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<div><span class="kw">theorem </span><a NAME="T30"><span class="comment"><font color="firebrick">:: FDIFF_12:30</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">x0</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">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>  st <font color="Olive" title="b2">f</font> <a href="fdiff_3.html#R1" title="FDIFF_3:pred.1">is_Lcontinuous_in</a> <font color="Olive" title="b1">x0</font> &amp; (  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="b3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x0</font> holds <br/> ex <font color="Olive" title="b4">g</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">g</font> &amp; <font color="Olive" title="b4">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x0</font> &amp; <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f</font> ) ) holds <br/>( <font color="Olive" title="b2">f</font> <a href="limfunc2.html#R1" title="LIMFUNC2:pred.1">is_left_convergent_in</a> <font color="Olive" title="b1">x0</font> &amp;  <a href="limfunc2.html#K1" title="LIMFUNC2:func.1">lim_left</a> (<font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font> )</div></div>
