<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">j</font> be    <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> ;<br/>



<b>set </b><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font>;<br/>
<a NAME="E1:7"/><i><font color="Green" title="E8">A1</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">j</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</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" title="E9">A2</font></i>: 
<font color="Maroon" title="c2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:7_1"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><b>consider </b><font color="Maroon" title="c4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:7_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="topreal6.html#E1:7_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><a NAME="E5:7_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font>
 <b>by </b><i><a class="txt" href="topreal6.html#E4:7_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:7_1"/><b>then </b><i><font color="Green" title="E12">A5</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">j</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:7_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="topreal6.html#E2:7_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E8:7_1"/>
 <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="topreal6.html#E1:7"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="topreal6.html#E6:7_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="rvsum_1.html#T133" title="RVSUM_1:th.133">RVSUM_1:133</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:7"/>
 <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">n</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> <b> such that </b><br/><a NAME="E5:7"/><i><font color="Green" title="E9">A6</font></i>: 
( <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="rvsum_1.html#T133" title="RVSUM_1:th.133">RVSUM_1:133</a></i>;<br/>
<b>thus </b><a NAME="E6:7"/>
( <font color="Maroon" title="c2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="topreal6.html#E1:7"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="topreal6.html#E5:7"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>


</div>
