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<b>take </b>
 <a href="int_3.html#K3" target="_self">absint</a> 
; <i><font color="Red">:: according to </font></i><a class="ref" href="int_3.html#D9" target="_self">INT_3:def 9</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> ;<br/>

<b>assume </b><a NAME="E1:28_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_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="E1:28_1_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b'</font> or <font color="Maroon">b'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_1"><b>case </b></a><a NAME="E1:28_1_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:28_1_1_1_1_1"/>
 ex <font color="Olive">q</font>, <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">q</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">r</font> &amp; ( <font color="Olive">r</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a>  or <a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> ) )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_2"><b>case </b></a><a NAME="E1:28_1_1_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">b'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:28_1_1_1_1_2"/><b>then </b><i><font color="Green">E23</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b'</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><b>reconsider </b><font color="Maroon">c</font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b'</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> <b> such that </b><br/><a NAME="E5:28_1_1_1_1_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r</font> &amp; ( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a>  or <a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> ) )
 <b>by </b><i><a class="txt" href="int_3.html#E2:28_1_1_1_1_2"><i><font color="Green">E23</font></i></a>, , </i>;<br/><b>reconsider </b><font color="Maroon">t</font> =  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  ;<br/><b>reconsider </b><font color="Maroon">t'</font> = <font color="Maroon">t</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font>, <font color="Maroon">r'</font> = <font color="Maroon">r</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">c</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><b>consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> <b> such that </b><br/><a NAME="E10:28_1_1_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font>
 ;<br/><b>consider </b><font color="Maroon">d'</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E12:28_1_1_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">d'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t'</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b'</font>
 ;<br/><b>consider </b><font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> <b> such that </b><br/><a NAME="E14:28_1_1_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font>
 ;<br/><b>consider </b><font color="Maroon">e'</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E16:28_1_1_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">e'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c'</font>
 ;<br/><a NAME="E17:28_1_1_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d'</font>
 <b>by </b><i>, <a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a>, </i>;<br/><a NAME="E18:28_1_1_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e'</font>
 <b>by </b><i>, , </i>;<br/><i><font color="Green">E48</font></i>: <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r</font> = 
<font color="Maroon">d'</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r'</font>
<b>by </b><i>, <a class="txt" href="int_3.html#E5"><i><font color="Green">Lemma29</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">q'</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b'</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r'</font>
<b>by </b><i><a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a>, </i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="int_3.html#E5:28_1_1_1_1_2"><i><font color="Green">E24</font></i></a>, , , <a class="txt" href="int_3.html#E6"><i><font color="Green">Lemma30</font></i></a>, </i>
;<br/><a NAME="E20:28_1_1_1_1_2"/>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a>  or <a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1_1_1_1_2_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a> 
 ;<br/>

<a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> = 
<a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i/>
<br/>.= 
 <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D2">EUCLID:def 2</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">b'</font>
<b>by </b><i><a class="txt" href="int_3.html#E2:28_1_1_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="int_2.html#K1">abs</a> <font color="Maroon">b'</font>
<b>by </b><i><a class="txt" href="int_3.html#E1:28_1_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font>
<b>by </b><i><a class="ref" href="euclid.html#D2">EUCLID:def 2</a></i>
<br/>.= 
<a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:28_1_1_1_1_2_2"/>
<a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="int_3.html#E5:28_1_1_1_1_2"><i><font color="Green">E24</font></i></a>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E21:28_1_1_1_1_2"/>
 ex <font color="Olive">q</font>, <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">q</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Olive">r</font> &amp; ( <font color="Olive">r</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a>  or <a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:28_1_1"/>
 ex <font color="Olive">q</font>, <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">q</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">r</font> &amp; ( <font color="Olive">r</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a>  or <a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> ) )
 ;<br/>


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