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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="measure5.html#NM1">Interval</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><b>suppose </b></a><a NAME="E1:19_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="measure5.html#NM1">Interval</a>
 <b>by </b><i><a class="ref" href="urysohn2.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2"><b>suppose </b></a><a NAME="E1:19_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_2_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V1">open_interval</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V2">closed_interval</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V3">right_open_interval</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V4">left_open_interval</a> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D9">MEASURE5:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_1"><b>case </b></a><a NAME="E1:19_1_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V1">open_interval</a>
 ;<br/><div class="add"><a NAME="E2:19_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V1">open_interval</a>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:19_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="urysohn2.html#T9" target="_self">Th9</a></i>;<br/><b>hence </b><a NAME="E3:19_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="measure5.html#NM1">Interval</a>
 <b>by </b><i><a class="ref" href="measure5.html#D9">MEASURE5:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_2"><b>case </b></a><a NAME="E1:19_1_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V2">closed_interval</a>
 ;<br/><div class="add"><a NAME="E2:19_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V2">closed_interval</a>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:19_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="urysohn2.html#T10" target="_self">Th10</a></i>;<br/><b>hence </b><a NAME="E3:19_1_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="measure5.html#NM1">Interval</a>
 <b>by </b><i><a class="ref" href="measure5.html#D9">MEASURE5:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_3"><b>case </b></a><a NAME="E1:19_1_2_1_1_3"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V3">right_open_interval</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_2_1_1_3_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="urysohn2.html#E1:19_1_2"><i><font color="Green">E19</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_3_1_1_1"><b>case </b></a><a NAME="E1:19_1_2_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:19_1_2_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V4">left_open_interval</a>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:19_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="urysohn2.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E3:19_1_2_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="measure5.html#NM1">Interval</a>
 <b>by </b><i><a class="ref" href="measure5.html#D9">MEASURE5:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_3_1_1_2"><b>case </b></a><a NAME="E1:19_1_2_1_1_3_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:19_1_2_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V3">right_open_interval</a>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:19_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="urysohn2.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E3:19_1_2_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="measure5.html#NM1">Interval</a>
 <b>by </b><i><a class="ref" href="measure5.html#D9">MEASURE5:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:19_1_2_1_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="measure5.html#NM1">Interval</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_4"><b>case </b></a><a NAME="E1:19_1_2_1_1_4"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V4">left_open_interval</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_2_1_1_4_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="urysohn2.html#E1:19_1_2"><i><font color="Green">E19</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_4_1_1_1"><b>case </b></a><a NAME="E1:19_1_2_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:19_1_2_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V3">right_open_interval</a>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:19_1_2_1_1_4"><i><font color="Green">E20</font></i></a>, <a class="ref" href="urysohn2.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E3:19_1_2_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="measure5.html#NM1">Interval</a>
 <b>by </b><i><a class="ref" href="measure5.html#D9">MEASURE5:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_4_1_1_2"><b>case </b></a><a NAME="E1:19_1_2_1_1_4_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:19_1_2_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V4">left_open_interval</a>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:19_1_2_1_1_4"><i><font color="Green">E20</font></i></a>, <a class="ref" href="urysohn2.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E3:19_1_2_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="measure5.html#NM1">Interval</a>
 <b>by </b><i><a class="ref" href="measure5.html#D9">MEASURE5:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:19_1_2_1_1_4"/>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="measure5.html#NM1">Interval</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:19_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="integra2.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="measure5.html#NM1">Interval</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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