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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>, <font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">Z</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">f</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">Z</font>;<br/>



<b>set </b><font color="Maroon">Xf</font> =  <a href="waybel26.html#K3" target="_self">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">f</font>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</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>thus </b><a NAME="E1:21"/>
 <a href="waybel26.html#K2" target="_self">pi</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="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel26.html#K2" target="_self">pi</a> <font color="Maroon">A</font>,<font color="Maroon">x</font></span>)</span>
  <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="21_1"><b>proof </b></a><div class="add">

<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="E1:21_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel26.html#K2" target="_self">pi</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="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:21_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <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="relset_1.html#K10">.:</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:21_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">g</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> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <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">g</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</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/>
<a NAME="E7:21_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:21_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel26.html#K2" target="_self">pi</a> <font color="Maroon">A</font>,<font color="Maroon">x</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, , <a class="ref" href="card_3.html#D6">CARD_3:def 6</a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E9:21_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel26.html#K2" target="_self">pi</a> <font color="Maroon">A</font>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</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="E2:21"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel26.html#K2" target="_self">pi</a> <font color="Maroon">A</font>,<font color="Maroon">x</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:21"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="waybel26.html#K2" target="_self">pi</a> <font color="Maroon">A</font>,<font color="Maroon">x</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">b</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:21"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</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/>
<a NAME="E8:21"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<a NAME="E9:21"/>
( <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <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">g</font> &amp; 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">Z</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E10:21"/><b>then </b>
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <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="relset_1.html#K10">.:</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E11:21"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel26.html#K2" target="_self">pi</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="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>


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