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

<b>let </b><font color="Maroon">c<sub>2</sub></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">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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">E47</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">c<sub>2</sub></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">E48</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="int_3.html#E1:64"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E5:64"/><i><font color="Green">E49</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<i><font color="Green">E50</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">c<sub>2</sub></font></span>)</span> = 
 <a href="funct_7.html#K1">In</a> 0,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
0
<b>by </b><i><a class="txt" href="int_3.html#E5:64"><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">E51</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">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="int_3.html#E1:64"><i><font color="Green">E46</font></i></a>, <a class="txt" href="int_3.html#E51"><i><font color="Green">Lemma44</font></i></a></i>;<br/>
<a NAME="E8:64"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>2</sub></font></span>)</span> holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_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_2_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_1_1"><b>case </b></a><a NAME="E1:64_2_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="int_3.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E1:64_2_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="int_3.html#T17" target="_self">Th17</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_1_2"><b>case </b></a><a NAME="E1:64_2_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="int_3.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E1:64_2_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="int_3.html#T17" target="_self">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_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:64"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>2</sub></font></span>)</span> holds <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font>
 
<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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:64_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="int_3.html#E4:64"><i><font color="Green">E48</font></i></a>, <a class="txt" href="int_3.html#E43"><i><font color="Green">Lemma38</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E4:64_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="int_3.html#T18" target="_self">Th18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E4:64_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="int_3.html#T18" target="_self">Th18</a></i>
;<br/>
<b>hence </b><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>

;<br/><br/>


</div><b>end;</b></div>
<a NAME="E10:64"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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">c<sub>2</sub></font></span>)</span> holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E8:64_4"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><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">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>11</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="int_3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E11:64_4"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64_4"><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_4"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>13</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="int_3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E14:64_4"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="int_3.html#E12:64_4"><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">E60</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">c<sub>6</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><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">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E11:64_4"><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">E61</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
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">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><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">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E14:64_4"><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">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E12:64_4"><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">c<sub>11</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>13</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E8:64_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="int_3.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="int_3.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="int_3.html#T14" target="_self">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">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E8:64_4"><i><font color="Green">E55</font></i></a>, <a class="txt" href="int_3.html#E9:64_4"><i><font color="Green">E56</font></i></a>, <a class="ref" href="int_3.html#T10" target="_self">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">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E15:64_4"><i><font color="Green">E60</font></i></a>, <a class="txt" href="int_3.html#E16:64_4"><i><font color="Green">E61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:64"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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">c<sub>2</sub></font></span>)</span> holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E8:64_5"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><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_5"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>
<a NAME="E10:64_5"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="int_3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E11:64_5"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64_5"><i><font color="Green">E57</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E12:64_5"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>
<a NAME="E13:64_5"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="int_3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E14:64_5"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="int_3.html#E12:64_5"><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">E61</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="int_3.html#D11" target="_self">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">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E11:64_5"><i><font color="Green">E58</font></i></a>, <a class="ref" href="int_3.html#D11" target="_self">Def11</a></i>
;<br/>
<i><font color="Green">E62</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<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">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="int_3.html#D11" target="_self">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E14:64_5"><i><font color="Green">E60</font></i></a>, <a class="ref" href="int_3.html#D11" target="_self">Def11</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E12:64_5"><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">c<sub>11</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>13</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E8:64_5"><i><font color="Green">E56</font></i></a>, <a class="ref" href="int_3.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="int_3.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="int_3.html#T15" target="_self">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">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E8:64_5"><i><font color="Green">E56</font></i></a>, <a class="txt" href="int_3.html#E9:64_5"><i><font color="Green">E57</font></i></a>, <a class="ref" href="int_3.html#T10" target="_self">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">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
;<br/>
<b>hence </b><a NAME="E18:64_5"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E15:64_5"><i><font color="Green">E61</font></i></a>, <a class="txt" href="int_3.html#E16:64_5"><i><font color="Green">E62</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:64"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font></span>)</span> holds <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span>;<br/>

<i><font color="Green">E57</font></i>: <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span></span>)</span> = 
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">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p3">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,0
<b>by </b><i><a class="txt" href="int_3.html#E6:64"><i><font color="Green">E50</font></i></a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E3:64_6"/><i><font color="Green">E58</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">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><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_6"/>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><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_6"/>
<font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="int_3.html#E1:64_6"><i><font color="Green">E57</font></i></a>, <a class="txt" href="int_3.html#E3:64_6"><i><font color="Green">E58</font></i></a>, <a class="ref" href="int_3.html#T17" target="_self">Th17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:64"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 
<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">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E2:64_7"/><i><font color="Green">E58</font></i>: 
( 0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> &amp; 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><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_7"/><i><font color="Green">E59</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">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="int_3.html#E1:64"><i><font color="Green">E46</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<b>then </b><i><font color="Green">E60</font></i>: <span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E2:64_7"><i><font color="Green">E58</font></i></a>, <a class="ref" href="int_3.html#T18" target="_self">Th18</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>

;<br/>
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,1 = 
<font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E2:64_7"><i><font color="Green">E58</font></i></a>, <a class="txt" href="int_3.html#E3:64_7"><i><font color="Green">E59</font></i></a>, <a class="ref" href="int_3.html#T18" target="_self">Th18</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>

;<br/>
<b>hence </b><a NAME="E6:64_7"/>
( <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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="int_3.html#E7:64"><i><font color="Green">E51</font></i></a>, <a class="txt" href="int_3.html#E4:64_7"><i><font color="Green">E60</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:64"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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">c<sub>2</sub></font></span>)</span> holds <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E8:64_8"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><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_8"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>
<a NAME="E10:64_8"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="int_3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E11:64_8"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="int_3.html#E9:64_8"><i><font color="Green">E60</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E12:64_8"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>
<a NAME="E13:64_8"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="int_3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E14:64_8"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="int_3.html#E12:64_8"><i><font color="Green">E62</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E15:64_8"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>
<a NAME="E16:64_8"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="int_3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E17:64_8"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="int_3.html#E15:64_8"><i><font color="Green">E64</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<i><font color="Green">E66</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="int_3.html#K7" target="_self">multint</a> <font color="Maroon">c<sub>2</sub></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">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><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">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E11:64_8"><i><font color="Green">E61</font></i></a>, <a class="ref" href="int_3.html#D11" target="_self">Def11</a></i>
;<br/>
<i><font color="Green">E67</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
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">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<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">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<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">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="int_3.html#D11" target="_self">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>2</sub></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">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="int_3.html#D11" target="_self">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">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></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<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E14:64_8"><i><font color="Green">E63</font></i></a>, <a class="txt" href="int_3.html#E17:64_8"><i><font color="Green">E65</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">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></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<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>15</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E12:64_8"><i><font color="Green">E62</font></i></a>, <a class="txt" href="int_3.html#E15:64_8"><i><font color="Green">E64</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">c<sub>13</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="int_3.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="int_3.html#T15" target="_self">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">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E8:64_8"><i><font color="Green">E59</font></i></a>, <a class="txt" href="int_3.html#E9:64_8"><i><font color="Green">E60</font></i></a>, <a class="ref" href="int_3.html#T10" target="_self">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">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
;<br/>
<b>hence </b><a NAME="E21:64_8"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E18:64_8"><i><font color="Green">E66</font></i></a>, <a class="txt" href="int_3.html#E19:64_8"><i><font color="Green">E67</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:64"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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">c<sub>2</sub></font></span>)</span> holds <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font></span>)</span>;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E53</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E14:64"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E53</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E9:64"><i><font color="Green">E53</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E16:64"/><i><font color="Green">E60</font></i>: 
 <a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">c<sub>2</sub></font> is <a href="rlvect_1.html#V6">right_complementable</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">c<sub>4</sub></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">c<sub>2</sub></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">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_10_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>6</sub></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_10_1_1"><b>suppose </b></a><a NAME="E1:64_10_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font></span>)</span>
;<br/><b>thus </b><a NAME="E2:64_10_1_1"/>
<font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E6:64"><i><font color="Green">E50</font></i></a>, <a class="txt" href="int_3.html#E12:64"><i><font color="Green">E56</font></i></a>, <a class="txt" href="int_3.html#E1:64_10_1_1"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_10_1_2"><b>suppose </b></a><a NAME="E1:64_10_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:64_10_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="int_3.html#E3:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E48"><i><font color="Green">Lemma41</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font></span>)</span> ;<br/><b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/><b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
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">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="int_3.html#E3:64"><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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E6:64"><i><font color="Green">E50</font></i></a>, <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">E61</font></i>: 
 <a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">c<sub>2</sub></font> is <a href="vectsp_1.html#V6">right_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">c<sub>4</sub></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">c<sub>2</sub></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_11"/>
<font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="int_3.html#E13:64"><i><font color="Green">E57</font></i></a></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">c<sub>2</sub></font> is <a href="vectsp_1.html#V8">left_unital</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></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_12"/>
<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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="int_3.html#E13:64"><i><font color="Green">E57</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">c<sub>2</sub></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#E8:64"><i><font color="Green">E52</font></i></a>, <a class="txt" href="int_3.html#E9:64"><i><font color="Green">E53</font></i></a>, <a class="txt" href="int_3.html#E10:64"><i><font color="Green">E54</font></i></a>, <a class="txt" href="int_3.html#E11:64"><i><font color="Green">E55</font></i></a>, <a class="txt" href="int_3.html#E12:64"><i><font color="Green">E56</font></i></a>, <a class="txt" href="int_3.html#E14:64"><i><font color="Green">E58</font></i></a>, <a class="txt" href="int_3.html#E15:64"><i><font color="Green">E59</font></i></a>, <a class="txt" href="int_3.html#E16:64"><i><font color="Green">E60</font></i></a>, <a class="txt" href="int_3.html#E17:64"><i><font color="Green">E61</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">c<sub>4</sub></font> is <a href="vectsp_1.html#V10">degenerated</a>
 
<b>by </b><i><a class="txt" href="int_3.html#E6:64"><i><font color="Green">E50</font></i></a>, <a class="txt" href="int_3.html#E7:64"><i><font color="Green">E51</font></i></a>, <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">c<sub>2</sub></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">c<sub>2</sub></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>
