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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:95"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="kurato_2.html#V3">convergent</a>
 ;<br/>

<i><font color="Green">E81</font></i>:  <a href="kurato_2.html#K4">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="setlim_2.html#T75" target="_self">Th75</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K3">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="setlim_2.html#E1:95"><i><font color="Green">E80</font></i></a>, <a class="ref" href="setlim_1.html#D6">SETLIM_1:def 6</a></i>
;<br/>
 <a href="kurato_2.html#K5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K5">lim_sup</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="setlim_2.html#T82" target="_self">Th82</a></i>
<br/>.= 
 <a href="kurato_2.html#K4">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="setlim_2.html#E2:95"><i><font color="Green">E81</font></i></a>, <a class="txt" href="setlim_2.html#E1:95"><i><font color="Green">E80</font></i></a>, <a class="ref" href="setlim_1.html#D6">SETLIM_1:def 6</a></i>
;<br/>
<a NAME="E4:95"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">c<sub>3</sub></font> is <a href="kurato_2.html#V3">convergent</a>
 
<b>by </b><i><a class="ref" href="kurato_2.html#D7">KURATO_2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E5:95"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">c<sub>3</sub></font> is <a href="kurato_2.html#V3">convergent</a> &amp;  <a href="setlim_1.html#K3">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K3">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="setlim_2.html#E2:95"><i><font color="Green">E81</font></i></a>, <a class="ref" href="setlim_1.html#D6">SETLIM_1:def 6</a></i>;<br/>


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