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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="xxreal_0.html#V1" target="_self">ext-real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T2" target="_self">Th2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1"><b>suppose </b></a><a NAME="E1:46_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><a NAME="E2:46_1_1"/><b>then </b>
(  <a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; (  <a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or  <a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="ref" href="xxreal_0.html#D9" target="_self">Def9</a>, <a class="ref" href="xxreal_0.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E3:46_1_1"/>
 <a href="xxreal_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2"><b>suppose </b></a><a NAME="E1:46_1_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><a NAME="E2:46_1_2"/><b>then </b>
(  <a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#D9" target="_self">Def9</a></i>;<br/><b>hence </b><a NAME="E3:46_1_2"/>
 <a href="xxreal_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_3"><b>suppose </b></a><a NAME="E1:46_1_3"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><a NAME="E2:46_1_3"/><b>then </b>
(  <a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; (  <a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or  <a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="xxreal_0.html#D9" target="_self">Def9</a>, <a class="ref" href="xxreal_0.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E3:46_1_3"/>
 <a href="xxreal_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" target="_self">max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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