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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span>)</span></span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_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"/>
( <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">c</font> or <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><b>suppose </b></a><a NAME="E1:49_1_1"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span>)</span></span><br/><div class="add"><a NAME="E2:49_1_1"/><b>then </b><i><font color="Green" title="E27">A2</font></i>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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">a</font>,<font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T26" target="_self" title="XXREAL_0:th.26">Th26</a></i>;<br/><b>thus </b> <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span>)</span> = 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>
<b>by </b><i><a class="txt" href="#E1:49_1_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="xxreal_0.html#D9" target="_self" title="XXREAL_0:def.9">Def8</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:49_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="xxreal_0.html#D9" target="_self" title="XXREAL_0:def.9">Def8</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2"><b>suppose </b></a><a NAME="E1:49_1_2"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span>)</span></span><br/><div class="add"><a NAME="E2:49_1_2"/><b>then </b><i><font color="Green" title="E27">A4</font></i>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</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">a</font>,<font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T26" target="_self" title="XXREAL_0:th.26">Th26</a></i>;<br/><b>thus </b> <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span>)</span> = 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>
<b>by </b><i><a class="txt" href="#E1:49_1_2"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="xxreal_0.html#D9" target="_self" title="XXREAL_0:def.9">Def8</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:49_1_2"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="xxreal_0.html#D9" target="_self" title="XXREAL_0:def.9">Def8</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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