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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:10"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="ndiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:10"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:10"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>2</sub></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">b<sub>3</sub></font> being <a href="funct_1.html#V3">V6</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>2</sub></font> st  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></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">b<sub>2</sub></font> <a href="ndiff_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></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">c<sub>3</sub></font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="ndiff_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lopban_1.html#K17">.</a> <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></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">c<sub>3</sub></font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="ndiff_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="ndiff_2.html#E1:10"><i><font color="Green">E6</font></i></a>, <a class="ref" href="ndiff_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:10"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font> being <a href="funct_1.html#V3">V6</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>2</sub></font> st  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></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">b<sub>1</sub></font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lopban_1.html#K17">.</a> <font color="Maroon">c<sub>5</sub></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">b<sub>1</sub></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">c<sub>3</sub></font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="ndiff_2.html#E4:10"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E6:10"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st  <a href="complex1.html#K18">abs</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></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">b<sub>3</sub></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">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lopban_1.html#K17">.</a> <font color="Maroon">c<sub>5</sub></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">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="ndiff_2.html#E3:10"><i><font color="Green">E7</font></i></a>, <a class="txt" href="ndiff_2.html#E5:10"><i><font color="Green">E9</font></i></a>, <a class="ref" href="ndiff_2.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E7:10"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ndiff_2.html#R1" target="_self">is_Gateaux_differentiable_in</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="ndiff_2.html#E3:10"><i><font color="Green">E7</font></i></a>, <a class="ref" href="ndiff_2.html#D3" target="_self">Def3</a></i>;<br/>

<b>hence </b><a NAME="E8:10"/>
 <a href="ndiff_2.html#K3" target="_self">Gateaux_diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3">diff</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lopban_1.html#K17">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="ndiff_2.html#E3:10"><i><font color="Green">E7</font></i></a>, <a class="txt" href="ndiff_2.html#E6:10"><i><font color="Green">E10</font></i></a>, <a class="ref" href="ndiff_2.html#D4" target="_self">Def4</a></i>;<br/>

<b>hence </b><a NAME="E9:10"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></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">b<sub>3</sub></font> being <a href="funct_1.html#V3">V6</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>2</sub></font> st  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></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">b<sub>2</sub></font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></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">c<sub>3</sub></font> <a href="nfcont_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></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">b<sub>2</sub></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">c<sub>3</sub></font> <a href="nfcont_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="ndiff_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="normsp_1.html#K3">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="ndiff_2.html#E3:10"><i><font color="Green">E7</font></i></a>, <a class="txt" href="ndiff_2.html#E5:10"><i><font color="Green">E9</font></i></a></i>;<br/>


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