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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="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">ft</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</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 <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>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#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/>









<b>assume </b><a NAME="E1:48"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">f</font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">x</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Gt</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/><a NAME="E1:48_1_1"/>
<font color="Maroon">Gt</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">Gt</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <b>by </b><i>, </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="E4:48_1_1"/><i><font color="Green">E42</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/><a NAME="E5:48_1_1"/>
<font color="Maroon">H</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">Ht</font> = <font color="Maroon">H</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">Ht</font> = <font color="Maroon">Ht</font>;<br/><b>thus </b><a NAME="E7:48_1_1"/>
<font color="Maroon">ft</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Ht</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Gt</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:48_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="E3:48"/><i><font color="Green">E43</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="48_2"><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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>;<br/><a NAME="E1:48_2"/>
<font color="Maroon">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">Gt</font> = <font color="Maroon">G</font> as    <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>consider </b><font color="Maroon">Ht</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">xt</font><b> such that </b><br/><a NAME="E4:48_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">ft</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Ht</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Gt</font>
 <b>by </b><i>, <a class="ref" href="tmap_1.html#D2">TMAP_1:def 2</a></i>;<br/><a NAME="E5:48_2"/>
<font color="Maroon">Ht</font> is   <a href="struct_0.html#NM2">Subset</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>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">H</font> = <font color="Maroon">Ht</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</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="E7:48_2"/>
<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>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:48"/>
<font color="Maroon">f</font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#D2">TMAP_1:def 2</a></i>;<br/>


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