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<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">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/>









<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1"/>
<font color="Maroon">f</font> <a href="ndiff_2.html#R1" target="_self">is_Gateaux_differentiable_in</a> <font color="Maroon">x0</font>,<font color="Maroon">z</font>
 ;<br/><b>then consider </b><font color="Maroon">N</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E3:9_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:9_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">h</font> being  <a href="real_1.html#NM1">Real</a>  st  <a href="complex1.html#K18">abs</a> <font color="Olive">h</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">d</font> &amp; <font color="Olive">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</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="Olive">h</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Olive">h</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="ndiff_2.html#K3" target="_self">Gateaux_diff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</font> ) )
 <b>by </b><i/>;<br/><a NAME="E5:9_1"/>
for <font color="Olive">h</font> being <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a><br/> for <font color="Olive">c</font> being <a href="funct_1.html#V3">V6</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">c</font></span>)</span></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="ndiff_2.html#K3" target="_self">Gateaux_diff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">c</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:9_1"/>
 ex <font color="Olive">N</font> being   <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> st <br/>( <font color="Olive">N</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">dv</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> st <br/>for <font color="Olive">h</font> being <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a><br/> for <font color="Olive">c</font> being <a href="funct_1.html#V3">V6</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">N</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">c</font></span>)</span></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Olive">dv</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">c</font></span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_2"/>
 ex <font color="Olive">N</font> being   <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> st <br/>( <font color="Olive">N</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">dv</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> st <br/>for <font color="Olive">h</font> being <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a><br/> for <font color="Olive">c</font> being <a href="funct_1.html#V3">V6</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">N</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">c</font></span>)</span></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Olive">dv</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">c</font></span>)</span></span>)</span></span>)</span> ) )
 ;<br/><b>then consider </b><font color="Maroon">N</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E3:9_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:9_2"/><i><font color="Green">E44</font></i>: 
 ex <font color="Olive">dv</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> st <br/>for <font color="Olive">h</font> being <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a><br/> for <font color="Olive">c</font> being <a href="funct_1.html#V3">V6</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">c</font></span>)</span></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Olive">dv</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">c</font></span>)</span></span>)</span></span>)</span> )
 ;<br/><b>consider </b><font color="Maroon">dv</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:9_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">h</font> being <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a><br/> for <font color="Olive">c</font> being <a href="funct_1.html#V3">V6</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">c</font></span>)</span></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">dv</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">c</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E7:9_2"/>
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">h</font> being  <a href="real_1.html#NM1">Real</a>  st  <a href="complex1.html#K18">abs</a> <font color="Olive">h</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">d</font> &amp; <font color="Olive">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</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="Olive">h</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Olive">h</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">dv</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</font> ) )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E8:9_2"/>
<font color="Maroon">f</font> <a href="ndiff_2.html#R1" target="_self">is_Gateaux_differentiable_in</a> <font color="Maroon">x0</font>,<font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:9"/>
( <font color="Maroon">f</font> <a href="ndiff_2.html#R1" target="_self">is_Gateaux_differentiable_in</a> <font color="Maroon">x0</font>,<font color="Maroon">z</font> iff  ex <font color="Olive">N</font> being   <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> st <br/>( <font color="Olive">N</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">dv</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> st <br/>for <font color="Olive">h</font> being <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a><br/> for <font color="Olive">c</font> being <a href="funct_1.html#V3">V6</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">N</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">c</font></span>)</span></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Olive">dv</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive">h</font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">c</font></span>)</span></span>)</span></span>)</span> ) ) )
 <b>by </b><i>, </i>;<br/>


</div>
