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<div><span class="kw">theorem </span><a NAME="T8"><span class="comment"><font color="firebrick">:: FDIFF_12:8</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">a</font>, <font color="Olive" title="b3">b</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b1">f</font> <a href="fdiff_12.html#R1" title="FDIFF_12:pred.1">is_differentiable_on_interval</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> holds <br/>( <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> <span class="p2"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">a</font>) &amp; <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> <span class="p2"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">b</font>) &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> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/><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> <span class="p2"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span></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>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b4">x</font>) ) )</div></div>
