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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:10"/><i><font color="Green">E10</font></i>: 
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:10"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>suppose </b></a><a NAME="E1:10_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:10_1_1"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:10"><i><font color="Green">E10</font></i></a>, <a class="txt" href="mesfunc5.html#E2:10"><i><font color="Green">E11</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E3:10_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/><a NAME="E4:10_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:10"><i><font color="Green">E11</font></i></a>, <a class="txt" href="mesfunc5.html#E1:10_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="supinf_2.html#T8">SUPINF_2:8</a></i>;<br/><b>hence </b><a NAME="E5:10_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:10_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2"><b>suppose </b></a><a NAME="E1:10_1_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:10_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/><a NAME="E3:10_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:10"><i><font color="Green">E10</font></i></a>, <a class="txt" href="mesfunc5.html#E2:10"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><a NAME="E4:10_1_2"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:10"><i><font color="Green">E11</font></i></a>, <a class="ref" href="measure6.html#T12">MEASURE6:12</a></i>;<br/><a NAME="E5:10_1_2"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T21">SUPINF_1:21</a></i>;<br/><a NAME="E6:10_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a>, <a class="txt" href="mesfunc5.html#E1:10_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="mesfunc5.html#E2:10"><i><font color="Green">E11</font></i></a></i>;<br/><a NAME="E7:10_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="measure6.html#T12">MEASURE6:12</a>, <a class="txt" href="mesfunc5.html#E2:10"><i><font color="Green">E11</font></i></a></i>;<br/><b>hence </b><a NAME="E8:10_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:10_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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