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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">Z</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><a NAME="E1:59"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Z</font> is <a href="nfcont_1.html#V3">open</a>
 ;<br/>

<b>let </b><font color="Maroon">f1</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">f2</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:59"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">Z</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">f1</font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">f2</font></span>)</span>
 <b>and </b><br/><a NAME="E3:59"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">f1</font> <a href="ndiff_1.html#R2" target="_self">is_differentiable_on</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E4:59"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f2</font> <a href="ndiff_1.html#R2" target="_self">is_differentiable_on</a> <font color="Maroon">Z</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:59_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><a NAME="E2:59_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">f1</font> <a href="ndiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:59"><i><font color="Green">E43</font></i></a>, <a class="txt" href="ndiff_1.html#E2:59"><i><font color="Green">E45</font></i></a>, </i>;<br/><a NAME="E3:59_1"/>
<font color="Maroon">f2</font> <a href="ndiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:59"><i><font color="Green">E43</font></i></a>, <a class="txt" href="ndiff_1.html#E3:59"><i><font color="Green">E46</font></i></a>, <a class="txt" href="ndiff_1.html#E4:59"><i><font color="Green">E47</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:59_1"/>
<font color="Maroon">f1</font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">f2</font> <a href="ndiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:59_1"><i><font color="Green">E48</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:59"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f1</font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">f2</font> <a href="ndiff_1.html#R2" target="_self">is_differentiable_on</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:59"><i><font color="Green">E43</font></i></a>, , </i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:59_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><a NAME="E2:59_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">f1</font> <a href="ndiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:59"><i><font color="Green">E43</font></i></a>, <a class="txt" href="ndiff_1.html#E2:59"><i><font color="Green">E45</font></i></a>, </i>;<br/><a NAME="E3:59_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">f2</font> <a href="ndiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:59"><i><font color="Green">E43</font></i></a>, <a class="txt" href="ndiff_1.html#E3:59"><i><font color="Green">E46</font></i></a>, , </i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">f2</font></span>)</span> <a href="ndiff_1.html#K4" target="_self">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> = 
 <a href="ndiff_1.html#K3" target="_self">diff</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E4:59"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="ndiff_1.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ndiff_1.html#K3" target="_self">diff</a> <font color="Maroon">f1</font>,<font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" target="_self">diff</a> <font color="Maroon">f2</font>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:59"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">f2</font></span>)</span> <a href="ndiff_1.html#K4" target="_self">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</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">f1</font>,<font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" target="_self">diff</a> <font color="Maroon">f2</font>,<font color="Olive">x</font></span>)</span>
 ;<br/>


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