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<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">Z</font> be  <a href="waybel25.html#V1">monotone-convergence</a> <a href="t_0topsp.html#NM1">T_0-TopSpace</a>;<br/><b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Z</font>;<br/>





<b>set </b><font color="Maroon">XY</font> =  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>
<b>set </b><font color="Maroon">XZ</font> =  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Z</font>;<br/>
<b>reconsider </b><font color="Maroon">uXZ</font> =  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Z</font> as  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V24">up-complete</a> <a href="orders_2.html#NM1">Poset</a> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">sXY</font> =  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</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 <font color="Maroon">uXZ</font> <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">f</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Z</font>,<font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E3:28"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">f</font> <a href="borsuk_1.html#NR1" target="_self">is_a_retraction</a> 
 ;<br/>

<b>set </b><font color="Maroon">F</font> =  <a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font>;<br/>
<b>reconsider </b><font color="Maroon">Y'</font> = <font color="Maroon">Y</font> as  <a href="waybel25.html#V1">monotone-convergence</a> <a href="t_0topsp.html#NM1">T_0-TopSpace</a> <b>by </b><i>, </i>;<br/>
<a NAME="E5:28"/>
 <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y'</font> is <a href="waybel_0.html#V24">up-complete</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:28"/>
 <a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font> is  <a href="waybel_0.html#V22">directed-sups-preserving</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">Z</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>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow16.html#D1">YELLOW16:def 1</a><br/>

<a NAME="E7:28"/><i><font color="Green">E46</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<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>)</span> <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="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<a NAME="E8:28"/><i><font color="Green">E59</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">sXY</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">uXZ</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E9:28"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</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> is   <a href="funct_2.html#NM1">Function</a> of 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>,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#T38">FUNCT_2:38</a></i>;<br/>
<a NAME="E10:28"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<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>)</span> <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#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:28_1"/><i><font color="Green">E62</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="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <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">Z</font></span>)</span> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">a</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">Z</font> <b>by </b><i/>;<br/><a NAME="E4:28_1"/><i><font color="Green">E64</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">x</font> is   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> )
 <b>by </b><i>, , , <a class="ref" href="yellow16.html#T46">YELLOW16:46</a></i>;<br/><a NAME="E5:28_1"/><b>then </b>
( <font color="Maroon">f</font> is <a href="quantal1.html#V11">idempotent</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp;  <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 <font color="Maroon">Z</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a>, <a class="ref" href="yellow16.html#T47">YELLOW16:47</a></i>;<br/><a NAME="E6:28_1"/><b>then </b>
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">a</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="yellow16.html#T4">YELLOW16:4</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:28"/>
<span class="p1">(<span class="default"><a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</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> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K7">id</a> <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_1.html#T9">FUNCT_1:9</a></i>;<br/>

<a NAME="E13:28"/><i><font color="Green">E66</font></i>: 
 <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="yellow_0.html#V4">full</a> <a href="waybel_0.html#V4">directed-sups-inheriting</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/>
<a NAME="E14:28"/><i><font color="Green">E67</font></i>: 
 <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Z</font> is  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="waybel_0.html#V4">directed-sups-inheriting</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">Z</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/>
<a NAME="E15:28"/>
 <a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font> is  <a href="yellow_0.html#V4">full</a> <a href="waybel_0.html#V4">directed-sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="waybel25.html#K1">Omega</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, , <a class="ref" href="waybel25.html#T17">WAYBEL25:17</a></i>;<br/>
<a NAME="E16:28"/><b>then </b>
<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> is  <a href="yellow_0.html#V4">full</a> <a href="waybel_0.html#V4">directed-sups-inheriting</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">Z</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#T41">YELLOW16:41</a>, <a class="ref" href="yellow16.html#T44">YELLOW16:44</a></i>;<br/>
<a NAME="E17:28"/><b>then </b>
 <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="waybel_0.html#V4">directed-sups-inheriting</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">Z</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>, <a class="ref" href="yellow16.html#T29">YELLOW16:29</a></i>;<br/>
<b>hence </b><a NAME="E18:28"/>
 <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="waybel_0.html#V4">directed-sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Z</font>
 <b>by </b><i>, , <a class="ref" href="yellow16.html#T30">YELLOW16:30</a></i>;<br/>


</div>
