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<b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">seq</font> is <a href="funct_1.html#V3">constant</a>
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">seq1</font> is <a href="seq_2.html#V4">convergent</a>
 ;<br/>

<a NAME="E3:50"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b> <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq1</font></span>)</span>
<b>by </b><i><a class="txt" href="seq_4.html#E1"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="seq_2.html#T20">SEQ_2:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq1</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/>

<b>thus </b> <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">seq1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq1</font></span>)</span>
<b>by </b><i><a class="txt" href="seq_4.html#E1"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="seq_4.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq1</font></span>)</span>
<b>by </b><i>, </i>
;<br/>

<b>thus </b> <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">seq</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span>
<b>by </b><i><a class="txt" href="seq_4.html#E1"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="seq_4.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> 0</span>)</span>
<b>by </b><i>, </i>
;<br/>

<b>thus </b> <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">seq1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq1</font></span>)</span>
<b>by </b><i><a class="txt" href="seq_4.html#E1"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="seq_4.html#E2"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="seq_2.html#T29">SEQ_2:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq1</font></span>)</span>
<b>by </b><i>, </i>
;<br/>


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