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<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">H</font> be    <a href="seqfunc.html#M1" target="_self">Functional_Sequence</a> of <font color="Maroon">D</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:59"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">H</font> <a href="seqfunc.html#R2" target="_self">is_point_conv_on</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:59"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">X</font> <a href="seqfunc.html#R1" target="_self">common_on_dom</a> <font color="Maroon">H</font> &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">H</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Olive">x</font> is <a href="seq_2.html#V4">convergent</a> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E3:59"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">X</font> <a href="seqfunc.html#R1" target="_self">common_on_dom</a> <font color="Maroon">r</font> <a href="seqfunc.html#K2" target="_self">(#)</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="txt" href="seqfunc.html#E2"><i><font color="Green">Lemma39</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>assume </b><a NAME="E1:59_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:59_1"/><b>then </b>
<font color="Maroon">H</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:59"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E3:59_1"/><b>then </b>
<font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#T21">SEQ_2:21</a></i>;<br/><b>hence </b><a NAME="E4:59_1"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seqfunc.html#K2" target="_self">(#)</a> <font color="Maroon">H</font></span>)</span> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E3:59"><i><font color="Green">E41</font></i></a>, <a class="txt" href="seqfunc.html#E5:59"><i><font color="Green">E43</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:59"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">r</font> <a href="seqfunc.html#K2" target="_self">(#)</a> <font color="Maroon">H</font> <a href="seqfunc.html#R2" target="_self">is_point_conv_on</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>

<i><font color="Green">E44</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H</font>,<font color="Maroon">X</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H</font>,<font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<font color="Maroon">X</font>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:59"><i><font color="Green">E40</font></i></a>, </i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>assume </b><a NAME="E1:59_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H</font>,<font color="Maroon">X</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:59_3"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H</font>,<font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/><a NAME="E3:59_3"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:59"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E4:59_3"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">H</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:59"><i><font color="Green">E40</font></i></a>, </i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E4"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:59"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seqfunc.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seqfunc.html#K2" target="_self">(#)</a> <font color="Maroon">H</font></span>)</span> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E3:59"><i><font color="Green">E41</font></i></a>, , </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:59"/>
 <a href="seqfunc.html#K11" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seqfunc.html#K2" target="_self">(#)</a> <font color="Maroon">H</font></span>)</span>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" target="_self">lim</a> <font color="Maroon">H</font>,<font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E5:59"><i><font color="Green">E43</font></i></a>, <a class="txt" href="seqfunc.html#E6:59"><i><font color="Green">E44</font></i></a>, </i>;<br/>


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