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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E44">Th53</font></span>: <a NAME="T52"><span class="comment"><font color="firebrick">:: NDIFF13:52</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">E</font>, <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">G</font> being   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/>  for <font color="Olive" title="b5">Z</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="prvect_3.html#K9" title="PRVECT_3:func.9">[:</a><span class="default"><font color="Olive" title="b2">E</font>,<font color="Olive" title="b3">F</font></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span><br/>  for <font color="Olive" title="b6">f</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1"><a href="prvect_3.html#K9" title="PRVECT_3:func.9">[:</a><span class="default"><font color="Olive" title="b2">E</font>,<font color="Olive" title="b3">F</font></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span>,<font color="Olive" title="b4">G</font>  st <font color="Olive" title="b6">f</font> <a href="ndiff_6.html#R1" title="NDIFF_6:pred.1">is_differentiable_on</a> <font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1,<font color="Olive" title="b5">Z</font> &amp;  <a href="ndiff_6.html#K6" title="NDIFF_6:func.6">diff</a> (<font color="Olive" title="b6">f</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b5">Z</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b5">Z</font> holds <br/>( <font color="Olive" title="b6">f</font> <a href="ndiff_7.html#K9" title="NDIFF_7:func.9">`partial`1|</a> <font color="Olive" title="b5">Z</font> <a href="ndiff_6.html#R1" title="NDIFF_6:pred.1">is_differentiable_on</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b5">Z</font> &amp;  <a href="ndiff_6.html#K6" title="NDIFF_6:func.6">diff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b6">f</font> <a href="ndiff_7.html#K9" title="NDIFF_7:func.9">`partial`1|</a> <font color="Olive" title="b5">Z</font></span>)</span>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b5">Z</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b5">Z</font> &amp; <font color="Olive" title="b6">f</font> <a href="ndiff_7.html#K10" title="NDIFF_7:func.10">`partial`2|</a> <font color="Olive" title="b5">Z</font> <a href="ndiff_6.html#R1" title="NDIFF_6:pred.1">is_differentiable_on</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b5">Z</font> &amp;  <a href="ndiff_6.html#K6" title="NDIFF_6:func.6">diff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b6">f</font> <a href="ndiff_7.html#K10" title="NDIFF_7:func.10">`partial`2|</a> <font color="Olive" title="b5">Z</font></span>)</span>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b5">Z</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b5">Z</font> )</div></div>
