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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>suppose </b></a><a NAME="E1:4_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:4_1_1"/><b>then </b>
<a href="supinf_2.html#K1">0.</a>  <a href="supinf_2.html#K4">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="supinf_2.html#T6">SUPINF_2:6</a></i>;<br/><b>hence </b><a NAME="E3:4_1_1"/>
<a href="supinf_2.html#K1">0.</a>  <a href="supinf_2.html#K4">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#D3">SUPINF_2:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2"><b>suppose </b></a><a NAME="E1:4_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_1_2"/>
<a href="supinf_2.html#K1">0.</a>  <a href="supinf_2.html#K4">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a>, <a class="ref" href="supinf_2.html#T7">SUPINF_2:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_3"><b>suppose </b></a><a NAME="E1:4_1_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a href="supinf_2.html#K1">0.</a>  <a href="supinf_2.html#K4">-</a> <font color="Maroon">x</font> = 
0 <a href="real_1.html#K5">-</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <font color="Maroon">a</font>

;<br/><b>hence </b><a NAME="E4:4_1_3"/>
<a href="supinf_2.html#K1">0.</a>  <a href="supinf_2.html#K4">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T3">SUPINF_2:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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