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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Si</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">S</font> being  <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Olive" title="b1">Si</font>  st <font color="Olive" title="b2">S</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  holds <br/>( <font color="Olive" title="b2">S</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="b2">S</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="b2">S</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">Si</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Si</font>  st <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  holds <br/>( <font color="Olive" title="b1">S</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">S</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">S</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Si</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">S</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  implies ( <font color="Maroon" title="c3">S</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="c3">S</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="c3">S</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:87"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
<font color="Maroon" title="c3">S</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">S</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="c3">S</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="c3">S</font> )</span><br/>

<a NAME="E2:87"/><span class="kw">then </span>
(  <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_sup</a> <font color="Maroon" title="c3">S</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="c3">S</font> &amp;  <a href="kurato_2.html#K3" title="KURATO_2:func.3">lim_inf</a> <font color="Maroon" title="c3">S</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="c3">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_1.html#T78" target="_self" title="SETLIM_1:th.78">Th78</a>, <a class="ref" href="setlim_1.html#T79" target="_self" title="SETLIM_1:th.79">Th79</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:87"/>
<font color="Maroon" title="c3">S</font> is  <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="kurato_2.html#D7" title="KURATO_2:def.7">KURATO_2:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c3">S</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="c3">S</font></span><br/>

<span class="kw">thus </span><a NAME="E4:87"/>
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c3">S</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="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="setlim_1.html#T78" target="_self" title="SETLIM_1:th.78">Th78</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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