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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E44">Th80</font></span>: <a NAME="T80"><span class="comment"><font color="firebrick">:: LIMFUNC1:80</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">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">f</font> is  <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a>  holds <br/>( <font color="Olive" title="b2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b1">f</font> is  <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a>  &amp;  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">lim_in+infty</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">lim_in+infty</a> <font color="Olive" title="b1">f</font></span>)</span> )</div></div>
