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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E29">Th30</font></span>: <a NAME="T30"><span class="comment"><font color="firebrick">:: MESFUN16:30</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">f</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"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></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="b4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X</font><br/>  for <font color="Olive" title="b5">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">Y</font> holds <br/> (  <a href="mesfun12.html#K2" title="MESFUN12:func.2">ProjPMap1</a> (<span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Olive" title="b3">f</font></span>)</span>,<font color="Olive" title="b4">x</font>) is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Olive" title="b2">Y</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> &amp;  <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="b3">f</font></span>)</span></span><a href="mesfunc1.html#K9" title="MESFUNC1:func.9">.|</a></span>,<font color="Olive" title="b4">x</font>) is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Olive" title="b2">Y</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> &amp;  <a href="mesfun12.html#K3" title="MESFUN12:func.3">ProjPMap2</a> (<span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Olive" title="b3">f</font></span>)</span>,<font color="Olive" title="b5">y</font>) is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Olive" title="b1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> &amp;  <a href="mesfun12.html#K3" title="MESFUN12:func.3">ProjPMap2</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="b3">f</font></span>)</span></span><a href="mesfunc1.html#K9" title="MESFUNC1:func.9">.|</a></span>,<font color="Olive" title="b5">y</font>) is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Olive" title="b1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> )</div></div>
