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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E50">Th58</font></span>: <a NAME="T57"><span class="comment"><font color="firebrick">:: NDIFF13:57</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">S</font>, <font color="Olive" title="b2">E</font>, <font color="Olive" title="b3">F</font> being   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/>  for <font color="Olive" title="b4">u</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">E</font><br/>  for <font color="Olive" title="b5">v</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <font color="Olive" title="b1">S</font>,<font color="Olive" title="b3">F</font><br/>  for <font color="Olive" title="b6">w</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <font color="Olive" title="b1">S</font>,<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="b7">Z</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Olive" title="b4">u</font>,<font color="Olive" title="b5">v</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> &amp; <font color="Olive" title="b4">u</font> <a href="ndiff_1.html#R2" title="NDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b7">Z</font> &amp; <font color="Olive" title="b5">v</font> <a href="ndiff_1.html#R2" title="NDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b7">Z</font> holds <br/>(  <a href="ndiff_6.html#K6" title="NDIFF_6:func.6">diff</a> (<font color="Olive" title="b6">w</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b7">Z</font>) <a href="ndiff_1.html#R2" title="NDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b7">Z</font> &amp;  ex <font color="Olive" title="b8">T</font> being   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">Lipschitzian</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1"><a href="prvect_3.html#K9" title="PRVECT_3:func.9">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ndiff_6.html#K3" title="NDIFF_6:func.3">diff_SP</a> (1,<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">E</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="ndiff_6.html#K3" title="NDIFF_6:func.3">diff_SP</a> (1,<font color="Olive" title="b1">S</font>,<font color="Olive" title="b3">F</font>)</span>)</span></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span>,<span class="p1">(<span class="default"><a href="ndiff_6.html#K3" title="NDIFF_6:func.3">diff_SP</a> (1,<font color="Olive" title="b1">S</font>,<span class="p2"><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>)</span>)</span> st <br/>( <font color="Olive" title="b8">T</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="ndiff13.html#K3" title="NDIFF13:func.3">CTP</a> (<font color="Olive" title="b1">S</font>,<span class="p1">(<span class="default"><a href="ndiff_6.html#K3" title="NDIFF_6:func.3">diff_SP</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">E</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="ndiff_6.html#K3" title="NDIFF_6:func.3">diff_SP</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b1">S</font>,<font color="Olive" title="b3">F</font>)</span>)</span>) &amp;  <a href="ndiff_6.html#K6" title="NDIFF_6:func.6">diff</a> (<font color="Olive" title="b6">w</font>,1,<font color="Olive" title="b7">Z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">T</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="ndiff_6.html#K6" title="NDIFF_6:func.6">diff</a> (<font color="Olive" title="b4">u</font>,1,<font color="Olive" title="b7">Z</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="ndiff_6.html#K6" title="NDIFF_6:func.6">diff</a> (<font color="Olive" title="b5">v</font>,1,<font color="Olive" title="b7">Z</font>)</span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> ) )</div></div>
