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<b>set </b><font color="Maroon">T</font> =  <a href="topgen_3.html#K2">Sorgenfrey-line</a> ;<br/>
<b>consider </b><font color="Maroon">B</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="E2:67"/><i><font color="Green">E42</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">B</font>
 <b>and </b><br/><a NAME="E3:67"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">B</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">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</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="E4:67"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a> 
 
<b>by </b><i>, <a class="ref" href="cantor_1.html#T1">CANTOR_1:1</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a> , <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="urysohn1.html#D9">URYSOHN1:def 9</a><br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font>, <font color="Maroon">b</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="topgen_3.html#D2">TOPGEN_3:def 2</a></i>;<br/>
<b>assume </b><a NAME="E6:67"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> or <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font> )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1"><b>suppose </b></a><a NAME="E1:67_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">w</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="E3:67_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="rat_1.html#T22">RAT_1:22</a></i>;<br/><a NAME="E4:67_1_1"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</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="E6:67_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">b</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">b</font> <a href="real_1.html#K3">+</a> 1 )
 <b>by </b><i><a class="ref" href="rat_1.html#T22">RAT_1:22</a></i>;<br/><a NAME="E7:67_1_1"/>
( <font color="Maroon">w</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">q</font> is   <a href="real_1.html#NM1">Real</a> )
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E8:67_1_1"/><b>then </b>
( <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">w</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">q</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E4:67"><i><font color="Green">E49</font></i></a>, <a class="txt" href="topgen_5.html#E6:67"><i><font color="Green">E53</font></i></a>, <a class="ref" href="topgen_5.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E9:67_1_1"/><b>then </b>
( <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">w</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a>  &amp; <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">q</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a>  )
 <b>by </b><i/>;<br/><b>then 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">w</font></span><a href="rcomp_2.html#K1">.[</a></span>, <font color="Maroon">V</font> = <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">q</font></span><a href="rcomp_2.html#K1">.[</a></span> as  <a href="pre_topc.html#V3">open</a> <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="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>take </b>
<font color="Maroon">U</font>
;<br/><b>take </b>
<font color="Maroon">V</font>
;<br/><b>thus </b><a NAME="E11:67_1_1"/>
( <font color="Maroon">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/><b>thus </b><a NAME="E12:67_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E6:67"><i><font color="Green">E53</font></i></a>, <a class="ref" href="topgen_5.html#T5" target="_self">Th5</a>, , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2"><b>suppose </b></a><a NAME="E1:67_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">w</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="E3:67_1_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="rat_1.html#T22">RAT_1:22</a></i>;<br/><a NAME="E4:67_1_2"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</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="E6:67_1_2"/><i><font color="Green">E79</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">a</font> <a href="real_1.html#K3">+</a> 1 )
 <b>by </b><i><a class="ref" href="rat_1.html#T22">RAT_1:22</a></i>;<br/><a NAME="E7:67_1_2"/>
( <font color="Maroon">w</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">q</font> is   <a href="real_1.html#NM1">Real</a> )
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E8:67_1_2"/><b>then </b>
( <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">w</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <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> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E4:67"><i><font color="Green">E49</font></i></a>, <a class="txt" href="topgen_5.html#E6:67"><i><font color="Green">E53</font></i></a>, <a class="ref" href="topgen_5.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E9:67_1_2"/><b>then </b>
( <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">w</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a>  &amp; <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> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a>  )
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">V</font> = <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">w</font></span><a href="rcomp_2.html#K1">.[</a></span>, <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="pre_topc.html#V3">open</a> <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="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>take </b>
<font color="Maroon">U</font>
;<br/><b>take </b>
<font color="Maroon">V</font>
;<br/><b>thus </b><a NAME="E11:67_1_2"/>
( <font color="Maroon">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/><b>thus </b><a NAME="E12:67_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E6:67"><i><font color="Green">E53</font></i></a>, <a class="ref" href="topgen_5.html#T5" target="_self">Th5</a>, , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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