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<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">U</font> be   <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>;<br/><b>let </b><font color="Maroon">Ut</font> be   <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>;<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="E1:47"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ut</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="E1:47_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">U</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font>
 ;<br/><b>consider </b><font color="Maroon">V</font> being   <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> such that </b><br/><a NAME="E3:47_1"/><i><font color="Green">E42</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">U</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/><b>reconsider </b><font color="Maroon">Vt</font> = <font color="Maroon">V</font> as  <a href="pre_topc.html#V3">open</a> <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>, , </i>;<br/><a NAME="E5:47_1"/>
( <font color="Maroon">Vt</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Vt</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ut</font> &amp; <font color="Maroon">xt</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Vt</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:47_1"/>
<font color="Maroon">Ut</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">xt</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#T8">CONNSP_2:8</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:47"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Ut</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">xt</font>
 ;<br/>

<b>consider </b><font color="Maroon">Vt</font> being   <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> such that </b><br/><a NAME="E5:47"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">Vt</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Vt</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ut</font> &amp; <font color="Maroon">xt</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Vt</font> )
 <b>by </b><i>, <a class="ref" href="connsp_2.html#T8">CONNSP_2:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">Vt</font> as  <a href="pre_topc.html#V3">open</a> <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>, , </i>;<br/>
<a NAME="E7:47"/>
( <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">U</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:47"/>
<font color="Maroon">U</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#T8">CONNSP_2:8</a></i>;<br/>


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