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<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:15"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:15"/><b>then </b><i><font color="Green">E49</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:15"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/>
<a NAME="E4:15"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:15"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w</font> =  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">r</font></span>)</span> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K5">0*</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">r</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E8:15"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/>


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