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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</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">Z</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>;<br/>





<b>assume </b><a NAME="E1:60"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c3">Z</font> is <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:60"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c3">Z</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>
 <b>and </b><br/><a NAME="E3:60"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c4">f1</font> <a href="ndiff_1.html#R2" title="NDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c3">Z</font>
 <b>and </b><br/><a NAME="E4:60"/><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c5">f2</font> <a href="ndiff_1.html#R2" title="NDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c3">Z</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" title="c6">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>;<br/><b>assume </b><a NAME="E1:60_1"/><i><font color="Green" title="E47">A5</font></i>: 
<font color="Maroon" title="c6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font>
 ;<br/><a NAME="E2:60_1"/><b>then </b><i><font color="Green" title="E48">A6</font></i>: 
<font color="Maroon" title="c4">f1</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x0</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:60"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="ndiff_1.html#E3:60"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="ndiff_1.html#T36" target="_self" title="NDIFF_1:th.36">Th36</a></i>;<br/><a NAME="E3:60_1"/>
<font color="Maroon" title="c5">f2</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x0</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:60"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="ndiff_1.html#E4:60"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="ndiff_1.html#E1:60_1"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="ndiff_1.html#T36" target="_self" title="NDIFF_1:th.36">Th36</a></i>;<br/><b>hence </b><a NAME="E4:60_1"/>
<font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f2</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x0</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E2:60_1"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="ndiff_1.html#T41" target="_self" title="NDIFF_1:th.41">Th41</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:60"/><i><font color="Green" title="E47">A7</font></i>: 
<font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f2</font> <a href="ndiff_1.html#R2" title="NDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c3">Z</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:60"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="ndiff_1.html#E2:60"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="ndiff_1.html#T36" target="_self" title="NDIFF_1:th.36">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" title="c6">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>;<br/><b>assume </b><a NAME="E1:60_2"/><i><font color="Green" title="E48">A8</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font>
 ;<br/><a NAME="E2:60_2"/><b>then </b><i><font color="Green" title="E49">A9</font></i>: 
<font color="Maroon" title="c4">f1</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:60"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="ndiff_1.html#E3:60"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="ndiff_1.html#T36" target="_self" title="NDIFF_1:th.36">Th36</a></i>;<br/><a NAME="E3:60_2"/><i><font color="Green" title="E50">A10</font></i>: 
<font color="Maroon" title="c5">f2</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:60"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="ndiff_1.html#E4:60"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="ndiff_1.html#E1:60_2"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="ndiff_1.html#T36" target="_self" title="NDIFF_1:th.36">Th36</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font> = 
 <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E6:60"><i><font color="Green" title="E47">A7</font></i></a>, <a class="txt" href="ndiff_1.html#E1:60_2"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="ndiff_1.html#D9" target="_self" title="NDIFF_1:def.9">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c6">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c6">x</font></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E2:60_2"><i><font color="Green" title="E49">A9</font></i></a>, <a class="txt" href="ndiff_1.html#E3:60_2"><i><font color="Green" title="E50">A10</font></i></a>, <a class="ref" href="ndiff_1.html#T41" target="_self" title="NDIFF_1:th.41">Th41</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:60"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</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="c4">f1</font>,<font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> <font color="Maroon" title="c5">f2</font>,<font color="Olive" title="b1">x</font></span>)</span>
 ;<br/>


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