<?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">H</font> be   <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>;<br/>





<b>assume </b><a NAME="E1:65"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">H</font> is <a href="tops_2.html#V1">open</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">P1</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K11" target="_self">Pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">P2</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K12" target="_self">Pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E4:65"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">P1</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_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>assume </b><a NAME="E1:65_1"/>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">D</font> being   <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><b> such that </b><br/><a NAME="E3:65_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; <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> )
 <b>by </b><i/>;<br/>
<a NAME="E4:65_1"/>
<font color="Maroon">D</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T16" target="_self">Th16</a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:65_1"/>
<font color="Maroon">X1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T16" target="_self">Th16</a>, <a class="ref" href="borsuk_1.html#T14" target="_self">Th14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:65"/>
<font color="Maroon">P2</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_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>assume </b><a NAME="E1:65_2"/>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">D</font> being   <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><b> such that </b><br/><a NAME="E3:65_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; <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> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E4:65_2"/>
<font color="Maroon">D</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T16" target="_self">Th16</a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:65_2"/>
<font color="Maroon">Y1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T16" target="_self">Th16</a>, <a class="ref" href="borsuk_1.html#T14" target="_self">Th14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:65"/>
( <span class="p1">(<span class="default"><a href="borsuk_1.html#K11" target="_self">Pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> is <a href="tops_2.html#V1">open</a> &amp; <span class="p1">(<span class="default"><a href="borsuk_1.html#K12" target="_self">Pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> is <a href="tops_2.html#V1">open</a> )
 <b>by </b><i/>;<br/>


</div>
