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

<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="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:60"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="nfcont_1.html#V3">open</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:60"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:60"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="ndiff_1.html#R2" target="_self">is_differentiable_on</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:60"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ndiff_1.html#R2" target="_self">is_differentiable_on</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:60_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:60_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="ndiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:60"><i><font color="Green">E43</font></i></a>, <a class="txt" href="ndiff_1.html#E3:60"><i><font color="Green">E45</font></i></a>, <a class="ref" href="ndiff_1.html#T36" target="_self">Th36</a></i>;<br/><a NAME="E3:60_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="ndiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:60"><i><font color="Green">E43</font></i></a>, <a class="txt" href="ndiff_1.html#E4:60"><i><font color="Green">E46</font></i></a>, <a class="txt" href="ndiff_1.html#E1:60_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="ndiff_1.html#T36" target="_self">Th36</a></i>;<br/><b>hence </b><a NAME="E4:60_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="ndiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E2:60_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="ndiff_1.html#T41" target="_self">Th41</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:60"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="ndiff_1.html#R2" target="_self">is_differentiable_on</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:60"><i><font color="Green">E43</font></i></a>, <a class="txt" href="ndiff_1.html#E2:60"><i><font color="Green">E44</font></i></a>, <a class="ref" href="ndiff_1.html#T36" target="_self">Th36</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:60_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:60_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="ndiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:60"><i><font color="Green">E43</font></i></a>, <a class="txt" href="ndiff_1.html#E3:60"><i><font color="Green">E45</font></i></a>, <a class="ref" href="ndiff_1.html#T36" target="_self">Th36</a></i>;<br/><a NAME="E3:60_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ndiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:60"><i><font color="Green">E43</font></i></a>, <a class="txt" href="ndiff_1.html#E4:60"><i><font color="Green">E46</font></i></a>, <a class="txt" href="ndiff_1.html#E1:60_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="ndiff_1.html#T36" target="_self">Th36</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ndiff_1.html#K4" target="_self">`|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>6</sub></font> = 
 <a href="ndiff_1.html#K3" target="_self">diff</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E6:60"><i><font color="Green">E47</font></i></a>, <a class="txt" href="ndiff_1.html#E1:60_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="ndiff_1.html#D9" target="_self">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ndiff_1.html#K3" target="_self">diff</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" target="_self">diff</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E2:60_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="ndiff_1.html#E3:60_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="ndiff_1.html#T41" target="_self">Th41</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:60"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ndiff_1.html#K4" target="_self">`|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" target="_self">diff</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" target="_self">diff</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>


</div>
