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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">B</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">B</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> holds <br/>( <font color="Olive" title="b1">B</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> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Olive" title="b1">B</font> )</span><br/><b>let </b><font color="Maroon" title="c2">B</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">B</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> implies ( <font color="Maroon" title="c2">B</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> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c2">B</font> ) )</span><br/>



<b>assume </b><a NAME="E1:69"/><i><font color="Green" title="E53">A0</font></i>: 
<font color="Maroon" title="c2">B</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</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">B</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> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c2">B</font> )</span><br/>

<a NAME="E2:69"/><b>then </b>
(  <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_sup</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c2">B</font> &amp;  <a href="kurato_2.html#K3" title="KURATO_2:func.3">lim_inf</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c2">B</font> )
 
<b>by </b><i><a class="ref" href="setlim_1.html#T43" target="_self" title="SETLIM_1:th.43">Th32</a>, <a class="ref" href="setlim_1.html#T46" target="_self" title="SETLIM_1:th.46">Th35</a></i>;<br/>
<b>hence </b><a NAME="E3:69"/>
<font color="Maroon" title="c2">B</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a>
 <b>by </b><i><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">  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c2">B</font></span><br/>

<b>thus </b><a NAME="E4:69"/>
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="#E1:69"><i><font color="Green" title="E53">A0</font></i></a>, <a class="ref" href="setlim_1.html#T43" target="_self" title="SETLIM_1:th.43">Th32</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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