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<div class="add">

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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 <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">a</font> <a href="group_1.html#NR1" title="GROUP_1:NR.1">is_not_of_order_0</a>  holds <br/> ex <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">i</font> )</span><br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c2">a</font> <a href="group_1.html#NR1" title="GROUP_1:NR.1">is_not_of_order_0</a>  holds <br/> ex <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">i</font> )</span><br/><b>let </b><font color="Maroon" title="c3">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="group_1.html#NR1" title="GROUP_1:NR.1">is_not_of_order_0</a>  implies  ex <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i</font> ) )</span><br/>





<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#NR1" title="GROUP_1:NR.1">is_not_of_order_0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i</font> )</span><br/>

<a NAME="E2:40"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="int_7.html#E1:40"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="group_1.html#D12" title="GROUP_1:def.12">GROUP_1:def 12</a></i>;<br/>
<a NAME="E3:40"/><b>then </b><i><font color="Green" title="E33">A2</font></i>: 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="xreal_1.html#T153" title="XREAL_1:th.153">XREAL_1:153</a></i>;<br/>
<a NAME="E4:40"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_1"/>
( <font color="Maroon" title="c3">i</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>  or  <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="c3">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1"><b>suppose </b></a><a NAME="E1:40_1_1_1"/>
<font color="Maroon" title="c3">i</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i</font></span><br/><div class="add"><a NAME="E2:40_1_1_1"/><b>then </b><i><font color="Green" title="E34">A3</font></i>: 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:40_1_1_1"/>
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="int_7.html#E3:40"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E4:40_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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="int_7.html#E2:40_1_1_1"><i><font color="Green" title="E34">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_1_2"/>
 <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="c3">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i</font></span><br/><div class="add"><b>hence </b><a NAME="E2:40_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T35" title="XREAL_1:th.35">XREAL_1:35</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:40"/><b>then </b><i><font color="Green" title="E34">A4</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">i</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T64" title="GROUP_1:th.64">GROUP_1:64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p3">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">i</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T50" title="GROUP_1:th.50">GROUP_1:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">i</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T82" title="GROUP_1:th.82">GROUP_1:82</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">i</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T42" title="GROUP_1:th.42">GROUP_1:42</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E7:40"/>
 ex <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i</font> )
 <b>by </b><i><a class="txt" href="int_7.html#E5:40"><i><font color="Green" title="E34">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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