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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">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>set </b><font color="Maroon">f</font> = <font color="Maroon">j</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font>;<br/>
<a NAME="E1:7"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T68">FINSEQ_2:68</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:7_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:7_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:7_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E5:7_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:7_1"/><b>then </b><i><font color="Green">E66</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:7_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T70">FINSEQ_2:70</a></i>;<br/><b>hence </b><a NAME="E8:7_1"/>
 <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="rvsum_1.html#T133">RVSUM_1:133</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:7"/>
 <a href="rvsum_1.html#K16">Product</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:7"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">n</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">j</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="newton.html#K1">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="rvsum_1.html#T133">RVSUM_1:133</a></i>;<br/>
<b>thus </b><a NAME="E6:7"/>
( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_2.html#T70">FINSEQ_2:70</a></i>;<br/>


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