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

<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E11">A1</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_1"/>
(  <a href="extreal2.html#K1" title="EXTREAL2:func.1">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> or  <a href="extreal2.html#K1" title="EXTREAL2:func.1">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1"><b>suppose </b></a><a NAME="E1:11_1_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
 <a href="extreal2.html#K1" title="EXTREAL2:func.1">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 ;<br/><div class="add"><a NAME="E2:11_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>;<br/><a NAME="E3:11_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:11"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="extreal1.html#T42" title="EXTREAL1:th.42">EXTREAL1:42</a></i>;<br/><b>hence </b><a NAME="E4:11_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="extreal2.html#K1" title="EXTREAL2:func.1">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal2.html#K1" title="EXTREAL2:func.1">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:11_1_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_2"><b>suppose </b></a><a NAME="E1:11_1_1_2"/><i><font color="Green" title="E12">A3</font></i>: 
 <a href="extreal2.html#K1" title="EXTREAL2:func.1">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 ;<br/><div class="add"><a NAME="E2:11_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>;<br/><a NAME="E3:11_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:11"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="extreal1.html#T42" title="EXTREAL1:th.42">EXTREAL1:42</a></i>;<br/><b>hence </b><a NAME="E4:11_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="extreal2.html#K1" title="EXTREAL2:func.1">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal2.html#K1" title="EXTREAL2:func.1">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:11_1_1_2"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:11"/>
<font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="extreal2.html#K1" title="EXTREAL2:func.1">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal2.html#K1" title="EXTREAL2:func.1">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_2"/>
(  <a href="extreal2.html#K2" title="EXTREAL2:func.2">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> or  <a href="extreal2.html#K2" title="EXTREAL2:func.2">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1"><b>suppose </b></a><a NAME="E1:11_2_1"/><i><font color="Green" title="E12">A4</font></i>: 
 <a href="extreal2.html#K2" title="EXTREAL2:func.2">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 ;<br/><div class="add"><a NAME="E2:11_2_1"/><b>then </b>
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a></i>;<br/><a NAME="E3:11_2_1"/><b>then </b>
<font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:11"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="extreal1.html#T42" title="EXTREAL1:th.42">EXTREAL1:42</a></i>;<br/><b>hence </b><a NAME="E4:11_2_1"/>
<font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="extreal2.html#K2" title="EXTREAL2:func.2">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal2.html#K2" title="EXTREAL2:func.2">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:11_2_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_2"><b>suppose </b></a><a NAME="E1:11_2_2"/><i><font color="Green" title="E12">A5</font></i>: 
 <a href="extreal2.html#K2" title="EXTREAL2:func.2">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 ;<br/><div class="add"><a NAME="E2:11_2_2"/><b>then </b>
<font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a></i>;<br/><a NAME="E3:11_2_2"/><b>then </b>
<font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:11"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="extreal1.html#T42" title="EXTREAL1:th.42">EXTREAL1:42</a></i>;<br/><b>hence </b><a NAME="E4:11_2_2"/>
<font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="extreal2.html#K2" title="EXTREAL2:func.2">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal2.html#K2" title="EXTREAL2:func.2">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:11_2_2"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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