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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E39">Th52</font></span>: <a NAME="T52"><span class="comment"><font color="firebrick">:: PDIFF_9:52</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b2">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Olive" title="b1">m</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b3">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b4">x</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Olive" title="b1">m</font>  st <font color="Olive" title="b2">f</font> <a href="pdiff_7.html#R1" title="PDIFF_7:pred.1">is_differentiable_in</a> <font color="Olive" title="b4">x</font> holds <br/>( <font color="Olive" title="b3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b2">f</font> <a href="pdiff_7.html#R1" title="PDIFF_7:pred.1">is_differentiable_in</a> <font color="Olive" title="b4">x</font> &amp;  <a href="pdiff_7.html#K1" title="PDIFF_7:func.1">diff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b2">f</font></span>)</span>,<font color="Olive" title="b4">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="pdiff_7.html#K1" title="PDIFF_7:func.1">diff</a> (<font color="Olive" title="b2">f</font>,<font color="Olive" title="b4">x</font>)</span>)</span> )</div></div>
