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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">S</font> be   <a href="struct_0.html#NM3">Subset-Family</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">St</font> be   <a href="struct_0.html#NM3">Subset-Family</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="E1:37"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">St</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xt</font> )
 ;<br/>

<a NAME="E2:37"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">St</font>
 
<b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">St</font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">xt</font>
 ;<br/><a NAME="E2:37_1"/><b>then </b>
<font color="Maroon">St</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</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><a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/><a NAME="E3:37_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</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="E4:37_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">S</font>
 <b>by </b><i>, , , <a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:37_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> )
 ;<br/><b>reconsider </b><font color="Maroon">Ut</font> = <font color="Maroon">U</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/><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="E4:37_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">Vt</font> <a href="hidden.html#R2">in</a> <font color="Maroon">St</font> &amp; <font color="Maroon">Vt</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ut</font> )
 <b>by </b><i>, , , <a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">Vt</font> as   <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>take </b><font color="Maroon">V</font> = <font color="Maroon">V</font>;<br/><b>thus </b><a NAME="E6:37_1_1"/>
( <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:37_1"/>
<font color="Maroon">S</font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:37"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">S</font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x</font>
 ;<br/>

<a NAME="E5:37"/><b>then </b>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</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><a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/>
<a NAME="E6:37"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">St</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</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="E7:37"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">xt</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">St</font>
 
<b>by </b><i>, , , <a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:37_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">Ut</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">xt</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ut</font> )
 ;<br/><b>reconsider </b><font color="Maroon">U</font> = <font color="Maroon">Ut</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/><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="E4:37_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 <b>by </b><i>, , , <a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">Vt</font> = <font color="Maroon">V</font> as   <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>take </b><font color="Maroon">Vt</font> = <font color="Maroon">Vt</font>;<br/><b>thus </b><a NAME="E6:37_2"/>
( <font color="Maroon">Vt</font> <a href="hidden.html#R2">in</a> <font color="Maroon">St</font> &amp; <font color="Maroon">Vt</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ut</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:37"/>
<font color="Maroon">St</font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">xt</font>
 <b>by </b><i>, , <a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/>


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