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

<b>let </b><font color="Maroon">S</font> be   <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:29"/><i><font color="Green">E42</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> #)
 ;<br/>

<i><font color="Green">E43</font></i>:  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">S</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">S</font></span>)</span> #) = 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #)
<b>by </b><i/>
<br/>.= 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">T</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">T</font></span>)</span> #)
<b>by </b><i>, </i>
;<br/>
<a NAME="E3:29"/>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">T</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a><br/>

<b>thus </b><a NAME="E1:29_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">S</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">T</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_2_1"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">S</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">a</font>, <font color="Maroon">s2</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">S</font></span>)</span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">s1</font>, <font color="Maroon">t2</font> = <font color="Maroon">s2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">T</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E4:29_2_1"/>
<font color="Maroon">s1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E6:29_2_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s2</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E8:29_2_1"/>
( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">t2</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">t1</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Z</font> )
 
<b>by </b><i>, , <a class="ref" href="tops_3.html#T80">TOPS_3:80</a></i>;<br/>
<a NAME="E9:29_2_1"/><b>then </b>
<font color="Maroon">t1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">t2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:29_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


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

<b>assume </b><a NAME="E2:29_2"/><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">T</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">a</font>, <font color="Maroon">s2</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">T</font></span>)</span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">s1</font>, <font color="Maroon">t2</font> = <font color="Maroon">s2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">S</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E5:29_2"/>
<font color="Maroon">s1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:29_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s2</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E9:29_2"/>
( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">t2</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">t1</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Z</font> )
 
<b>by </b><i>, , <a class="ref" href="tops_3.html#T80">TOPS_3:80</a></i>;<br/>
<a NAME="E10:29_2"/><b>then </b>
<font color="Maroon">t1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">t2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:29_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:29"/>
 <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">T</font>
 <b>by </b><i/>;<br/>


</div>
