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<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>, <font color="Maroon">Y</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">ft</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">Y</font></span>)</span>;<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">xt</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/>









<b>assume </b><a NAME="E1:27"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ft</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xt</font> )
 ;<br/>

<a NAME="E2:27"/><i><font color="Green">E41</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 <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:27"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">ft</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:27_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="nfcont_1.html#R1">is_continuous_in</a> <font color="Maroon">x</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>now </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">ft</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xt</font>;<br/><b>reconsider </b><font color="Maroon">N1</font> = <font color="Maroon">G</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/><a NAME="E2:27_1_1"/>
<font color="Maroon">N1</font> is    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">N</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x</font><b> such that </b><br/><a NAME="E4:27_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</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="nfcont_1.html#T17">NFCONT_1:17</a></i>;<br/><b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">N</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">xt</font> <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">H</font> = <font color="Maroon">H</font>;<br/><b>thus </b><a NAME="E6:27_1_1"/>
<font color="Maroon">ft</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>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:27_1"/>
<font color="Maroon">ft</font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">xt</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#D2">TMAP_1:def 2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:27"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">ft</font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">xt</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">N1</font> be    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>;<br/><b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">N1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/><a NAME="E2:27_2"/>
<font color="Maroon">G</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">ft</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xt</font>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">H</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">xt</font><b> such that </b><br/><a NAME="E4:27_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">ft</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>reconsider </b><font color="Maroon">N</font> = <font color="Maroon">H</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/><b>reconsider </b><font color="Maroon">N</font> = <font color="Maroon">N</font> as    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x</font> <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">N</font> = <font color="Maroon">N</font>;<br/><b>thus </b><a NAME="E7:27_2"/>
<font color="Maroon">f</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>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:27"/>
<font color="Maroon">f</font> <a href="nfcont_1.html#R1">is_continuous_in</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="nfcont_1.html#T17">NFCONT_1:17</a></i>;<br/>


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