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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">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">X1</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>, <font color="Maroon">X2</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">g</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">x1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X1</font>;<br/><b>let </b><font color="Maroon">x2</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X2</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>;<br/>













<b>assume </b><a NAME="E1:130"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 ;<br/>

<a NAME="E2:130"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<b>hence </b><a NAME="E3:130"/>
( <font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font> implies ( <font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x2</font> ) )
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E4:130"/>
( <font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x2</font> implies <font color="Maroon">g</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="130_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:130_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x2</font> )
 ;<br/>

<a NAME="E2:130_1"/>
for <font color="Olive">G</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>  ex <font color="Olive">H</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font> st <font color="Maroon">g</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="130_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>;<br/>

<a NAME="E1:130_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">H1</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x1</font><b> such that </b><br/><a NAME="E3:130_1_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="txt" href="tmap_1.html#E1:130_1_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">H2</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x2</font><b> such that </b><br/><a NAME="E5:130_1_1"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="tmap_1.html#T20" target="_self">Th20</a>, <a class="txt" href="tmap_1.html#E1:130_1_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">H</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font><b> such that </b><br/><a NAME="E7:130_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H2</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<a NAME="E8:130_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">S1</font> = <font color="Maroon">H1</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:130_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">S2</font> = <font color="Maroon">H2</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:130_1_1"/>
( <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">S1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">S2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="tmap_1.html#T21" target="_self">Th21</a>, , </i>;<br/>
<a NAME="E13:130_1_1"/><b>then </b>
( <font color="Maroon">S1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> &amp; <font color="Maroon">S2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E14:130_1_1"/><b>then </b>
<font color="Maroon">S1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">S2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E15:130_1_1"/><b>then </b>
<font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T22" target="_self">Th22</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E16:130_1_1"/>
<font color="Maroon">g</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/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:130_1"/>
<font color="Maroon">g</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>
