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

<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c4">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>;<br/>







<b>assume </b><a NAME="E1:67"/>
<font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x0</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">N</font> being    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c4">x0</font><b> such that </b><br/><a NAME="E3:67"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E4:67"/><i><font color="Green" title="E45">A2</font></i>: 
 ex <font color="Olive" title="b1">R</font> being  <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font> st <br/>for <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ndiff_1.html#D7" target="_self" title="NDIFF_1:def.7">Def7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">N</font>
;<br/>

<a NAME="E5:67"/>
 ex <font color="Olive" title="b1">R</font> being  <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">T</font> &amp; <font color="Olive" title="b1">R</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font> &amp; ( for <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c6">R</font> being   <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E2:67_1"/><i><font color="Green" title="E46">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E4:67"><i><font color="Green" title="E45">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">R</font>
;<br/>

<a NAME="E3:67_1"/>
 <a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#G1" title="NORMSP_1:aggr.1">NORMSTR</a>(# <span class="p1">(<span class="default"><a href="lopban_1.html#K10" title="LOPBAN_1:func.10">BoundedLinearOperators</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K10" title="RSSPACE:func.10">Zero_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10" title="LOPBAN_1:func.10">BoundedLinearOperators</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8" title="LOPBAN_1:func.8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K8" title="RSSPACE:func.8">Add_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10" title="LOPBAN_1:func.10">BoundedLinearOperators</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8" title="LOPBAN_1:func.8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K9" title="RSSPACE:func.9">Mult_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10" title="LOPBAN_1:func.10">BoundedLinearOperators</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8" title="LOPBAN_1:func.8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="lopban_1.html#K15" title="LOPBAN_1:func.15">BoundedLinearOperatorsNorm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="lopban_1.html#D15" title="LOPBAN_1:def.15">LOPBAN_1:def 15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">L</font> =  <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">x0</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lopban_1.html#K10" title="LOPBAN_1:func.10">BoundedLinearOperators</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font> ;<br/>
<i><font color="Green" title="E47">A4</font></i>: <font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K9" title="LOPBAN_1:func.9">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <font color="Maroon" title="c7">L</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#D12" title="LOPBAN_1:def.12">LOPBAN_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <font color="Maroon" title="c7">L</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K13" title="LOPBAN_1:func.13">modetrans</a> <font color="Maroon" title="c7">L</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#D6" title="LOPBAN_1:def.6">LOPBAN_1:def 6</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">T</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>
;<br/>
<a NAME="E6:67_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K9" title="LOPBAN_1:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E2:67_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="nfcont_1.html#T4" title="NFCONT_1:th.4">NFCONT_1:4</a></i>;<br/>
<a NAME="E7:67_1"/><b>then </b>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K9" title="LOPBAN_1:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>;<br/>
<a NAME="E8:67_1"/><b>then </b>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K9" title="LOPBAN_1:func.9">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x0</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>;<br/>
<a NAME="E9:67_1"/><b>then </b>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K9" title="LOPBAN_1:func.9">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>;<br/>
<b>hence </b><a NAME="E10:67_1"/><i><font color="Green" title="E48">A5</font></i>: 
<font color="Maroon" title="c6">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E5:67_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>;<br/>

<a NAME="E11:67_1"/><i><font color="Green" title="E49">A6</font></i>: 
( <font color="Maroon" title="c6">R</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; ( for <font color="Olive" title="b1">h</font> being <a href="ndiff_1.html#V2" title="NDIFF_1:attr.2">convergent_to_0</a> <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Olive" title="b1">h</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">h</font></span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Olive" title="b1">h</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">T</font> ) ) )
 
<b>by </b><i><a class="ref" href="ndiff_1.html#D5" target="_self" title="NDIFF_1:def.5">Def5</a></i>;<br/>
<div><i><font color="Green" title="E50">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">s1</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>;<br/><b>assume </b><a NAME="E1:67_1_2"/><i><font color="Green" title="E51">A8</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">R</font> &amp; <font color="Maroon" title="c8">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c8">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c8">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font> ) )
 ;<br/><a NAME="E2:67_1_2"/><b>then </b>
<font color="Maroon" title="c8">s1</font> <a href="ndiff_1.html#NR1" title="NDIFF_1:NR.1">is_not_0</a> 
 <b>by </b><i><a class="ref" href="ndiff_1.html#T7" target="_self" title="NDIFF_1:th.7">Th7</a></i>;<br/><a NAME="E3:67_1_2"/><b>then </b>
<font color="Maroon" title="c8">s1</font> is  <a href="ndiff_1.html#V2" title="NDIFF_1:attr.2">convergent_to_0</a> <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:67_1_2"><i><font color="Green" title="E51">A8</font></i></a>, <a class="ref" href="ndiff_1.html#D4" target="_self" title="NDIFF_1:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E4:67_1_2"/>
( <font color="Maroon" title="c6">R</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c8">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span> )
 <b>by </b><i><a class="txt" href="ndiff_1.html#E10:67_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="ndiff_1.html#T27" target="_self" title="NDIFF_1:th.27">Th27</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:67_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E11:67_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E14:67_1"/>
( <font color="Maroon" title="c6">R</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="ndiff_1.html#E2:67_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="ndiff_1.html#E12:67_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="ndiff_1.html#T32" target="_self" title="NDIFF_1:th.32">Th32</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:67"/>
( <font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp;  ex <font color="Olive" title="b1">R</font> being  <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">T</font> &amp; <font color="Olive" title="b1">R</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font> &amp; ( for <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">x0</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="ndiff_1.html#E3:67"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/>


</div>
