<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c2">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:64"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/>

<a NAME="E2:64"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:64"/><b>then </b><i><font color="Green" title="E47">A2</font></i>: 
1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:64"/><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="int_3.html#E1:64"><i><font color="Green" title="E46">A1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">F</font> =  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font>;<br/>
<a NAME="E5:64"/><i><font color="Green" title="E49">A4</font></i>: 
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<a NAME="E6:64"/><i><font color="Green" title="E50">A5</font></i>: 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="int_3.html#E5:64"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/>
<a NAME="E7:64"/><i><font color="Green" title="E51">A6</font></i>: 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="int_3.html#E1:64"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="int_3.html#E51"><i><font color="Green" title="E44">Lm18</font></i></a></i>;<br/>
<a NAME="E8:64"/><i><font color="Green" title="E52">A7</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c4">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">b'</font> = <font color="Maroon" title="c5">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_1"/>
( <font color="Maroon" title="c6">a'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">b'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> or <font color="Maroon" title="c6">a'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">b'</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1"><b>case </b></a><a NAME="E1:64_1_1_1_1"/><i><font color="Green" title="E53">A8</font></i>: 
<font color="Maroon" title="c6">a'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">b'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">addint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> = 
<font color="Maroon" title="c6">a'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">b'</font>
<b>by </b><i><a class="ref" href="int_3.html#T17" target="_self" title="INT_3:th.17">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">addint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="int_3.html#E1:64_1_1_1_1"><i><font color="Green" title="E53">A8</font></i></a>, <a class="ref" href="int_3.html#T17" target="_self" title="INT_3:th.17">Th17</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_2"><b>case </b></a><a NAME="E1:64_1_1_1_2"/><i><font color="Green" title="E53">A9</font></i>: 
<font color="Maroon" title="c6">a'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">b'</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">addint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">b'</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="int_3.html#T17" target="_self" title="INT_3:th.17">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">addint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="int_3.html#E1:64_1_1_1_2"><i><font color="Green" title="E53">A9</font></i></a>, <a class="ref" href="int_3.html#T17" target="_self" title="INT_3:th.17">Th17</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:64_1"/>
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c4">a</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:64"/><i><font color="Green" title="E53">A10</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c4">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">b'</font> = <font color="Maroon" title="c5">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font> ;<br/>
<b>consider </b><font color="Maroon" title="c8">k</font> being    <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> <b> such that </b><br/><a NAME="E4:64_2"/><i><font color="Green" title="E54">A11</font></i>: 
( <font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">a'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">b'</font> &amp; <font color="Maroon" title="c6">a'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">b'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E4:64"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="int_3.html#E43"><i><font color="Green" title="E38">Lm14</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><a href="int_3.html#K7" title="INT_3:func.7">multint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c7">b'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">b'</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="int_3.html#E4:64_2"><i><font color="Green" title="E54">A11</font></i></a>, <a class="ref" href="int_3.html#T18" target="_self" title="INT_3:th.18">Th18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_3.html#K7" title="INT_3:func.7">multint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">b'</font>,<font color="Maroon" title="c6">a'</font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="int_3.html#E4:64_2"><i><font color="Green" title="E54">A11</font></i></a>, <a class="ref" href="int_3.html#T18" target="_self" title="INT_3:th.18">Th18</a></i>
;<br/>
<b>hence </b><a NAME="E6:64_2"/>
<font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:64"/><i><font color="Green" title="E54">A12</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b3">c</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c8">b'</font> = <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c9">c'</font> = <font color="Maroon" title="c6">c</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">aa</font> = <font color="Maroon" title="c7">a'</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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">aa</font> = <font color="Maroon" title="c10">aa</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">bb</font> = <font color="Maroon" title="c8">b'</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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">bb</font> = <font color="Maroon" title="c12">bb</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">cc</font> = <font color="Maroon" title="c9">c'</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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c15">cc</font> = <font color="Maroon" title="c14">cc</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E8:64_3"/><i><font color="Green" title="E55">A13</font></i>: 
( <font color="Maroon" title="c11">aa</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c11">aa</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c13">bb</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c13">bb</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c15">cc</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c15">cc</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<a NAME="E9:64_3"/><i><font color="Green" title="E56">A14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">bb</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>;<br/>
<a NAME="E10:64_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">bb</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">bb</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="int_3.html#T9" target="_self" title="INT_3:th.9">Th9</a></i>;<br/>
<a NAME="E11:64_3"/><b>then </b><i><font color="Green" title="E57">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64_3"><i><font color="Green" title="E56">A14</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<a NAME="E12:64_3"/><i><font color="Green" title="E58">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>;<br/>
<a NAME="E13:64_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="int_3.html#T9" target="_self" title="INT_3:th.9">Th9</a></i>;<br/>
<a NAME="E14:64_3"/><b>then </b><i><font color="Green" title="E59">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E12:64_3"><i><font color="Green" title="E58">A16</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<i><font color="Green" title="E60">A18</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c6">c</font> = 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">addint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c9">c'</font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#D5" title="GR_CY_1:def.5">GR_CY_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="int_3.html#E11:64_3"><i><font color="Green" title="E57">A15</font></i></a>, <a class="ref" href="gr_cy_1.html#D5" title="GR_CY_1:def.5">GR_CY_1:def 5</a></i>
;<br/>
<i><font color="Green" title="E61">A19</font></i>: <font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c6">c</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">addint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">a'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#D5" title="GR_CY_1:def.5">GR_CY_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="int_3.html#E14:64_3"><i><font color="Green" title="E59">A17</font></i></a>, <a class="ref" href="gr_cy_1.html#D5" title="GR_CY_1:def.5">GR_CY_1:def 5</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E12:64_3"><i><font color="Green" title="E58">A16</font></i></a>, <a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E8:64_3"><i><font color="Green" title="E55">A13</font></i></a>, <a class="ref" href="int_3.html#T10" target="_self" title="INT_3:th.10">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">cc</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="ref" href="int_3.html#T14" target="_self" title="INT_3:th.14">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">bb</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">bb</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">cc</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="ref" href="int_3.html#T14" target="_self" title="INT_3:th.14">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E8:64_3"><i><font color="Green" title="E55">A13</font></i></a>, <a class="txt" href="int_3.html#E9:64_3"><i><font color="Green" title="E56">A14</font></i></a>, <a class="ref" href="int_3.html#T10" target="_self" title="INT_3:th.10">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>
;<br/>
<b>hence </b><a NAME="E18:64_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c6">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E15:64_3"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="int_3.html#E16:64_3"><i><font color="Green" title="E61">A19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:64"/><i><font color="Green" title="E55">A20</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">c</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c8">b'</font> = <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c9">c'</font> = <font color="Maroon" title="c6">c</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">aa</font> = <font color="Maroon" title="c7">a'</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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">aa</font> = <font color="Maroon" title="c10">aa</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">bb</font> = <font color="Maroon" title="c8">b'</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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">bb</font> = <font color="Maroon" title="c12">bb</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">cc</font> = <font color="Maroon" title="c9">c'</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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c15">cc</font> = <font color="Maroon" title="c14">cc</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E8:64_4"/><i><font color="Green" title="E56">A21</font></i>: 
( <font color="Maroon" title="c11">aa</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c11">aa</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c13">bb</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c13">bb</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c15">cc</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c15">cc</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<a NAME="E9:64_4"/><i><font color="Green" title="E57">A22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">bb</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>;<br/>
<a NAME="E10:64_4"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">bb</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">bb</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="int_3.html#T9" target="_self" title="INT_3:th.9">Th9</a></i>;<br/>
<a NAME="E11:64_4"/><b>then </b><i><font color="Green" title="E58">A23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64_4"><i><font color="Green" title="E57">A22</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<a NAME="E12:64_4"/><i><font color="Green" title="E59">A24</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>;<br/>
<a NAME="E13:64_4"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="int_3.html#T9" target="_self" title="INT_3:th.9">Th9</a></i>;<br/>
<a NAME="E14:64_4"/><b>then </b><i><font color="Green" title="E60">A25</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E12:64_4"><i><font color="Green" title="E59">A24</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<i><font color="Green" title="E61">A26</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">c</font> = 
<span class="p1">(<span class="default"><a href="int_3.html#K7" title="INT_3:func.7">multint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c9">c'</font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="int_3.html#D11" target="_self" title="INT_3:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="int_3.html#E11:64_4"><i><font color="Green" title="E58">A23</font></i></a>, <a class="ref" href="int_3.html#D11" target="_self" title="INT_3:def.11">Def11</a></i>
;<br/>
<i><font color="Green" title="E62">A27</font></i>: <font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">c</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="int_3.html#K7" title="INT_3:func.7">multint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">a'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="int_3.html#D11" target="_self" title="INT_3:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="int_3.html#E14:64_4"><i><font color="Green" title="E60">A25</font></i></a>, <a class="ref" href="int_3.html#D11" target="_self" title="INT_3:def.11">Def11</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E12:64_4"><i><font color="Green" title="E59">A24</font></i></a>, <a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E8:64_4"><i><font color="Green" title="E56">A21</font></i></a>, <a class="ref" href="int_3.html#T10" target="_self" title="INT_3:th.10">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">cc</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="ref" href="int_3.html#T15" target="_self" title="INT_3:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">bb</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">bb</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">cc</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="ref" href="int_3.html#T15" target="_self" title="INT_3:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E8:64_4"><i><font color="Green" title="E56">A21</font></i></a>, <a class="txt" href="int_3.html#E9:64_4"><i><font color="Green" title="E57">A22</font></i></a>, <a class="ref" href="int_3.html#T10" target="_self" title="INT_3:th.10">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>
;<br/>
<b>hence </b><a NAME="E18:64_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E15:64_4"><i><font color="Green" title="E61">A26</font></i></a>, <a class="txt" href="int_3.html#E16:64_4"><i><font color="Green" title="E62">A27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:64"/><i><font color="Green" title="E56">A28</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/>

<a NAME="E1:64_5"/><i><font color="Green" title="E57">A29</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">addint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">a</font>,0
 
<b>by </b><i><a class="txt" href="int_3.html#E6:64"><i><font color="Green" title="E50">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">a'</font> = <font color="Maroon" title="c4">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font> ;<br/>
<a NAME="E3:64_5"/><i><font color="Green" title="E58">A30</font></i>: 
0 is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<a NAME="E4:64_5"/>
<font color="Maroon" title="c5">a'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<b>hence </b><a NAME="E5:64_5"/>
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="int_3.html#E1:64_5"><i><font color="Green" title="E57">A29</font></i></a>, <a class="txt" href="int_3.html#E3:64_5"><i><font color="Green" title="E58">A30</font></i></a>, <a class="ref" href="int_3.html#T17" target="_self" title="INT_3:th.17">Th17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:64"/><i><font color="Green" title="E57">A31</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span> holds <br/> ( <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">a'</font> = <font color="Maroon" title="c4">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font> ;<br/>
<a NAME="E2:64_6"/><i><font color="Green" title="E58">A32</font></i>: 
( 0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">a'</font> &amp; 1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">a'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<a NAME="E3:64_6"/><i><font color="Green" title="E59">A33</font></i>: 
1 is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E1:64"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<b>then </b><i><font color="Green" title="E60">A34</font></i>: <span class="p1">(<span class="default"><a href="int_3.html#K7" title="INT_3:func.7">multint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c4">a</font> = 
<font color="Maroon" title="c5">a'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E2:64_6"><i><font color="Green" title="E58">A32</font></i></a>, <a class="ref" href="int_3.html#T18" target="_self" title="INT_3:th.18">Th18</a></i>
<br/>.= 
<font color="Maroon" title="c5">a'</font>

;<br/>
<span class="p1">(<span class="default"><a href="int_3.html#K7" title="INT_3:func.7">multint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">a</font>,1 = 
<font color="Maroon" title="c5">a'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E2:64_6"><i><font color="Green" title="E58">A32</font></i></a>, <a class="txt" href="int_3.html#E3:64_6"><i><font color="Green" title="E59">A33</font></i></a>, <a class="ref" href="int_3.html#T18" target="_self" title="INT_3:th.18">Th18</a></i>
<br/>.= 
<font color="Maroon" title="c5">a'</font>

;<br/>
<b>hence </b><a NAME="E6:64_6"/>
( <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E7:64"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="int_3.html#E4:64_6"><i><font color="Green" title="E60">A34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:64"/><i><font color="Green" title="E58">A35</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">c</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c8">b'</font> = <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c9">c'</font> = <font color="Maroon" title="c6">c</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">aa</font> = <font color="Maroon" title="c7">a'</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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">aa</font> = <font color="Maroon" title="c10">aa</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">bb</font> = <font color="Maroon" title="c8">b'</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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">bb</font> = <font color="Maroon" title="c12">bb</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">cc</font> = <font color="Maroon" title="c9">c'</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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c15">cc</font> = <font color="Maroon" title="c14">cc</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E8:64_7"/><i><font color="Green" title="E59">A36</font></i>: 
( <font color="Maroon" title="c11">aa</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c11">aa</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c13">bb</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c13">bb</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c15">cc</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c15">cc</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<a NAME="E9:64_7"/><i><font color="Green" title="E60">A37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>;<br/>
<a NAME="E10:64_7"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="int_3.html#T9" target="_self" title="INT_3:th.9">Th9</a></i>;<br/>
<a NAME="E11:64_7"/><b>then </b><i><font color="Green" title="E61">A38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64_7"><i><font color="Green" title="E60">A37</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<a NAME="E12:64_7"/><i><font color="Green" title="E62">A39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">aa</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">a'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>;<br/>
<a NAME="E13:64_7"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">aa</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">aa</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="int_3.html#T9" target="_self" title="INT_3:th.9">Th9</a></i>;<br/>
<a NAME="E14:64_7"/><b>then </b><i><font color="Green" title="E63">A40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">a'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E12:64_7"><i><font color="Green" title="E62">A39</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<a NAME="E15:64_7"/><i><font color="Green" title="E64">A41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">cc</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">aa</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">c'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">a'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>;<br/>
<a NAME="E16:64_7"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c15">cc</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">aa</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">cc</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">aa</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="int_3.html#T9" target="_self" title="INT_3:th.9">Th9</a></i>;<br/>
<a NAME="E17:64_7"/><b>then </b><i><font color="Green" title="E65">A42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">c'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">a'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E15:64_7"><i><font color="Green" title="E64">A41</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<i><font color="Green" title="E66">A43</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font> = 
<span class="p1">(<span class="default"><a href="int_3.html#K7" title="INT_3:func.7">multint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">a'</font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#D5" title="GR_CY_1:def.5">GR_CY_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">a'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="int_3.html#E11:64_7"><i><font color="Green" title="E61">A38</font></i></a>, <a class="ref" href="int_3.html#D11" target="_self" title="INT_3:def.11">Def11</a></i>
;<br/>
<i><font color="Green" title="E67">A44</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">addint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_3.html#K7" title="INT_3:func.7">multint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">c'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">a'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="int_3.html#D11" target="_self" title="INT_3:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">addint</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">a'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">c'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">a'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="int_3.html#D11" target="_self" title="INT_3:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">a'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">c'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">a'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="int_3.html#E14:64_7"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="int_3.html#E17:64_7"><i><font color="Green" title="E65">A42</font></i></a>, <a class="ref" href="gr_cy_1.html#D5" title="GR_CY_1:def.5">GR_CY_1:def 5</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">a'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">c'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">a'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">aa</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">cc</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">aa</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E12:64_7"><i><font color="Green" title="E62">A39</font></i></a>, <a class="txt" href="int_3.html#E15:64_7"><i><font color="Green" title="E64">A41</font></i></a>, <a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">aa</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">cc</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">aa</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="ref" href="int_3.html#T14" target="_self" title="INT_3:th.14">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">aa</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">bb</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">cc</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">aa</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="ref" href="int_3.html#T15" target="_self" title="INT_3:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">aa</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E8:64_7"><i><font color="Green" title="E59">A36</font></i></a>, <a class="txt" href="int_3.html#E9:64_7"><i><font color="Green" title="E60">A37</font></i></a>, <a class="ref" href="int_3.html#T10" target="_self" title="INT_3:th.10">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">c'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">a'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>
;<br/>
<b>hence </b><a NAME="E21:64_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E18:64_7"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="int_3.html#E19:64_7"><i><font color="Green" title="E67">A44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:64"/><i><font color="Green" title="E59">A45</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">c</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/>

<b>thus </b><font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c6">c</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green" title="E53">A10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E14:64"><i><font color="Green" title="E58">A35</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green" title="E53">A10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">c</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green" title="E53">A10</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E16:64"/><i><font color="Green" title="E60">A46</font></i>: 
 <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font> is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon" title="c5">a'</font> = <font color="Maroon" title="c4">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c5">a'</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>  ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_9_1"/>
( <font color="Maroon" title="c6">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c6">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_9_1_1"><b>suppose </b></a><a NAME="E1:64_9_1_1"/><i><font color="Green" title="E61">A47</font></i>: 
<font color="Maroon" title="c6">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>take </b>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span>
;<br/><b>thus </b><a NAME="E2:64_9_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E6:64"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="int_3.html#E12:64"><i><font color="Green" title="E56">A28</font></i></a>, <a class="txt" href="int_3.html#E1:64_9_1_1"><i><font color="Green" title="E61">A47</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_9_1_2"><b>suppose </b></a><a NAME="E1:64_9_1_2"/>
<font color="Maroon" title="c6">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:64_9_1_2"/><b>then </b>
<font color="Maroon" title="c6">a'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">a'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c2">n</font> <b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="int_3.html#E48"><i><font color="Green" title="E41">Lm15</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">v</font> = <font color="Maroon" title="c7">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span> ;<br/><b>take </b>
<font color="Maroon" title="c8">v</font>
;<br/><b>thus </b><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c8">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#D5" title="GR_CY_1:def.5">GR_CY_1:def 5</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E6:64"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E17:64"/><i><font color="Green" title="E61">A48</font></i>: 
 <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font> is <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">right_unital</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a><br/>

<b>thus </b><a NAME="E1:64_10"/>
<font color="Maroon" title="c4">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="int_3.html#E13:64"><i><font color="Green" title="E57">A31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:64"/>
 <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font> is <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">left_unital</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a><br/>

<b>thus </b><a NAME="E1:64_11"/>
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="int_3.html#E13:64"><i><font color="Green" title="E57">A31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">F</font> =  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font> as  <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> <b>by </b><i><a class="txt" href="int_3.html#E8:64"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="int_3.html#E9:64"><i><font color="Green" title="E53">A10</font></i></a>, <a class="txt" href="int_3.html#E10:64"><i><font color="Green" title="E54">A12</font></i></a>, <a class="txt" href="int_3.html#E11:64"><i><font color="Green" title="E55">A20</font></i></a>, <a class="txt" href="int_3.html#E12:64"><i><font color="Green" title="E56">A28</font></i></a>, <a class="txt" href="int_3.html#E14:64"><i><font color="Green" title="E58">A35</font></i></a>, <a class="txt" href="int_3.html#E15:64"><i><font color="Green" title="E59">A45</font></i></a>, <a class="txt" href="int_3.html#E16:64"><i><font color="Green" title="E60">A46</font></i></a>, <a class="txt" href="int_3.html#E17:64"><i><font color="Green" title="E61">A48</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a>, <a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a>, <a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a>, <a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></i>;<br/>
<a NAME="E20:64"/>
not <font color="Maroon" title="c4">F</font> is <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">degenerated</a>
 
<b>by </b><i><a class="txt" href="int_3.html#E6:64"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="int_3.html#E7:64"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="vectsp_1.html#D21" title="VECTSP_1:def.21">VECTSP_1:def 21</a></i>;<br/>
<b>hence </b><a NAME="E21:64"/>
( not  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font> is <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">degenerated</a> &amp;  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">n</font> is  <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">distributive</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> )
 ;<br/>


</div>
