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<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">g</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">x1</font> be   <a href="real_1.html#NM1">Real</a>;<br/>







<b>assume </b><a NAME="E1:11"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">f</font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> )
 ;<br/>

<a NAME="E2:11"/><i><font color="Green">E38</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:11"/>
for <font color="Olive">N1</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font>  ex <font color="Olive">N</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x1</font> st <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">N</font> <a href="tarski.html#R1">c=</a> <font color="Olive">N1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">N1</font> be    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font>;<br/>

<b>reconsider </b><font color="Maroon">fx</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K3">R^1</a>  ;<br/>
<b>reconsider </b><font color="Maroon">N1'</font> = <font color="Maroon">N1</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">N2</font> = <font color="Maroon">N1</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E4:11_1"/>
<font color="Maroon">N2</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <a href="metric_1.html#K8">RealSpace</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:11_1"/><b>then </b>
<font color="Maroon">N2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <a href="metric_1.html#K8">RealSpace</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E6:11_1"/><b>then </b>
( <font color="Maroon">fx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1'</font> &amp; <font color="Maroon">N1'</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a>, <a class="ref" href="rcomp_1.html#T37">RCOMP_1:37</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">N1'</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">fx</font> <b>by </b><i><a class="ref" href="connsp_2.html#T5">CONNSP_2:5</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="E9:11_1"/><i><font color="Green">E39</font></i>: 
<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#D2">TMAP_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E11:11_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> &amp; <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/>
<a NAME="E12:11_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E13:11_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <a href="metric_1.html#K8">RealSpace</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">V</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E15:11_1"/>
<font color="Maroon">V1</font> is <a href="rcomp_1.html#V3">open</a>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">N</font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x1</font><b> such that </b><br/><a NAME="E17:11_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font>
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T39">RCOMP_1:39</a></i>;<br/>
<a NAME="E18:11_1"/>
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E19:11_1"/><b>then </b>
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E20:11_1"/><b>then </b>
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E21:11_1"/>
 ex <font color="Olive">N</font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">x1</font> st <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:11"/>
<font color="Maroon">g</font> <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">x1</font>
 <b>by </b><i>, , <a class="ref" href="fcont_1.html#T5">FCONT_1:5</a></i>;<br/>


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