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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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">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="b2">B</font> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  &amp;  <a href="funct_1.html#K3" title="FUNCT_1:func.3">the_value_of</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> holds <br/>( <font color="Olive" title="b2">B</font> is  <a href="kurato_0.html#V3" title="KURATO_0:attr.3">convergent</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp;  <a href="kurato_0.html#K3" title="KURATO_0:func.3">lim_inf</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp;  <a href="kurato_0.html#K4" title="KURATO_0:func.4">lim_sup</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  &amp;  <a href="funct_1.html#K3" title="FUNCT_1:func.3">the_value_of</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> holds <br/>( <font color="Olive" title="b1">B</font> is  <a href="kurato_0.html#V3" title="KURATO_0: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> <font color="Maroon" title="c2">A</font> &amp;  <a href="kurato_0.html#K3" title="KURATO_0:func.3">lim_inf</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="kurato_0.html#K4" title="KURATO_0:func.4">lim_sup</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">B</font> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  &amp;  <a href="funct_1.html#K3" title="FUNCT_1:func.3">the_value_of</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> implies ( <font color="Maroon" title="c3">B</font> is  <a href="kurato_0.html#V3" title="KURATO_0:attr.3">convergent</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="kurato_0.html#K3" title="KURATO_0:func.3">lim_inf</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="kurato_0.html#K4" title="KURATO_0:func.4">lim_sup</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:68"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
( <font color="Maroon" title="c3">B</font> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  &amp;  <a href="funct_1.html#K3" title="FUNCT_1:func.3">the_value_of</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">B</font> is  <a href="kurato_0.html#V3" title="KURATO_0:attr.3">convergent</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="kurato_0.html#K3" title="KURATO_0:func.3">lim_inf</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="kurato_0.html#K4" title="KURATO_0:func.4">lim_sup</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<a NAME="E2:68"/><span class="kw">then </span>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="setlim_1.html#K4" title="SETLIM_1:func.4">superior_setsequence</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_1.html#T39" target="_self" title="SETLIM_1:th.39">Th39</a></span>;<br/>
<a NAME="E3:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A2</font></span>: 
 <a href="kurato_0.html#K4" title="KURATO_0:func.4">lim_sup</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_1.html#T11" target="_self" title="SETLIM_1:th.11">Th11</a></span>;<br/>
<a NAME="E4:68"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="setlim_1.html#T38" target="_self" title="SETLIM_1:th.38">Th38</a></span>;<br/>
<a NAME="E5:68"/><span class="kw">then </span>
 <a href="kurato_0.html#K3" title="KURATO_0:func.3">lim_inf</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_1.html#T10" target="_self" title="SETLIM_1:th.10">Th10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:68"/>
( <font color="Maroon" title="c3">B</font> is  <a href="kurato_0.html#V3" title="KURATO_0:attr.3">convergent</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="kurato_0.html#K3" title="KURATO_0:func.3">lim_inf</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="kurato_0.html#K4" title="KURATO_0:func.4">lim_sup</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="kurato_0.html#D5" title="KURATO_0:def.5">KURATO_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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