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<b>let </b><font color="Maroon">T</font> be  non <a href="realset2.html#V3">trivial</a> <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="realset2.html#V3">trivial</a> <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">S</font>,<font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/>







<b>thus </b><a NAME="E1:41"/>
( <font color="Maroon">f</font> <a href="nfcont_1.html#R1">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">s1</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">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="Olive">s1</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp; ( 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="Olive">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> ) holds <br/>( <font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">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="nfcont_1.html#K3">*</a> <font color="Olive">s1</font></span>)</span> ) ) ) )
 <b>by </b><i><a class="ref" href="nfcont_1.html#D9">NFCONT_1:def 9</a></i>;<br/>

<b>assume </b><a NAME="E2:41"/><i><font color="Green">E43</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">s1</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">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="Olive">s1</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp; ( 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="Olive">s1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> ) holds <br/>( <font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">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="nfcont_1.html#K3">*</a> <font color="Olive">s1</font></span>)</span> ) ) )
 ;<br/>

<b>hence </b><a NAME="E3:41"/>
<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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="nfcont_1.html#D9">NFCONT_1:def 9</a><br/>

<b>let </b><font color="Maroon">s2</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E4:41"/><i><font color="Green">E45</font></i>: 
(  <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> &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#R1">=</a> <font color="Maroon">x0</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1"/>
(  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> or 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">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 ;<br/><div class="add"><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="E3:41_1_1_1"/><i><font color="Green">E46</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/><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 ;<br/><a NAME="E4:41_1_1_1"/><i><font color="Green">E47</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  <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">N</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_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="E1:41_1_1_1_1"/>
<font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E2:41_1_1_1_1"/><b>then </b>
<font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:41_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">N</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="lopban_3.html#D6">LOPBAN_3:def 6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:41_1_1_1"/>
<font color="Maroon">s2</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">N</font> is    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">s2</font>
 <b>by </b><i><a class="ref" href="lopban_3.html#T13">LOPBAN_3:13</a></i>;<br/><a NAME="E6:41_1_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">N</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s2</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T7">NFCONT_1:7</a></i>;<br/><a NAME="E7:41_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">N</font></span>)</span> <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/><div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_2"><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="E1:41_1_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>take </b><font color="Maroon">n</font> = 0;<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="E2:41_1_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<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="nfcont_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">N</font></span>)</span></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> = 
<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"><span class="p5">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">N</font></span>)</span> <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>
<b>by </b><i>, <a class="ref" href="nfcont_1.html#T8">NFCONT_1:8</a></i>
<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="Maroon">x0</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>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="normsp_1.html#T5">NORMSP_1:5</a></i>
;<br/><b>hence </b><a NAME="E4:41_1_1_1_2"/>
<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="nfcont_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">N</font></span>)</span></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#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E9:41_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">N</font></span>)</span> 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/><a NAME="E10:41_1_1_1"/>
<font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">N</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">N</font>
 <b>by </b><i>, </i>;<br/><a NAME="E11:41_1_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s2</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"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s2</font></span>)</span> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">N</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="lopban_3.html#T15">LOPBAN_3:15</a>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/><b>hence </b><a NAME="E12:41_1_1_1"/>
( <font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s2</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="nfcont_1.html#K3">*</a> <font color="Maroon">s2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="lopban_3.html#T14">LOPBAN_3:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_2"/><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">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m1</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="E3:41_1_1_2"/><i><font color="Green">E55</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> &amp; <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</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> ] <b>means </b><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font>;<br/><a NAME="E4:41_1_1_2"/><i><font color="Green">E60</font></i>: 
 ex <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">S<sub>1</sub></font>[<font color="Olive">m</font>]
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">M</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="E6:41_1_1_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">M</font>] &amp; ( for <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">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i><div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_1"><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/><b>consider </b><font color="Maroon">m</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="E2:41_1_1_2_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> )
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">m</font>;<br/><b>thus </b><a NAME="E3:41_1_1_2_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> )
 <b>by </b><i><a class="ref" href="ndiff_1.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">n</font>, <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">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> holds <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">m</font> &amp; <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> holds <br/><font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) );<br/><a NAME="E8:41_1_1_2"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_2"><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/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> );<br/>
<a NAME="E1:41_1_1_2_2"/><i><font color="Green">E66</font></i>: 
 ex <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">S<sub>3</sub></font>[<font color="Olive">m</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">l</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="E3:41_1_1_2_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">l</font>] &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<b>take </b>
<font color="Maroon">l</font>
;<br/>

<b>thus </b><a NAME="E4:41_1_1_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Maroon">l</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">M'</font> = <font color="Maroon">M</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E11:41_1_1_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">M'</font> &amp; ( 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>2</sub></font>[<font color="Olive">n</font>,<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>,<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>(<a class="ref" href="ndiff_1.html#T1" target="_self">Th1</a>);<br/></i><a NAME="E12:41_1_1_2"/><i><font color="Green">E69</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</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="E13:41_1_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/><div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_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="E1:41_1_1_2_3"/>
<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E2:41_1_1_2_3"/>
<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_4"><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="E1:41_1_1_2_4"/>
( <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:41_1_1_2_4"/>
<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a> <b>by </b><i>, <a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a>, <a class="ref" href="seqm_3.html#D11">SEQM_3:def 11</a></i>;<br/><a NAME="E18:41_1_1_2"/><i><font color="Green">E71</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>   st <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> holds <br/> ex <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">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_5"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<b>assume </b><a NAME="E1:41_1_1_2_5"/><i><font color="Green">E117</font></i>: 
 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">S<sub>3</sub></font>[<font color="Olive">n</font>]
 ;<br/>

<b>consider </b><font color="Maroon">M1</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="E3:41_1_1_2_5"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">M1</font>] &amp; ( for <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">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">M1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="ndiff_1.html#E2:41"><i><font color="Green">E43</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> &amp;  ex <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">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E4:41_1_1_2_5"/><i><font color="Green">E119</font></i>: 
 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">S<sub>4</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_5_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">M</font>
;<br/>

<a NAME="E1:41_1_1_2_5_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:41_1_1_2_5_1"/>
<font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:41_1_1_2_5_1"/>
<font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">M1</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E4:41_1_1_2_5_1"/>
<font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font>
 <b>by </b><i/>;<br/>

<b>take </b>
0
;<br/>

<b>thus </b><a NAME="E5:41_1_1_2_5_1"/>
<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:41_1_1_2_5"/><i><font color="Green">E121</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>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M1</font>
 
;<br/>
<b>consider </b><font color="Maroon">MX</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="E7:41_1_1_2_5"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">MX</font>] &amp; ( for <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">S<sub>4</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">MX</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, <a class="txt" href="ndiff_1.html#E4:41"><i><font color="Green">E45</font></i></a>);<br/></i>
<a NAME="E8:41_1_1_2_5"/><i><font color="Green">E123</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">MX</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">M1</font> holds <br/><font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_5_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:41_1_1_2_5_2"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">MX</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_5_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_5_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1_2_5_2_1_1"/>
(  ex <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">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_5_2_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_2_5_2_1_1_1"/>
 ex <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">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_1_2_5_2_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_5_2_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_2_5_2_1_1_2"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_1_2_5_2_1_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:41_1_1_2_5_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">m</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="E10:41_1_1_2_5"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MX</font>
 <b>by </b><i/>;<br/>
<a NAME="E11:41_1_1_2_5"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">MX</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">MX</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> holds <br/><font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:41_1_1_2_5"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M1</font>
 
<b>by </b><i>, , , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_5_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_1_2_5_3"/>
<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M1</font>
 ;<br/><a NAME="E2:41_1_1_2_5_3"/><b>then </b>
<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">M1</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:41_1_1_2_5_3"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:41_1_1_2_5"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E19:41_1_1_2"/><i><font color="Green">E128</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 <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">F</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_6"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">F</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font>;<br/>
<a NAME="E1:41_1_1_2_6"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i>, , <a class="ref" href="lopban_3.html#T7">LOPBAN_3:7</a></i>;<br/>
<a NAME="E2:41_1_1_2_6"/><i><font color="Green">E130</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_6_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:41_1_1_2_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">F</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font>
 ;<br/>

<a NAME="E2:41_1_1_2_6_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">k</font>,<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 
<b>by </b><i/>;<br/>
<a NAME="E3:41_1_1_2_6_1"/><b>then </b>
<font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font>
 
;<br/>
<b>hence </b><a NAME="E4:41_1_1_2_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">F</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="lopban_3.html#T7">LOPBAN_3:7</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:41_1_1_2_6"/>
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>3</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="txt" href="ndiff_1.html#E4:41"><i><font color="Green">E45</font></i></a>);</i><br/>


</div><b>end;</b></div><a NAME="E20:41_1_1_2"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">F</font> is    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">s2</font>
 <b>by </b><i><a class="ref" href="bhsp_3.html#D4">BHSP_3:def 4</a></i>;<br/><a NAME="E21:41_1_1_2"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">F</font> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i>, <a class="ref" href="lopban_3.html#T11">LOPBAN_3:11</a></i>;<br/><a NAME="E22:41_1_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s2</font>
 <b>by </b><i>, <a class="ref" href="nfcont_1.html#T7">NFCONT_1:7</a></i>;<br/><a NAME="E23:41_1_1_2"/><b>then </b><i><font color="Green">E153</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">F</font></span>)</span> <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="E24:41_1_1_2"/>
 <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 <b>by </b><i>, , <a class="ref" href="lopban_3.html#T12">LOPBAN_3:12</a></i>;<br/><a NAME="E25:41_1_1_2"/><b>then </b><i><font color="Green">E154</font></i>: 
( <font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">F</font></span>)</span> 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="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="ndiff_1.html#E2:41"><i><font color="Green">E43</font></i></a>, <a class="txt" href="ndiff_1.html#E4:41"><i><font color="Green">E45</font></i></a>, </i>;<br/><div><i><font color="Green">E155</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_7"><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="E1:41_1_1_2_7"/><i><font color="Green">E156</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<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="E3:41_1_1_2_7"/><i><font color="Green">E157</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="nfcont_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">F</font></span>)</span></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#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <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="E4:41_1_1_2_7"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1_2_7_1_1"/>
( <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> or <font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_7_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_2_7_1_1_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 ;<br/><div class="add"><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="nfcont_1.html#K3">*</a> <font color="Maroon">s2</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> = 
<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">s2</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>
<b>by </b><i>, <a class="ref" href="nfcont_1.html#T8">NFCONT_1:8</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="normsp_1.html#T5">NORMSP_1:5</a></i>
;<br/><b>hence </b><a NAME="E3:41_1_1_2_7_1_1_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="nfcont_1.html#K3">*</a> <font color="Maroon">s2</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#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_7_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_2_7_1_1_2"/>
<font color="Maroon">s2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">l</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="E3:41_1_1_2_7_1_1_2"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">l</font>
 <b>by </b><i/>;<br/><a NAME="E4:41_1_1_2_7_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i>, , <a class="ref" href="seqm_3.html#T7">SEQM_3:7</a></i>;<br/><a NAME="E5:41_1_1_2_7_1_1_2"/><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"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</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#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><a NAME="E6:41_1_1_2_7_1_1_2"/><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"><span class="p5">(<span class="default"><font color="Maroon">s2</font> <a href="lopban_3.html#K5">*</a> <font color="Maroon">F</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</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#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="nfcont_1.html#T8">NFCONT_1:8</a></i>;<br/><a NAME="E7:41_1_1_2_7_1_1_2"/><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">s2</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#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="lopban_3.html#T7">LOPBAN_3:7</a></i>;<br/><b>hence </b><a NAME="E8:41_1_1_2_7_1_1_2"/>
<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="nfcont_1.html#K3">*</a> <font color="Maroon">s2</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#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="nfcont_1.html#T8">NFCONT_1:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:41_1_1_2_7"/>
<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="nfcont_1.html#K3">*</a> <font color="Maroon">s2</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#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E27:41_1_1_2"/>
<font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s2</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="E28:41_1_1_2"/>
<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="nfcont_1.html#K3">*</a> <font color="Maroon">s2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:41"/>
( <font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s2</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="nfcont_1.html#K3">*</a> <font color="Maroon">s2</font></span>)</span> )
 ;<br/>


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