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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<a NAME="E1:12"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:12"/><b>then </b>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> 
 
;<br/>
<b>then </b><i><font color="Green" title="E10">A1</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
<b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></i>
;<br/>
<a NAME="E4:12"/><i><font color="Green" title="E11">A2</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">b</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></i>;<br/>
<a NAME="E5:12"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">b</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> holds <br/><span class="p1">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">b</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">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" title="c3">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:12_2"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">b</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 ;<br/>

<a NAME="E2:12_2"/><i><font color="Green" title="E13">A4</font></i>: 
( <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> &amp; <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E3:12_2"/><i><font color="Green" title="E14">A5</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">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" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="topreal6.html#E4:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="topreal6.html#E1:12_2"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">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" title="E15">A6</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">a</font>
<b>by </b><i><a class="txt" href="topreal6.html#E3:12"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="topreal6.html#E2:12_2"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="topreal6.html#E3:12_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="topreal6.html#T17" target="_self" title="TOPREAL6:th.17">Th17</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">b</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="topreal6.html#E1:12_2_1_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">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" title="E15">A7</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">b</font>
<b>by </b><i><a class="txt" href="topreal6.html#E3:12"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="topreal6.html#E2:12_2"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="topreal6.html#E3:12_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="topreal6.html#T17" target="_self" title="TOPREAL6:th.17">Th17</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">b</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="topreal6.html#E1:12_2_1_2"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">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" title="EUCLID:func.3">abs</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">b</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="topreal6.html#E3:12"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="topreal6.html#E4:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div>
