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



<a NAME="E1:12"/>
 <a href="relset_1.html#K4">dom</a> <a href="euclid.html#K2">absreal</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:12"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="euclid.html#K2">absreal</a> 
 
;<br/>
<i><font color="Green">E57</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29">FINSEQ_3:29</a></i>
;<br/>
<a NAME="E4:12"/><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29">FINSEQ_3:29</a></i>;<br/>
<a NAME="E5:12"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> holds <br/><span class="p1">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:12_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 ;<br/>

<a NAME="E2:12_2"/><i><font color="Green">E66</font></i>: 
( <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="seq_1.html#K2">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E3:12_2"/><i><font color="Green">E67</font></i>: 
( 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span> &amp; 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1"><b>suppose </b></a><a NAME="E1:12_2_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> = 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_2"><b>suppose </b></a><a NAME="E1:12_2_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="euclid.html#K3">abs</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> = 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">b</font>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E6:12"/>
 <a href="euclid.html#K3">abs</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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