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<b>take </b><font color="Maroon">f</font> = <font color="Maroon">cf</font>;<br/>

<a NAME="E1:7_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>hence </b><a NAME="E2:7_1_1"/>
<font color="Maroon">f</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="fdiff_1.html#D3" target="_self">FDIFF_1:def 3</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">h</font> be  <a href="fdiff_1.html#V1" target="_self">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1"><b>now </b></a><div class="add"><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/><a NAME="E1:7_1_1_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> 0
<b>by </b><i/>
<br/>.= 
0

;<br/></div><b>end;</b></div><a NAME="E2:7_1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span> is <a href="funct_1.html#V3">constant</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D6">SEQM_3:def 6</a></i>;<br/><a NAME="E3:7_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:7_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="seq_4.html#T39">SEQ_4:39</a>, <a class="ref" href="seq_4.html#T40">SEQ_4:40</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:7_1_1"/>
for <font color="Olive">h</font> being <a href="fdiff_1.html#V1" target="_self">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Olive">h</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfunct_2.html#K1">*</a> <font color="Olive">h</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>


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