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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:35"/><i><font color="Green">E30</font></i>: 
( not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 ;<br/>

<a NAME="E2:35"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:35_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K2" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E4:35_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="supinf_2.html#E3:35_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="supinf_1.html#T76">SUPINF_1:76</a></i>;<br/>
<b>hence </b><a NAME="E5:35_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="supinf_2.html#E1:35"><i><font color="Green">E30</font></i></a>, <a class="txt" href="supinf_2.html#E3:35_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="supinf_2.html#T14" target="_self">Th14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:35"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is    <a href="supinf_1.html#M1">majorant</a> of <font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:35"/>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/>


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