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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>] <b>means </b> ex <font color="Olive">S</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T</font> st <br/>(  <a href="relset_1.html#K5">rng</a> <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Olive">S</font> );<br/>
<b>reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>

<b>thus </b><a NAME="E3:29_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies  ex <font color="Olive">S</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T</font> st <br/>(  <a href="relset_1.html#K5">rng</a> <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Olive">S</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x'</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:29_1_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> &amp;  ex <font color="Olive">S</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T</font> st <br/>(  <a href="relset_1.html#K5">rng</a> <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Olive">S</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:29_1_1_1"/>
 ex <font color="Olive">S</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T</font> st <br/>(  <a href="relset_1.html#K5">rng</a> <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Olive">S</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E4:29_1_1"/>
 ex <font color="Olive">S</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T</font> st <br/>(  <a href="relset_1.html#K5">rng</a> <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Olive">S</font> )
 ;<br/>

<b>hence </b><a NAME="E5:29_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>


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