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<b>let </b><font color="Maroon">t</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><a NAME="E1:6"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font>, <font color="Maroon">t</font> = <font color="Maroon">t</font>, <font color="Maroon">s</font> = <font color="Maroon">s</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:6"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1"/>
( <font color="Maroon">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1"><b>suppose </b></a><a NAME="E1:6_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:6_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:6_1_1"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E4:6_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:6_1_1"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="square_1.html#T50">SQUARE_1:50</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2"><b>suppose </b></a><a NAME="E1:6_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:6_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="real_1.html#K1">-</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:6_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E4:6_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E5:6_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>hence </b><a NAME="E6:6_1_2"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="square_1.html#T50">SQUARE_1:50</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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