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<b>let </b><font color="Maroon" title="c1">seq</font> be   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a>;<br/><b>let </b><font color="Maroon" title="c2">rseq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E21">AS</font></i>: 
( <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">rseq</font> &amp; <font color="Maroon" title="c2">rseq</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> )
 ;<br/>

<div><i><font color="Green" title="E22">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c4">n</font> = <font color="Maroon" title="c3">x</font> as    <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>  ;<br/><b>consider </b><font color="Maroon" title="c5">Y1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green" title="E23">AS1</font></i>: 
( <font color="Maroon" title="c5">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  &amp; <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c5">Y1</font> )
 <b>by </b><i><a class="ref" href="rinfsup2.html#D6" target="_self" title="RINFSUP2:def.6">Definferior</a></i>;<br/><a NAME="E5:20_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:20_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span> 
 ;<br/><a NAME="E2:20_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">k</font> being   <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>  st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> )
 ;<br/><b>hence </b><a NAME="E3:20_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c6">Y2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  ;<br/><a NAME="E7:20_1"/>
<font color="Maroon" title="c2">rseq</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a>
 <b>by </b><i><a class="txt" href="rinfsup2.html#E1:20"><i><font color="Green" title="E21">AS</font></i></a>, <a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a></i>;<br/><a NAME="E8:20_1"/><b>then </b>
<font color="Maroon" title="c6">Y2</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T32" title="RINFSUP1:th.32">RINFSUP1:32</a></i>;<br/><a NAME="E9:20_1"/><b>then </b>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c5">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c6">Y2</font>
 <b>by </b><i><a class="ref" href="rinfsup2.html#T3" target="_self" title="RINFSUP2:th.3">Lm022</a>, <a class="txt" href="rinfsup2.html#E1:20"><i><font color="Green" title="E21">AS</font></i></a>, <a class="txt" href="rinfsup2.html#E4:20_1"><i><font color="Green" title="E23">AS1</font></i></a></i>;<br/><b>hence </b><a NAME="E10:20_1"/>
<span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</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="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">rseq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="rinfsup2.html#E4:20_1"><i><font color="Green" title="E23">AS1</font></i></a>, <a class="ref" href="rinfsup1.html#D4" title="RINFSUP1:def.4">RINFSUP1:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:20"/>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <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="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <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="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">rseq</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:20"/><b>then </b><i><font color="Green" title="E23">P3</font></i>: 
 <a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">rseq</font>
 
<b>by </b><i><a class="txt" href="rinfsup2.html#E2:20"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E5:20"/>
 <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">rseq</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 
<b>by </b><i><a class="txt" href="rinfsup2.html#E1:20"><i><font color="Green" title="E21">AS</font></i></a>, <a class="ref" href="rinfsup1.html#T58" title="RINFSUP1:th.58">RINFSUP1:58</a></i>;<br/>
<a NAME="E6:20"/><b>then </b>
 <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">rseq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a></i>;<br/>
<a NAME="E7:20"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">rseq</font></span>)</span> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T5" title="RINFSUP1:th.5">RINFSUP1:5</a></i>;<br/>
<b>hence </b><a NAME="E8:20"/>
(  <a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">rseq</font> &amp;  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c2">rseq</font> )
 <b>by </b><i><a class="txt" href="rinfsup2.html#E4:20"><i><font color="Green" title="E23">P3</font></i></a>, <a class="ref" href="rinfsup2.html#T1" target="_self" title="RINFSUP2:th.1">Lm021</a></i>;<br/>


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