<?xml version="1.0"?>
<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">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>let </b><font color="Maroon">N</font> be    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font>;<br/>









<b>assume </b><a NAME="E1:54"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">f</font> <a href="ndiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">x0</font> &amp; <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> )
 ;<br/>

<b>let </b><font color="Maroon">h</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/><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><a NAME="E2:54"/><i><font color="Green">E45</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> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <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/>

<b>consider </b><font color="Maroon">N1</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E4:54"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">N1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">L</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> ex <font color="Olive">R</font> being  <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> st <br/>for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">L</font> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">L</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>, <font color="Maroon">R</font> being   <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:54"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">N2</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E8:54"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">N2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font> &amp; <font color="Maroon">N2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font> )
 <b>by </b><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="E10:54"/><i><font color="Green">E50</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> &amp; <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">N2</font> )
 <b>by </b><i><a class="ref" href="nfcont_1.html#D3">NFCONT_1:def 3</a></i>;<br/>
<a NAME="E11:54"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 
<b>by </b><i><a class="ref" href="nfcont_1.html#T4">NFCONT_1:4</a></i>;<br/>
<a NAME="E12:54"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Olive">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">N2</font> &amp;  <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="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span> <a href="lopban_3.html#K3">^\</a> <font color="Olive">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">N2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<a NAME="E1:54_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c</font> is <a href="normsp_1.html#V4">convergent</a>
 
<b>by </b><i/>;<br/>
<a NAME="E2:54_1"/>
<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="E3:54_1"/><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="E4:54_1"/><b>then </b><i><font color="Green">E54</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/>;<br/>
<a NAME="E5:54_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">h</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">h</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/>
<b>then </b><i><font color="Green">E60</font></i>:  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <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="txt" href="ndiff_1.html#E6:54"><i><font color="Green">E47</font></i></a>, <a class="txt" href="ndiff_1.html#E8:54"><i><font color="Green">E48</font></i></a>, <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="E7:54_1"/>
<font color="Maroon">h</font> <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="txt" href="ndiff_1.html#E6:54"><i><font color="Green">E47</font></i></a>, , <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="E9:54_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <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="txt" href="ndiff_1.html#E2:54"><i><font color="Green">E45</font></i></a>, <a class="txt" href="ndiff_1.html#E10:54"><i><font color="Green">E50</font></i></a>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/>
<a NAME="E10:54_1"/>
<font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font> is    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:54_1"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <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>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E12:54_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">N2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2"><b>proof </b></a><div class="add">

<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="E1:54_1_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"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<b>hence </b><a NAME="E2:54_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E4:54"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<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="E13:54_1"/>
<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"><font color="Maroon">h</font> <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>
 ;<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="E15:54_1"/><i><font color="Green">E64</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"><font color="Maroon">h</font> <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>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T6">NFCONT_1:6</a></i>;<br/>
<a NAME="E16:54_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E17:54_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <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="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E18:54_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <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><a class="txt" href="ndiff_1.html#E11:54"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E19:54_1"/><b>then </b><i><font color="Green">E65</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"><font color="Maroon">h</font> <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> <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="lopban_3.html#D6">LOPBAN_3:def 6</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><a class="txt" href="ndiff_1.html#E1:54_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E21:54_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</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">z</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><a class="txt" href="ndiff_1.html#E1:54_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="ndiff_1.html#E4:54_1"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>hence </b><a NAME="E22:54_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E2:54"><i><font color="Green">E45</font></i></a></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="E14:54"/><i><font color="Green">E66</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">N2</font> &amp;  <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="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="tarski.html#R1">c=</a> <font color="Maroon">N2</font> )
 ;<br/>
<a NAME="E15:54"/><i><font color="Green">E67</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </b></a><div class="add">

<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/>

<a NAME="E1:54_2"/>
<font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font> is    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:54_2"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E3:54_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"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<a NAME="E4:54_2"/><b>then </b>
<font color="Maroon">y</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/>
<a NAME="E5:54_2"/><b>then </b>
<font color="Maroon">y</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/>
<b>hence </b><a NAME="E6:54_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E16:54"/><i><font color="Green">E69</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="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="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><b>proof </b></a><div class="add">

<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="E1:54_3"/>
<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"><font color="Maroon">h</font> <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>
 ;<br/>

<a NAME="E2:54_3"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E6:54"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E3:54_3"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:54_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E17:54"/><i><font color="Green">E70</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4"><b>proof </b></a><div class="add">

<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="E1:54_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c</font>
 ;<br/>

<a NAME="E2:54_4"/><b>then </b>
<font color="Maroon">y</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/>
<a NAME="E3:54_4"/><b>then </b>
<font color="Maroon">y</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/>
<b>hence </b><a NAME="E4:54_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E18:54"/><i><font color="Green">E71</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5"><b>proof </b></a><div class="add">

<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="E1:54_5"/>
<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"><font color="Maroon">h</font> <a href="normsp_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span>
 ;<br/>

<a NAME="E2:54_5"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:54_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E19:54"/><i><font color="Green">E117</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <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="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="Olive">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_6"><b>proof </b></a><div class="add">

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

<a NAME="E1:54_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <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">k</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"><font color="Maroon">h</font> <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>
 
<b>by </b><i><a class="ref" href="nfcont_1.html#T6">NFCONT_1:6</a></i>;<br/>
<a NAME="E2:54_6"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <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">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E6:54"><i><font color="Green">E47</font></i></a></i>;<br/>
<i><font color="Green">E119</font></i>: <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="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">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <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">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</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="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></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"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</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"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</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="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</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">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<a NAME="E4:54_6"/><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nfcont_1.html#T6">NFCONT_1:6</a></i>;<br/>
<a NAME="E5:54_6"/>
<font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font> is    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:54_6"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:54_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</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/>
<b>hence </b><a NAME="E8:54_6"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <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="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">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:54"><i><font color="Green">E43</font></i></a>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:54"/>
 <a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#G1">NORMSTR</a>(# <span class="p1">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K10">Zero_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="lopban_1.html#D15">LOPBAN_1:def 15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">L</font> = <font color="Maroon">L</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> ;<br/>
<a NAME="E22:54"/><i><font color="Green">E121</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <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="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E23:54"/><b>then </b><i><font color="Green">E122</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
 
;<br/>
<a NAME="E24:54"/>
<font color="Maroon">R</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i/>;<br/>
<a NAME="E25:54"/><b>then </b>
 <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="E26:54"/><b>then </b><i><font color="Green">E123</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font>
 
;<br/>
<b>reconsider </b><font color="Maroon">LP</font> =  <a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font> as   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> ;<br/>
<a NAME="E28:54"/><i><font color="Green">E124</font></i>: 
<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="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></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">LP</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="normsp_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><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="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></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> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> = 
<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="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></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</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"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</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"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <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="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">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></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"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <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="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">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E8:54"><i><font color="Green">E48</font></i></a>, <a class="ref" href="nfcont_1.html#T8">NFCONT_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K9">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#D12">LOPBAN_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">LP</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">LP</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</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"><font color="Maroon">LP</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></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"><font color="Maroon">LP</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="normsp_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</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>
<b>hence </b><a NAME="E2:54_7"/>
<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="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></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">LP</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="normsp_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:54"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</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"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</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/>;<br/>
<a NAME="E30:54"/><i><font color="Green">E126</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <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="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E127</font></i>: <font color="Maroon">LP</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
0 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#D6">LOPBAN_1:def 6</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
;<br/>
<a NAME="E32:54"/>
<font color="Maroon">LP</font> <a href="nfcont_1.html#R5">is_Lipschitzian_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_9"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">LL</font> =  <a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>
<b>thus </b><a NAME="E1:54_9"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">LP</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nfcont_1.html#D13">NFCONT_1:def 13</a><br/>

<b>consider </b><font color="Maroon">K</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:54_9"/><i><font color="Green">E128</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> &amp; ( for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">S</font> holds  <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"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">x</font></span><a href="normsp_1.html#K1">.||</a></span> ) )
 <b>by </b><i><a class="ref" href="lopban_1.html#D9">LOPBAN_1:def 9</a></i>;<br/>
<a NAME="E4:54_9"/><i><font color="Green">E129</font></i>: 
0 <a href="real_1.html#K3">+</a> <font color="Maroon">K</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K3">+</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E5:54_9"/><i><font color="Green">E130</font></i>: 
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K3">+</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>take </b>
<font color="Maroon">K</font> <a href="real_1.html#K3">+</a> 1
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_9_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>, <font color="Maroon">x2</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:54_9_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 ;<br/><a NAME="E2:54_9_1"/><i><font color="Green">E151</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"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x2</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">K</font> <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">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i/>;<br/><a NAME="E3:54_9_1"/>
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">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></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="E4:54_9_1"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">K</font> <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">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="real_1.html#K3">+</a> 1</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">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></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/><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">LP</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">LP</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</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"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">LP</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</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"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</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"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29">RLVECT_1:29</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"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="lopban_1.html#D6">LOPBAN_1:def 6</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"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="lopban_1.html#D5">LOPBAN_1:def 5</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"><a href="lopban_1.html#K13">modetrans</a> <font color="Maroon">L</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x2</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#T29">RLVECT_1:29</a></i>
;<br/><b>hence </b><a NAME="E6:54_9_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">LP</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">LP</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="real_1.html#K3">+</a> 1</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">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <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="E7:54_9"/>
( not <font color="Maroon">K</font> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> or <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">LP</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">LP</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="real_1.html#K3">+</a> 1</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="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> ) ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E33:54"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">LP</font> <a href="nfcont_1.html#R3">is_continuous_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="nfcont_1.html#T52">NFCONT_1:52</a></i>;<br/>
<a NAME="E34:54"/>
( <font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</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/>;<br/>
<a NAME="E35:54"/><b>then </b>
( <font color="Maroon">LP</font> <a href="nfcont_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</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"><font color="Maroon">LP</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</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="nfcont_1.html#T25">NFCONT_1:25</a></i>;<br/>
<a NAME="E36:54"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">LP</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="normsp_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></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"><font color="Maroon">LP</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="normsp_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="normsp_1.html#T34">NORMSP_1:34</a>, <a class="ref" href="normsp_1.html#T42">NORMSP_1:42</a></i>;<br/>
<a NAME="E37:54"/><b>then </b><i><font color="Green">E154</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">LP</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="normsp_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></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"><font color="Maroon">LP</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="normsp_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></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="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>
<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="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></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span> = 
<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"><font color="Maroon">h</font> <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#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E11:54"><i><font color="Green">E51</font></i></a>, </i>
<br/>.= 
<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"><font color="Maroon">h</font> <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#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="lopban_3.html#K3">^\</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E10:54"><i><font color="Green">E50</font></i></a>, </i>
<br/>.= 
<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"><font color="Maroon">h</font> <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="lopban_3.html#K3">^\</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E39:54"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <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="p1">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">c</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"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <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="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="ndiff_1.html#T2" target="_self">Th2</a>, , <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>
