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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V2">TopSpace-like</a>  <a href="waybel_9.html#L1">TopRelStr</a> ;<br/>

<b>assume </b><a NAME="E1:64_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="orders_2.html#V2">reflexive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="realset2.html#V3">trivial</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V2">TopSpace-like</a> <a href="realset2.html#V3">trivial</a> <a href="orders_2.html#V2">reflexive</a>  <a href="waybel_9.html#L1">TopRelStr</a>  ;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:64_1_1"/><i><font color="Green">E24</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tex_1.html#D1">TEX_1:def 1</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow13.html#D5" target="_self">YELLOW13:def 5</a><br/>

<b>assume </b><a NAME="E5:64_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R1">=</a>  <a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:64_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_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="E1:64_1_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="yellow13.html#E1:64_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="tdlat_3.html#T21">TDLAT_3:21</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:64_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T171">RELAT_1:171</a></i>;<br/><b>hence </b><a NAME="E3:64_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><i><font color="Green">E28</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<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">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>
;<br/><a NAME="E3:64_1_1_1_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K12">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span><a href="yellow_3.html#K12">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:64_1_1_1_1_2_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K12">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span><a href="yellow_3.html#K12">:]</a></span>
 <b>by </b><i><a class="txt" href="yellow13.html#E4:64_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yellow13.html#E2:64_1_1_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></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="E2:64_1_1_1_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K12">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span><a href="yellow_3.html#K12">:]</a></span>
 ;<br/>

<a NAME="E3:64_1_1_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="yellow_3.html#K7">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T35">ZFMISC_1:35</a></i>;<br/>
<a NAME="E4:64_1_1_1_1_2_2"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green">E31</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="yellow_3.html#K7">]</a></span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="yellow13.html#E5:64_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="waybel_2.html#D4">WAYBEL_2:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
;<br/>
<a NAME="E6:64_1_1_1_1_2_2"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</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">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E7:64_1_1_1_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow13.html#E1:64_1_1_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="yellow13.html#E2:64_1_1_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow13.html#E4:64_1_1_1_1_2_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="yellow13.html#E5:64_1_1_1_1_2_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div> <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K12">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span><a href="yellow_3.html#K12">:]</a></span>
<b>by </b><i><a class="txt" href="yellow13.html#E4:64_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>
;<br/><b>hence </b><a NAME="E5:64_1_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="yellow13.html#E3:64_1_1_1_1_2"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:64_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>


</div>
