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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E27">Th83</font></span>: <a NAME="T83"><span class="comment"><font color="firebrick">:: RVSUM_1:83</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">Number</a> <br/>  for <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <font color="Olive">b<sub>1</sub></font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st (  for <font color="Olive" title="b4">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K1" title="FINSEQ_1:func.1">Seg</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Olive" title="b2">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">R2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> ) &amp;  ex <font color="Olive" title="b4">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b4">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K1" title="FINSEQ_1:func.1">Seg</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">R2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> ) holds <br/> <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Olive" title="b2">R1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Olive" title="b3">R2</font></div></div>
