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

<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:64"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/>

<a NAME="E2:64"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:64"/><b>then </b><i><font color="Green">E22</font></i>: 
1 <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:64"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">F</font> =  <a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font>;<br/>
<a NAME="E5:64"/><i><font color="Green">E24</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E6:64"/><i><font color="Green">E39</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="int_3.html#E10:64"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>;<br/>
<a NAME="E7:64"/><i><font color="Green">E40</font></i>: 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="txt" href="int_3.html#E8:64"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E8:64"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span> holds  <font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font> ;<br/>
<b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">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">a'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">a'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">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">E45</font></i>: 
<font color="Maroon">a'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<font color="Maroon">a'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b'</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
<b>by </b><i>, </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">E46</font></i>: 
<font color="Maroon">a'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></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">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:64"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span> holds  <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="group_1.html#K1">*</a> <font color="Olive">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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font> ;<br/>
<b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font> ;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:64_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">a'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E4:64_2"><i><font color="Green">E48</font></i></a>, </i>;<br/>
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, , <a class="txt" href="int_3.html#E6:64"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, , <a class="txt" href="int_3.html#E6:64"><i><font color="Green">E39</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:64_2"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:64"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font>, <font color="Maroon">c'</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font> ;<br/>
<b>reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">a'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">aa</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">b'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">bb</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">cc</font> = <font color="Maroon">c'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">cc</font> = <font color="Maroon">cc</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E8:64_3"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">aa</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">aa</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">bb</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">bb</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">cc</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">cc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E9:64_3"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">bb</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>
<a NAME="E10:64_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">bb</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">bb</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<a NAME="E11:64_3"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E14:64_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E12:64_3"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>
<a NAME="E13:64_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<a NAME="E14:64_3"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E16:64_3"><i><font color="Green">E58</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<i><font color="Green">E57</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<font color="Maroon">c</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">c'</font>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="gr_cy_1.html#D5">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">a'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E15:64_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="gr_cy_1.html#D5">GR_CY_1:def 5</a></i>
;<br/>
<i><font color="Green">E58</font></i>: <font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="gr_cy_1.html#D5">GR_CY_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E11:64"><i><font color="Green">E59</font></i></a>, <a class="ref" href="gr_cy_1.html#D5">GR_CY_1:def 5</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E16:64_3"><i><font color="Green">E58</font></i></a>, <a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">aa</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E12:64_3"><i><font color="Green">E55</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">cc</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">aa</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">bb</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">aa</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">bb</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">cc</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E12:64_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="int_3.html#E14:64_3"><i><font color="Green">E56</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
;<br/>
<b>hence </b><a NAME="E18:64_3"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="int_3.html#E9"><i><font color="Green">Lemma61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:64"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="group_1.html#K1">*</a> <font color="Olive">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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font>, <font color="Maroon">c'</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font> ;<br/>
<b>reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">a'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">aa</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">b'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">bb</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">cc</font> = <font color="Maroon">c'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">cc</font> = <font color="Maroon">cc</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E8:64_4"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">aa</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">aa</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">bb</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">bb</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">cc</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">cc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E9:64_4"/><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">bb</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>
<a NAME="E10:64_4"/>
( <span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">bb</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">bb</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<a NAME="E11:64_4"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E12:64_4"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>
<a NAME="E13:64_4"/>
( <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<a NAME="E14:64_4"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E11:64"><i><font color="Green">E59</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<i><font color="Green">E104</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">c'</font>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, , </i>
;<br/>
<i><font color="Green">E105</font></i>: <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E8:64_4"><i><font color="Green">E60</font></i></a>, </i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E11:64"><i><font color="Green">E59</font></i></a>, <a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">aa</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">cc</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">aa</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">bb</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">aa</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">bb</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">cc</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
;<br/>
<b>hence </b><a NAME="E18:64_4"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E9"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="int_3.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:64"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span> holds  <font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>;<br/>

<a NAME="E1:64_5"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,0
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font> ;<br/>
<a NAME="E3:64_5"/><i><font color="Green">E108</font></i>: 
0 is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E4:64_5"/>
<font color="Maroon">a'</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<b>hence </b><a NAME="E5:64_5"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:64"/><i><font color="Green">E109</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span> holds <br/> ( <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font> ;<br/>
<a NAME="E2:64_6"/><i><font color="Green">E110</font></i>: 
( 0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font> &amp; 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E3:64_6"/><i><font color="Green">E141</font></i>: 
1 is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<b>then </b><i><font color="Green">E142</font></i>: <span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">a</font> = 
<font color="Maroon">a'</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="txt" href="int_3.html#E6:64"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">a'</font>

;<br/>
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,1 = 
<font color="Maroon">a'</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="txt" href="int_3.html#E11:64"><i><font color="Green">E59</font></i></a>, <a class="txt" href="int_3.html#E6:64"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">a'</font>

;<br/>
<b>hence </b><a NAME="E6:64_6"/>
( <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:64"/><i><font color="Green">E143</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">c</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="group_1.html#K1">*</a> <font color="Olive">a</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">c</font> <a href="group_1.html#K1">*</a> <font color="Olive">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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font>, <font color="Maroon">c'</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font> ;<br/>
<b>reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">a'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">aa</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">b'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">bb</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">cc</font> = <font color="Maroon">c'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">cc</font> = <font color="Maroon">cc</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E8:64_7"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">aa</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">aa</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">bb</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">bb</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">cc</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">cc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E9:64_7"/><i><font color="Green">E145</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>
<a NAME="E10:64_7"/>
( <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<a NAME="E11:64_7"/><b>then </b><i><font color="Green">E146</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E12:64_7"/><i><font color="Green">E147</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>
<a NAME="E13:64_7"/>
( <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<a NAME="E14:64_7"/><b>then </b><i><font color="Green">E148</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="int_3.html#D4" target="_self">Def4</a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E15:64_7"/><i><font color="Green">E149</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">cc</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>
<a NAME="E16:64_7"/>
( <span class="p1">(<span class="default"><font color="Maroon">cc</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">cc</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<a NAME="E17:64_7"/><b>then </b><i><font color="Green">E171</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E11"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<i><font color="Green">E172</font></i>: <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<font color="Maroon">a</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">a'</font>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="gr_cy_1.html#D5">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">b'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E9"><i><font color="Green">Lemma61</font></i></a>, </i>
;<br/>
<i><font color="Green">E173</font></i>: <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="int_3.html#D5" target="_self">Def5</a>, <a class="ref" href="gr_cy_1.html#D5">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">b'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">cc</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="int_3.html#D4" target="_self">Def4</a>, <a class="txt" href="int_3.html#E11"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">cc</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">cc</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">bb</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">cc</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">aa</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">aa</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E11:64"><i><font color="Green">E59</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
;<br/>
<b>hence </b><a NAME="E21:64_7"/>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="int_3.html#D6" target="_self">Def6</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:64"/><i><font color="Green">E174</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span> holds  <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>thus </b><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E16:64"/><i><font color="Green">E175</font></i>: 
 <a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font> is <a href="rlvect_1.html#V6">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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font> ;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">a'</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a'</font> <a href="hidden.html#NR2" target="_self">&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">E176</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>take </b>
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>
;<br/><b>thus </b><a NAME="E2:64_9_1_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="int_3.html#E9"><i><font color="Green">Lemma61</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">a'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:64_9_1_2"/><b>then </b>
<font color="Maroon">a'</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">n</font> <b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span> ;<br/><b>take </b>
<font color="Maroon">v</font>
;<br/><b>thus </b><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font> = 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">v</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="gr_cy_1.html#D5">GR_CY_1:def 5</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="nat_1.html#T77">NAT_1:77</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">E177</font></i>: 
 <a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font> is <a href="vectsp_1.html#V6">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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a><br/>

<b>thus </b><a NAME="E1:64_10"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E18:64"/>
 <a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font> is <a href="vectsp_1.html#V8">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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D19">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">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> =  <a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font> as  <a href="group_1.html#V7">commutative</a> <a href="funcsdom.html#NM1">Ring</a> <b>by </b><i>, , , , , , <a class="txt" href="int_3.html#E11:64"><i><font color="Green">E59</font></i></a>, , <a class="txt" href="int_3.html#E9"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a>, <a class="ref" href="group_1.html#D16">GROUP_1:def 16</a>, <a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a>, <a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a></i>;<br/>
<a NAME="E20:64"/>
not <font color="Maroon">F</font> is <a href="vectsp_1.html#V10">degenerated</a>
 
<b>by </b><i>, , <a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/>
<b>hence </b><a NAME="E21:64"/>
( not  <a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font> is <a href="vectsp_1.html#V10">degenerated</a> &amp;  <a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">n</font> is  <a href="group_1.html#V2">unital</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="funcsdom.html#NM1">Ring</a> )
 ;<br/>


</div>
