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<span class="kw">let </span><font color="Maroon" title="c1">k</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c1">k</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  &amp; <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>,<font color="Maroon" title="c1">k</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  )</span><br/>

<a NAME="E1:19"/><span class="lab"><font color="Green" title="E10">A</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">R</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>,<font color="Maroon" title="c1">k</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/>
<a NAME="E2:19"/><span class="lab"><font color="Green" title="E11">FJ20</font></span>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>,<font color="Maroon" title="c1">k</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c3">L</font> =  <a href="prob_1.html#K10" title="PROB_1:func.10">halfline</a> <font color="Maroon" title="c1">k</font>;<br/>
<a NAME="E1:19_1"/><span class="lab"><font color="Green" title="E12">FG2</font></span>: 
(  <a href="prob_1.html#K10" title="PROB_1:func.10">halfline</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">Family_of_halflines</a>  &amp;  <a href="prob_1.html#K10" title="PROB_1:func.10">halfline</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>,<font color="Maroon" title="c1">k</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E10">A</font></span></a></span>;<br/>
<a NAME="E2:19_1"/>
 <a href="prob_1.html#K11" title="PROB_1:func.11">Family_of_halflines</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <a href="prob_1.html#K11" title="PROB_1:func.11">Family_of_halflines</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:19_1"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>,<font color="Maroon" title="c1">k</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1"><span class="lab"><font color="Green" title="E12">FG2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:19"/><span class="kw">then </span>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>,<font color="Maroon" title="c1">k</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D1" title="PROB_1:def.1">PROB_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:19"/>
( <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c1">k</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  &amp; <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>,<font color="Maroon" title="c1">k</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance1.html#T2" target="_self" title="FINANCE1:th.2">AJF0</a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E11">FJ20</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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