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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:50"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="pscomp_1.html#V4" target="_self">closed</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rcomp_1.html#D4">RCOMP_1:def 4</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:50"/><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:50"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><b>suppose </b></a><a NAME="E1:50_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:50_1_1"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:50_1_1"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:50"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2"><b>suppose </b></a><a NAME="E1:50_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_1_2_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:50_1_2_1"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:50_1_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:50_1_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:50"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pscomp_1.html#E1:50_1_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="pscomp_1.html#E1:50_1_2_1"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E4:50_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="rcomp_1.html#V2">closed</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:50"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pscomp_1.html#E1:50_1_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pscomp_1.html#D6" target="_self">Def6</a></i>;<br/><b>hence </b><a NAME="E5:50_1_2_1"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:50"><i><font color="Green">E33</font></i></a>, <a class="txt" href="pscomp_1.html#E2:50_1_2_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rcomp_1.html#D4">RCOMP_1:def 4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:50_1_2"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:50_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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