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<b>let </b><font color="Maroon" title="c1">a</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>set </b><font color="Maroon" title="c2">F</font> = <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>;<br/>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <b> such that </b><br/><a NAME="E2:22"/><i><font color="Green" title="E14">A1</font></i>: 
 <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>and </b><br/><a NAME="E4:22"/><i><font color="Green" title="E16">A3</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> holds <br/><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="convfun1.html#D5" target="_self" title="CONVFUN1:def.5">Def5</a></i>;<br/>
<a NAME="E5:22"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E6:22"/><b>then </b>
(  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> 1 &amp; <font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:22"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="convfun1.html#E4:22"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/>
<b>then </b> <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> = 
<span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> 1
<b>by </b><i><a class="txt" href="convfun1.html#E3:22"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>
<br/>.= 
<font color="Maroon" title="c1">a</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/>
<b>hence </b><a NAME="E8:22"/>
 <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 ;<br/>


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