NAME="E1:64_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font>
 ;<br/>

<b>set </b><font color="Maroon">P</font> =  <a href="funct_3.html#K2">.:</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">PK</font> = <span class="p1">(<span class="default"><a href="funct_3.html#K2">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">PK</font> = <font color="Maroon">PK</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E4:64_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">PK</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">X1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/>
<b>assume </b><a NAME="E2:64_1_1"/>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PK</font>
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being   <a href="subset_1.html#NM2">Subset</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">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E4:64_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E5:64_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K2">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">X2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">Y2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E7:64_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">Y2</font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> &amp; <font color="Maroon">X2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Y2</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:64_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K2">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <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">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:64_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T17" target="_self">Th17</a>, <a class="ref" href="funct_3.html#T8">FUNCT_3:8</a></i>;<br/>
<a NAME="E10:64_1_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
;<br/>
<a NAME="E11:64_1_1"/><b>then </b>
( <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X2</font> or <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, , , <a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>;<br/>
<b>hence </b><a NAME="E12:64_1_1"/>
<font color="Maroon">X1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:64_1"/>
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K2">.:</a> <span class="p3">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T16" target="_self">Th16</a>, </i>;<br/>
<b>hence </b><a NAME="E6:64_1"/>
<font color="Maroon">X1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/>


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

<b>set </b><font color="Maroon">p</font> =  <a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>;<br/>
<b>assume </b><a NAME="E8:64"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font>
 ;<br/>

<b>set </b><font color="Maroon">P</font> =  <a href="funct_3.html#K2">.:</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">PK</font> = <span class="p1">(<span class="default"><a href="funct_3.html#K2">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/>
<b>reconsider </b><font color="Maroon">PK</font> = <font color="Maroon">PK</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E11:64"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">PK</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">Y1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/>
<b>assume </b><a NAME="E2:64_2"/>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PK</font>
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being   <a href="subset_1.html#NM2">Subset</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">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E4:64_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E5:64_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K2">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">X2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">Y2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E7:64_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">Y2</font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> &amp; <font color="Maroon">X2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Y2</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:64_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K2">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <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">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:64_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T17" target="_self">Th17</a>, <a class="ref" href="funct_3.html#T8">FUNCT_3:8</a></i>;<br/>
<a NAME="E10:64_2"/>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
;<br/>
<a NAME="E11:64_2"/><b>then </b>
( <font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y2</font> or <font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, , , <a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>;<br/>
<b>hence </b><a NAME="E12:64_2"/>
<font color="Maroon">Y1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:64"/>
<font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K2">.:</a> <span class="p3">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T16" target="_self">Th16</a>, </i>;<br/>
<b>hence </b><a NAME="E13:64"/>
<font color="Maroon">Y1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/>


</div>
