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<b>let </b><font color="Maroon">T</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">S</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="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:47_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><a NAME="E2:47_1"/><i><font color="Green">E34</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:47_1"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 ;<br/><a NAME="E4:47_1"/><b>then </b>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:47_1"/><b>then </b>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/><a NAME="E6:47_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/><a NAME="E7:47_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:47_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <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"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/><a NAME="E9:47_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T145">FUNCT_1:145</a></i>;<br/><b>hence </b><a NAME="E10:47_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:47"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> holds  <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Olive">P</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">P</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/><a NAME="E1:47_2"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 ;<br/><a NAME="E2:47_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:47_2"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/><a NAME="E4:47_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T145">FUNCT_1:145</a></i>;<br/><a NAME="E5:47_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P1</font> )
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/><a NAME="E6:47_2"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:47_2"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P1</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/><b>hence </b><a NAME="E8:47_2"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:47"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i/>;<br/>


</div>
