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<b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>reconsider </b><font color="Maroon">V</font> =  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_3.html#K2" target="_self">Sorgenfrey-line</a>  <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topgen_3.html#K2" target="_self">Sorgenfrey-line</a> ;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/><b>assume </b><a NAME="E2:21_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/><a NAME="E3:21_1"/><b>then </b>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T17">BORSUK_5:17</a></i>;<br/><b>then consider </b><font color="Maroon">q</font> being  <a href="rat_1.html#V1">rational</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:21_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="rat_1.html#T22">RAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">U</font> = <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">q</font></span><a href="rcomp_2.html#K1">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_3.html#K2" target="_self">Sorgenfrey-line</a>  <b>by </b><i/>;<br/><b>take </b><font color="Maroon">U</font> = <font color="Maroon">U</font>;<br/><a NAME="E7:21_1"/>
( <font color="Maroon">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  )
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>hence </b><a NAME="E8:21_1"/>
<font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E9:21_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font>
 <b>by </b><i>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><a NAME="E10:21_1"/>
( <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">q</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp; <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">q</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a>, <a class="ref" href="rcomp_2.html#T5">RCOMP_2:5</a>, <a class="ref" href="limfunc1.html#T22">LIMFUNC1:22</a></i>;<br/><b>hence </b><a NAME="E11:21_1"/>
<font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
 <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">x</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_3.html#K2" target="_self">Sorgenfrey-line</a> 
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/>


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