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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">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>;<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">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X0</font>,<font color="Maroon">Y</font>;<br/>









<b>assume </b><a NAME="E1:85"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X0</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">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X0</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>assume </b><b>that </b><br/><a NAME="E2:85"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>and </b><br/><a NAME="E3:85"/><i><font color="Green">E35</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">X1</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> )
 ;<br/>

<b>thus </b><a NAME="E4:85"/>
( <font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x0</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> )
 <b>by </b><i>, <a class="txt" href="tmap_1.html#E12"><i><font color="Green">Lemma69</font></i></a>, </i>;<br/>

<b>thus </b><a NAME="E5:85"/>
( <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> implies <font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<a NAME="E1:85_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 <font color="Maroon">X0</font>
 
<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">B</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font> <b>by </b><i><a class="txt" href="tmap_1.html#E12"><i><font color="Green">Lemma69</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:85_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> )
 
<b>by </b><i>, <a class="txt" href="tmap_1.html#E12"><i><font color="Green">Lemma69</font></i></a>, <a class="ref" href="tops_2.html#T33">TOPS_2:33</a></i>;<br/>
<b>assume </b><a NAME="E4:85_1"/>
<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>
 ;<br/>

<b>hence </b><a NAME="E5:85_1"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="txt" href="tmap_1.html#E12"><i><font color="Green">Lemma69</font></i></a>, , </i>;<br/>


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


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