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<div class="add">

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="topgen_3.html#K2">Sorgenfrey-line</a> ;<br/>
<a NAME="E1:73"/><i><font color="Green">E52</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>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="E3:73"/><i><font color="Green">E53</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="E4:73"/><i><font color="Green">E54</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="E5:73"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#T1">CANTOR_1:1</a></i>;<br/>
<a NAME="E6:73"/><b>then </b>
<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#E1:73"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topgen_5.html#E3:73"><i><font color="Green">E53</font></i></a>, <a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="cantor_1.html#M2">prebasis</a> of  <a href="topgen_3.html#K2">Sorgenfrey-line</a>  <b>by </b><i><a class="ref" href="yellow_9.html#T27">YELLOW_9:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a> ;<br/><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="E1:73_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</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="E3:73_1"/><i><font color="Green">E56</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#E4:73"><i><font color="Green">E54</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a> ,<a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E5:73_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a>  holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</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> implies <font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</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> implies <font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:73_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="topgen_5.html#T62" target="_self">Th62</a></i>;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font>;<br/><b>thus </b><a NAME="E6:73_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:73_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topgen_5.html#E3:73_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="topgen_5.html#E5:73_1"><i><font color="Green">E57</font></i></a></i>;<br/><b>thus </b><a NAME="E7:73_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</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="E1:73_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topgen_3.html#K2">Sorgenfrey-line</a> <b> such that </b><br/><a NAME="E3:73_1_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:73_1_1"/>
not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</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>
 
<b>by </b><i><a class="txt" href="topgen_5.html#E3:73_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="topgen_5.html#E3:73_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:73_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="topgen_5.html#E5:73_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="topgen_5.html#E3:73_1_1"><i><font color="Green">E58</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:73_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:73"/>
 <a href="topgen_3.html#K2">Sorgenfrey-line</a>  is <a href="topgen_5.html#V1" target="_self">Tychonoff</a>
 <b>by </b><i><a class="ref" href="topgen_5.html#T57" target="_self">Th57</a>, <a class="ref" href="topgen_5.html#T56" target="_self">Th56</a></i>;<br/>


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