<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>









<b>assume </b><a NAME="E1:34"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:34"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>3</sub></font> st  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> holds <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Olive">b<sub>1</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) ) )
 
<b>by </b><i><a class="ref" href="nfcont_1.html#D9" target="_self">Def9</a></i>;<br/>
<b>hence </b><a NAME="E3:34"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="vfunct_1.html#K4">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D4">VFUNCT_1:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nfcont_1.html#D9" target="_self">NFCONT_1:def 9</a><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E4:34"/><i><font color="Green">E31</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="vfunct_1.html#K4">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E5:34"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="vfunct_1.html#D4">VFUNCT_1:def 4</a></i>;<br/>
<a NAME="E6:34"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="nfcont_1.html#E1:34"><i><font color="Green">E29</font></i></a>, <a class="txt" href="nfcont_1.html#E4:34"><i><font color="Green">E31</font></i></a>, <a class="ref" href="nfcont_1.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E7:34"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> is <a href="normsp_1.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T37">NORMSP_1:37</a></i>;<br/>
<b>hence </b><a NAME="E8:34"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="vfunct_1.html#K4">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E5:34"><i><font color="Green">E32</font></i></a>, <a class="ref" href="nfcont_1.html#T20" target="_self">Th20</a></i>;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="vfunct_1.html#K4">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E3:34"><i><font color="Green">E30</font></i></a>, <a class="ref" href="vfunct_1.html#D4">VFUNCT_1:def 4</a></i>
<br/>.= 
 <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="normsp_1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E6:34"><i><font color="Green">E33</font></i></a>, <a class="ref" href="normsp_1.html#T45">NORMSP_1:45</a></i>
<br/>.= 
 <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="vfunct_1.html#K4">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E5:34"><i><font color="Green">E32</font></i></a>, <a class="ref" href="nfcont_1.html#T20" target="_self">Th20</a></i>
;<br/>


</div>
