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<a NAME="E1:54"/><i><font color="Green">E42</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="topgen_3.html#D2">TOPGEN_3:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>1</sub></font> =  <a href="numbers.html#K3">RAT</a>  as   <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a>  <b>by </b><i><a class="ref" href="numbers.html#T12">NUMBERS:12</a>, <a class="ref" href="topgen_3.html#D2">TOPGEN_3:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:54"/><i><font color="Green">E43</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a>  <a href="hidden.html#R1">=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:54"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="rat_1.html#V1">rational</a> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="topgen_3.html#D2">TOPGEN_3:def 2</a></i>;<br/>
<a NAME="E6:54"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="cantor_1.html#M1">Basis</a> of  <a href="topgen_3.html#K2">Sorgenfrey-line</a> 
 
<b>by </b><i><a class="txt" href="topgen_5.html#E1:54"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topgen_5.html#E4:54"><i><font color="Green">E43</font></i></a>, <a class="ref" href="yellow_9.html#T22">YELLOW_9:22</a></i>;<br/>
<a NAME="E7:54"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="tops_1.html#V1">dense</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:54_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>,<a class="ref" href="tops_3.html#D2">TOPS_3:def 2</a><br/>

<b>let </b><font color="Maroon">c<sub>3</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="E2:54_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a> ;<br/><b>assume </b><a NAME="E1:54_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:54_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="rat_1.html#V1">rational</a> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:54"><i><font color="Green">E44</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="rat_1.html#V1">rational</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:54_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:54_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="rat_1.html#T22">RAT_1:22</a></i>;<br/><b>assume </b><a NAME="E6:54_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E7:54_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:54_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topgen_5.html#E5:54_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="rat_1.html#D2">RAT_1:def 2</a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><b>hence </b><a NAME="E8:54_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:54_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E6:54"><i><font color="Green">E45</font></i></a>, <a class="ref" href="yellow_9.html#T37">YELLOW_9:37</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:54"/>
 <a href="numbers.html#K3">RAT</a>  is  <a href="tops_1.html#V1">dense</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a> 
 ;<br/>


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