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

<b>let </b><font color="Maroon">CNS</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><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="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">RNS</font>,<font color="Maroon">CNS</font>;<br/><b>let </b><font color="Maroon">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font>;<br/>







<b>thus </b><a NAME="E1:57"/>
( <font color="Maroon">f</font> <a href="ncfcont1.html#R3" target="_self">is_continuous_in</a> <font color="Maroon">x0</font> implies ( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">N1</font> being   <a href="ncfcont1.html#M1" target="_self">Neighbourhood</a> of <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font>  ex <font color="Olive">N</font> being   <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> st <br/>for <font color="Olive">x1</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> holds <br/><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">N1</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="ncfcont1.html#R3" target="_self">is_continuous_in</a> <font color="Maroon">x0</font>
 ;<br/>

<b>hence </b><a NAME="E2:57_1"/>
<font color="Maroon">x0</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/>;<br/>

<b>let </b><font color="Maroon">N1</font> be    <a href="ncfcont1.html#M1" target="_self">Neighbourhood</a> of <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font>;<br/>

<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:57_1"/><i><font color="Green">E35</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS</font> : <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:57_1"/><i><font color="Green">E54</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">x1</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> ) )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E6:57_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="ncfcont1.html#T3" target="_self">Th3</a>, <a class="ref" href="ncfcont1.html#D4" target="_self">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">N</font> = <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">z</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> </span>}</span>  as    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> <b>by </b><i><a class="txt" href="ncfcont1.html#E8:57_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="nfcont_1.html#T3">NFCONT_1:3</a></i>;<br/>
<b>take </b>
<font color="Maroon">N</font>
;<br/>

<b>let </b><font color="Maroon">x1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font>;<br/>

<b>assume </b><a NAME="E8:57_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x1</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">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 ;<br/>

<b>consider </b><font color="Maroon">z</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font><b> such that </b><br/><a NAME="E10:57_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E11:57_1"/><i><font color="Green">E110</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E12:57_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E8:57_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="ncfcont1.html#T4" target="_self">Th4</a>, <a class="ref" href="ncfcont1.html#D6" target="_self">Def6</a>, <a class="ref" href="ncfcont1.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E13:57_1"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS</font> : <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E14:57_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T3" target="_self">Th3</a></i>;<br/>


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

<b>assume </b><a NAME="E2:57"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">N1</font> being   <a href="ncfcont1.html#M1" target="_self">Neighbourhood</a> of <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font>  ex <font color="Olive">N</font> being   <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> st <br/>for <font color="Olive">x1</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> holds <br/><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">N1</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:57_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/><b>then reconsider </b><font color="Maroon">N1</font> = <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS</font> : <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span>  as    <a href="ncfcont1.html#M1" target="_self">Neighbourhood</a> of <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">N2</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E4:57_2"/><i><font color="Green">E113</font></i>: 
for <font color="Olive">x1</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font> holds <br/><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:57_2"/><i><font color="Green">E114</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> &amp; <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">z</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">N2</font> )
 <b>by </b><i><a class="ref" href="nfcont_1.html#D3">NFCONT_1:def 3</a></i>;<br/><b>take </b><font color="Maroon">s</font> = <font color="Maroon">s</font>;<br/><a NAME="E7:57_2"/>
for <font color="Olive">x1</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font>;<br/>

<b>assume </b><a NAME="E1:57_2_1"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> )
 ;<br/>

<a NAME="E2:57_2_1"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">z</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> </span>}</span> 
 
;<br/>
<a NAME="E3:57_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 
<b>by </b><i><a class="ref" href="ncfcont1.html#T3" target="_self">Th3</a>, , <a class="ref" href="ncfcont1.html#T4" target="_self">Th4</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS</font><b> such that </b><br/><a NAME="E5:57_2_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E6:57_2_1"/><i><font color="Green">E118</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/>
<b>thus </b><a NAME="E7:57_2_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#D6" target="_self">Def6</a>, <a class="ref" href="ncfcont1.html#D9" target="_self">Def9</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:57_2"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">x1</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:57"/>
<font color="Maroon">f</font> <a href="ncfcont1.html#R3" target="_self">is_continuous_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="ncfcont1.html#D4" target="_self">Def4</a></i>;<br/>


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