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

<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>assume </b><a NAME="E1:4"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="ndiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x0</font>
 ;<br/>

<b>let </b><font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">h</font> be  <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">c</font> be  <a href="funct_1.html#V3">V6</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E2:4"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E3:4"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>consider </b><font color="Maroon">N</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E5:4"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E6:4"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font><br/> for <font color="Olive">h</font> being <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a><br/> for <font color="Olive">c</font> being <a href="funct_1.html#V3">V6</a> <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">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Olive">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Olive">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">c</font></span>)</span></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font></span>)</span> <a href="lopban_1.html#K17">.</a> <font color="Olive">z</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="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Olive">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">c</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:4"/><i><font color="Green">E44</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E9:4"/><i><font color="Green">E45</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#D3">NFCONT_1:def 3</a></i>;<br/>
<a NAME="E10:4"/><i><font color="Green">E46</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">c</font> <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="4_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="E1:4_1"/>
<font color="Maroon">c</font> <a href="normsp_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">c</font>
 
<b>by </b><i><a class="ref" href="nfcont_1.html#T6">NFCONT_1:6</a></i>;<br/>
<b>hence </b><a NAME="E2:4_1"/>
<font color="Maroon">c</font> <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="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:4"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Olive">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<a NAME="E1:4_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c</font> is <a href="normsp_1.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="ndiff_1.html#T21">NDIFF_1:21</a></i>;<br/>
<a NAME="E2:4_2"/>
<font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="nfcont_1.html#T6">NFCONT_1:6</a></i>;<br/>
<a NAME="E3:4_2"/><b>then </b>
<font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:4_2"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="ref" href="ndiff_1.html#T21">NDIFF_1:21</a></i>;<br/>
<a NAME="E5:4_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:4_2_1_1"/><i><font color="Green">E53</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:4_2_1_1"/>
 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">z</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="4_2_1_1_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">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> 1</span>)</span>;<br/>

<a NAME="E1:4_2_1_1_1"/><i><font color="Green">E54</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">z</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="E2:4_2_1_1_1"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</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="E3:4_2_1_1_1"/><b>then </b><i><font color="Green">E56</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">z</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="E4:4_2_1_1_1"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z</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">z</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="E5:4_2_1_1_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">z</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="E6:4_2_1_1_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">z</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">z</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="E7:4_2_1_1_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">z</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="E4:4_2_1_1"/><i><font color="Green">E58</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">z</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="E5:4_2_1_1"/>
( <font color="Maroon">h</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:4_2_1_1"/><i><font color="Green">E59</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/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">pp</font>
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</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="E8:4_2_1_1"/><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="E9:4_2_1_1"/><i><font color="Green">E62</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">pp</font>
 <b>by </b><i>, </i>;<br/><i><font color="Green">E63</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"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</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"><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> = 
<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">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</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><a class="ref" href="ndiff_1.html#D3">NDIFF_1:def 3</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">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</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"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</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">z</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/><a NAME="E11:4_2_1_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">z</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="E12:4_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</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">z</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">z</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="E13:4_2_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">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</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"><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>
<b>hence </b><a NAME="E2:4_2_1"/>
<font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</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="E3:4_2_1"/>
 <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <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/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E64</font></i>:  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font>
<b>by </b><i>, , <a class="ref" href="normsp_1.html#T42">NORMSP_1:42</a></i>
<br/>.= 
<font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<a NAME="E7:4_2"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font> is <a href="normsp_1.html#V4">convergent</a>
 
<b>by </b><i>, , <a class="ref" href="normsp_1.html#T34">NORMSP_1:34</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:4_2"/><i><font color="Green">E65</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"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</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>
;<br/>

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

<b>assume </b><a NAME="E10:4_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span>
 ;<br/>

<b>then 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="E12:4_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T6">NFCONT_1:6</a></i>;<br/>
<a NAME="E13:4_2"/>
<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">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E14:4_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"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:4_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"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D5">NORMSP_1:def 5</a></i>;<br/>
<a NAME="E16:4_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"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<a NAME="E17:4_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"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<a NAME="E18:4_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"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="ndiff_1.html#D3">NDIFF_1:def 3</a></i>;<br/>
<a NAME="E19:4_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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/>
<a NAME="E20:4_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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="ndiff_1.html#D3">NDIFF_1:def 3</a></i>;<br/>
<a NAME="E21:4_2"/><b>then </b><i><font color="Green">E67</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">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D5">NORMSP_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E23:4_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">w</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E24:4_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<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="E13:4"/><i><font color="Green">E68</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 ;<br/>
<a NAME="E14:4"/><i><font color="Green">E69</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font></span>)</span> <a href="lopban_1.html#K17">.</a> <font color="Maroon">z</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"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, , </i>;<br/>
<div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>now </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>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <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 class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <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">m</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6">LOPBAN_3:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D5">NORMSP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="ndiff_1.html#D3">NDIFF_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="ndiff_1.html#D3">NDIFF_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="normsp_1.html#D5">NORMSP_1:def 5</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E16:4"/>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="nfcont_1.html#T4">NFCONT_1:4</a></i>;<br/>
<a NAME="E17:4"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c</font> <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="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E18:4"/><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</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="E19:4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_1"><b>now </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/><i><font color="Green">E75</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <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 class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <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">m</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6">LOPBAN_3:def 6</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="nfcont_1.html#T8">NFCONT_1:8</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <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>
<b>by </b><i><a class="ref" href="lopban_3.html#D6">LOPBAN_3:def 6</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i/>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <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"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <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 class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <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">m</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6">LOPBAN_3:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ndiff_1.html#D2">NDIFF_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</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="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</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="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D6">NORMSP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</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="p1">(<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"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <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> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6">LOPBAN_3:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</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="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</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="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="nfcont_1.html#T8">NFCONT_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</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="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</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="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</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="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</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="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</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="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</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="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="nfcont_1.html#T8">NFCONT_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</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="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</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="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="nfcont_1.html#T8">NFCONT_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</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="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D6">NORMSP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</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="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="ndiff_1.html#D2">NDIFF_1:def 2</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:4_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:4"/>
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font></span>)</span> <a href="lopban_1.html#K17">.</a> <font color="Maroon">z</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">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="lopban_3.html#T15">LOPBAN_3:15</a>, <a class="ref" href="lopban_3.html#T16">LOPBAN_3:16</a></i>;<br/>


</div>
