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<a NAME="E1:17_1_1"/>
 <a href="relat_1.html#K1">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="E2:17_1_1"/><b>then </b><i><font color="Green">E11</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="relat_1.html#K7">|</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> 
 
<b>by </b><i><a class="ref" href="numbers.html#T15">NUMBERS:15</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E3:17_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="relat_1.html#K7">|</a> <a href="numbers.html#K4">INT</a> </span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:17_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="relat_1.html#K7">|</a> <a href="numbers.html#K4">INT</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_1_1_1"/><i><font color="Green">E12</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <a href="euclid.html#K2">absreal</a>  <a href="relat_1.html#K7">|</a> <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<a NAME="E4:17_1_1_1"/><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="relat_1.html#K7">|</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="int_3.html#E3:17_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E5:17_1_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="relat_1.html#K7">|</a> <a href="numbers.html#K4">INT</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="int_3.html#E3:17_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="txt" href="int_3.html#E2:17_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E7:17_1_1_1"/><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="relat_1.html#K7">|</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="int_3.html#E2:17_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="int_3.html#E5:17_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_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="E1:17_1_1_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> or  not 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:17_1_1_1_1_1_1"/><i><font color="Green">E16</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="relat_1.html#K7">|</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E7:17_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="euclid.html#D2">EUCLID:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E1:17_1_1_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><b>hence </b><a NAME="E3:17_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="relat_1.html#K7">|</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="txt" href="int_3.html#E1:17_1_1_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:17_1_1_1_1_1_2"/><i><font color="Green">E16</font></i>: 
not 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><i><font color="Green">E17</font></i>: <span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="relat_1.html#K7">|</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E7:17_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="euclid.html#D2">EUCLID:def 2</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E1:17_1_1_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><a NAME="E3:17_1_1_1_1_1_2"/>
 <a href="binop_2.html#K7">-</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="int_3.html#E1:17_1_1_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="relat_1.html#K7">|</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="txt" href="int_3.html#E2:17_1_1_1_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:17_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="int_3.html#E4:17_1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:17_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="relat_1.html#K7">|</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <a href="euclid.html#K2">absreal</a>  <a href="relat_1.html#K7">|</a> <a href="numbers.html#K4">INT</a>  as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="int_3.html#E2:17_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>thus </b><a NAME="E6:17_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


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