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<b>set </b><font color="Maroon" title="c2">A</font> =  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c1">a</font>;<br/>
<b>thus </b><a NAME="E1:30_3_1"/>
 <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c1">a</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c1">a</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a><br/>

<b>thus </b><a NAME="E1:30_3_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c1">a</font> or <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="borsuk_5.html#T15" title="BORSUK_5:th.15">BORSUK_5:15</a></i>;<br/>


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

<a NAME="E2:30_3_1"/><i><font color="Green" title="E18">A6</font></i>: 
 <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="limfunc1.html#T10" title="LIMFUNC1:th.10">LIMFUNC1:10</a></i>;<br/>
<a NAME="E3:30_3_1"/>
not  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c1">a</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E23"><i><font color="Green" title="E17">Lm3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:30_3_1"/>
not  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c1">a</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E2:30_3_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="rcomp_1.html#T4" title="RCOMP_1:th.4">RCOMP_1:4</a></i>;<br/>

<b>let </b><font color="Maroon" title="c3">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="jct_misc.html#D1" title="JCT_MISC:def.1">JCT_MISC:def 1</a><br/><b>let </b><font color="Maroon" title="c4">s</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>assume </b><a NAME="E5:30_3_1"/>
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c1">a</font>
 ;<br/>

<a NAME="E6:30_3_1"/><b>then </b><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T15" title="BORSUK_5:th.15">BORSUK_5:15</a></i>;<br/>
<b>assume </b><a NAME="E7:30_3_1"/>
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c1">a</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E8:30_3_1"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E10:30_3_1"/>
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E8:30_3_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E11:30_3_1"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E6:30_3_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E12:30_3_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T15" title="BORSUK_5:th.15">BORSUK_5:15</a></i>;<br/>


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