<?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="E1:10"/><i><font color="Green" title="E6">A1</font></i>: 
<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>let </b><font color="Maroon" title="c5">z</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>consider </b><font color="Maroon" title="c6">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="E3:10"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c6">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="E4:10"/><i><font color="Green" title="E8">A3</font></i>: 
for <font color="Olive" title="b1">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">h</font> being <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/> for <font color="Olive" title="b3">c</font> being <a href="funct_1.html#V3" title="FUNCT_1:attr.3">V6</a> <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">N</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">h</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Olive" title="b3">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b3">c</font></span>)</span></span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <span class="p1">(<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> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><font color="Olive" title="b2">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b2">h</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Olive" title="b3">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b3">c</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="ndiff_2.html#E1:10"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="ndiff_2.html#T1" target="_self" title="NDIFF_2:th.1">Th1</a></i>;<br/>
<a NAME="E5:10"/><i><font color="Green" title="E9">A4</font></i>: 
for <font color="Olive" title="b1">h</font> being <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/> for <font color="Olive" title="b2">c</font> being <a href="funct_1.html#V3" title="FUNCT_1:attr.3">V6</a> <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Olive" title="b2">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">N</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">h</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <span class="p1">(<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> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><font color="Olive" title="b1">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">h</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="ndiff_2.html#E4:10"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/>
<a NAME="E6:10"/><b>then </b><i><font color="Green" title="E10">A5</font></i>: 
for <font color="Olive" title="b1">e</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> ex <font color="Olive" title="b2">d</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b3">h</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Olive" title="b3">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">d</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Olive" title="b3">h</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">N</font> holds <br/><span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">h</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Olive" title="b3">h</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p5">(<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></span>)</span></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<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> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">e</font> ) )
 
<b>by </b><i><a class="txt" href="ndiff_2.html#E3:10"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="ndiff_2.html#T3" target="_self" title="NDIFF_2:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E7:10"/>
<font color="Maroon" title="c3">f</font> <a href="ndiff_2.html#R1" title="NDIFF_2:pred.1">is_Gateaux_differentiable_in</a> <font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="txt" href="ndiff_2.html#E3:10"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="ndiff_2.html#D3" target="_self" title="NDIFF_2:def.3">Def3</a></i>;<br/>

<b>hence </b><a NAME="E8:10"/>
 <a href="ndiff_2.html#K3" title="NDIFF_2:func.3">Gateaux_diff</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="txt" href="ndiff_2.html#E3:10"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="ndiff_2.html#E6:10"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="ndiff_2.html#D4" target="_self" title="NDIFF_2:def.4">Def4</a></i>;<br/>

<b>hence </b><a NAME="E9:10"/>
 ex <font color="Olive" title="b1">N</font> being   <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c4">x0</font> st <br/>( <font color="Olive" title="b1">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; ( for <font color="Olive" title="b2">h</font> being <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/> for <font color="Olive" title="b3">c</font> being <a href="funct_1.html#V3" title="FUNCT_1:attr.3">V6</a> <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">N</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">h</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Olive" title="b3">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b3">c</font></span>)</span></span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="ndiff_2.html#K3" title="NDIFF_2:func.3">Gateaux_diff</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><font color="Olive" title="b2">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b2">h</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Olive" title="b3">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b3">c</font></span>)</span></span>)</span></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="ndiff_2.html#E3:10"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="ndiff_2.html#E5:10"><i><font color="Green" title="E9">A4</font></i></a></i>;<br/>


</div>
