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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> ; <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">F1</font>, <font color="Olive" title="b2">F2</font> being  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="Olive" title="b2">F2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Olive" title="b1">F1</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; <font color="Olive" title="b2">F2</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font></span>)</span> ) holds <br/>(  <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Olive" title="b1">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Olive" title="b2">F2</font></span>)</span> &amp;  <a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Olive" title="b1">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Olive" title="b2">F2</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">F1</font>, <font color="Maroon" title="c3">F2</font> be   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="c3">F2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Maroon" title="c2">F1</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; <font color="Maroon" title="c3">F2</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; ( 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">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) implies (  <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c3">F2</font></span>)</span> &amp;  <a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c3">F2</font></span>)</span> ) )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green" title="E16">A1</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="c3">F2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Maroon" title="c2">F1</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; <font color="Maroon" title="c3">F2</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> )
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green" title="E17">A2</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">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c3">F2</font></span>)</span> &amp;  <a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c3">F2</font></span>)</span> )</span><br/>

<a NAME="E3:15"/><i><font color="Green" title="E18">A3</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#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F1</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">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp;  <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_sup</a> <font color="Maroon" title="c3">F2</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="c3">F2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp;  <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_sup</a> <font color="Maroon" title="c2">F1</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">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c3">F2</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="c3">F2</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" title="MESFUNC8:def.8">MESFUNC8:def 8</a>, <a class="ref" href="mesfunc8.html#D9" title="MESFUNC8:def.9">MESFUNC8:def 9</a></i>;<br/>
<a NAME="E4:15"/><i><font color="Green" title="E19">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c3">F2</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_sup</a> <font color="Maroon" title="c3">F2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D7" title="MESFUNC1:def.7">MESFUNC1:def 7</a></i>;<br/>
<div><i><font color="Green" title="E20">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="c4">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"><font color="Maroon" title="c2">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> implies for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">F2</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E21">A61</font></i>: 
<font color="Maroon" title="c4">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"><font color="Maroon" title="c2">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 ; <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">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">F2</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F2</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><br/><a NAME="E2:15_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c5">n</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">F1</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="txt" href="#E1:15"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/><a NAME="E3:15_1"/><b>then </b><i><font color="Green" title="E22">A62</font></i>: 
<font color="Maroon" title="c4">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="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:15_1"><i><font color="Green" title="E21">A61</font></i></a>, <a class="txt" href="#E2:15"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/><a NAME="E4:15_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/><a NAME="E5:15_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:15_1"><i><font color="Green" title="E22">A62</font></i></a>, <a class="ref" href="mesfunc1.html#D7" title="MESFUNC1:def.7">MESFUNC1:def 7</a></i>;<br/><a NAME="E6:15_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F2</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/><b>hence </b><a NAME="E7:15_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F2</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>; <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="15_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="c4">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#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F1</font></span>)</span> implies <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c3">F2</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><b>assume </b><a NAME="E1:15_2"/><i><font color="Green" title="E21">P01</font></i>: 
<font color="Maroon" title="c4">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#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c3">F2</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font></span><br/><a NAME="E2:15_2"/><b>then </b><i><font color="Green" title="E22">P07</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  <span class="p1">(<span class="default"><font color="Maroon" title="c3">F2</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="#E5:15"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/><a NAME="E3:15_2"/><i><font color="Green" title="E23">P08</font></i>: 
( <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">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">F1</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c3">F2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F2</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:15_2"><i><font color="Green" title="E21">P01</font></i></a>, <a class="txt" href="#E3:15"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="#E1:15"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="mesfunc8.html#D8" title="MESFUNC8:def.8">MESFUNC8:def 8</a>, <a class="ref" href="mesfunc8.html#D9" title="MESFUNC8:def.9">MESFUNC8:def 9</a></i>;<br/><a NAME="E4:15_2"/>
<font color="Maroon" title="c4">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="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c3">F2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:15_2"><i><font color="Green" title="E21">P01</font></i></a>, <a class="txt" href="#E3:15"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="#E1:15"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="mesfunc1.html#D7" title="MESFUNC1:def.7">MESFUNC1:def 7</a></i>;<br/><a NAME="E5:15_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c3">F2</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c3">F2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D7" title="MESFUNC1:def.7">MESFUNC1:def 7</a></i>;<br/><b>hence </b><a NAME="E6:15_2"/>
<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c3">F2</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E3:15_2"><i><font color="Green" title="E23">P08</font></i></a>, <a class="txt" href="#E2:15_2"><i><font color="Green" title="E22">P07</font></i></a>, <a class="ref" href="mesfun10.html#T15" target="_self" title="MESFUN10:th.15">supinfa</a></i>; <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="E7:15"/>
 <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c3">F2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:15"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E3:15"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="#E1:15"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c3">F2</font></span>)</span></span><br/>

<a NAME="E8:15"/><i><font color="Green" title="E21">B5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c3">F2</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#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c3">F2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D7" title="MESFUNC1:def.7">MESFUNC1:def 7</a></i>;<br/>
<a NAME="E9:15"/>
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_sup</a> <font color="Maroon" title="c2">F1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F1</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="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c3">F2</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="c4">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_sup</a> <font color="Maroon" title="c2">F1</font></span>)</span> implies <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c3">F2</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> )</span><br/>

<b>assume </b><a NAME="E1:15_3"/><i><font color="Green" title="E22">Q01</font></i>: 
<font color="Maroon" title="c4">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_sup</a> <font color="Maroon" title="c2">F1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c3">F2</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font></span><br/>

<a NAME="E2:15_3"/><b>then </b><i><font color="Green" title="E23">Q07</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  <span class="p1">(<span class="default"><font color="Maroon" title="c3">F2</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="#E5:15"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/>
<a NAME="E3:15_3"/><i><font color="Green" title="E24">Q08</font></i>: 
( <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c3">F2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">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="c3">F2</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c4">x</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:15_3"><i><font color="Green" title="E22">Q01</font></i></a>, <a class="txt" href="#E1:15"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E3:15"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="mesfunc8.html#D8" title="MESFUNC8:def.8">MESFUNC8:def 8</a>, <a class="ref" href="mesfunc8.html#D9" title="MESFUNC8:def.9">MESFUNC8:def 9</a></i>;<br/>
<a NAME="E4:15_3"/>
<font color="Maroon" title="c4">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="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c3">F2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:15_3"><i><font color="Green" title="E22">Q01</font></i></a>, <a class="txt" href="#E1:15"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E3:15"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="mesfunc1.html#D7" title="MESFUNC1:def.7">MESFUNC1:def 7</a></i>;<br/>
<a NAME="E5:15_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c3">F2</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c3">F2</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D7" title="MESFUNC1:def.7">MESFUNC1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E6:15_3"/>
<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c3">F2</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E3:15_3"><i><font color="Green" title="E24">Q08</font></i></a>, <a class="txt" href="#E2:15_3"><i><font color="Green" title="E23">Q07</font></i></a>, <a class="ref" href="mesfun10.html#T15" target="_self" title="MESFUN10:th.15">supinfa</a></i>; <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="E10:15"/>
 <a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c3">F2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E3:15"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="#E8:15"><i><font color="Green" title="E21">B5</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
