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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E23">Th24</font></span>: <a NAME="T24"><span class="comment"><font color="firebrick">:: MESFUN16:24</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">S</font> being   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/>  for <font color="Olive" title="b3">E</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b2">S</font><br/>  for <font color="Olive" title="b4">f</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <font color="Olive" title="b2">S</font>,<a href="dualsp02.html#K3" title="DUALSP02:func.3">RNS_Real</a>  st <font color="Olive" title="b4">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b3">E</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">E</font> holds <br/> ex <font color="Olive" title="b5">H</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b2">S</font> st <br/>( <font color="Olive" title="b5">H</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Olive" title="b1">r</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Olive" title="b5">H</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  )</div></div>
