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<b>set </b><font color="Maroon" title="c2">g</font> = <a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">x</font>;<br/>
<a NAME="E1:3_1_2"/><i><font color="Green" title="E2">A1</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="valued_1.html#NK57" title="VALUED_1:NK.57">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="ref" href="valued_1.html#D11" title="VALUED_1:def.11">VALUED_1:def 11</a></i>;<br/>
<a NAME="E2:3_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="E3:3_1_2"/><b>then </b>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> 
 
;<br/>
<a NAME="E4:3_1_2"/><b>then </b><i><font color="Green" title="E3">A2</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#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_2_1"/><i><font color="Green" title="E4">A3</font></i>: 
<font color="Maroon" title="c3">a</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="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="valued_1.html#NK57" title="VALUED_1:NK.57">.|</a></span>
 ;<br/><b>thus </b><span class="p1"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="valued_1.html#NK57" title="VALUED_1:NK.57">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">a</font> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">a</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></i>
<br/>.= 
<a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">a</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D2" target="_self" title="EUCLID:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="euclid.html#E1:3_1_2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="euclid.html#E4:3_1_2"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="euclid.html#E1:3_1_2_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:3_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/> ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="valued_1.html#K54" title="VALUED_1:func.54">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="valued_1.html#K54" title="VALUED_1:func.54">.|</a></span> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">x</font> )
 <b>by </b><i><a class="txt" href="euclid.html#E1:3_1_2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="euclid.html#E4:3_1_2"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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