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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E37">Th45</font></span>: <a NAME="T44"><span class="comment"><font color="firebrick">:: NDIFF13:44</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">E</font>, <font color="Olive" title="b2">F</font>, <font color="Olive" title="b3">G</font> being   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/>  for <font color="Olive" title="b4">Z</font> being   <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="b1">E</font>,<font color="Olive" title="b2">F</font></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span><br/>  for <font color="Olive" title="b5">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="b1">E</font>,<font color="Olive" title="b2">F</font></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span>,<font color="Olive" title="b3">G</font>  st <font color="Olive" title="b5">f</font> <a href="ndiff_1.html#R2" title="NDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b4">Z</font> holds <br/>( <font color="Olive" title="b5">f</font> <a href="ndiff_7.html#R3" title="NDIFF_7:pred.3">is_partial_differentiable_on`1</a> <font color="Olive" title="b4">Z</font> &amp; <font color="Olive" title="b5">f</font> <a href="ndiff_7.html#R4" title="NDIFF_7:pred.4">is_partial_differentiable_on`2</a> <font color="Olive" title="b4">Z</font> &amp; (  for <font color="Olive" title="b6">z</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K9" title="PRVECT_3:func.9">[:</a><span class="default"><font color="Olive" title="b1">E</font>,<font color="Olive" title="b2">F</font></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span>  st <font color="Olive" title="b6">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">Z</font> holds <br/>( (  for <font color="Olive" title="b7">dx</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">E</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">f</font> <a href="ndiff_7.html#K9" title="NDIFF_7:func.9">`partial`1|</a> <font color="Olive" title="b4">Z</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b6">z</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b7">dx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Olive" title="b4">Z</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b6">z</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b7">dx</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b2">F</font></span>)</span>) ) &amp; (  for <font color="Olive" title="b7">dy</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b2">F</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">f</font> <a href="ndiff_7.html#K10" title="NDIFF_7:func.10">`partial`2|</a> <font color="Olive" title="b4">Z</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b6">z</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b7">dy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Olive" title="b4">Z</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b6">z</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">E</font></span>)</span>,<font color="Olive" title="b7">dy</font>) ) ) ) )</div></div>
