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<div><span class="kw">theorem </span><a NAME="T40"><span class="comment"><font color="firebrick">:: RVSUM_3:40</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">f</font> being   <a href="rvsum_3.html#NV4" title="RVSUM_3:NV.4">heterogeneous</a>   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>   <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><br/>  for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b3">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b2">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_3.html#K6" title="RVSUM_3:func.6">MeanMore</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b3">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_3.html#K5" title="RVSUM_3:func.5">MeanLess</a> <font color="Olive" title="b1">f</font> holds <br/> ex <font color="Olive" title="b4">g</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>   <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_3.html#K7" title="RVSUM_3:func.7">Replace</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>,<span class="p1">(<span class="default"><a href="rvsum_3.html#K1" title="RVSUM_3:func.1">Mean</a> <font color="Olive" title="b1">f</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="rvsum_3.html#K1" title="RVSUM_3:func.1">Mean</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span>) &amp;  <a href="rvsum_3.html#K2" title="RVSUM_3:func.2">GMean</a> <font color="Olive" title="b1">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="rvsum_3.html#K2" title="RVSUM_3:func.2">GMean</a> <font color="Olive" title="b4">g</font> )</div></div>
