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



<a NAME="E1:49"/><i><font color="Green">E18</font></i>: 
(  <a href="supinf_2.html#K3">-</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> implies <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E19</font></i>: 
(  <a href="supinf_2.html#K3">-</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> or  not 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_1_1_1"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="extreal2.html#T15" target="_self">Th15</a>, <a class="ref" href="extreal1.html#D3">EXTREAL1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_2"/><i><font color="Green">E20</font></i>: 
not 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:49_1_1_2"/><b>then </b>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K3">-</a> <font color="Maroon">x</font> &amp;  <a href="supinf_2.html#K3">-</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="extreal2.html#T15" target="_self">Th15</a>, <a class="ref" href="extreal2.html#T9" target="_self">Th9</a>, <a class="ref" href="extreal1.html#T25">EXTREAL1:25</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1_2"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="extreal2.html#T17" target="_self">Th17</a>, <a class="ref" href="extreal1.html#D3">EXTREAL1:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:49"/>
( <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> implies (  <a href="supinf_2.html#K3">-</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E24</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_2_1"/>
( <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> )
 </span><b>by </b><i><a class="ref" href="extreal2.html#T15" target="_self">Th15</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1"><b>suppose </b></a><a NAME="E1:49_2_1_1"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_2_1_1"/>
(  <a href="supinf_2.html#K3">-</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="extreal2.html#E17"><i><font color="Green">Lemma29</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_2"><b>suppose </b></a><a NAME="E1:49_2_1_2"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_2_1_2"/>
(  <a href="supinf_2.html#K3">-</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="extreal2.html#E19"><i><font color="Green">Lemma30</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:49"/>
( (  <a href="supinf_2.html#K3">-</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> ) iff <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="extreal2.html#T14" target="_self">Th14</a></i>;<br/>


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