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<span class="kw">reconsider </span><font color="Maroon" title="c4">b9</font> = <font color="Maroon" title="c3">b</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_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="E1:20_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">b9</font> or <font color="Maroon" title="c4">b9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:20_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">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font> )</span><br/><div class="add"><a NAME="E2:20_1_1_1_1_1"/><span class="kw">then </span>
 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#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">r</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="int_3.html#E14"><span class="lab"><font color="Green" title="E10">Lm8</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:20_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#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="20_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:20_1_1_1_1_2"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c4">b9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font> )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">c</font> =  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <font color="Maroon" title="c4">b9</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/><a NAME="E3:20_1_1_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <font color="Maroon" title="c4">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="xreal_1.html#T60" title="XREAL_1:th.60">XREAL_1:60</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">q</font>, <font color="Maroon" title="c7">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> <span class="kw"> such that </span><br/><a NAME="E5:20_1_1_1_1_2"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<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="c6">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">r</font>
 <span class="kw">and </span><br/><a NAME="E6:20_1_1_1_1_2"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r</font>
 <span class="kw">and </span><br/><a NAME="E7:20_1_1_1_1_2"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c5">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="int_3.html#E6"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a>, <a class="txt" href="int_3.html#E14"><span class="lab"><font color="Green" title="E10">Lm8</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">t</font> =  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">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/><a NAME="E9:20_1_1_1_1_2"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A3</font></span></a></span>;<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="c5">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="c5">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_3.html#D6" target="_self" title="INT_3:def.6">Def6</a></span>
<br/>.= 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <font color="Maroon" title="c4">b9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><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">b9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
<br/>.= 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c4">b9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
<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="c4">b9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D2" title="EUCLID:def.2">EUCLID:def 2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_3.html#D6" target="_self" title="INT_3:def.6">Def6</a></span>
<br/>.= 
 <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_3.html#T1" target="_self" title="INT_3:th.1">Th1</a></span>
;<br/><a NAME="E11:20_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_1_1_1_1_2"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="int_3.html#T1" target="_self" title="INT_3:th.1">Th1</a></span>;<br/><span class="kw">hence </span><a NAME="E12:20_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#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_1_1_1_1_2"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E9:20_1_1_1_1_2"><span class="lab"><font color="Green" title="E18">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E3:20_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#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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