<?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  <a href="waybel25.html#V1">monotone-convergence</a> <a href="t_0topsp.html#NM1">T_0-TopSpace</a>;<br/>



<b>assume </b><a NAME="E1:40"/><i><font color="Green">E45</font></i>: 
(  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> is <a href="lattice3.html#V3">complete</a> &amp;  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> is <a href="waybel_3.html#V3">continuous</a> )
 ;<br/>

<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>
<b>consider </b><font color="Maroon">F</font> being  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="waybel_1.html#V6">projection</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span><b> such that </b><br/><a NAME="E4:40"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">f</font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> holds  <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E5:40"/>
 ex <font color="Olive">h</font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Olive">h</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">X</font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a>  <a href="waybel26.html#K4" target="_self">oContMaps</a> <font color="Olive">h</font>,<font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:40"/>
 <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> is  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="orders_2.html#NM1">Poset</a>
 
<b>by </b><i/>;<br/>
<a NAME="E7:40"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">F</font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i><a class="ref" href="waybel15.html#T17">WAYBEL15:17</a>, <a class="ref" href="yellow_2.html#T37">YELLOW_2:37</a></i>;<br/>
<a NAME="E8:40"/>
 <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="waybel24.html#D3">WAYBEL24:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">imF</font> =  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">F</font> as  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="yellow16.html#T28">YELLOW16:28</a></i>;<br/>
<a NAME="E10:40"/><i><font color="Green">E60</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">imF</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E11:40"/><i><font color="Green">E62</font></i>: 
 <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">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/>
<a NAME="E12:40"/><i><font color="Green">E64</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">Omega</a> <font color="Maroon">Y</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span> #) <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">Y</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> #)
 
<b>by </b><i><a class="ref" href="waybel25.html#D2">WAYBEL25:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:40_1_1"/>
<font color="Maroon">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">imF</font>
 ;<br/><b>then consider </b><font color="Maroon">h</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">h</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">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <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">Omega</a> <font color="Maroon">Y</font></span>)</span> <b>by </b><i/>;<br/><font color="Maroon">a</font> = 
<font color="Maroon">X</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">x</font>

;<br/><b>hence </b><a NAME="E7:40_1_1"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Olive">x</font>
 ;<br/></div>
<b>end;</b></div><b>given </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span><b> such that </b><a NAME="E2:40_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E3:40_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E4:40_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">a</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">a</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/><a NAME="E6:40_1"/>
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">X</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#R2">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E7:40_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:40_1"/>
<font color="Maroon">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">imF</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:40"/><b>then </b>
 <a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font>,<font color="Maroon">imF</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="yellow16.html#T52">YELLOW16:52</a></i>;<br/>
<a NAME="E15:40"/><b>then </b>
<font color="Maroon">imF</font>, <a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="waybel_1.html#T7">WAYBEL_1:7</a></i>;<br/>
<b>hence </b><a NAME="E16:40"/>
(  <a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font> is <a href="lattice3.html#V3">complete</a> &amp;  <a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font> is <a href="waybel_3.html#V3">continuous</a> )
 <b>by </b><i>, <a class="ref" href="waybel15.html#T11">WAYBEL15:11</a>, <a class="ref" href="waybel20.html#T18">WAYBEL20:18</a></i>;<br/>


</div>
