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<div class="add">

<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">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">R</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:35"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">R</font> is <a href="partfun1.html#V1">total</a>
 ;<br/>

<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">R</font> is <a href="ndiff_1.html#V3" target="_self">REST-like</a>
 ;<br/><b>assume </b><a NAME="E2:35_1"/><i><font color="Green">E47</font></i>: 
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a>  holds <br/>(  not <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or  ex <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">d</font> &amp; not <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> ) ) ) )
 ;<br/><b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:35_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E5:35_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">d</font> &amp; not <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<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">S</font>] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">a<sub>2</sub></font></span><a href="normsp_1.html#K1">.||</a></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">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; not <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">a<sub>2</sub></font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> );<br/><a NAME="E6:35_1"/><i><font color="Green">E51</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">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">z</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_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="E1:35_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E2:35_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E3:35_1_1"/><b>then </b>
0 <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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E4:35_1_1"/><b>then </b>
0 <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="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/>
<b>hence </b><a NAME="E5:35_1_1"/>
 ex <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">z</font></span><a href="normsp_1.html#K1">.||</a></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="nat_1.html#K1">+</a> 1</span>)</span> &amp; not <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">s</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:35_1"/><i><font color="Green">E53</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">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i><div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_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="E1:35_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:35_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</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="E4:35_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">p</font> <a href="real_1.html#K2">"</a>  <a href="xxreal_0.html#R1">&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/><a NAME="E5:35_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p</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">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E6:35_1_2"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</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">p</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="E7:35_1_2"/><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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T218">XCMPLX_1:218</a></i>;<br/><b>take </b><font color="Maroon">n</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="E8:35_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E9:35_1_2"/><b>then </b><i><font color="Green">E62</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> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E10:35_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E11:35_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E12:35_1_2"/><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="nat_1.html#K1">+</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T120">XREAL_1:120</a></i>;<br/><a NAME="E13:35_1_2"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></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="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E14:35_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">s</font> <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"><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> <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="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>;<br/><a NAME="E15:35_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">s</font> <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"><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> <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="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="ndiff_1.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E16:35_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"><font color="Maroon">s</font> <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"><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> <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><b>end;</b></div><a NAME="E10:35_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">s</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/><a NAME="E11:35_1"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/><a NAME="E12:35_1"/>
<font color="Maroon">s</font> <a href="ndiff_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as  <a href="ndiff_1.html#V2" target="_self">convergent_to_0</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font> <b>by </b><i>, , </i>;<br/><a NAME="E14:35_1"/><i><font color="Green">E67</font></i>: 
( <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font> )
 <b>by </b><i>, </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="E16:35_1"/><i><font color="Green">E68</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"><span class="p5">(<span class="default"><span class="p0"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</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"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</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">r</font>
 <b>by </b><i>, , <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/><a NAME="E17:35_1"/><i><font color="Green">E69</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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></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"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</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">r</font>
 <b>by </b><i/>;<br/><a NAME="E18:35_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/><a NAME="E19:35_1"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font>
 ;<br/><i><font color="Green">E71</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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></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"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</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"><span class="p4">(<span class="default"><span class="p5"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</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"><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"><span class="p5"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</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"><span class="p4">(<span class="default"><span class="p5"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 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"><span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</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"><span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></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/><a NAME="E21:35_1"/>
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/><a NAME="E22:35_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/><a NAME="E23:35_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/><a NAME="E24:35_1"/><b>then </b><i><font color="Green">E117</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></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/><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"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></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"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></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/><b>hence </b><a NAME="E26:35_1"/>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">d</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> ) )
 <b>by </b><i>, <a class="ref" href="ndiff_1.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_2"/><i><font color="Green">E118</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">d</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> ) )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be  <a href="ndiff_1.html#V2" target="_self">convergent_to_0</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>;<br/><a NAME="E1:35_2_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">s</font> <a href="ndiff_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i/>;<br/><a NAME="E2:35_2_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">s</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> )
 <b>by </b><i/>;<br/><div><i><font color="Green">E121</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:35_2_1_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>consider </b><font color="Maroon">d</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:35_2_1_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E4:35_2_1_1"/><i><font color="Green">E124</font></i>: 
for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </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="E6:35_2_1_1"/><i><font color="Green">E125</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"><font color="Maroon">s</font> <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"><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> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</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">n</font> = <font color="Maroon">n</font>;<br/><b>thus </b><a NAME="E7:35_2_1_1"/>
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"><span class="p5">(<span class="default"><span class="p0"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</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"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</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">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_1_1"><b>proof </b></a><div class="add">

<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="E1:35_2_1_1_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:35_2_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"><font color="Maroon">s</font> <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"><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> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:35_2_1_1_1"/><b>then </b><i><font color="Green">E127</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>;<br/>
<a NAME="E4:35_2_1_1_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:35_2_1_1_1"/>
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:35_2_1_1_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/>
<a NAME="E7:35_2_1_1_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/>
<a NAME="E8:35_2_1_1_1"/><b>then </b><i><font color="Green">E129</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></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="E9:35_2_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E10:35_2_1_1_1"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font>
 
;<br/>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="ndiff_1.html#T1" target="_self">Th1</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</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"><span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></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"><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"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</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"><span class="p4">(<span class="default"><span class="p5"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</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"><span class="p4">(<span class="default"><span class="p5"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 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"><span class="p4">(<span class="default"><span class="p5"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</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"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span></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/>
<b>hence </b><a NAME="E12:35_2_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"><span class="p5">(<span class="default"><span class="p0"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p5">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</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"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</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">r</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:35_2_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</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/><b>hence </b><a NAME="E5:35_2_1"/>
 <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font>
 <b>by </b><i>, <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="E3:35_2"/>
<font color="Maroon">R</font> is <a href="ndiff_1.html#V3" target="_self">REST-like</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:35"/>
( <font color="Maroon">R</font> is <a href="ndiff_1.html#V3" target="_self">REST-like</a> iff for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">d</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> ) ) )
 <b>by </b><i/>;<br/>


</div>
