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<b>let </b><font color="Maroon" title="c1">y</font>, <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">z</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>





<a NAME="E1:67"/><i><font color="Green" title="E43">A1</font></i>: 
( <font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> implies  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">y</font>,<font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1"/><i><font color="Green" title="E44">A2</font></i>: 
( <font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_1_1"/>
(  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">y</font>,<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">y</font> or  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">y</font>,<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> )
 </span><b>by </b><i><a class="ref" href="extreal2.html#T95" target="_self" title="EXTREAL2:th.95">Th95</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_1_1_1"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">y</font>,<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:67_1_1_1_1"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">y</font>,<font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="extreal2.html#E1:67_1"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_1_1_2"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">y</font>,<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:67_1_1_1_2"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">y</font>,<font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="extreal2.html#E1:67_1"><i><font color="Green" title="E44">A2</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="E3:67_1"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">y</font>,<font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:67"/>
(  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">y</font>,<font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> implies ( <font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_2"/><i><font color="Green" title="E44">A3</font></i>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">y</font>,<font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font>
 ;<br/>

<a NAME="E2:67_2"/>
( <font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">y</font>,<font color="Maroon" title="c3">z</font> &amp; <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">y</font>,<font color="Maroon" title="c3">z</font> )
 
<b>by </b><i><a class="ref" href="extreal2.html#T92" target="_self" title="EXTREAL2:th.92">Th92</a></i>;<br/>
<b>hence </b><a NAME="E3:67_2"/>
( <font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> )
 <b>by </b><i><a class="txt" href="extreal2.html#E1:67_2"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:67"/>
( ( <font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> ) iff  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">y</font>,<font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> )
 <b>by </b><i><a class="txt" href="extreal2.html#E1:67"><i><font color="Green" title="E43">A1</font></i></a></i>;<br/>


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