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<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>thus </b><a NAME="E1:1"/>
(  not  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> 1 or ( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1 ) or ( <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="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1"/><i><font color="Green">E39</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font> or <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> or <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1"><b>case </b></a><a NAME="E1:1_1_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E3:1_1_1_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_1_1"/>
( ( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1 ) or ( <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="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1 ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2"><b>case </b></a><a NAME="E1:1_1_1_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_3"><b>case </b></a><a NAME="E1:1_1_1_1_3"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_3"/><b>then </b><i><font color="Green">E42</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>1 = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><b>hence </b><a NAME="E4:1_1_1_1_3"/>
( ( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1 ) or ( <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="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1 ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:1_1"/>
( ( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1 ) or ( <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="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1 ) )
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:1"/><i><font color="Green">E43</font></i>: 
( ( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1 ) or ( <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="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1 ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_2"/>
( ( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1 ) or ( <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_2_1"><b>suppose </b></a><a NAME="E1:1_2_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_2_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><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="1_2_2"><b>suppose </b></a><a NAME="E1:1_2_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 ;<br/><div class="add"><b>thus </b> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

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