<?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">CNS</font>,<font color="Maroon">RNS</font>;<br/><b>let </b><font color="Maroon">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS</font>;<br/>







<b>thus </b><a NAME="E1:50"/>
( <font color="Maroon">f</font> <a href="ncfcont1.html#R2" 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">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">x1</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS</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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> ) ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="ncfcont1.html#R2" target="_self">is_continuous_in</a> <font color="Maroon">x0</font>
 ;<br/>

<b>hence </b><a NAME="E2:50_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>given </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E3:50_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; ( for <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  holds <br/>(  not 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> or  ex <font color="Olive">x1</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS</font> st <br/>( <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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; not <span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> ) ) ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS</font>] <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; <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="real_1.html#K3">+</a> 1</span>)</span> &amp; not <span class="p1"><a href="normsp_1.html#K1">||.</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">a<sub>2</sub></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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> );<br/>
<a NAME="E4:50_1"/><i><font color="Green">E54</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">CNS</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="50_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="E1:50_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E2:50_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K2">"</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E3:50_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</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">CNS</font><b> such that </b><br/><a NAME="E5:50_1_1"/><i><font color="Green">E56</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; <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span> &amp; not <span class="p1"><a href="normsp_1.html#K1">||.</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">p</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E6:50_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">p</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">s1</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CNS</font><b> such that </b><br/><a NAME="E6:50_1"/><i><font color="Green">E109</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">s1</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>(<a class="ref" href="ncfcont1.html#D5" target="_self">Def5</a>);<br/></i>
<b>reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s1</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">CNS</font> ;<br/>
<a NAME="E8:50_1"/><i><font color="Green">E110</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:50_1_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font>
 ;<br/>

<a NAME="E2:50_1_2"/>
 <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="E3:50_1_2"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:50_1_2"/>
<font color="Maroon">v</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/>


</div><b>end;</b></div>
<div><i><font color="Green">E113</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_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="E1:50_1_3"/>
not <span class="p1"><a href="normsp_1.html#K1">||.</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> <span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:50_1_3"/>
not <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="ncfcont1.html#K6" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E114</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:50_1_4"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E2:50_1_4"/><b>then </b><i><font color="Green">E115</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> <a href="real_1.html#K2">"</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:50_1_4"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">s</font> <a href="real_1.html#K2">"</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T10">SEQ_4:10</a></i>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E5:50_1_4"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E6:50_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="ncfcont1.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E7:50_1_4"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E4:50_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xreal_1.html#T78">XREAL_1:78</a></i>;<br/><a NAME="E8:50_1_4"/><b>then </b><i><font color="Green">E119</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T218">XCMPLX_1:218</a></i>;<br/><a NAME="E9:50_1_4"/><i><font color="Green">E120</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E10:50_1_4"/>
<font color="Maroon">k</font> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:50_1_4"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K3">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="ncfcont1.html#D6" target="_self">Def6</a>, <a class="ref" href="xreal_1.html#T120">XREAL_1:120</a></i>;<br/><a NAME="E12:50_1_4"/><b>then </b><i><font color="Green">E121</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</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>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:50_1_4"/>
<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">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E14:50_1_4"/>
<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">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="clvect_1.html#K5">.||</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#D9" target="_self">Def9</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:50_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">s1</font> is <a href="clvect_1.html#V6">convergent</a>
 
<b>by </b><i><a class="ref" href="clvect_1.html#D15">CLVECT_1:def 15</a></i>;<br/>
<a NAME="E12:50_1"/><b>then </b>
 <a href="clvect_1.html#K11">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 
<b>by </b><i>, <a class="ref" href="clvect_1.html#D17">CLVECT_1:def 17</a></i>;<br/>
<a NAME="E13:50_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="ncfcont1.html#K6" 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> <font color="Maroon">x0</font> <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="ncfcont1.html#K6" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="ncfcont1.html#D4" target="_self">Def4</a>, , <a class="txt" href="ncfcont1.html#E4:50_1"><i><font color="Green">E54</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E15:50_1"/><i><font color="Green">E123</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="ncfcont1.html#K6" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</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="normsp_1.html#K1">.||</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="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/>
<a NAME="E16:50_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="ncfcont1.html#K6" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</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="normsp_1.html#K1">.||</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#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E17:50_1"/>
contradiction
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E2:50"/><i><font color="Green">E124</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">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">x1</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">CNS</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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> ) ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><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">CNS</font>;<br/><b>assume </b><a NAME="E1:50_2"/><i><font color="Green">E125</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> &amp; <font color="Maroon">s1</font> is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> )
 ;<br/><div><i><font color="Green">E126</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:50_2_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 ;<br/><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:50_2_1"/><i><font color="Green">E127</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">CNS</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="clvect_1.html#K5">||.</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="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> ) )
 <b>by </b><i><a class="ref" href="ncfcont1.html#D4" target="_self">Def4</a></i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:50_2_1"/><i><font color="Green">E128</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</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">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="clvect_1.html#K5">.||</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="clvect_1.html#D17">CLVECT_1:def 17</a></i>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E6:50_2_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E7:50_2_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="E8:50_2_1"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</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/><a NAME="E9:50_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">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/><a NAME="E10:50_2_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</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> <span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E11:50_2_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="ncfcont1.html#K6" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</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="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E3:50_2"/><b>then </b>
<font color="Maroon">f</font> <a href="ncfcont1.html#K6" target="_self">*</a> <font color="Maroon">s1</font> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="normsp_1.html#D9">NORMSP_1:def 9</a></i>;<br/><b>hence </b><a NAME="E4:50_2"/>
( <font color="Maroon">f</font> <a href="ncfcont1.html#K6" 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> <font color="Maroon">x0</font> <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="ncfcont1.html#K6" target="_self">*</a> <font color="Maroon">s1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="ncfcont1.html#D5" target="_self">Def5</a>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:50"/>
<font color="Maroon">f</font> <a href="ncfcont1.html#R2" 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/>


</div>
