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<span class="kw">consider </span><font color="Maroon" title="c2">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> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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>  implies for <font color="Olive" title="b1">b9</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp;  <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="Olive" title="b1">b9</font> holds <br/> ex <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">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="c3">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="c4">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">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="b3">r</font> &amp; <font color="Olive" title="b3">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="c4">b</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:17"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b9</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp;  <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="Olive" title="b1">b9</font> holds <br/> ex <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">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="c3">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="c4">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">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="b3">r</font> &amp; <font color="Olive" title="b3">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="c4">b</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">a9</font> = <font color="Maroon" title="c3">a</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">b9</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp;  <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="c6">b9</font> implies  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="c3">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="c4">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="c4">b</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E4:17"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c6">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  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="c6">b9</font> or  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="c3">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="c4">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="c4">b</font> ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">s</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c6">b9</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E5:17"/><span class="lab"><font color="Green" title="E13">A3</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="c6">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" 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="c3">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="c4">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="c4">b</font> )</span><br/>

<a NAME="E6:17"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_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="E1:17_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="c5">a9</font> or <font color="Maroon" title="c5">a9</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="17_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:17_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="c5">a9</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" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">a9</font> = <font color="Maroon" title="c5">a9</font> as    <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="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><a NAME="E3:17_1_1_1_1"/>
<font color="Maroon" title="c7">a9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c6">b9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a9</font>
 ;<br/><span class="kw">hence </span><a NAME="E4:17_1_1_1_1"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</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="17_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:17_1_1_1_2"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<font color="Maroon" title="c5">a9</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" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]</span><br/><div class="add"><a NAME="E2:17_1_1_1_2"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <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="c6">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="int_3.html#E6"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></span>;<br/><a NAME="E3:17_1_1_1_2"/><span class="kw">then </span>
1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">m</font> = <font color="Maroon" title="c6">b9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 as    <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="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">n</font> =  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <font color="Maroon" title="c5">a9</font> as    <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="#E1:17_1_1_1_2"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><a NAME="E6:17_1_1_1_2"/>
( <font color="Maroon" title="c5">a9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a9</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c6">b9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <font color="Maroon" title="c5">a9</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> &amp; <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">m</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>  )
 ;<br/><span class="kw">hence </span><a NAME="E7:17_1_1_1_2"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:17_1"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</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>
<a NAME="E7:17"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] &amp; ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n</font>] holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> ) )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E6:17"><span class="lab"><font color="Green" title="E14">A4</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">k9</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:17"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
 ex <font color="Olive" title="b1">s</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c7">k9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c6">b9</font></span>)</span> &amp; ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st  ex <font color="Olive" title="b3">s9</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">s9</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c6">b9</font></span>)</span> holds <br/><font color="Maroon" title="c7">k9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> ) )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">l9</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E11:17"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
<font color="Maroon" title="c7">k9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">l9</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c6">b9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17"><span class="lab"><font color="Green" title="E15">A6</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">k</font> = <font color="Maroon" title="c7">k9</font>, <font color="Maroon" title="c10">l</font> = <font color="Maroon" title="c8">l9</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/>
<a NAME="E13:17"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
( <font color="Maroon" title="c7">k9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c7">k9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b9</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:17_2"/>
<font color="Maroon" title="c7">k9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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"> <font color="Maroon" title="c7">k9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b9</font></span><br/>

<span class="kw">assume </span><a NAME="E2:17_2"/>
<font color="Maroon" title="c6">b9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">k</font> = <font color="Maroon" title="c7">k9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c6">b9</font> as    <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="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/>
<a NAME="E4:17_2"/><span class="lab"><font color="Green" title="E18">A9</font></span>: 
<font color="Maroon" title="c7">k9</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c12">b9</font> = <font color="Maroon" title="c6">b9</font> as    <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="#E5:17"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:17_2_1"/>
<font color="Maroon" title="c7">k9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:17_2_1"/><span class="lab"><font color="Green" title="E19">A10</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k9</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/>
<a NAME="E5:17_2_1"/>
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_2_1"><span class="lab"><font color="Green" title="E19">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:17_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="int_3.html#E5"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</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>
<a NAME="E5:17_2"/>
<font color="Maroon" title="c7">k9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c6">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">l9</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1</span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c6">b9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:17"><span class="lab"><font color="Green" title="E16">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:17_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E4:17_2"><span class="lab"><font color="Green" title="E18">A9</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>
<a NAME="E14:17"/><span class="lab"><font color="Green" title="E18">A11</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="c9">k</font> &amp; <font color="Maroon" title="c9">k</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="c4">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c11">b9</font> = <font color="Maroon" title="c6">b9</font> as    <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="#E5:17"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">k9</font> = <font color="Maroon" title="c7">k9</font> as    <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="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="lab"><font color="Green" title="E19">A12</font></span>: <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="c4">b</font> = 
<a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <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> <font color="Maroon" title="c11">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/>.= 
<font color="Maroon" title="c11">b9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_3_2_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="E1:17_3_2_1"/>
( <font color="Maroon" title="c12">k9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c12">k9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">b9</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:17"><span class="lab"><font color="Green" title="E17">A8</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_3_2_1_1"><span class="kw">case </span></a><a NAME="E1:17_3_2_1_1"/>
<font color="Maroon" title="c12">k9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> (  <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="c9">k</font> &amp; <font color="Maroon" title="c9">k</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="c4">b</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:17_3_2_1_1"/>
(  <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="c9">k</font> &amp; <font color="Maroon" title="c9">k</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="c4">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E3:17_3"><span class="lab"><font color="Green" title="E19">A12</font></span></a>, <a class="txt" href="int_3.html#E6"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a>, <a class="ref" href="int_3.html#T1" target="_self" title="INT_3:th.1">Th1</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_3_2_1_2"><span class="kw">case </span></a><a NAME="E1:17_3_2_1_2"/>
<font color="Maroon" title="c12">k9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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="c9">k</font> &amp; <font color="Maroon" title="c9">k</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="c4">b</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:17_3_2_1_2"/>
(  <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="c9">k</font> &amp; <font color="Maroon" title="c9">k</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="c4">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_3"><span class="lab"><font color="Green" title="E19">A12</font></span></a>, <a class="txt" href="int_3.html#E5"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a>, <a class="ref" href="int_3.html#T1" target="_self" title="INT_3:th.1">Th1</a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</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="E5:17_3"/>
(  <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="c9">k</font> &amp; <font color="Maroon" title="c9">k</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="c4">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>
<a NAME="E15:17"/>
<font color="Maroon" title="c9">k</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E11:17"><span class="lab"><font color="Green" title="E16">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:17"/>
 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="c3">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="c4">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="c4">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:17"><span class="lab"><font color="Green" title="E18">A11</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/>


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