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<b>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">T</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">S</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> implies ( <font color="Maroon" title="c2">S</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> ) )</span><br/>

<b>assume </b><a NAME="E1:30_1_1"/>
<font color="Maroon" title="c2">S</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</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">S</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> )</span><br/>

<b>then consider </b><font color="Maroon" title="c3">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:30_1_1"/><i><font color="Green" title="E16">A1</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="ref" href="valued_0.html#D18" title="VALUED_0:def.18">VALUED_0:def 18</a></i>;<br/>
<a NAME="E4:30_1_1"/><i><font color="Green" title="E17">A2</font></i>: 
 <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_sup</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="#E3:30_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="kurato_2.html#T17" target="_self" title="KURATO_2:th.17">Th17</a></i>;<br/>
<a NAME="E5:30_1_1"/><i><font color="Green" title="E18">A3</font></i>: 
 <a href="kurato_2.html#K3" title="KURATO_2:func.3">lim_inf</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="#E3:30_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="kurato_2.html#T18" target="_self" title="KURATO_2:th.18">Th18</a></i>;<br/>
<div><i><font color="Green" title="E19">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><a NAME="E1:30_1_1_1"/>
( <font color="Maroon" title="c2">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c2">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="txt" href="#E3:30_1_1"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E2:30_1_1_1"/>
<font color="Maroon" title="c2">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span><br/><a NAME="E1:30_1_1_2"/>
( <font color="Maroon" title="c2">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c2">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="txt" href="#E3:30_1_1"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E2:30_1_1_2"/>
<font color="Maroon" title="c2">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:30_1_1"/>
( <font color="Maroon" title="c2">S</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> )
 <b>by </b><i><a class="txt" href="#E4:30_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="#E5:30_1_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="#E6:30_1_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="kurato_2.html#D5" target="_self" title="KURATO_2:def.5">Def5</a>, <a class="ref" href="kurato_2.html#D6" target="_self" title="KURATO_2:def.6">Def6</a>, <a class="ref" href="kurato_2.html#D7" target="_self" title="KURATO_2:def.7">Def7</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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