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<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>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>assume </b><a NAME="E1:12"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1"/>
( <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><b>suppose </b></a><a NAME="E1:12_1_1"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:12_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
(  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">s</font> &amp;  <a href="xcmplx_0.html#K4">-</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="E3:12_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E4:12_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E5:12_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E6:12_1_1"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2"><b>suppose </b></a><a NAME="E1:12_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:12_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">s</font> &amp; <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="E3:12_1_2"/>
0 <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="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E4:12_1_2"/><b>then </b>
<font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E5:12_1_2"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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