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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1"><b>suppose </b></a><a NAME="E1:9_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/><font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>
<br/>.= 
0

;<br/><b>hence </b><a NAME="E4:9_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2"><b>suppose </b></a><a NAME="E1:9_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> = 
<a href="supinf_1.html#K1">+infty</a>  <a href="supinf_2.html#K4">-</a> <a href="supinf_1.html#K1">+infty</a> 

<br/>.= 
<a href="supinf_1.html#K1">+infty</a>  <a href="supinf_2.html#K2">+</a> <a href="supinf_1.html#K2">-infty</a> 
<b>by </b><i><a class="ref" href="supinf_2.html#D3">SUPINF_2:def 3</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="extreal2.html#E14"><i><font color="Green">Lemma4</font></i></a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_3"><b>suppose </b></a><a NAME="E1:9_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> = 
<a href="supinf_1.html#K2">-infty</a>  <a href="supinf_2.html#K4">-</a> <a href="supinf_1.html#K2">-infty</a> 

<br/>.= 
<a href="supinf_1.html#K2">-infty</a>  <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K2">-infty</a> </span>)</span>

<br/>.= 
<a href="supinf_1.html#K2">-infty</a>  <a href="supinf_2.html#K2">+</a> <a href="supinf_1.html#K1">+infty</a> 
<b>by </b><i><a class="ref" href="supinf_2.html#D3">SUPINF_2:def 3</a>, <a class="ref" href="xxreal_0.html#T11">XXREAL_0:11</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="extreal2.html#E14"><i><font color="Green">Lemma4</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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