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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="toprns_1.html#NM1">Real_Sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:55"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="toprns_1.html#V3" target="_self">convergent</a>
 <b>and </b><br/><a NAME="E2:55"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="toprns_1.html#V3" target="_self">convergent</a>
 ;<br/>

<a NAME="E3:55"/><i><font color="Green">E43</font></i>: 
 <a href="toprns_1.html#K3" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> is <a href="toprns_1.html#V3" target="_self">convergent</a>
 
<b>by </b><i><a class="txt" href="toprns_1.html#E2:55"><i><font color="Green">E42</font></i></a>, <a class="ref" href="toprns_1.html#T45" target="_self">Th45</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">c<sub>2</sub></font> <a href="toprns_1.html#K4" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="toprns_1.html#K7" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="toprns_1.html#K1" target="_self">+</a> <span class="p2">(<span class="default"><a href="toprns_1.html#K3" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprns_1.html#E1:55"><i><font color="Green">E41</font></i></a>, <a class="txt" href="toprns_1.html#E3:55"><i><font color="Green">E43</font></i></a>, <a class="ref" href="toprns_1.html#T42" target="_self">Th42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprns_1.html#E2:55"><i><font color="Green">E42</font></i></a>, <a class="ref" href="toprns_1.html#T46" target="_self">Th46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprns_1.html#K7" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
;<br/>


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