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<b>let </b><font color="Maroon">TX</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">TY</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">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TY</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">TX</font>,<span class="p1">(<span class="default"><font color="Maroon">TY</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>;<br/>





<i><font color="Green">E37</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">TY</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">TY</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>

<br/>.= 
<font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>hence </b><a NAME="E2:3"/>
<font color="Maroon">f</font> is   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">TX</font>,<font color="Maroon">TY</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>

<b>let </b><font color="Maroon">f2</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">TX</font>,<font color="Maroon">TY</font>;<br/>

<b>assume </b><a NAME="E3:3"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">f2</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> )
 ;<br/>

<b>let </b><font color="Maroon">P1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TY</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a><br/>

<b>assume </b><a NAME="E4:3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">P1</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">P3</font> = <font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">TY</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <b>by </b><i><a class="ref" href="tops_2.html#T38">TOPS_2:38</a></i>;<br/>
<a NAME="E6:3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">P3</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E52</font></i>: <font color="Maroon">f2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">TX</font>
<b>by </b><i>, , <a class="ref" href="funct_2.html#T48">FUNCT_2:48</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E8:3"/>
<font color="Maroon">f2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/>
<b>then </b><font color="Maroon">f2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P3</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
;<br/>
<b>hence </b><a NAME="E10:3"/>
<font color="Maroon">f2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>


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