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<b>let </b><font color="Maroon">T</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>









<b>assume </b><a NAME="E1:47"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="nfcont_1.html#R3" target="_self">is_continuous_on</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:47"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:47"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">X</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">r</font> <a href="vfunct_1.html#K4">(#)</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vfunct_1.html#D4">VFUNCT_1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:47_1"/><i><font color="Green">E57</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">s1</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/><a NAME="E2:47_1"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:47_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">f</font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:47_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">r</font> <a href="normsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</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/><span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="vfunct_1.html#K4">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s1</font></span>)</span> = 
<font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</font></span>)</span></span>)</span>
<b>by </b><i>, , , <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">r</font> <a href="normsp_1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</font></span>)</span></span>)</span>
<b>by </b><i>, <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">r</font> <a href="vfunct_1.html#K4">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</font></span>)</span>
<b>by </b><i><a class="ref" href="nfcont_1.html#D1" target="_self">Def1</a>, </i>
;<br/><b>hence </b><a NAME="E6:47_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="vfunct_1.html#K4">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</font> is <a href="normsp_1.html#V4">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="vfunct_1.html#K4">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="vfunct_1.html#K4">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nfcont_1.html#D1" target="_self">Def1</a>, <a class="txt" href="nfcont_1.html#E1:47"><i><font color="Green">E43</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:47"/>
<font color="Maroon">r</font> <a href="vfunct_1.html#K4">(#)</a> <font color="Maroon">f</font> <a href="nfcont_1.html#R3" target="_self">is_continuous_on</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


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