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<b>let </b><font color="Maroon">RNS</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">RNS</font>, <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:129"/><i><font color="Green">E34</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> is <a href="nfcont_1.html#V1">compact</a>
 <b>and </b><br/><a NAME="E2:129"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="ncfcont1.html#R12" target="_self">is_continuous_on</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s1</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>assume </b><a NAME="E1:129_1"/><i><font color="Green">E54</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#K5">rng</a> <font color="Maroon">f</font>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E2:129_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">p</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:129_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:129_1_1"/><b>then </b>
<font color="Maroon">s1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font><b> such that </b><br/><a NAME="E4:129_1_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">s1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E5:129_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">p</font>]
 <b>by </b><i><a class="ref" href="ncfcont1.html#T26" target="_self">Th26</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">q1</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">RNS</font><b> such that </b><br/><a NAME="E4:129_1"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">q1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:129_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">q1</font>
 ;<br/><a NAME="E2:129_1_2"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T6">NFCONT_1:6</a></i>;<br/><b>hence </b><a NAME="E3:129_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T26" target="_self">Th26</a></i>;<br/></div><b>end;</b></div><a NAME="E6:129_1"/><b>then </b><i><font color="Green">E112</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">q1</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="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:129_1_3"/>
( <font color="Maroon">q1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="ncfcont1.html#T26" target="_self">Th26</a></i>;<br/><b>hence </b><a NAME="E2:129_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ncfcont1.html#K9" target="_self">*</a> <font color="Maroon">q1</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T27" target="_self">Th27</a>, </i>;<br/></div><b>end;</b></div><a NAME="E8:129_1"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">f</font> <a href="ncfcont1.html#K9" target="_self">*</a> <font color="Maroon">q1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>consider </b><font color="Maroon">s2</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">RNS</font><b> such that </b><br/><a NAME="E10:129_1"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">s2</font> is    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">q1</font> &amp; <font color="Maroon">s2</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="ncfcont1.html#T24" target="_self">Th24</a>, <a class="ref" href="ncfcont1.html#T27" target="_self">Th27</a>, <a class="ref" href="nfcont_1.html#D4">NFCONT_1:def 4</a></i>;<br/><a NAME="E11:129_1"/>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="ncfcont1.html#R6" target="_self">is_continuous_in</a>  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s2</font>
 <b>by </b><i>, <a class="ref" href="ncfcont1.html#T29" target="_self">Th29</a>, <a class="ref" href="ncfcont1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E12:129_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">f</font> <a href="ncfcont1.html#R6" target="_self">is_continuous_in</a>  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s2</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/><a NAME="E13:129_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s2</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">q1</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T29" target="_self">Th29</a>, <a class="ref" href="nfcont_1.html#T7">NFCONT_1:7</a></i>;<br/><a NAME="E14:129_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s2</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="ncfcont1.html#T27" target="_self">Th27</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E15:129_1"/><b>then </b><i><font color="Green">E116</font></i>: 
( <font color="Maroon">f</font> <a href="ncfcont1.html#K9" target="_self">*</a> <font color="Maroon">s2</font> is <a href="comseq_2.html#V2">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">s2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ncfcont1.html#K9" target="_self">*</a> <font color="Maroon">s2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="ncfcont1.html#T29" target="_self">Th29</a>, , </i>;<br/><b>take </b><font color="Maroon">q2</font> = <font color="Maroon">f</font> <a href="ncfcont1.html#K9" target="_self">*</a> <font color="Maroon">s2</font>;<br/><b>thus </b><a NAME="E16:129_1"/>
( <font color="Maroon">q2</font> is    <a href="comseq_3.html#M1">subsequence</a> of <font color="Maroon">s1</font> &amp; <font color="Maroon">q2</font> is <a href="comseq_2.html#V2">convergent</a> &amp;  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="ncfcont1.html#T27" target="_self">Th27</a>, <a class="ref" href="ncfcont1.html#T28" target="_self">Th28</a>, <a class="ref" href="ncfcont1.html#T29" target="_self">Th29</a>, , , <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:129"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="cfcont_1.html#V1">compact</a>
 <b>by </b><i><a class="ref" href="cfcont_1.html#D6">CFCONT_1:def 6</a></i>;<br/>


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