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



<b>assume </b><a NAME="E1:53"/>
<font color="Maroon">seq</font> is <a href="toprns_1.html#V3" target="_self">convergent</a>
 ;<br/>

<b>then </b> <a href="toprns_1.html#K7" target="_self">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="toprns_1.html#K2" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="euclid.html#K19">-</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T44">EUCLID:44</a></i>
<br/>.= 
 <a href="euclid.html#K19">-</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/>
<b>hence </b><a NAME="E3:53"/>
 <a href="toprns_1.html#K7" target="_self">lim</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K3" target="_self">-</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K19">-</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span>
 <b>by </b><i/>;<br/>


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