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<b>let </b><font color="Maroon">a</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>set </b><font color="Maroon">F</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E2:22"/><i><font color="Green">E34</font></i>: 
 <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>and </b><br/><a NAME="E4:22"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span> holds <br/><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E5:22"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E6:22"/><b>then </b>
(  <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> 1 &amp; <font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="supinf_2.html#K13">.</a> 0</span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<b>then </b> <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span> = 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="supinf_2.html#K13">.</a> 1
<b>by </b><i>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<b>hence </b><a NAME="E8:22"/>
 <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>


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