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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">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<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>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>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">X</font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">x</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">X</font> ;<br/>
<b>set </b><font color="Maroon">F</font> =  <a href="waybel26.html#K4" target="_self">oContMaps</a> <font color="Maroon">f</font>,<font color="Maroon">Y</font>;<br/>
<a NAME="E2:39"/>
 <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">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> = 
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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T23">FUNCOP_1:23</a></i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E4:39"/><b>then </b>
<font color="Maroon">f</font> is <a href="quantal1.html#V11">idempotent</a>
 
<b>by </b><i><a class="ref" href="quantal1.html#D9">QUANTAL1:def 9</a></i>;<br/>
<a NAME="E5:39"/><b>then </b>
 <a href="waybel26.html#K4" target="_self">oContMaps</a> <font color="Maroon">f</font>,<font color="Maroon">Y</font> is  <a href="quantal1.html#V11">idempotent</a> <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">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>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> =  <a href="waybel26.html#K4" target="_self">oContMaps</a> <font color="Maroon">f</font>,<font color="Maroon">Y</font> as  <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>by </b><i><a class="ref" href="waybel_1.html#D13">WAYBEL_1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">h</font> be  <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>;<br/><a NAME="E1:39_2"/><i><font color="Green">E45</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>thus </b><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> = 
<font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default">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></span>)</span>
<b>by </b><i/>
<br/>.= 
<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">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T23">FUNCOP_1:23</a></i>
;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E8:39"/>
 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">x</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> )
 ;<br/>


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