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

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Olive" title="b1">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a></span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:19_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><span class="kw">suppose </span></a><a NAME="E1:19_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:19_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn2.html#T8" target="_self" title="URYSOHN2:th.8">Th8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2"><span class="kw">suppose </span></a><a NAME="E1:19_1_2"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:19_1_2_1_1"/>
( <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  or <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  or <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a>  or <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D9" title="MEASURE5:def.9">MEASURE5:def 9</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:19_1_2_1_1_1"/>
<font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a></span><br/><div class="add"><a NAME="E2:19_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_2"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="urysohn2.html#T9" target="_self" title="URYSOHN2:th.9">Th9</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_2_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D9" title="MEASURE5:def.9">MEASURE5:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:19_1_2_1_1_2"/>
<font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a></span><br/><div class="add"><a NAME="E2:19_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_2"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="urysohn2.html#T10" target="_self" title="URYSOHN2:th.10">Th10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_2_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D9" title="MEASURE5:def.9">MEASURE5:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_3"><span class="kw">case </span></a><a NAME="E1:19_1_2_1_1_3"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:19_1_2_1_1_3_1"/>
( <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_2"><span class="lab"><font color="Green" title="E14">A1</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_3_1_1"><span class="kw">case </span></a><a NAME="E1:19_1_2_1_1_3_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a></span><br/><div class="add"><a NAME="E2:19_1_2_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_2_1_1_3"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="urysohn2.html#T12" target="_self" title="URYSOHN2:th.12">Th12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_2_1_1_3_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D9" title="MEASURE5:def.9">MEASURE5:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_3_1_2"><span class="kw">case </span></a><a NAME="E1:19_1_2_1_1_3_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a></span><br/><div class="add"><a NAME="E2:19_1_2_1_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_2_1_1_3"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="urysohn2.html#T11" target="_self" title="URYSOHN2:th.11">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_2_1_1_3_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D9" title="MEASURE5:def.9">MEASURE5:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_4"><span class="kw">case </span></a><a NAME="E1:19_1_2_1_1_4"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_1_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:19_1_2_1_1_4_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_2"><span class="lab"><font color="Green" title="E14">A1</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_4_1_1_1"><span class="kw">case </span></a><a NAME="E1:19_1_2_1_1_4_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a></span><br/><div class="add"><a NAME="E2:19_1_2_1_1_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_2_1_1_4"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="urysohn2.html#T14" target="_self" title="URYSOHN2:th.14">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_2_1_1_4_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D9" title="MEASURE5:def.9">MEASURE5:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_4_1_1_2"><span class="kw">case </span></a><a NAME="E1:19_1_2_1_1_4_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a></span><br/><div class="add"><a NAME="E2:19_1_2_1_1_4_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_2_1_1_4"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="urysohn2.html#T13" target="_self" title="URYSOHN2:th.13">Th13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_2_1_1_4_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D9" title="MEASURE5:def.9">MEASURE5:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:19_1_2_1_1_4"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:19_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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