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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="mesfunc8.html#M1" title="MESFUNC8:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ;<br/>



<a NAME="E1:23"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K2" title="MESFUNC8:func.2">sup</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc8.html#D4" target="_self" title="MESFUNC8:def.4">Def4</a></i>;<br/>
<a NAME="E2:23"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K2" title="MESFUNC8:func.2">sup</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc8.html#D5" target="_self" title="MESFUNC8:def.5">Def5</a></i>;<br/>
<a NAME="E3:23"/><b>then </b><i><font color="Green" title="E18">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K2" title="MESFUNC8:func.2">sup</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_inf</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc8.html#D8" target="_self" title="MESFUNC8:def.8">Def8</a></i>;<br/>
<div><i><font color="Green" title="E19">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:23_1"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_inf</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/><a NAME="E2:23_1"/><b>then </b><i><font color="Green" title="E21">A4</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_inf</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c3">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc8.html#D8" target="_self" title="MESFUNC8:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E3:23_1"/>
<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_inf</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K1" title="RINFSUP2:func.1">sup</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
 ;<br/><a NAME="E4:23_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_inf</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc8.html#D8" target="_self" title="MESFUNC8:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E5:23_1"/>
<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_inf</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K1" title="RINFSUP2:func.1">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c3">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc8.html#E1:23_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="mesfunc8.html#E2:23_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="mesfunc8.html#T7" target="_self" title="MESFUNC8:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E6:23_1"/>
<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_inf</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K2" title="MESFUNC8:func.2">sup</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="mesfunc8.html#E3:23"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc8.html#E1:23_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="mesfunc8.html#D4" target="_self" title="MESFUNC8:def.4">Def4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:23"/><b>then </b>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_inf</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_inf</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K2" title="MESFUNC8:func.2">sup</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 
;<br/>
<b>hence </b><a NAME="E6:23"/>
( ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_inf</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_inf</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K1" title="RINFSUP2:func.1">sup</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_inf</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K1" title="RINFSUP2:func.1">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_inf</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K2" title="MESFUNC8:func.2">sup</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> ) ) &amp;  <a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_inf</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc8.html#K2" title="MESFUNC8:func.2">sup</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K3" title="MESFUNC8:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc8.html#E3:23"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc8.html#E4:23"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>


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