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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E53">Th51</font></span>: <a NAME="T51"><span class="comment"><font color="firebrick">:: LPSPACE1:51</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">X</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">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Olive" title="b2">X</font><br/>  for <font color="Olive" title="b4">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b3">S</font><br/>  for <font color="Olive" title="b5">f</font>, <font color="Olive" title="b6">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Olive" title="b2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b7">x</font>, <font color="Olive" title="b8">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspace1.html#K20" title="LPSPACE1:func.20">L-1-Space</a> <font color="Olive" title="b4">M</font></span>)</span> holds <br/> ( ( <font color="Olive" title="b5">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b7">x</font> &amp; <font color="Olive" title="b6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b8">y</font> implies <font color="Olive" title="b5">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b8">y</font> ) &amp; ( <font color="Olive" title="b5">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b7">x</font> implies <font color="Olive" title="b1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b5">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b7">x</font> ) )</div></div>
