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<b>let </b><font color="Maroon">N</font> be    <a href="subset_1.html#M2">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>let </b><font color="Maroon">seq</font> be   <a href="toprns_1.html#NM1">Real_Sequence</a> of <font color="Maroon">N</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">seq</font> is <a href="toprns_1.html#V1" target="_self">non-zero</a>
 <b>and </b><br/><a NAME="E3:27"/><i><font color="Green">E49</font></i>: 
not <font color="Maroon">r</font> <a href="toprns_1.html#K2" target="_self">*</a> <font color="Maroon">seq</font> is <a href="toprns_1.html#V1" target="_self">non-zero</a>
 ;<br/>

<b>consider </b><font color="Maroon">n1</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="E5:27"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="toprns_1.html#K2" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">N</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:27"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">N</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:27"/>
<font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">N</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:27"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="euclid.html#T35">EUCLID:35</a></i>;<br/>


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