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

<b>let </b><font color="Maroon">N</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_2.html#V2">meet-continuous</a> <a href="waybel19.html#V2">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>assume </b><a NAME="E1:39"/><i><font color="Green">E32</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">N</font></span>)</span> is <a href="waybel_3.html#V3">continuous</a>
 ;<br/>

<a NAME="E2:39"/><i><font color="Green">E34</font></i>: 
<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">N</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> 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">N</font>,<font color="Maroon">N</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">N</font> is <a href="compts_1.html#V3">being_T2</a>
 ;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>assume </b><a NAME="E2:39_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font>
 ;<br/><b>set </b><font color="Maroon">INF</font> =  <a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">N</font>;<br/><b>set </b><font color="Maroon">f</font> = <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><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">N</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">N</font></span>)</span></span><a href="funct_3.html#K13">:&gt;</a></span>;<br/><a NAME="E3:39_1"/><i><font color="Green">E37</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="yellow_3.html#K3">:]</a></span> <a href="hidden.html#R1">=</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">N</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">f</font> = <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><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">N</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">N</font></span>)</span></span><a href="funct_3.html#K13">:&gt;</a></span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i>, <a class="ref" href="funct_3.html#T78">FUNCT_3:78</a></i>;<br/><a NAME="E5:39_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/>

<a NAME="E1:39_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K4">dom</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">N</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</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">N</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</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="E2:39_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</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">N</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</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="E3:39_1_1"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</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">N</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</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><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7">]</a></span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><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">N</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">N</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="waybel30.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_3.html#T69">FUNCT_3:69</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">N</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
<b>by </b><i><a class="ref" href="waybel_2.html#D4">WAYBEL_2:def 4</a></i>
;<br/><br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">D</font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i/>;<br/><a NAME="E7:39_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">D</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="yellow12.html#T46">YELLOW12:46</a></i>;<br/><b>reconsider </b><font color="Maroon">INF</font> =  <a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">N</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">N</font> <b>by </b><i>, </i>;<br/><a NAME="E9:39_1"/>
<font color="Maroon">N</font> is <a href="yellow13.html#V2">topological_semilattice</a>
 <b>by </b><i>, </i>;<br/><a NAME="E10:39_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">INF</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="yellow13.html#D5">YELLOW13:def 5</a></i>;<br/><a NAME="E11:39_1"/>
 <a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="yellow12.html#T39">YELLOW12:39</a></i>;<br/><a NAME="E12:39_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="yellow12.html#T41">YELLOW12:41</a></i>;<br/><a NAME="E13:39_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2"><b>proof </b></a><div class="add">

<div><b>hereby </b> <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>
<div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_2_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font>, <font color="Maroon">t</font> = <font color="Maroon">t</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/><a NAME="E5:39_1_2_1"/>
<font color="Maroon">s</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E6:39_1_2_1"/><b>then </b>
<font color="Maroon">s</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>;<br/><a NAME="E7:39_1_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">s</font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E8:39_1_2_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/><a NAME="E9:39_1_2_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 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">N</font>,<font color="Maroon">N</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/><a NAME="E10:39_1_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E11:39_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E1:39"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">a</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="E2:39_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E3:39_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:39_1_2"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font>, <font color="Maroon">t</font> = <font color="Maroon">t</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i><a class="txt" href="waybel30.html#E1:39"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E7:39_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">t</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i>, <a class="txt" href="waybel30.html#E1:39"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E8:39_1_2"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<a NAME="E9:39_1_2"/><b>then </b>
<font color="Maroon">s</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>;<br/>
<b>hence </b><a NAME="E10:39_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="waybel30.html#E1:39"><i><font color="Green">E32</font></i></a>, <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="E14:39_1"/>
<font color="Maroon">X</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="waybel30.html#T1" target="_self">Th1</a>, , <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:39"/><i><font color="Green">E122</font></i>: 
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="borsuk_1.html#K5">:]</a></span>  st <font color="Olive">X</font> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font> holds <br/><font color="Olive">X</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E5:39"/><i><font color="Green">E123</font></i>: 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font> <a href="subset_1.html#K9">/\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="lattice3.html#K7">~</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:39_2"/>
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font> <a href="subset_1.html#K9">/\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="lattice3.html#K7">~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="yellow12.html#T22">YELLOW12:22</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:39_2"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font> <a href="subset_1.html#K9">/\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="lattice3.html#K7">~</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="yellow12.html#T23">YELLOW12:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:39"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="borsuk_1.html#K5">:]</a></span>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> holds <br/><font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>assume </b><a NAME="E1:39_3"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">X</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font>, <font color="Maroon">Y</font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="lattice3.html#K7">~</a> </span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font>, <font color="Maroon">Y</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <span class="p1"><a href="funct_3.html#K14">&lt;:</a><span class="default"><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">N</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font></span>)</span>,<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">N</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font></span>)</span></span><a href="funct_3.html#K14">:&gt;</a></span> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">N</font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="yellow12.html#T42">YELLOW12:42</a></i>;<br/>
<a NAME="E5:39_3"/><i><font color="Green">E125</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</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">N</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow12.html#T4">YELLOW12:4</a></i>;<br/>
<a NAME="E6:39_3"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">X</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E7:39_3"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:39_3_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_3_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_3_1_1"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:39_3_1_1"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:39_3_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">x1</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:39_3_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/>


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

<b>let </b><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:39_3_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:39_3_1"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E5:39_3_1"/><i><font color="Green">E134</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y2</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/>
<a NAME="E6:39_3_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y2</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:39_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:39_3"/>
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="yellow12.html#T43">YELLOW12:43</a></i>;<br/>
<a NAME="E9:39_3"/><b>then </b>
<font color="Maroon">Y</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="waybel30.html#T1" target="_self">Th1</a>, , <a class="ref" href="tops_2.html#T72">TOPS_2:72</a></i>;<br/>
<b>hence </b><a NAME="E10:39_3"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, , <a class="ref" href="waybel30.html#T1" target="_self">Th1</a>, <a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:39"/>
<font color="Maroon">N</font> is <a href="compts_1.html#V3">being_T2</a>
 <b>by </b><i><a class="ref" href="yellow12.html#T46">YELLOW12:46</a></i>;<br/>


</div>
