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

<b>let </b><font color="Maroon">T</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">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">T</font>;<br/>



<a NAME="E1:34"/><i><font color="Green">E42</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/>
<a NAME="E2:34"/><i><font color="Green">E43</font></i>: 
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> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:34"/><b>then </b><i><font color="Green">E44</font></i>: 
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> <a href="tarski.html#R1">c=</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>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:34"/><i><font color="Green">E47</font></i>: 
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="tarski.html#R1">c=</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="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a><br/>

<b>assume </b><a NAME="E1:34_1"/><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</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/>

<a NAME="E2:34_1"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</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> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</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> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">o1</font> = <font color="Maroon">x</font>, <font color="Maroon">o2</font> = <font color="Maroon">y</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/>
<a NAME="E4:34_1"/>
<font color="Maroon">o1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">o2</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">Y2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E6:34_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o2</font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E7:34_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">o1</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Y2</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">s2</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">t2</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i>, , </i>;<br/>
<a NAME="E10:34_1"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s2</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T47">PRE_TOPC:47</a></i>;<br/>
<a NAME="E11:34_1"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s2</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t2</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">o3</font> = <font color="Maroon">x</font>, <font color="Maroon">o4</font> = <font color="Maroon">y</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>, , , </i>;<br/>
<a NAME="E13:34_1"/>
<font color="Maroon">o3</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">o4</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E14:34_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</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>
<a NAME="E5:34"/>
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> <a href="toler_1.html#K1">|_2</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</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="E1:34_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</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">x</font>,<font color="Maroon">y</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> <a href="toler_1.html#K1">|_2</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> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<b>assume </b><a NAME="E2:34_2"/><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</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> <a href="toler_1.html#K1">|_2</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>
 ;<br/>

<a NAME="E3:34_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:34_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</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> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</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> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">o3</font> = <font color="Maroon">x</font>, <font color="Maroon">o4</font> = <font color="Maroon">y</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">s1</font> = <font color="Maroon">x</font>, <font color="Maroon">s2</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">x</font>, <font color="Maroon">t2</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">o1</font> = <font color="Maroon">x</font>, <font color="Maroon">o2</font> = <font color="Maroon">y</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>, , , </i>;<br/>
<a NAME="E9:34_2"/>
<font color="Maroon">o1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">o2</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="E11:34_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t2</font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E12:34_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">t1</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/>
<a NAME="E13:34_2"/><i><font color="Green">E75</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s2</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T47">PRE_TOPC:47</a></i>;<br/>
<a NAME="E14:34_2"/>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s2</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E15:34_2"/><b>then </b>
<font color="Maroon">o3</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">o4</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E16:34_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</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="E6:34"/>
 <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">S</font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">T</font>
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a>, <a class="ref" href="yellow_0.html#D14">YELLOW_0:def 14</a></i>;<br/>


</div>
