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<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:30_1_1"/><i><font color="Green">E18</font></i>: 
 <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="limfunc1.html#T15">LIMFUNC1:15</a></i>;<br/>
<a NAME="E2:30_1_1"/>
not  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:30_1_1"/>
not  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:30_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="rcomp_1.html#T3">RCOMP_1:3</a></i>;<br/>

<b>thus </b><a NAME="E4:30_1_1"/>
 <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V1">bounded_above</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>1</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D1">SEQ_4:def 1</a><br/>

<b>thus </b><a NAME="E1:30_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="borsuk_5.html#T16">BORSUK_5:16</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="jct_misc.html#D1">JCT_MISC:def 1</a><br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E5:30_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E6:30_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E7:30_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T16">BORSUK_5:16</a></i>;<br/>
<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="E8:30_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E10:30_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E8:30_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E11:30_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E7:30_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E12:30_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T16">BORSUK_5:16</a></i>;<br/>


</div>
