<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1"><b>case </b></a><a NAME="E1:4_1_1_1"/><i><font color="Green">E3</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><i><font color="Green">E4</font></i>:  <a href="rfunct_3.html#K2" target="_self">max+</a> <font color="Maroon">c<sub>1</sub></font> = 
 <a href="xxreal_0.html#K4">max</a> 0,<font color="Maroon">c<sub>1</sub></font>

<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="rfunct_3.html#E1:4_1_1_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>
;<br/><a NAME="E3:4_1_1_1"/><i><font color="Green">E5</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:4_1_1_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/> <a href="rfunct_3.html#K3" target="_self">max-</a> <font color="Maroon">c<sub>1</sub></font> = 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,0

<br/>.= 
0
<b>by </b><i><a class="txt" href="rfunct_3.html#E3:4_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>
;<br/><b>hence </b><a NAME="E5:4_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K2" target="_self">max+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K3" target="_self">max-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:4_1_1_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="rfunct_3.html#E2:4_1_1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_2"><b>case </b></a><a NAME="E1:4_1_1_2"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><i><font color="Green">E4</font></i>:  <a href="rfunct_3.html#K2" target="_self">max+</a> <font color="Maroon">c<sub>1</sub></font> = 
 <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,0

<br/>.= 
0
<b>by </b><i><a class="txt" href="rfunct_3.html#E1:4_1_1_2"><i><font color="Green">E3</font></i></a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>
;<br/><a NAME="E3:4_1_1_2"/><i><font color="Green">E5</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:4_1_1_2"><i><font color="Green">E3</font></i></a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/> <a href="rfunct_3.html#K3" target="_self">max-</a> <font color="Maroon">c<sub>1</sub></font> = 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,0

<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="rfunct_3.html#E3:4_1_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>
;<br/><b>hence </b><a NAME="E5:4_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K2" target="_self">max+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K3" target="_self">max-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:4_1_1_2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="rfunct_3.html#E2:4_1_1_2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:4"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K2" target="_self">max+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K3" target="_self">max-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>


</div>
