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<div class="add">

<b>set </b><font color="Maroon" title="c1">X</font> =  <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> ;<br/>
<a NAME="E1:65"/><i><font color="Green" title="E48">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="topgen_3.html#D2" title="TOPGEN_3:def.2">TOPGEN_3:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">B</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:65"/><i><font color="Green" title="E49">A2</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E4:65"/><i><font color="Green" title="E50">A3</font></i>: 
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">q</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">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>  : ( <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b2">q</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="topgen_3.html#D2" title="TOPGEN_3:def.2">TOPGEN_3:def 2</a></i>;<br/>
<a NAME="E5:65"/>
<font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#T1" title="CANTOR_1:th.1">CANTOR_1:1</a></i>;<br/>
<a NAME="E6:65"/><b>then </b>
<font color="Maroon" title="c2">B</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of  <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> 
 
<b>by </b><i><a class="txt" href="#E1:65"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="#E3:65"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="cantor_1.html#D2" title="CANTOR_1:def.2">CANTOR_1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">B</font> = <font color="Maroon" title="c2">B</font> as    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of  <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <b>by </b><i><a class="ref" href="yellow_9.html#T27" title="YELLOW_9:th.27">YELLOW_9:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <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>   st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> holds <br/> ex <font color="Olive" title="b2">f</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Olive" title="b2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/><b>let </b><font color="Maroon" title="c5">V</font> be   <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V</font> &amp; <font color="Maroon" title="c5">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> implies  ex <font color="Olive" title="b1">f</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) )</span><br/><b>assume </b><a NAME="E1:65_1"/><i><font color="Green" title="E51">A4</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V</font> &amp; <font color="Maroon" title="c5">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/><b>then consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">q</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:65_1"/><i><font color="Green" title="E52">A5</font></i>: 
( <font color="Maroon" title="c5">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">q</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> &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="c7">q</font> &amp; <font color="Maroon" title="c7">q</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a> )
 <b>by </b><i><a class="txt" href="#E4:65"><i><font color="Green" title="E50">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c8">f</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E5:65_1"/><i><font color="Green" title="E53">A6</font></i>: 
for <font color="Olive" title="b1">b</font> being  <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>  holds <br/> ( ( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">q</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> implies <font color="Maroon" title="c8">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( not <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">q</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> implies <font color="Maroon" title="c8">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <b>by </b><i><a class="txt" href="#E3:65_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="topgen_5.html#T62" target="_self" title="TOPGEN_5:th.62">Th62</a></i>;<br/><b>take </b><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">f</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c9">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/><b>thus </b><a NAME="E6:65_1"/>
<font color="Maroon" title="c9">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:65_1"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="#E3:65_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="#E5:65_1"><i><font color="Green" title="E53">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><b>thus </b><a NAME="E7:65_1"/>
<font color="Maroon" title="c9">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> or <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<b>assume </b><a NAME="E1:65_1_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<b>then consider </b><font color="Maroon" title="c11">b</font> being   <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> <b> such that </b><br/><a NAME="E3:65_1_1"/><i><font color="Green" title="E54">A7</font></i>: 
( <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">b</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:65_1_1"/>
not <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">q</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 
<b>by </b><i><a class="txt" href="#E3:65_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="#E3:65_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E5:65_1_1"/><b>then </b>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="#E5:65_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="#E3:65_1_1"><i><font color="Green" title="E54">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:65_1_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:65"/>
 <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  is <a href="topgen_5.html#V1" title="TOPGEN_5:attr.1">Tychonoff</a>
 <b>by </b><i><a class="ref" href="topgen_5.html#T56" target="_self" title="TOPGEN_5:th.56">Th56</a>, <a class="ref" href="topgen_5.html#T57" target="_self" title="TOPGEN_5:th.57">Th57</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
