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<b>let </b><font color="Maroon">Y</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  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">X</font>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">X0</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</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 <font color="Maroon">X</font>;<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">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X0</font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:66"/><i><font color="Green">E23</font></i>: 
<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">X0</font>
 <b>and </b><br/><a NAME="E2:66"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">A</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E3:66"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 ;<br/>

<b>thus </b><a NAME="E4:66"/>
( <font color="Maroon">f</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font> implies <font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x0</font> )
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E5:66"/>
( <font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x0</font> implies <font color="Maroon">f</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x0</font>
 ;<br/>

<a NAME="E2:66_1"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font> holds <br/> ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">H</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">H</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:66_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:66_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E3:66_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i>, , <a class="txt" href="tmap_1.html#E7:66_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">H0</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font><b> such that </b><br/><a NAME="E5:66_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">H0</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:66_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H0</font>
 <b>and </b><br/><a NAME="E7:66_1_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="txt" href="tmap_1.html#E1:66_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:66_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E10:66_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E11:66_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="connsp_2.html#T8">CONNSP_2:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">V0</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font> <b>by </b><i>, <a class="ref" href="tmap_1.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:66_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">H0</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E14:66_1_1"/>
<font color="Maroon">V0</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T33">TOPS_2:33</a></i>;<br/>
<a NAME="E15:66_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">H0</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V0</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H0</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<a NAME="E17:66_1_1"/>
for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E1:66_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:66_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E4:66_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font>;<br/>
<a NAME="E5:66_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:66_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E7:66_1_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E4:66_1_1_1"><i><font color="Green">E51</font></i></a>, , <a class="txt" href="tmap_1.html#E7:66_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E8:66_1_1_1"/>
<font color="Maroon">H0</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H0</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E9:66_1_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H0</font>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E10:66_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:66_1_1"/>
( <font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 <b>by </b><i>, , , , , <a class="ref" href="tmap_1.html#T7" target="_self">Th7</a>, <a class="ref" href="subset_1.html#T7">SUBSET_1:7</a>, <a class="ref" href="tsep_1.html#T9">TSEP_1:9</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:66_1"/>
<font color="Maroon">f</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>


</div>
