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

<b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</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>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green">E43</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green">E45</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x0</font>
 ;<br/>

<b>take </b><font color="Maroon">g</font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="normsp_1.html#D9">NORMSP_1:def 9</a><br/>

<b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E3:26"/><i><font color="Green">E46</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E4:26"/>
 ex <font color="Olive">pp</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">pp</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">pp</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">pp</font> = <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> 1</span>)</span>;<br/>

<a NAME="E1:26_1"/><i><font color="Green">E47</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/>
<a NAME="E2:26_1"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E3:26_1"/><b>then </b><i><font color="Green">E50</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/>
<a NAME="E4:26_1"/><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E5:26_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/>
<a NAME="E6:26_1"/><b>then </b>
<font color="Maroon">pp</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">pp</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T99">XREAL_1:99</a></i>;<br/>
<b>hence </b><a NAME="E7:26_1"/>
( <font color="Maroon">pp</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">pp</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></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="xreal_1.html#T141">XREAL_1:141</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">pp</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:26"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">pp</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">pp</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> )
 ;<br/>
<a NAME="E7:26"/><i><font color="Green">E54</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">pp</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">k1</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="E9:26"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">pp</font> <a href="real_1.html#K2">"</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k1</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">n</font> = <font color="Maroon">k1</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="E10:26"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E11:26"/>
<span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E12:26"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></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">pp</font> <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T78">XREAL_1:78</a></i>;<br/>
<a NAME="E13:26"/><b>then </b><i><font color="Green">E61</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/>
<a NAME="E14:26"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:26"/><b>then </b><i><font color="Green">E62</font></i>: 
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E16:26"/>
<font color="Maroon">n</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E17:26"/><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> <font color="Maroon">r</font></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">n</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T120">XREAL_1:120</a></i>;<br/>
<a NAME="E18:26"/><b>then </b><i><font color="Green">E64</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> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">pp</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:26"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E20:26"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E21:26"/><b>then </b><i><font color="Green">E65</font></i>: 
0 <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> <font color="Maroon">r</font></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">m</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<i><font color="Green">E66</font></i>: <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">seq</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">g</font></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"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x0</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</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"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<a NAME="E23:26"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/>
<a NAME="E24:26"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pp</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x0</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<b>hence </b><a NAME="E25:26"/>
<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">seq</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">g</font></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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div>
