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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A1</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a> &amp; <font color="Olive" title="b2">A2</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a> holds <br/>( <font color="Olive" title="b1">A1</font> <a href="setlim_2.html#K2" title="SETLIM_2:func.2">(\/)</a> <font color="Olive" title="b2">A2</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a> &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A1</font> <a href="setlim_2.html#K2" title="SETLIM_2:func.2">(\/)</a> <font color="Olive" title="b2">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b2">A2</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c3">A2</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A1</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a> &amp; <font color="Maroon" title="c3">A2</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a> implies ( <font color="Maroon" title="c2">A1</font> <a href="setlim_2.html#K2" title="SETLIM_2:func.2">(\/)</a> <font color="Maroon" title="c3">A2</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a> &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="setlim_2.html#K2" title="SETLIM_2:func.2">(\/)</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c3">A2</font></span>)</span> ) )</span><br/>





<b>assume </b><a NAME="E1:91"/><i><font color="Green" title="E80">A1</font></i>: 
( <font color="Maroon" title="c2">A1</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a> &amp; <font color="Maroon" title="c3">A2</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A1</font> <a href="setlim_2.html#K2" title="SETLIM_2:func.2">(\/)</a> <font color="Maroon" title="c3">A2</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a> &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="setlim_2.html#K2" title="SETLIM_2:func.2">(\/)</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c3">A2</font></span>)</span> )</span><br/>

<b>then </b><i><font color="Green" title="E81">A2</font></i>:  <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="setlim_2.html#K2" title="SETLIM_2:func.2">(\/)</a> <font color="Maroon" title="c3">A2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_inf</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_inf</a> <font color="Maroon" title="c3">A2</font></span>)</span>
<b>by </b><i><a class="ref" href="setlim_2.html#T63" target="_self" title="SETLIM_2:th.63">Th63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_inf</a> <font color="Maroon" title="c3">A2</font></span>)</span>
<b>by </b><i><a class="txt" href="setlim_2.html#E1:91"><i><font color="Green" title="E80">A1</font></i></a>, <a class="ref" href="kurato_2.html#D7" title="KURATO_2:def.7">KURATO_2:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c3">A2</font></span>)</span>
<b>by </b><i><a class="txt" href="setlim_2.html#E1:91"><i><font color="Green" title="E80">A1</font></i></a>, <a class="ref" href="kurato_2.html#D7" title="KURATO_2:def.7">KURATO_2:def 7</a></i>
;<br/>
 <a href="kurato_2.html#K5" title="KURATO_2:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="setlim_2.html#K2" title="SETLIM_2:func.2">(\/)</a> <font color="Maroon" title="c3">A2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="kurato_2.html#K5" title="KURATO_2:func.5">lim_sup</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K5" title="KURATO_2:func.5">lim_sup</a> <font color="Maroon" title="c3">A2</font></span>)</span>
<b>by </b><i><a class="ref" href="setlim_2.html#T68" target="_self" title="SETLIM_2:th.68">Th68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c3">A2</font></span>)</span>

;<br/>
<b>hence </b><a NAME="E4:91"/>
( <font color="Maroon" title="c2">A1</font> <a href="setlim_2.html#K2" title="SETLIM_2:func.2">(\/)</a> <font color="Maroon" title="c3">A2</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a> &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="setlim_2.html#K2" title="SETLIM_2:func.2">(\/)</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c3">A2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="setlim_2.html#E2:91"><i><font color="Green" title="E81">A2</font></i></a>, <a class="ref" href="kurato_2.html#D7" title="KURATO_2:def.7">KURATO_2:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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