<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</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>reconsider </b><font color="Maroon" title="c4">V</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <b>by </b><i><a class="txt" href="topgen_3.html#E15"><i><font color="Green" title="E13">Lm3</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> ;<br/><b>reconsider </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c5">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="topgen_3.html#E15"><i><font color="Green" title="E13">Lm3</font></i></a></i>;<br/><b>assume </b><a NAME="E2:20_1"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font>
 ;<br/><a NAME="E3:20_1"/><b>then </b><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">q</font> being  <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E5:20_1"/><i><font color="Green" title="E19">A2</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c7">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="ref" href="rat_1.html#T22" title="RAT_1:th.22">RAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">U</font> = <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">q</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <b>by </b><i><a class="txt" href="topgen_3.html#E15"><i><font color="Green" title="E13">Lm3</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c9">U</font> = <font color="Maroon" title="c8">U</font>;<br/><a NAME="E7:20_1"/>
<font color="Maroon" title="c7">q</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>hence </b><a NAME="E8:20_1"/>
<font color="Maroon" title="c9">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">BB</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E5:20_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="topgen_3.html#E18"><i><font color="Green" title="E15">Lm5</font></i></a></i>;<br/><b>thus </b><a NAME="E9:20_1"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">U</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E5:20_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="rcomp_2.html#T3" title="RCOMP_2:th.3">RCOMP_2:3</a></i>;<br/><b>thus </b><a NAME="E10:20_1"/>
<font color="Maroon" title="c9">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">V</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:20_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="topgen_3.html#E5:20_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="topreala.html#T14" title="TOPREALA:th.14">TOPREALA:14</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> 
 <b>by </b><i><a class="txt" href="topgen_3.html#E20"><i><font color="Green" title="E16">Lm6</font></i></a>, <a class="ref" href="yellow_9.html#T31" title="YELLOW_9:th.31">YELLOW_9:31</a></i>;<br/>


</div>
