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<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">fr</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#K3">F_Real</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">fp</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <a href="vectsp_1.html#K3">F_Real</a> ;<br/>







<b>assume </b><a NAME="E1:17"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fr</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fp</font> )
 ;<br/>

<a NAME="E2:17"/><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font>
 
<b>by </b><i>, <a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>;<br/>
<a NAME="E3:17"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">fp</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:17"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">p</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">p</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">fp</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:17_1"/><i><font color="Green">E56</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>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">p</font></span>)</span> )
 ;<br/>

<a NAME="E2:17_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:17_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fp</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:17_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">fp</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fp</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">fp</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#K3">F_Real</a>  <b>by </b><i/>;<br/>
<a NAME="E6:17_1"/><i><font color="Green">E59</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"><font color="Maroon">fr</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">fp</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, , , </i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">p</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
<br/>.= 
<font color="Maroon">fr</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">fp</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="fvsum_1.html#T62">FVSUM_1:62</a>, </i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:17"/>
<font color="Maroon">r</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fr</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">fp</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


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