<?xml version="1.0"?>
<div class="add">

<a NAME="E1:13_1_1"/>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:13_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T15" title="NUMBERS:th.15">NUMBERS:15</a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E3:13_1_1"/>
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span> implies <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:13_1_1_1"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:13_1_1_1"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></span>;<br/>
<a NAME="E4:13_1_1_1"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13_1_1_1"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="funct_1.html#T1" title="FUNCT_1:th.1">FUNCT_1:1</a></span>;<br/>
<a NAME="E5:13_1_1_1"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13_1_1_1"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/>
<a NAME="E7:13_1_1_1"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</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#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E5:13_1_1_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:13_1_1_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> or  not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:13_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><div class="add"><span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:13_1_1_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="euclid.html#D2" title="EUCLID:def.2">EUCLID:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:13_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:13_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E11">A7</font></span>: 
 not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><div class="add"><span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:13_1_1_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="euclid.html#D2" title="EUCLID:def.2">EUCLID:def 2</a></span>
<br/>.= 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:13_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:13_1_1_1"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13_1_1_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:13_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">f</font> = <a href="euclid.html#K2" title="EUCLID:func.2">absreal</a> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a> holds  <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</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#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font></span><br/>

<span class="kw">thus </span><a NAME="E6:13_1_1"/>
 for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a> holds  <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</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#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
