<?xml version="1.0"?>
<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E20">Th19</font></span>: <a NAME="T19"><span class="comment"><font color="firebrick">:: MESFUN17:19</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/>  for <font color="Olive" title="b3">f</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1"><a href="prvect_3.html#K9" title="PRVECT_3:func.9">[:</a><span class="default"><span class="p2"><a href="prvect_3.html#K9" title="PRVECT_3:func.9">[:</a><span class="default"><a href="dualsp02.html#K3" title="DUALSP02:func.3">RNS_Real</a>,<a href="dualsp02.html#K3" title="DUALSP02:func.3">RNS_Real</a></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span>,<a href="dualsp02.html#K3" title="DUALSP02:func.3">RNS_Real</a></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span>,<a href="dualsp02.html#K3" title="DUALSP02:func.3">RNS_Real</a><br/>  for <font color="Olive" title="b4">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b5">Pg1</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="b3">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">f</font> &amp; <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> &amp; <font color="Olive" title="b5">Pg1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfun12.html#K2" title="MESFUN12:func.2">ProjPMap1</a> (<span class="p1"><a href="mesfunc1.html#K9" title="MESFUNC1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Olive" title="b4">g</font></span>)</span></span><a href="mesfunc1.html#K9" title="MESFUNC1:func.9">.|</a></span>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>) holds <br/><font color="Olive" title="b5">Pg1</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> </div></div>
