<?xml version="1.0"?>
<div class="add">

<a NAME="E1:37"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</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>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:37_1"/>
 <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</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>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:37_1_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a>  implies <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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><br/>

<span class="kw">assume </span><a NAME="E1:37_1_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">r</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E3:37_1_1_1"/><span class="lab"><font color="Green" title="E18">B1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">right_closed_halfline</a> <font color="Maroon" title="c2">r</font>
 ;<br/>
<a NAME="E4:37_1_1_1"/>
<font color="Maroon" title="c1">x</font> 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#T3" onmouseover="rs('finance1/T3')" onmouseout="rh()">FINANCE1:3</a>, <a class="txt" href="#E3:37_1_1_1"><span class="lab"><font color="Green" title="E18">B1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:37_1_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">hence </span><a NAME="E2:37_1_1"/>
 <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">hence </span><a NAME="E2:37_1"/>
 <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</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" onmouseover="rs('prob_1/D9')" onmouseout="rh()">PROB_1:def 9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:37"/>
 <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> <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="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:37_2"/>
 <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="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:37_2_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</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>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</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>  implies <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_2_1_1"/>
<font color="Maroon" title="c1">x</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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">r</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E3:37_2_1_1"/><span class="lab"><font color="Green" title="E18">B1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">halfline</a> <font color="Maroon" title="c2">r</font>
 ;<br/>
<a NAME="E4:37_2_1_1"/>
 <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">right_closed_halfline</a> <font color="Maroon" title="c2">r</font> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c3">L</font> =  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">right_closed_halfline</a> <font color="Maroon" title="c2">r</font>;<br/>
<a NAME="E1:37_2_1_1_1"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
(  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">right_closed_halfline</a> <font color="Maroon" title="c2">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a>  &amp;  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">right_closed_halfline</a> <font color="Maroon" title="c2">r</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="c2">r</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> )
 
;<br/>
<a NAME="E2:37_2_1_1_1"/>
 <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</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="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D9" onmouseover="rs('prob_1/D9')" onmouseout="rh()">PROB_1:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:37_2_1_1_1"/>
 <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">right_closed_halfline</a> <font color="Maroon" title="c2">r</font> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_2_1_1_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E5:37_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="prob_1.html#K10" title="PROB_1:func.10">halfline</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance1.html#T2" onmouseover="rs('finance1/T2')" onmouseout="rh()">FINANCE1:2</a></span>;<br/>
<a NAME="E6:37_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_1.html#K10" title="PROB_1:func.10">halfline</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T15" onmouseover="rs('prob_1/T15')" onmouseout="rh()">PROB_1:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:37_2_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_2_1_1"><span class="lab"><font color="Green" title="E18">B1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">hence </span><a NAME="E2:37_2_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="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">hence </span><a NAME="E2:37_2"/>
 <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> <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="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D9" onmouseover="rs('prob_1/D9')" onmouseout="rh()">PROB_1:def 9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">hence </span><a NAME="E3:37"/>
 <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <a href="finance2.html#K1" title="FINANCE2:func.1">Family_of_halflines2</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
