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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>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="E1:55"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">seq</font> is <a href="toprns_1.html#V3" target="_self">convergent</a>
 <b>and </b><br/><a NAME="E2:55"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">seq'</font> is <a href="toprns_1.html#V3" target="_self">convergent</a>
 ;<br/>

<a NAME="E3:55"/><i><font color="Green">E49</font></i>: 
 <a href="toprns_1.html#K3" target="_self">-</a> <font color="Maroon">seq'</font> is <a href="toprns_1.html#V3" target="_self">convergent</a>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>thus </b> <a href="toprns_1.html#K7" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="toprns_1.html#K4" target="_self">-</a> <font color="Maroon">seq'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <span class="p2">(<span class="default"><a href="toprns_1.html#K3" target="_self">-</a> <font color="Maroon">seq'</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</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="toprns_1.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="euclid.html#K20">-</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#T45">EUCLID:45</a></i>
;<br/>


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