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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E27">Th34</font></span>: <a NAME="T33"><span class="comment"><font color="firebrick">:: NDIFF13:33</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>, <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">B</font> being   <a href="lopban_9.html#V1" title="LOPBAN_9:attr.1">Lipschitzian</a>  <a href="lopban_8.html#NM2" title="LOPBAN_8:NM.2">BilinearOperator</a> of <font color="Olive" title="b2">E</font>,<font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">G</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>,<font color="Olive" title="b4">G</font><br/>  for <font color="Olive" title="b7">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="b8">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="b9">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="b10">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b8">u</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b10">x</font> &amp; <font color="Olive" title="b9">v</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b10">x</font> &amp; <font color="Olive" title="b10">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b7">w</font> &amp; <font color="Olive" title="b6">W</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">B</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b7">w</font> &amp; <font color="Olive" title="b7">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="b8">u</font>,<font color="Olive" title="b9">v</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> holds <br/>( <font color="Olive" title="b6">W</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b10">x</font> &amp; <font color="Olive" title="b7">w</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b10">x</font> &amp;  <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Olive" title="b6">W</font>,<font color="Olive" title="b10">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="Olive" title="b5">B</font>,<span class="p2"><a href="prvect_3.html#K10" title="PRVECT_3:func.10">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b8">u</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b10">x</font></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b9">v</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b10">x</font></span>)</span></span><a href="prvect_3.html#K10" title="PRVECT_3:func.10">]</a></span>)</span>)</span> <a href="ndiff_2.html#K2" title="NDIFF_2:func.2">*</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Olive" title="b7">w</font>,<font color="Olive" title="b10">x</font>)</span>)</span> &amp;  <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Olive" title="b7">w</font>,<font color="Olive" title="b10">x</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"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Olive" title="b8">u</font>,<font color="Olive" title="b10">x</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Olive" title="b9">v</font>,<font color="Olive" title="b10">x</font>)</span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> &amp; (  for <font color="Olive" title="b11">ds</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">S</font> holds  <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Olive" title="b6">W</font>,<font color="Olive" title="b10">x</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b11">ds</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">B</font> <a href="lopban_8.html#K2" title="LOPBAN_8:func.2">.</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Olive" title="b8">u</font>,<font color="Olive" title="b10">x</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b11">ds</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b9">v</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b10">x</font></span>)</span>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">B</font> <a href="lopban_8.html#K2" title="LOPBAN_8:func.2">.</a> (<span class="p2">(<span class="default"><font color="Olive" title="b8">u</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b10">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Olive" title="b9">v</font>,<font color="Olive" title="b10">x</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b11">ds</font></span>)</span>)</span>)</span> ) )</div></div>
