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<div><span class="kw">theorem </span><a NAME="T23"><span class="comment"><font color="firebrick">:: FDIFF_12:23</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b2">I</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a><br/>  for <font color="Olive" title="b3">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">f</font> <a href="fdiff_12.html#R1" title="FDIFF_12:pred.1">is_differentiable_on_interval</a> <font color="Olive" title="b2">I</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="b1">f</font> <a href="fdiff_12.html#R1" title="FDIFF_12:pred.1">is_differentiable_on_interval</a> <font color="Olive" title="b2">I</font> &amp; <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="b1">f</font></span>)</span> <a href="fdiff_12.html#K1" title="FDIFF_12:func.1">`\</a> <font color="Olive" title="b2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><font color="Olive" title="b1">f</font> <a href="fdiff_12.html#K1" title="FDIFF_12:func.1">`\</a> <font color="Olive" title="b2">I</font></span>)</span> &amp; (  for <font color="Olive" title="b4">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">I</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<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="b1">f</font></span>)</span> <a href="fdiff_12.html#K1" title="FDIFF_12:func.1">`\</a> <font color="Olive" title="b2">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="fdiff_12.html#K1" title="FDIFF_12:func.1">`\</a> <font color="Olive" title="b2">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font></span>)</span> ) )</div></div>
