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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:42_1"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c1">A</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 ;<br/><a NAME="E2:42_1"/><b>then </b>
( <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> )
 <b>by </b><i><a class="ref" href="measure5.html#D9" title="MEASURE5:def.9">MEASURE5:def 9</a></i>;<br/><a NAME="E3:42_1"/><b>then </b>
(  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" title="MEASURE5:def.5">MEASURE5:def 5</a>, <a class="ref" href="measure5.html#D6" title="MEASURE5:def.6">MEASURE5:def 6</a>, <a class="ref" href="measure5.html#D7" title="MEASURE5:def.7">MEASURE5:def 7</a>, <a class="ref" href="measure5.html#D8" title="MEASURE5:def.8">MEASURE5:def 8</a></i>;<br/><b>then consider </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E5:42_1"/><i><font color="Green" title="E31">A2</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> )
 ;<br/><b>let </b><font color="Maroon" title="c4">F</font>, <font color="Maroon" title="c5">G</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E6:42_1"/><i><font color="Green" title="E32">A3</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ;<br/><b>let </b><font color="Maroon" title="c6">c</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E7:42_1"/><i><font color="Green" title="E33">A4</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c6">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">G</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1_1"/>
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> )
 </span><b>by </b><i><a class="txt" href="measure6.html#E5:42_1"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1"><b>case </b></a><a NAME="E1:42_1_1_1_1"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_1_1"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:42_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="measure6.html#E6:42_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="measure6.html#E7:42_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="measure6.html#T38" target="_self" title="MEASURE6:th.38">Th38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_2"><b>case </b></a><a NAME="E1:42_1_1_1_2"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_1_2"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:42_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="measure6.html#E6:42_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="measure6.html#E7:42_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="measure6.html#T39" target="_self" title="MEASURE6:th.39">Th39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_3"><b>case </b></a><a NAME="E1:42_1_1_1_3"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_1_3"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:42_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="measure6.html#E6:42_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="measure6.html#E7:42_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="measure6.html#T40" target="_self" title="MEASURE6:th.40">Th40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_4"><b>case </b></a><a NAME="E1:42_1_1_1_4"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_1_4"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:42_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="measure6.html#E6:42_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="measure6.html#E7:42_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="measure6.html#T41" target="_self" title="MEASURE6:th.41">Th41</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:42_1"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:42"/><i><font color="Green" title="E30">A5</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>for <font color="Olive" title="b3">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>assume </b><a NAME="E3:42"/><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon" title="c1">A</font> is  not   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 ;<br/>

<a NAME="E4:42"/><b>then </b><i><font color="Green" title="E32">A7</font></i>: 
( not <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> &amp; not <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> &amp; not <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> &amp; not <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> )
 
<b>by </b><i><a class="ref" href="measure5.html#D9" title="MEASURE5:def.9">MEASURE5:def 9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_2"/>
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  or <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1"><b>suppose </b></a><a NAME="E1:42_2_1"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure6.html#E3:42"><i><font color="Green" title="E31">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_2"><b>suppose </b></a><a NAME="E1:42_2_2"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c2">A</font> = <font color="Maroon" title="c1">A</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  <b>by </b><i><a class="ref" href="membered.html#T2" title="MEMBERED:th.2">MEMBERED:2</a></i>;<br/><b>set </b><font color="Maroon" title="c3">m</font> =  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">A</font>;<br/><b>set </b><font color="Maroon" title="c4">M</font> =  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">A</font>;<br/><b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:42_2_2"/><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="txt" href="measure6.html#E4:42_2_2"><i><font color="Green" title="E33">A8</font></i></a></i>;<br/><a NAME="E6:42_2_2"/>
(  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E4:42_2_2"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="supinf_1.html#T76" title="SUPINF_1:th.76">SUPINF_1:76</a>, <a class="ref" href="supinf_1.html#T79" title="SUPINF_1:th.79">SUPINF_1:79</a></i>;<br/><a NAME="E7:42_2_2"/><b>then </b><i><font color="Green" title="E34">A9</font></i>: 
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_2_2_1_1"/>
( (  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> ) or (  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; not  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> ) or ( not  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> ) or ( not  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; not  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_2_1_1_1"><b>case </b></a><a NAME="E1:42_2_2_1_1_1"/>
(  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 ;<br/><div class="add"><a NAME="E2:42_2_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">A</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="measure6.html#E2:42"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="measure6.html#T43" target="_self" title="MEASURE6:th.43">Th43</a></i>;<br/><b>hence </b><a NAME="E3:42_2_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure6.html#E4:42"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="measure6.html#E7:42_2_2"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="measure5.html#D6" title="MEASURE5:def.6">MEASURE5:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_2_1_1_2"><b>case </b></a><a NAME="E1:42_2_2_1_1_2"/>
(  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; not  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 ;<br/><div class="add"><a NAME="E2:42_2_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">A</font></span>)</span></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span>
 <b>by </b><i><a class="txt" href="measure6.html#E2:42"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="measure6.html#T44" target="_self" title="MEASURE6:th.44">Th44</a></i>;<br/><b>hence </b><a NAME="E3:42_2_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure6.html#E4:42"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="measure6.html#E7:42_2_2"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="measure5.html#D7" title="MEASURE5:def.7">MEASURE5:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_2_1_1_3"><b>case </b></a><a NAME="E1:42_2_2_1_1_3"/>
( not  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 ;<br/><div class="add"><a NAME="E2:42_2_2_1_1_3"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">A</font></span>)</span></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span>
 <b>by </b><i><a class="txt" href="measure6.html#E2:42"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="measure6.html#T45" target="_self" title="MEASURE6:th.45">Th45</a></i>;<br/><b>hence </b><a NAME="E3:42_2_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="measure6.html#E4:42"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="measure6.html#E7:42_2_2"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="measure5.html#D8" title="MEASURE5:def.8">MEASURE5:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_2_1_1_4"><b>case </b></a><a NAME="E1:42_2_2_1_1_4"/>
( not  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; not  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 ;<br/><div class="add"><a NAME="E2:42_2_2_1_1_4"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c2">A</font></span>)</span></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="measure6.html#E2:42"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="measure6.html#T42" target="_self" title="MEASURE6:th.42">Th42</a></i>;<br/><b>hence </b><a NAME="E3:42_2_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="measure6.html#E4:42"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="measure6.html#E7:42_2_2"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="measure5.html#D5" title="MEASURE5:def.5">MEASURE5:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:42_2_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
