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<b>let </b><font color="Maroon" title="c1">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> &amp; <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> implies  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <font color="Maroon" title="c1">seq</font> )</span><br/>

<b>assume </b><a NAME="E1:82"/><i><font color="Green" title="E72">A1</font></i>: 
( <font color="Maroon" title="c1">seq</font> is <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> &amp; <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <font color="Maroon" title="c1">seq</font></span><br/>

<a NAME="E2:82"/><b>then </b>
 <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">V6</a>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T75" target="_self" title="RINFSUP1:th.75">Th75</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:82"/><i><font color="Green" title="E73">A2</font></i>: 
 <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="c1">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c2">r1</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#T188" title="FUNCT_2:th.188">FUNCT_2:188</a></i>;<br/>
<a NAME="E5:82"/>
<font color="Maroon" title="c2">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c1">seq</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:82"><i><font color="Green" title="E73">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:82"/><b>then </b>
 ex <font color="Olive" title="b1">n</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 <font color="Maroon" title="c2">r1</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="c1">seq</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T190" title="FUNCT_2:th.190">FUNCT_2:190</a></i>;<br/>
<b>then </b> <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> = 
 <a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <font color="Maroon" title="c1">seq</font></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
<b>by </b><i><a class="txt" href="#E1:82"><i><font color="Green" title="E72">A1</font></i></a>, <a class="txt" href="#E4:82"><i><font color="Green" title="E73">A2</font></i></a>, <a class="ref" href="rinfsup1.html#T75" target="_self" title="RINFSUP1:th.75">Th75</a></i>
<br/>.= 
 <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <font color="Maroon" title="c1">seq</font>
<b>by </b><i><a class="ref" href="seq_4.html#T22" title="SEQ_4:th.22">SEQ_4:22</a></i>
;<br/>
<b>hence </b><a NAME="E8:82"/>
 <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <font color="Maroon" title="c1">seq</font>
 ; <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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