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<b>set </b><font color="Maroon" title="c4">I</font> =  <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">b'</font> = <font color="Maroon" title="c3">b</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_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="E1:25_1_1_1_1"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">b'</font> or <font color="Maroon" title="c5">b'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1_1"><b>case </b></a><a NAME="E1:25_1_1_1_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">b'</font>
 ;<br/><div class="add"><a NAME="E2:25_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">r</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>  st <br/>( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">r</font> &amp;  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  <a href="int_3.html#R1" title="INT_3:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b2">r</font> <a href="int_3.html#NR3" title="INT_3:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="int_3.html#E18"><i><font color="Green" title="E16">Lm11</font></i></a></i>;<br/><b>hence </b><a NAME="E3:25_1_1_1_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive" title="b2">q</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>  st <br/>( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  <a href="int_3.html#R1" title="INT_3:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="int_3.html#NR3" title="INT_3:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1_2"><b>case </b></a><a NAME="E1:25_1_1_1_1_2"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c5">b'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:25_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c5">b'</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T60" title="XREAL_1:th.60">XREAL_1:60</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">c</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c5">b'</font> as   <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>  <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">q</font>, <font color="Maroon" title="c8">r</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> <b> such that </b><br/><a NAME="E5:25_1_1_1_1_2"/><i><font color="Green" title="E22">A4</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c8">r</font> &amp;  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  <a href="int_3.html#R1" title="INT_3:pred.1">&lt;=</a> <font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c8">r</font> <a href="int_3.html#NR3" title="INT_3:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c6">c</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E2:25_1_1_1_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="int_3.html#E5"><i><font color="Green" title="E3">Lm2</font></i></a>, <a class="txt" href="int_3.html#E18"><i><font color="Green" title="E16">Lm11</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">t</font> =  <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c7">q</font> as   <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>  ;<br/><b>reconsider </b><font color="Maroon" title="c10">t'</font> = <font color="Maroon" title="c9">t</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">q'</font> = <font color="Maroon" title="c7">q</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">r'</font> = <font color="Maroon" title="c8">r</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">c'</font> = <font color="Maroon" title="c6">c</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><b>consider </b><font color="Maroon" title="c14">d</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> <b> such that </b><br/><a NAME="E12:25_1_1_1_1_2"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c14">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">b</font>
 ;<br/><b>consider </b><font color="Maroon" title="c15">d'</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E14:25_1_1_1_1_2"/><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c15">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">b'</font>
 ;<br/><b>consider </b><font color="Maroon" title="c16">e</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> <b> such that </b><br/><a NAME="E16:25_1_1_1_1_2"/><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c16">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">c</font>
 ;<br/><b>consider </b><font color="Maroon" title="c17">e'</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E18:25_1_1_1_1_2"/><i><font color="Green" title="E26">A8</font></i>: 
<font color="Maroon" title="c17">e'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">c'</font>
 ;<br/><a NAME="E19:25_1_1_1_1_2"/><i><font color="Green" title="E27">A9</font></i>: 
<font color="Maroon" title="c14">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d'</font>
 <b>by </b><i><a class="txt" href="int_3.html#E12:25_1_1_1_1_2"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="int_3.html#E14:25_1_1_1_1_2"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="int_3.html#E6"><i><font color="Green" title="E4">Lm3</font></i></a></i>;<br/><a NAME="E20:25_1_1_1_1_2"/><i><font color="Green" title="E28">A10</font></i>: 
<font color="Maroon" title="c16">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">e'</font>
 <b>by </b><i><a class="txt" href="int_3.html#E16:25_1_1_1_1_2"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="int_3.html#E18:25_1_1_1_1_2"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="int_3.html#E6"><i><font color="Green" title="E4">Lm3</font></i></a></i>;<br/><i><font color="Green" title="E29">A11</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c8">r</font> = 
<font color="Maroon" title="c15">d'</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">r'</font>
<b>by </b><i><a class="txt" href="int_3.html#E12:25_1_1_1_1_2"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="int_3.html#E19:25_1_1_1_1_2"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="int_3.html#E16"><i><font color="Green" title="E14">Lm9</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c11">q'</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">b'</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">r'</font>
<b>by </b><i><a class="txt" href="int_3.html#E14:25_1_1_1_1_2"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="int_3.html#E9"><i><font color="Green" title="E7">Lm6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="int_3.html#E5:25_1_1_1_1_2"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="int_3.html#E16:25_1_1_1_1_2"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="int_3.html#E18:25_1_1_1_1_2"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="int_3.html#E20:25_1_1_1_1_2"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="int_3.html#E16"><i><font color="Green" title="E14">Lm9</font></i></a></i>
;<br/><a NAME="E22:25_1_1_1_1_2"/>
(  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  <a href="int_3.html#R1" title="INT_3:pred.1">&lt;=</a> <font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c8">r</font> <a href="int_3.html#NR3" title="INT_3:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<a href="int_3.html#K3" title="INT_3:func.3">absint</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">c</font> = 
<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="c6">c</font>
<b>by </b><i><a class="ref" href="int_3.html#D6" target="_self" title="INT_3:def.6">Def6</a></i>
<br/>.= 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c5">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D2" title="EUCLID:def.2">EUCLID:def 2</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c5">b'</font>
<b>by </b><i><a class="txt" href="int_3.html#E2:25_1_1_1_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c5">b'</font>
<b>by </b><i><a class="txt" href="int_3.html#E1:25_1_1_1_1_2"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<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="c5">b'</font>
<b>by </b><i><a class="ref" href="euclid.html#D2" title="EUCLID:def.2">EUCLID:def 2</a></i>
<br/>.= 
<a href="int_3.html#K3" title="INT_3:func.3">absint</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="ref" href="int_3.html#D6" target="_self" title="INT_3:def.6">Def6</a></i>
<br/>.= 
 <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="ref" href="int_3.html#T1" target="_self" title="INT_3:th.1">Th1</a></i>
;<br/>
<b>hence </b><a NAME="E2:25_1_1_1_1_2_2"/>
(  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  <a href="int_3.html#R1" title="INT_3:pred.1">&lt;=</a> <font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c8">r</font> <a href="int_3.html#NR3" title="INT_3:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E5:25_1_1_1_1_2"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="int_3.html#T1" target="_self" title="INT_3:th.1">Th1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E23:25_1_1_1_1_2"/>
 ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive" title="b2">q</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>  st <br/>( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  <a href="int_3.html#R1" title="INT_3:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="int_3.html#NR3" title="INT_3:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E21:25_1_1_1_1_2"><i><font color="Green" title="E29">A11</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:25_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive" title="b2">q</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>  st <br/>( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  <a href="int_3.html#R1" title="INT_3:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="int_3.html#NR3" title="INT_3:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font> )
 ;<br/>


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